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Elementary and Intermediate Algebra - With CD - 02 edition
Summary: With a strong emphasis on mathematics of the real world, this text unites the concepts of elementary algebra and intermediate algebra in one volume.
The content, pedagogy, and design of Elementary and Intermediate Algebra are based on those of the successful copyright 2000 Dugopolski algebra series.
A midtext diagnostic review contains exercises based on the material contained in the first half of the text. The exercises are organized by chapte...show morer and section so that students can identify any chapters and sections that need further review.
Features :
Midtext Diagnostic Review
Margin notes
Exercises
End of chapter Wrap-up
NetTutor is a revolutionary new web-based learning environment for the live dissemination of mathematical content.
4.1 Graphing Lines in the Coordinate Plane 4.2 Slope 4.3 Equations of Lines in Slope-Intercept and Standard Form 4.4 The Point-Slope Form 4.5 Applications of Linear Equations 4.6 Introduction to Functions84 |
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The Philosopher's Toolkit: How to Be the Most Rational Person in Any Room ProfessorGet a new perspective on two of the greatest achievements of human culture in the 12 dazzling lectures of How Music and Mathematics Relate, taught by award-winning mathematician and musician David Kung of St. Mary's College of Maryland. Understanding the connections between music and mathematics helps you appreciate both, even if you have no special ability in either field. By exploring the mathematics of music, you learn why non-Western music sounds so different, gain insight into the technology of modern sound reproduction, and start to hear the world around you in exciting new ways. No expertise in either music or higher-level mathematics is required to appreciate this astonishing alliance between art and scienceSave up toSave up to $275
Understanding Calculus II: Problems, Solutions, and Tips Professor systemsWith record-breaking highs one minute and economy-rocking lows the next—not to mention arcane rules and ambiguous terminology—it can seem as if there's no way to feel truly comfortable in the financial markets. But there is, and it's more accessible than you might think. Financial Literacy: Finding Your Way in the Financial Markets is an essential primer on this domain, from its functions, strengths, and possibilities to its weaknesses and vulnerabilities. In 24 eye-opening lectures, award-winning Professor Connel Fullenkamp of Duke University reveals the interconnected workings of the financial markets and how society's financial strength—and your own—depend on money continuing to move through these channels. You'll come to understand just how interdependent the world's markets have become; feel newly at ease in the realm of stocks, bonds, derivatives, and credit; and be better able to make informed decisions for your financial future. |
Essential Mathematics for Economic Analysis - 2nd edition
Summary: Essential Mathematics for Economic Analysis provides an invaluable introduction to mathematical analysis and linear algebra for economists. Its main purpose is to help students acquire the mathematical skills they need in order to read the less technical literature associated with economic problems. The coverage is comprehensive, ranging from elementary algebra to more advanced material, whilst focusing on all the core topics usually taught in undergraduate course...show mores on mathematics for economists. Features
Large number of examples throughout the book help to link abstract mathematics with real life
Extremely clear writing style without sacrificing mathematical precision and rigour ensures that students gain a thorough understanding of the use of mathematics to analyse economic problems.
Extensive number of problems and exercises at the end of each section with solutions to odd-numbered questions at the back of the book, allowing students to constantly practice what they are learning to reinforce their understanding.
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Extensive resources for instructors and students on the companion website at including Instructors Manual with tests; Excel supplement with exercises; Excel supp Answers for lecturers; Downloadable Exam style problems which can be set as assignments for14 |
The ATETV project delivers web-based videos to connect students to careers in advanced technology. In this episode of ATETV, students will see how important a strong foundation in mathematics is for those entering technical fields. This video would be particularly useful for students who struggle to see the importance of learning mathematics- they will get a glimpse at some of the real world applications of what they are learning.
Running time for the episode is 2:55.
Job opportunities in statistics are projected to remain favorable in the future. As the U.S. economy continues to produce jobs using quantitative literacy and analyses, increased numbers of statisticians are required. |
1
00:00:04 --> 00:00:09
OK, here we go with,
quiz review for the third quiz
2
00:00:09 --> 00:00:12
that's coming on Friday.
3
00:00:12 --> 00:00:18
So, one key point is that the
quiz covers through chapter six.
4
00:00:18 --> 00:00:24
Chapter seven on linear
transformations will appear on
5
00:00:24 --> 00:00:29
the final exam,
but not on the quiz.
6
00:00:29 --> 00:00:33
So I won't review linear
transformations today,
7
00:00:33 --> 00:00:39
but they'll come into the full
course review on the very last
8
00:00:39 --> 00:00:40
lecture.
9
00:00:40 --> 00:00:46
So today, I'm reviewing chapter
six, and I'm going to take some
10
00:00:46 --> 00:00:52
old exams, and I'm always ready
to answer questions.
11
00:00:52 --> 00:00:55
And I thought,
kind of help our memories if I
12
00:00:55 --> 00:00:59
write down the main topics in
chapter six.
13
00:00:59 --> 00:01:01
So, already,
on the previous quiz,
14
00:01:01 --> 00:01:04.98
we knew how to find eigenvalues
and eigenvectors.
15
00:01:04.98 --> 00:01:09
Well, we knew how to find them
by that determinant of A minus
16
00:01:09 --> 00:01:10
lambda I equals zero.
17
00:01:10 --> 00:01:15
But, of course,
there could be shortcuts.
18
00:01:15 --> 00:01:19
There could be,
like, useful information about
19
00:01:19 --> 00:01:23
the eigenvalues that we can
speed things up with.
20
00:01:23 --> 00:01:24
OK.
21
00:01:24 --> 00:01:29
Then, the new stuff starts out
with a differential equation,
22
00:01:29 --> 00:01:32
so I'll do a problem.
23
00:01:32 --> 00:01:35
I'll do a differential equation
problem first.
24
00:01:35 --> 00:01:37
What's special about symmetric
matrices?
25
00:01:37 --> 00:01:39
Can we just say that in words?
26
00:01:39 --> 00:01:42
I'd better write it down,
though.
27
00:01:42 --> 00:01:44
What's special about symmetric
matrices?
28
00:01:44 --> 00:01:46
Their eigenvalues are real.
29
00:01:46 --> 00:01:50
The eigenvalues of a symmetric
matrix always come out real,
30
00:01:50 --> 00:01:54.37
and there always are enough
eigenvectors.
31
00:01:54.37 --> 00:01:58
Even if there are repeated
eigenvalues, there are enough
32
00:01:58 --> 00:02:00
eigenvectors,
and we can choose those
33
00:02:00 --> 00:02:03
eigenvectors to be orthogonal.
34
00:02:03 --> 00:02:07
So if A equals A transposed,
the big fact will be that we
35
00:02:07 --> 00:02:10
can diagonalize it,
and those eigenvector matrix,
36
00:02:10 --> 00:02:14
with the eigenvectors in the
column, can be an orthogonal
37
00:02:14 --> 00:02:16
matrix.
38
00:02:16 --> 00:02:22
So we get a Q lambda Q
transpose.
39
00:02:22 --> 00:02:32
That, in three symbols,
expresses a wonderful fact,
40
00:02:32 --> 00:02:42
a fundamental fact for
symmetric matrices.
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OK.
42
00:02:42 --> 00:02:47
Then, we went beyond that fact
to ask about positive definite
43
00:02:47 --> 00:02:51
matrices, when the eigenvalues
were positive.
44
00:02:51 --> 00:02:53
I'll do an example of that.
45
00:02:53 --> 00:02:55
Now we've left symmetry.
46
00:02:55 --> 00:03:00
Similar matrices are any square
matrices, but two matrices are
47
00:03:00 --> 00:03:03
similar if they're related that
way.
48
00:03:03 --> 00:03:07
And what's the key point about
similar matrices?
49
00:03:07 --> 00:03:12
Somehow, those matrices are
representing the same thing in
50
00:03:12 --> 00:03:16
different basis,
in chapter seven language.
51
00:03:16 --> 00:03:20
In chapter six language,
what's up with these similar
52
00:03:20 --> 00:03:22.52
matrices?
53
00:03:22.52 --> 00:03:27
What's the key fact,
the key positive fact about
54
00:03:27 --> 00:03:28
similar matrices?
55
00:03:28 --> 00:03:31
They have the same eigenvalues.
56
00:03:31 --> 00:03:33
Same eigenvalues.
57
00:03:33 --> 00:03:37
So if one of them grows,
the other one grows.
58
00:03:37 --> 00:03:44
If one of them decays to zero,
the other one decays to zero.
59
00:03:44 --> 00:03:49
Powers of A will look like
powers of B, because powers of A
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00:03:49 --> 00:03:55
and powers of B only differ by
an M inverse and an M way on the
61
00:03:55 --> 00:03:55
outside.
62
00:03:55 --> 00:04:00
So if these are similar,
then B to the k-th power is M
63
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inverse A to the k-th power M.
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00:04:04 --> 00:04:09
And that's why I say,
eh, this M, it does change the
65
00:04:09 --> 00:04:12
eigenvectors,
but it doesn't change the
66
00:04:12 --> 00:04:14
eigenvalues.
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So same lambdas.
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And then, finally,
I've got to review the point
69
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about the SVD,
the Singular Value
70
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Decomposition.
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OK.
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00:04:26 --> 00:04:29
So that's what this quiz has
got to cover,
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00:04:29 --> 00:04:34
and now I'll just take problems
from earlier exams,
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00:04:34 --> 00:04:37
starting with a differential
equation.
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OK.
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And always ready for questions.
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So here is an exam from about
the year zero,
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and it has a three by three.
79
00:04:48 --> 00:04:53
So that was -- but it's a
pretty special-looking matrix,
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it's got zeroes on the
diagonal, it's got minus ones
81
00:04:58 --> 00:05:02
above, and it's got plus ones
like that.
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00:05:02 --> 00:05:04
So that's the matrix A.
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00:05:04 --> 00:05:05
OK.
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00:05:05 --> 00:05:09
Step one is,
well, I want to solve that
85
00:05:09 --> 00:05:09
equation.
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00:05:09 --> 00:05:14
I want to find the general
solution.
87
00:05:14 --> 00:05:18
I haven't given you a u(0)
here, so I'm looking for the
88
00:05:18 --> 00:05:22
general solution,
so now what's the form of the
89
00:05:22 --> 00:05:23
general solution?
90
00:05:23 --> 00:05:27
With three arbitrary constants
going to be inside it,
91
00:05:27 --> 00:05:31
because those will be used to
match the initial condition.
92
00:05:31 --> 00:05:35
So the general form is u at
time t is some multiple of the
93
00:05:35 --> 00:05:38
first special solution.
94
00:05:38 --> 00:05:43
The first special solution will
be growing like the eigenvalue,
95
00:05:43 --> 00:05:46
and it's the eigenvector.
96
00:05:46 --> 00:05:51
So that's a pure exponential
solution, just staying with that
97
00:05:51 --> 00:05:52
eigenvector.
98
00:05:52 --> 00:05:57.19
Of course, I haven't found,
yet, the eigenvalues and
99
00:05:57.19 --> 00:05:58
eigenvectors.
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00:05:58 --> 00:06:02
That's, normally,
the first job.
101
00:06:02 --> 00:06:06
Now, there will be second one,
growing like e to the lambda
102
00:06:06 --> 00:06:10
two, and a third one growing
like e to the lambda three.
103
00:06:10 --> 00:06:15
So we're all done -- well,
we haven't done anything yet,
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00:06:15 --> 00:06:15
actually.
105
00:06:15 --> 00:06:19
I've got to find the
eigenvalues and eigenvectors,
106
00:06:19 --> 00:06:23
and then I would match u(0) by
choosing the right three
107
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constants.
108
00:06:25 --> 00:06:25
OK.
109
00:06:25 --> 00:06:30
So now I ask -- ask you about
the eigenvalues and
110
00:06:30 --> 00:06:35
eigenvectors,
and you look at this matrix and
111
00:06:35 --> 00:06:39
what do you see in that matrix?
112
00:06:39 --> 00:06:44
Um, well, I guess we might ask
ourselves right away,
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00:06:44 --> 00:06:46
is it singular?
114
00:06:46 --> 00:06:49
Is it singular?
115
00:06:49 --> 00:06:51
Because, if so,
then we really have a head
116
00:06:51 --> 00:06:54
start, we know one of the
eigenvalues is zero.
117
00:06:54 --> 00:06:56
Is that matrix singular?
118
00:06:56 --> 00:06:59
Eh, I don't know,
do you take the determinant to
119
00:06:59 --> 00:06:59
find out?
120
00:06:59 --> 00:07:03
Or maybe you look at the first
row and third row and say,
121
00:07:03 --> 00:07:06
hey, the first row and third
row are just opposite signs,
122
00:07:06 --> 00:07:09
they're linear-dependent?
123
00:07:09 --> 00:07:13
The first column and third
column are dependent -- it's
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00:07:13 --> 00:07:13
singular.
125
00:07:13 --> 00:07:15.74
So one eigenvalue is zero.
126
00:07:15.74 --> 00:07:17
Let's make that lambda one.
127
00:07:17 --> 00:07:20
Lambda one, then,
will be zero.
128
00:07:20 --> 00:07:20
OK.
129
00:07:20 --> 00:07:24
Now we've got a couple of other
eigenvalues to find,
130
00:07:24 --> 00:07:29
and, I suppose the simplest way
is to look at A minus lambda I
131
00:07:29 --> 00:07:35
So let me just put minus lambda
in here, minus ones above,
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00:07:35 --> 00:07:36
ones below.
133
00:07:36 --> 00:07:40
But, actually,
before I do it,
134
00:07:40 --> 00:07:44
that matrix is not symmetric,
for sure, right?
135
00:07:44 --> 00:07:49
In fact, it's the very opposite
of symmetric.
136
00:07:49 --> 00:07:55
That matrix A transpose,
how is A transpose connected to
137
00:07:55 --> 00:07:57
A?
138
00:07:57 --> 00:07:58
It's negative A.
139
00:07:58 --> 00:08:03
It's an anti-symmetric matrix,
skew-symmetric matrix.
140
00:08:03 --> 00:08:07
And we've met,
maybe, a two-by-two example of
141
00:08:07 --> 00:08:11
skew-symmetric matrices,
and let me just say,
142
00:08:11 --> 00:08:15
what's the deal with their
eigenvalues?
143
00:08:15 --> 00:08:18
They're pure imaginary.
144
00:08:18 --> 00:08:21
They'll be on the imaginary
axis, there be some multiple of
145
00:08:21 --> 00:08:24
I if it's an anti-symmetric,
skew-symmetric matrix.
146
00:08:24 --> 00:08:27
So I'm looking for multiples of
I, and of course,
147
00:08:27 --> 00:08:30
that's zero times I,
that's on the imaginary axis,
148
00:08:30 --> 00:08:32
but maybe I just do it out,
here.
149
00:08:32 --> 00:08:35.44
Lambda cubed.
well, maybe that's minus lambda
150
00:08:35.44 --> 00:08:38
cubed, and then a zero and a
zero.
151
00:08:38 --> 00:08:43
Zero, and then maybe I have a
plus a lambda,
152
00:08:43 --> 00:08:50
and another plus lambda,
but those go with a minus sign.
153
00:08:50 --> 00:08:55
Am I getting minus two lambda
equals zero?
154
00:08:55 --> 00:08:55.52
So.
155
00:08:55.52 --> 00:09:03.16
So I'm solving lambda cube plus
two lambda equals zero.
156
00:09:03.16 --> 00:09:08
So one root factors out lambda,
and the the rest is lambda
157
00:09:08 --> 00:09:10
squared plus two.
158
00:09:10 --> 00:09:10
OK.
159
00:09:10 --> 00:09:14
This is going the way we
expect, right?
160
00:09:14 --> 00:09:19
Because this gives the root
lambda equals zero,
161
00:09:19 --> 00:09:26
and gives the other two roots,
which are lambda equal what?
162
00:09:26 --> 00:09:31
The solutions of when is lambda
squared plus two equals zero
163
00:09:31 --> 00:09:35
then the eigenvalues those guys,
what are they?
164
00:09:35 --> 00:09:39
They're a multiple of i,
they're just square root of two
165
00:09:39 --> 00:09:39
i.
166
00:09:39 --> 00:09:44.91
When I set this equals to zero,
I have lambda squared equal to
167
00:09:44.91 --> 00:09:46.32
minus two, right?
168
00:09:46.32 --> 00:09:48
To make that zero?
169
00:09:48 --> 00:09:54
And the roots are square root
of two i and minus the square
170
00:09:54 --> 00:09:55
root of two i.
171
00:09:55 --> 00:09:58
So now I know what those are.
172
00:09:58 --> 00:10:00
I'll put those in,
now.
173
00:10:00 --> 00:10:03
Either the zero t is just a
one.
174
00:10:03 --> 00:10:04
That's just a one.
175
00:10:04 --> 00:10:10
This is square root of two I
and this is minus square root of
176
00:10:10 --> 00:10:12
two I.
177
00:10:12 --> 00:10:15
So, is the solution decaying to
zero?
178
00:10:15 --> 00:10:19
Is this a completely stable
problem where the solution is
179
00:10:19 --> 00:10:20
going to zero?
180
00:10:20 --> 00:10:21
No.
181
00:10:21 --> 00:10:25
In fact, all these things are
staying the same size.
182
00:10:25 --> 00:10:29
This thing is getting
multiplied by this number.
183
00:10:29 --> 00:10:33.35
e to the I something t,
that's a number that has
184
00:10:33.35 --> 00:10:37
magnitude one,
and sort of wanders around the
185
00:10:37 --> 00:10:39
unit circle.
186
00:10:39 --> 00:10:41
Same for this.
187
00:10:41 --> 00:10:48
So that the solution doesn't
blow up, and it doesn't go to
188
00:10:48 --> 00:10:48
zero.
189
00:10:48 --> 00:10:49
OK.
190
00:10:49 --> 00:10:55
And to find out what it
actually is, we would have to
191
00:10:55 --> 00:11:00
plug in initial conditions.
192
00:11:00 --> 00:11:03
But actually,
the next question I ask is,
193
00:11:03 --> 00:11:07
when does the solution return
to its initial value?
194
00:11:07 --> 00:11:10
I won't even say what's the
initial value.
195
00:11:10 --> 00:11:15
This is a case in which I think
this solution is periodic after.
196
00:11:15 --> 00:11:17
At t equals zero,
it starts with c1,
197
00:11:17 --> 00:11:22.14
c2, and c3, and then at some
value of t, it comes back to
198
00:11:22.14 --> 00:11:23
that.
199
00:11:23 --> 00:11:29.09
So that's a very special
question, Well,
200
00:11:29.09 --> 00:11:37
let's just take three seconds,
because that special question
201
00:11:37 --> 00:11:42.64
isn't likely to be on the quiz.
202
00:11:42.64 --> 00:11:46
But it comes back to the start,
when?
203
00:11:46 --> 00:11:51
Well, whenever we have e to the
two pi i, that's one,
204
00:11:51 --> 00:11:54
and we've come back again.
205
00:11:54 --> 00:11:57
So it comes back to the start.
206
00:11:57 --> 00:12:02
It's periodic,
when this square root of two i
207
00:12:02 --> 00:12:07
-- shall I call it capital T,
for the period?
208
00:12:07 --> 00:12:12
For that particular T,
if that equals two pi i,
209
00:12:12 --> 00:12:18
then e to this thing is one,
and we've come around again.
210
00:12:18 --> 00:12:21.93
So the period is T is
determined here,
211
00:12:21.93 --> 00:12:26.31
cancel the i-s,
and T is pi times the square
212
00:12:26.31 --> 00:12:27
root of two.
213
00:12:27 --> 00:12:30
So that's pretty neat.
214
00:12:30 --> 00:12:34
We get all the information
about all solutions,
215
00:12:34 --> 00:12:38.29
we haven't fixed on only one
particular solution,
216
00:12:38.29 --> 00:12:40
but it comes around again.
217
00:12:40 --> 00:12:44.99
So this was probably my first
chance to say something about
218
00:12:44.99 --> 00:12:47
the whole family of
anti-symmetric,
219
00:12:47 --> 00:12:50
skew-symmetric matrices.
220
00:12:50 --> 00:12:50
OK.
221
00:12:50 --> 00:12:55
And then, finally,
I asked, take two eigenvectors
222
00:12:55 --> 00:13:01
(again, I haven't computed the
eigenvectors) and it turns out
223
00:13:01 --> 00:13:03
they're orthogonal.
224
00:13:03 --> 00:13:04
They're orthogonal.
225
00:13:04 --> 00:13:10
The eigenvectors of a symmetric
matrix, or a skew-symmetric
226
00:13:10 --> 00:13:14
matrix, are always orthogonal.
227
00:13:14 --> 00:13:19
I guess may conscience makes me
tell you, what are all the
228
00:13:19 --> 00:13:23
matrices that have orthogonal
eigenvectors?
229
00:13:23 --> 00:13:27
And symmetric is the most
important class,
230
00:13:27 --> 00:13:31
so that's the one we've spoken
about.
231
00:13:31 --> 00:13:35
But let me just put that little
fact down, here.
232
00:13:35 --> 00:13:39
Orthogonal x-s.
eigenvectors.
233
00:13:39 --> 00:13:42
A matrix has orthogonal
eigenvectors,
234
00:13:42 --> 00:13:47
the exact condition -- it's
quite beautiful that I can tell
235
00:13:47 --> 00:13:49
you exactly when that happens.
236
00:13:49 --> 00:13:54
It happens when A times A
transpose equals A transpose
237
00:13:54 --> 00:13:54
times A.
238
00:13:54 --> 00:14:00
Any time that's the condition
for orthogonal eigenvectors.
239
00:14:00 --> 00:14:04
And because we're interested in
special families of vectors,
240
00:14:04 --> 00:14:07
tell me some special families
that fit.
241
00:14:07 --> 00:14:09
This is the whole requirement.
242
00:14:09 --> 00:14:13
That's a pretty special
requirement most matrices have.
243
00:14:13 --> 00:14:17
So the average three-by-three
matrix has three eigenvectors,
244
00:14:17 --> 00:14:19
but not orthogonal.
245
00:14:19 --> 00:14:24
But if it happens to commute
with its transpose,
246
00:14:24 --> 00:14:29
then, wonderfully,
the eigenvectors are
247
00:14:29 --> 00:14:30.34
orthogonal.
248
00:14:30.34 --> 00:14:36
Now, do you see how symmetric
matrices pass this test?
249
00:14:36 --> 00:14:38
Of course.
250
00:14:38 --> 00:14:42
If A transpose equals A,
then both sides are A squared,
251
00:14:42 --> 00:14:43
we've got it.
252
00:14:43 --> 00:14:47
How do anti-symmetric matrices
pass this test?
253
00:14:47 --> 00:14:51
If A transpose equals minus A,
then we've got it again,
254
00:14:51 --> 00:14:55
because we've got minus A
squared on both sides.
255
00:14:55 --> 00:14:58.24
So that's another group.
256
00:14:58.24 --> 00:15:01
And finally,
let me ask you about our other
257
00:15:01 --> 00:15:04
favorite family,
orthogonal matrices.
258
00:15:04 --> 00:15:09
Do orthogonal matrices pass
this test, if A is a Q,
259
00:15:09 --> 00:15:13
do they pass the test for
orthogonal eigenvectors.
260
00:15:13 --> 00:15:16
Well, if A is Q,
an orthogonal matrix,
261
00:15:16 --> 00:15:18
what is Q transpose Q?
262
00:15:18 --> 00:15:19
It's I.
263
00:15:19 --> 00:15:22
And what is Q Q transpose?
264
00:15:22 --> 00:15:26
It's I, we're talking square
matrices here.
265
00:15:26 --> 00:15:29
So yes, it passes the test.
266
00:15:29 --> 00:15:34
So the special cases are
symmetric, anti-symmetric (I'll
267
00:15:34 --> 00:15:38
say skew-symmetric,) and
orthogonal.
268
00:15:38 --> 00:15:44
Those are the three important
special classes that are in this
269
00:15:44 --> 00:15:45
family.
270
00:15:45 --> 00:15:46
OK.
271
00:15:46 --> 00:15:52
That's like a comment that,
could have been made back in,
272
00:15:52 --> 00:15:54
section six point four.
273
00:15:54 --> 00:15:59
OK, I can pursue the
differential equations,
274
00:15:59 --> 00:16:03
also this question,
didn't ask you to tell me,
275
00:16:03 --> 00:16:09
how would I find this matrix
exponential, e to the At?
276
00:16:09 --> 00:16:12
So can I erase this?
277
00:16:12 --> 00:16:16.33
I'll just stay with this
same...
278
00:16:16.33 --> 00:16:19
how would I find e to the At?
279
00:16:19 --> 00:16:23.09
Because, how does that come in?
280
00:16:23.09 --> 00:16:28
That's the key matrix for a
differential equation,
281
00:16:28 --> 00:16:36
because the solution is -- the
solution is u(t) is e^(At) u(0).
282
00:16:36 --> 00:16:41
So this is like the fundamental
matrix that multiplies the given
283
00:16:41 --> 00:16:44
function and gives the answer.
284
00:16:44 --> 00:16:47
And how would we compute it if
we wanted that?
285
00:16:47 --> 00:16:52
We don't always have to find e
to the At, because I can go
286
00:16:52 --> 00:16:56
directly to the answer without
any e to the At-s,
287
00:16:56 --> 00:17:02
but hiding here is an e to the
At, and how would I compute it?
288
00:17:02 --> 00:17:05.66
Well, if A is diagonalizable.
289
00:17:05.66 --> 00:17:11
So I'm now going to put in my
usual if A can be diagonalized
290
00:17:11 --> 00:17:16
(and everybody remember that
there is an if there,
291
00:17:16 --> 00:17:21
because it might not have
enough eigenvectors) this
292
00:17:21 --> 00:17:27
example does have enough,
random matrices have enough.
293
00:17:27 --> 00:17:33
So if we can diagonalize,
then we get a nice formula for
294
00:17:33 --> 00:17:38
this, because an S comes way out
at the beginning,
295
00:17:38 --> 00:17:44
and S inverse comes way out at
the end, and we only have to
296
00:17:44 --> 00:17:48
take the exponential of lambda.
297
00:17:48 --> 00:17:52
And that's just a diagonal
matrix, so that's just e the
298
00:17:52 --> 00:17:55
lambda one t,
these guys are showing up,
299
00:17:55 --> 00:17:58
now, in e to the lambda nt.
300
00:17:58 --> 00:17:58.27
OK?
301
00:17:58.27 --> 00:18:01
That's a really quick review of
that formula.
302
00:18:01 --> 00:18:06
It's something we can compute
it quickly if we have done the S
303
00:18:06 --> 00:18:07
and lambda part.
304
00:18:07 --> 00:18:12
If we know S and lambda,
then it's not hard to take that
305
00:18:12 --> 00:18:13
step.
306
00:18:13 --> 00:18:17
OK, that's some comments on
differential equations.
307
00:18:17 --> 00:18:21
I would like to go on to a next
question that I started here.
308
00:18:21 --> 00:18:25
And it's, got several parts,
and I can just read it out.
309
00:18:25 --> 00:18:28
What we're given is a
three-by-three matrix,
310
00:18:28 --> 00:18:32
and we're told its eigenvalues,
except one of these is,
311
00:18:32 --> 00:18:35
like, we don't know,
and we're told the
312
00:18:35 --> 00:18:37
eigenvectors.
313
00:18:37 --> 00:18:41
And I want to ask you about the
matrix.
314
00:18:41 --> 00:18:41
OK.
315
00:18:41 --> 00:18:43
So, first question.
316
00:18:43 --> 00:18:46
Is the matrix diagonalizable?
317
00:18:46 --> 00:18:51.19
And I really mean for which c,
because I don't know c,
318
00:18:51.19 --> 00:18:56
so my questions will all be,
for which is there a condition
319
00:18:56 --> 00:18:59
on c, does one c work.
320
00:18:59 --> 00:19:06
But your answer should tell me
all the c-s that work.
321
00:19:06 --> 00:19:13
I'm not asking for you to tell
me, well, c equal four,
322
00:19:13 --> 00:19:16
yes, that checks out.
323
00:19:16 --> 00:19:23
I want to know all the c-s that
make it diagonalizable.
324
00:19:23 --> 00:19:24
OK?
325
00:19:24 --> 00:19:28
What's the real on
diagonalizable?
326
00:19:28 --> 00:19:32
We need enough eigenvectors,
right?
327
00:19:32 --> 00:19:38
We don't care what those
eigenvalues are,
328
00:19:38 --> 00:19:43
it's eigenvectors that count
for
329
00:19:43 --> 00:19:47.1
diagonalizable,
and we need three independent
330
00:19:47.1 --> 00:19:50
ones, and are those three guys
independent?
331
00:19:50 --> 00:19:50
Yes.
332
00:19:50 --> 00:19:53
Actually, let's look at them
for a moment.
333
00:19:53 --> 00:19:57
What do you see about those
three vectors right away?
334
00:19:57 --> 00:19:59
They're more than independent.
335
00:19:59 --> 00:20:03
Can you see why those three got
chosen?
336
00:20:03 --> 00:20:08
Because it will come up in the
next part, they're orthogonal.
337
00:20:08 --> 00:20:11
Those eigenvectors are
orthogonal.
338
00:20:11 --> 00:20:13
They're certainly independent.
339
00:20:13 --> 00:20:17
So the answer to diagonalizable
is, yes, all c,
340
00:20:17 --> 00:20:18
all c.
341
00:20:18 --> 00:20:21
Doesn't matter.
c could be a repeated guy,
342
00:20:21 --> 00:20:24
but we've got enough
eigenvectors,
343
00:20:24 --> 00:20:28
so that's what we care about.
344
00:20:28 --> 00:20:30
OK, second question.
345
00:20:30 --> 00:20:34.65
For which values of c is it
symmetric?
346
00:20:34.65 --> 00:20:38.43
OK, what's the answer to that
one?
347
00:20:38.43 --> 00:20:44
If we know the same setup if we
know that much about it,
348
00:20:44 --> 00:20:50
we know those eigenvectors,
and we've noticed they're
349
00:20:50 --> 00:20:56
orthogonal, then which c-s will
work?
350
00:20:56 --> 00:21:02
So the eigenvalues of that
symmetric matrix have to be
351
00:21:02 --> 00:21:02
real.
352
00:21:02 --> 00:21:04
So all real c.
353
00:21:04 --> 00:21:10
If c was i, the matrix wouldn't
have been symmetric.
354
00:21:10 --> 00:21:15
But if c is a real number,
then we've got real
355
00:21:15 --> 00:21:21
eigenvalues, we've got
orthogonal eigenvectors,
356
00:21:21 --> 00:21:25.59
that matrix is symmetric.
357
00:21:25.59 --> 00:21:27
OK, positive definite.
358
00:21:27 --> 00:21:31
OK, now this is a sub-case of
symmetric, so we need c to be
359
00:21:31 --> 00:21:35
real, so we've got a symmetric
matrix, but we also want the
360
00:21:35 --> 00:21:37
thing to be positive definite.
361
00:21:37 --> 00:21:41.31
Now, we're looking at
eigenvalues, we've got a lot of
362
00:21:41.31 --> 00:21:44
tests for positive definite,
but eigenvalues,
363
00:21:44 --> 00:21:47
if we know them,
is certainly a good,
364
00:21:47 --> 00:21:49.24
quick, clean test.
365
00:21:49.24 --> 00:21:52
Could this matrix be positive
definite?
366
00:21:52 --> 00:21:53
No.
367
00:21:53 --> 00:21:56
No, because it's got an
eigenvalue zero.
368
00:21:56 --> 00:21:59
It could be positive
semi-definite,
369
00:21:59 --> 00:22:05
you know, like consolation
prize, if c was greater or equal
370
00:22:05 --> 00:22:10
to zero, it would be positive
semi-definite.
371
00:22:10 --> 00:22:12
But it's not,
no.
372
00:22:12 --> 00:22:18.79
Semi-definite,
if I put that comment in,
373
00:22:18.79 --> 00:22:25
semi-definite,
that the condition would be c
374
00:22:25 --> 00:22:29
greater or equal to zero.
375
00:22:29 --> 00:22:32
That would be all right.
376
00:22:32 --> 00:22:33
OK.
377
00:22:33 --> 00:22:34
Next part.
378
00:22:34 --> 00:22:38
Is it a Markov matrix?
379
00:22:38 --> 00:22:40
Hm.
380
00:22:40 --> 00:22:46
Could this matrix be,
if I choose the number c
381
00:22:46 --> 00:22:49
correctly, a Markov matrix?
382
00:22:49 --> 00:22:55
Well, what do we know about
Markov matrices?
383
00:22:55 --> 00:23:01
Mainly, we know something about
their eigenvalues.
384
00:23:01 --> 00:23:08
One eigenvalue is always one,
and the other eigenvalues are
385
00:23:08 --> 00:23:09
smaller.
386
00:23:09 --> 00:23:12
Not larger.
387
00:23:12 --> 00:23:15
So an eigenvalue two can't
happen.
388
00:23:15 --> 00:23:19
So the answer is,
no, not a ma- that's never a
389
00:23:19 --> 00:23:20
Markov matrix.
390
00:23:20 --> 00:23:20.73
OK?
391
00:23:20.73 --> 00:23:23
And finally,
could one half of A be a
392
00:23:23 --> 00:23:25.36
projection matrix?
393
00:23:25.36 --> 00:23:29
So could it- could this --
eh-eh could this be twice a
394
00:23:29 --> 00:23:31
projection matrix?
395
00:23:31 --> 00:23:33
So let me write it this way.
396
00:23:33 --> 00:23:38
Could A over two be a
projection matrix?
397
00:23:38 --> 00:23:41
OK, what are projection
matrices?
398
00:23:41 --> 00:23:42
They're real.
399
00:23:42 --> 00:23:48
I mean, th- they're symmetric,
so their eigenvalues are real.
400
00:23:48 --> 00:23:52
But more than that,
we know what those eigenvalues
401
00:23:52 --> 00:23:53
have to be.
402
00:23:53 --> 00:24:00
What do the eigenvalues of a
projection matrix have to be?
403
00:24:00 --> 00:24:03
See, that any nice matrix we've
got an idea about its
404
00:24:03 --> 00:24:04
eigenvalues.
405
00:24:04 --> 00:24:07
So the eigenvalues of
projection matrices are zero and
406
00:24:07 --> 00:24:07
one.
407
00:24:07 --> 00:24:09
Zero and one,
only.
408
00:24:09 --> 00:24:12
Because P squared equals P,
let me call this matrix P,
409
00:24:12 --> 00:24:15
so P squared equals P,
so lambda squared equals
410
00:24:15 --> 00:24:19
lambda, because eigenvalues of P
squared are lambda squared,
411
00:24:19 --> 00:24:23
and we must have that,
so lambda equals zero or one.
412
00:24:23 --> 00:24:23.88
OK.
413
00:24:23.88 --> 00:24:30
Now what value of c will work
there?
414
00:24:30 --> 00:24:39
So, then, there are some value
that will work,
415
00:24:39 --> 00:24:47
and what will work?
c equals zero will work,
416
00:24:47 --> 00:24:54
or what else will work?
c equal to two.
417
00:24:54 --> 00:24:58
Because if c is two,
then when we divide by two,
418
00:24:58 --> 00:25:02
this Eigenvalue of two will
drop to one, and so will the
419
00:25:02 --> 00:25:04
other one, so,
or c equal to two.
420
00:25:04 --> 00:25:08
OK, those are the guys that
will work, and it was the fact
421
00:25:08 --> 00:25:12
that those eigenvectors were
orthogonal, the fact that those
422
00:25:12 --> 00:25:16.63
eigenvectors were orthogonal
carried us a lot of the way,
423
00:25:16.63 --> 00:25:17
here.
424
00:25:17 --> 00:25:22
If they weren't orthogonal,
then symmetric would have been
425
00:25:22 --> 00:25:26
dead, positive definite would
have been dead,
426
00:25:26 --> 00:25:29
projection would have been
dead.
427
00:25:29 --> 00:25:34.08
But those eigenvectors were
orthogonal, so it came down to
428
00:25:34.08 --> 00:25:36
the eigenvalues.
429
00:25:36 --> 00:25:44
OK, that was like a chance to
review a lot of this chapter.
430
00:25:44 --> 00:25:51.87
Shall I jump to the singular
value decomposition,
431
00:25:51.87 --> 00:25:58
then, as the third,
topic for, for the review?
432
00:25:58 --> 00:26:04
OK, so I'm going to.
jump to this.
433
00:26:04 --> 00:26:04
OK.
434
00:26:04 --> 00:26:10
So this is the singular value
decomposition,
435
00:26:10 --> 00:26:14
known to everybody as the SVD.
436
00:26:14 --> 00:26:22
And that's a factorization of A
into orthogonal times diagonal
437
00:26:22 --> 00:26:24
times orthogonal.
438
00:26:24 --> 00:26:31
And we always call those U and
sigma and V transpose.
439
00:26:31 --> 00:26:32
OK.
440
00:26:32 --> 00:26:38
And the key to that -- this is
for every matrix,
441
00:26:38 --> 00:26:42
every A, every A.
442
00:26:42 --> 00:26:45
Rectangular,
doesn't matter,
443
00:26:45 --> 00:26:49
whatever, has this
decomposition.
444
00:26:49 --> 00:26:52
So it's really important.
445
00:26:52 --> 00:26:58
And the key to it is to look at
things like A transpose A.
446
00:26:58 --> 00:27:05
Can we remember what happens
with A transpose A?
447
00:27:05 --> 00:27:12
If I just transpose that I get
V sigma transpose U transpose,
448
00:27:12 --> 00:27:17
that's multiplying A,
which is U, sigma V transpose,
449
00:27:17 --> 00:27:24
and the result is V on the
outside, s- U transpose U is the
450
00:27:24 --> 00:27:30
identity, because it's an
orthogonal matrix.
451
00:27:30 --> 00:27:35
So I'm just left with sigma
transpose sigma in the middle,
452
00:27:35 --> 00:27:39
that's a diagonal,
possibly rectangular diagonal
453
00:27:39 --> 00:27:42
by its transpose,
so the result,
454
00:27:42 --> 00:27:46
this is orthogonal,
diagonal, orthogonal.
455
00:27:46 --> 00:27:50
So, I guess,
actually, this is the SVD for A
456
00:27:50 --> 00:27:51
transpose A.
457
00:27:51 --> 00:27:57
Here I see orthogonal,
diagonal, and orthogonal.
458
00:27:57 --> 00:27:57
Great.
459
00:27:57 --> 00:28:01.54
But a little more is happening.
460
00:28:01.54 --> 00:28:05
For A transpose A,
the difference is,
461
00:28:05 --> 00:28:09
the orthogonal guys are the
same.
462
00:28:09 --> 00:28:12
It's V and V transpose.
463
00:28:12 --> 00:28:14
What I seeing here?
464
00:28:14 --> 00:28:20
I'm seeing the factorization
for a symmetric matrix.
465
00:28:20 --> 00:28:25
This thing is symmetric.
466
00:28:25 --> 00:28:28
So in a symmetric case,
U is the same as V.
467
00:28:28 --> 00:28:31
U is the same as V for this
symmetric matrix,
468
00:28:31 --> 00:28:34
and, of course,
we see it happening.
469
00:28:34 --> 00:28:34
OK.
470
00:28:34 --> 00:28:37
So that tells us,
right away, what V is.
471
00:28:37 --> 00:28:40
V is the eigenvector matrix for
A transpose A.
472
00:28:40 --> 00:28:40
OK.
473
00:28:40 --> 00:28:44
Now, if you were here when I
lectured about this topic,
474
00:28:44 --> 00:28:48
when I gave the topic on
singular value decompositions,
475
00:28:48 --> 00:28:53.22
you'll remember that I got into
trouble.
476
00:28:53.22 --> 00:29:00.65
I'm sorry to remember that
myself, but it happened.
477
00:29:00.65 --> 00:29:01
OK.
478
00:29:01 --> 00:29:03
How did it happen?
479
00:29:03 --> 00:29:10
I was in great shape for a
while, cruising along.
480
00:29:10 --> 00:29:17
So I found the eigenvectors for
A transpose A.
481
00:29:17 --> 00:29:19
Good.
482
00:29:19 --> 00:29:23
I found the singular values,
what were they?
483
00:29:23 --> 00:29:25
What were the singular values?
484
00:29:25 --> 00:29:29
The singular value number i,
or -- these are the guys in
485
00:29:29 --> 00:29:33
sigma -- this is diagonal with
the number sigma in it.
486
00:29:33 --> 00:29:37
This diagonal is sigma one,
sigma two, up to the rank,
487
00:29:37 --> 00:29:41
sigma r, those are the non-zero
ones.
488
00:29:41 --> 00:29:45
So I found those,
and what are they?
489
00:29:45 --> 00:29:47
Remind me about that?
490
00:29:47 --> 00:29:52.9
Well, here, I'm seeing them
squared, so their squares are
491
00:29:52.9 --> 00:29:56
the eigenvalues of A transpose
A.
492
00:29:56 --> 00:29:56
Good.
493
00:29:56 --> 00:30:02
So I just take the square root,
if I want the eigenvalues of A
494
00:30:02 --> 00:30:08
transpose --
If I want the sigmas and I know
495
00:30:08 --> 00:30:13
these, I take the square root,
the positive square root.
496
00:30:13 --> 00:30:14
OK.
497
00:30:14 --> 00:30:17
Where did I run into trouble?
498
00:30:17 --> 00:30:20
Well, then, my final step was
to find U.
499
00:30:20 --> 00:30:23
And I didn't read the book.
500
00:30:23 --> 00:30:28
So, I did something that was
practically right,
501
00:30:28 --> 00:30:33
but --
well, I guess practically right
502
00:30:33 --> 00:30:35
is not quite the same.
503
00:30:35 --> 00:30:41.1
OK, so I thought,
OK, I'll look at A A transpose.
504
00:30:41.1 --> 00:30:45
What happened when I looked at
A A transpose?
505
00:30:45 --> 00:30:51
Let me just put it here,
and then I can feel it.
506
00:30:51 --> 00:30:55.14
OK, so here's A A transpose.
507
00:30:55.14 --> 00:31:00
So that's U sigma V transpose,
that's A, and then the
508
00:31:00 --> 00:31:06.49
transpose is V sigma transpose,
U sigma transpose.
509
00:31:06.49 --> 00:31:07
Fine.
510
00:31:07 --> 00:31:12.11
And then, in the middle is the
identity again,
511
00:31:12.11 --> 00:31:14
so it looks great.
512
00:31:14 --> 00:31:18
U sigma sigma transpose,
U transpose.
513
00:31:18 --> 00:31:19
Fine.
514
00:31:19 --> 00:31:25.05
All good, and now these columns
of U are the eigenvectors,
515
00:31:25.05 --> 00:31:29
that's U is the eigenvector
matrix for this guy.
516
00:31:29 --> 00:31:32
That was correct,
so I did that fine.
517
00:31:32 --> 00:31:35
Where did something go wrong?
518
00:31:35 --> 00:31:36.62
A sign went wrong.
519
00:31:36.62 --> 00:31:41
A sign went wrong because --
and now -- now I see,
520
00:31:41 --> 00:31:44
actually, somebody told me
right after class,
521
00:31:44 --> 00:31:48
we can't tell from this
description which sign to give
522
00:31:48 --> 00:31:50.23
the eigenvectors.
523
00:31:50.23 --> 00:31:53
If these are the eigenvectors
of this matrix,
524
00:31:53 --> 00:31:57
well, if you give me an
eigenvector and I change all its
525
00:31:57 --> 00:32:01
signs, we've still got another
eigenvector.
526
00:32:01 --> 00:32:05
So what I wasn't able to
determine (and I had a
527
00:32:05 --> 00:32:09
fifty-fifty change and life let
me down,) the signs I just
528
00:32:09 --> 00:32:11
happened to pick for the
eigenvectors,
529
00:32:11 --> 00:32:14
one of them I should have
reversed the sign.
530
00:32:14 --> 00:32:17
So, from this,
I can't tell whether the
531
00:32:17 --> 00:32:22
eigenvector or its negative is
the right one to use in there.
532
00:32:22 --> 00:32:28
So the right way to do it is
to, having settled on the signs,
533
00:32:28 --> 00:32:32.26
the Vs also,
I don't know which sign to
534
00:32:32.26 --> 00:32:34
choose, but I choose one.
535
00:32:34 --> 00:32:36
I choose one.
536
00:32:36 --> 00:32:41
And then, instead,
I should have used the one that
537
00:32:41 --> 00:32:47
tells me what sign to choose,
the rule that A times a V is
538
00:32:47 --> 00:32:50
sigma times the U.
539
00:32:50 --> 00:32:55
So, having decided on the V,
I multiply by A,
540
00:32:55 --> 00:33:02
I'll notice the factor sigma
coming out, and there will be a
541
00:33:02 --> 00:33:08
unit vector there,
and I now know exactly what it
542
00:33:08 --> 00:33:14
is, and not only up to a change
of sign.
543
00:33:14 --> 00:33:19
So that's the good and,
of course, this is the main
544
00:33:19 --> 00:33:21
point about the SVD.
545
00:33:21 --> 00:33:25
That's the point that we've
diagonalized,
546
00:33:25 --> 00:33:31
that's A times the matrix of Vs
equals U times the diagonal
547
00:33:31 --> 00:33:33
matrix of sigmas.
548
00:33:33 --> 00:33:36
That's the same as that.
549
00:33:36 --> 00:33:36
OK.
550
00:33:36 --> 00:33:40
So that's, like,
correcting the wrong sign from
551
00:33:40 --> 00:33:42
that earlier lecture.
552
00:33:42 --> 00:33:48
And that would complete that,
so that's how you would compute
553
00:33:48 --> 00:33:48
the SVD.
554
00:33:48 --> 00:33:52
Now, on the quiz,
I going to ask -- well,
555
00:33:52 --> 00:33:55
maybe on the final.
556
00:33:55 --> 00:33:58
So we've got quiz and final
ahead.
557
00:33:58 --> 00:34:03
Sometimes, you might be asked
to find the SVD if I give you
558
00:34:03 --> 00:34:08
the matrix -- let me come back,
now, to the main board -- or,
559
00:34:08 --> 00:34:11
I might give you the pieces.
560
00:34:11 --> 00:34:15
And I might ask you something
about the matrix.
561
00:34:15 --> 00:34:18
For example,
suppose I ask you,
562
00:34:18 --> 00:34:21
oh, let's say,
if I tell you what sigma is --
563
00:34:21 --> 00:34:23
OK.
564
00:34:23 --> 00:34:27
Let's take one example.
565
00:34:27 --> 00:34:36
Suppose sigma is -- so all
that's how we would compute
566
00:34:36 --> 00:34:37
them.
567
00:34:37 --> 00:34:43.26
But now, suppose I give you
these.
568
00:34:43.26 --> 00:34:52
Suppose I give you sigma is,
say, three two.
569
00:34:52 --> 00:34:55
And I tell you that U has a
couple of columns,
570
00:34:55 --> 00:34:57
and V has a couple of columns.
571
00:34:57 --> 00:34:58
OK.
572
00:34:58 --> 00:35:02
Those are orthogonal columns,
of course, because U and V are
573
00:35:02 --> 00:35:03
orthogonal.
574
00:35:03 --> 00:35:07
I'm just sort of,
like, getting you to think
575
00:35:07 --> 00:35:10
about the SVD,
because we only had that one
576
00:35:10 --> 00:35:12
lecture about it,
and one homework,
577
00:35:12 --> 00:35:17.44
and, what kind of a matrix have
I got here?
578
00:35:17.44 --> 00:35:21
What do I know about this
matrix?
579
00:35:21 --> 00:35:27
All I really know right now is
that its singular values,
580
00:35:27 --> 00:35:34.01
those sigmas are three and two,
and the only thing interesting
581
00:35:34.01 --> 00:35:40
that I can see in that is that
they're not zero.
582
00:35:40 --> 00:35:43
I know that this matrix is
non-singular,
583
00:35:43 --> 00:35:44
right?
584
00:35:44 --> 00:35:47
That's invertible,
I don't have any zero
585
00:35:47 --> 00:35:52
eigenvalues, and zero singular
values, that's invertible,
586
00:35:52 --> 00:35:57
there's a typical SVD for a
nice two-by-two non-singular
587
00:35:57 --> 00:36:00
invertible good matrix.
588
00:36:00 --> 00:36:05
If I actually gave you a
matrix, then you'd have to find
589
00:36:05 --> 00:36:08
the Us and the Vs as we just
spoke.
590
00:36:08 --> 00:36:09
But, there.
591
00:36:09 --> 00:36:13
Now, what if the two wasn't a
two but it was -- well,
592
00:36:13 --> 00:36:18
let me make an extreme case,
here -- suppose it was minus
593
00:36:18 --> 00:36:19
five.
594
00:36:19 --> 00:36:21.43
That's wrong,
right away.
595
00:36:21.43 --> 00:36:24
That's not a singular value
decomposition,
596
00:36:24 --> 00:36:25
right?
597
00:36:25 --> 00:36:27
The singular values are not
negative.
598
00:36:27 --> 00:36:31
So that's not a singular value
decomposition,
599
00:36:31 --> 00:36:32
and forget it.
600
00:36:32 --> 00:36:32
OK.
601
00:36:32 --> 00:36:35
So let me ask you about that
one.
602
00:36:35 --> 00:36:39
What can you tell me about that
matrix?
603
00:36:39 --> 00:36:41
It's singular,
right?
604
00:36:41 --> 00:36:47
It's got a singular matrix
there in the middle,
605
00:36:47 --> 00:36:51
and, let's see,
so, OK, it's singular,
606
00:36:51 --> 00:36:55
maybe you can tell me,
its rank?
607
00:36:55 --> 00:36:58
What's the rank of A?
608
00:36:58 --> 00:37:03.46
It's clearly -- somebody just
say it -- one,
609
00:37:03.46 --> 00:37:05
thanks.
610
00:37:05 --> 00:37:08
The rank is one,
so the null space,
611
00:37:08 --> 00:37:12
what's the dimension of the
null space?
612
00:37:12 --> 00:37:12.68
One.
613
00:37:12.68 --> 00:37:13
Right?
614
00:37:13 --> 00:37:18
We've got a two-by-two matrix
of rank one, so of all that
615
00:37:18 --> 00:37:23
stuff from the beginning of the
course is still with us.
616
00:37:23 --> 00:37:27
The dimensions of those
fundamental spaces is still
617
00:37:27 --> 00:37:31
central, and a basis for them.
618
00:37:31 --> 00:37:37
Now, can you tell me a vector
that's in the null space?
619
00:37:37 --> 00:37:43
And then that will be my last
point to make about the SVD.
620
00:37:43 --> 00:37:49
Can you tell me a vector that's
in the null space?
621
00:37:49 --> 00:37:55.15
So what would I multiply by and
get zero, here?
622
00:37:55.15 --> 00:37:58
I think the answer is probably
v2.
623
00:37:58 --> 00:38:03
I think probably v2 is in the
null space, because I think that
624
00:38:03 --> 00:38:08
must be the eigenvector going
with this zero eigenvalue.
625
00:38:08 --> 00:38:09
Yes.
626
00:38:09 --> 00:38:10
Have a look at that.
627
00:38:10 --> 00:38:16
And I could ask you the null
space of A transpose.
628
00:38:16 --> 00:38:19
And I could ask you the column
space.
629
00:38:19 --> 00:38:21
All that stuff.
630
00:38:21 --> 00:38:25.26
Everything is sitting there in
the SVD.
631
00:38:25.26 --> 00:38:29
The SVD takes a little more
time to compute,
632
00:38:29 --> 00:38:34
but it displays all the good
stuff about a matrix.
633
00:38:34 --> 00:38:34
OK.
634
00:38:34 --> 00:38:38
Any question about the SVD?
635
00:38:38 --> 00:38:44
Let me keep going with further
topics.
636
00:38:44 --> 00:38:47
Now, let's see.
637
00:38:47 --> 00:38:56
Similar matrices we've talked
about, let me see if I've got
638
00:38:56 --> 00:38:58.91
another, -- OK.
639
00:38:58.91 --> 00:39:05
Here's a true false,
so we can do that,
640
00:39:05 --> 00:39:06
easily.
641
00:39:06 --> 00:39:06
So.
642
00:39:06 --> 00:39:11
Question, A given.
643
00:39:11 --> 00:39:14
A is symmetric and orthogonal.
644
00:39:14 --> 00:39:15
OK.
645
00:39:15 --> 00:39:21
So beautiful matrices like that
don't come along every day.
646
00:39:21 --> 00:39:26
But what can we say first about
its eigenvalues?
647
00:39:26 --> 00:39:28
Actually, of course.
648
00:39:28 --> 00:39:34
Here are our two most important
classes of matrices,
649
00:39:34 --> 00:39:39
and we're looking at the
intersection.
650
00:39:39 --> 00:39:44
So those really are neat
matrices, and what can you tell
651
00:39:44 --> 00:39:48
me about what could the possible
eigenvalues be?
652
00:39:48 --> 00:39:51
Eigenvalues can be what?
653
00:39:51 --> 00:39:55.55
What do I know about the
eigenvalues of a symmetric
654
00:39:55.55 --> 00:39:56.17
matrix?
655
00:39:56.17 --> 00:39:57
Lambda is real.
656
00:39:57 --> 00:40:02
What do I know about the
eigenvalues of an orthogonal
657
00:40:02 --> 00:40:03
matrix?
658
00:40:03 --> 00:40:04
Ha.
659
00:40:04 --> 00:40:05
Maybe nothing.
660
00:40:05 --> 00:40:08.23
But, no, that can't be.
661
00:40:08.23 --> 00:40:14
What do I know about the
eigenvalues of an orthogonal
662
00:40:14 --> 00:40:14
matrix?
663
00:40:14 --> 00:40:17
Well, what feels right?
664
00:40:17 --> 00:40:23
Basing mathematics on just a
little gut instinct here,
665
00:40:23 --> 00:40:29
the eigenvalues of an
orthogonal matrix ought to have
666
00:40:29 --> 00:40:31
magnitude one.
667
00:40:31 --> 00:40:38
Orthogonal matrices are like
rotations, they're not changing
668
00:40:38 --> 00:40:44
the length, so orthogonal,
the eigenvalues are one.
669
00:40:44 --> 00:40:47.16
Let me just show you why.
670
00:40:47.16 --> 00:40:47
Why?
671
00:40:47 --> 00:40:52
So the matrix,
can I call it Q for orthogonal
672
00:40:52 --> 00:40:55
for the moment?
673
00:40:55 --> 00:40:59
If I look at Q x equal lambda
x, how do I see that this thing
674
00:40:59 --> 00:41:01
has magnitude one?
675
00:41:01 --> 00:41:03
I take the length of both
sides.
676
00:41:03 --> 00:41:06
This is taking lengths,
taking lengths,
677
00:41:06 --> 00:41:10
this is whatever the magnitude
is times the length of x.
678
00:41:10 --> 00:41:14
And what's the length of Q x if
Q is an orthogonal matrix?
679
00:41:14 --> 00:41:16
This is something you should
know.
680
00:41:16 --> 00:41:20
It's the same as the length of
x.
681
00:41:20 --> 00:41:24
Orthogonal matrices don't
change lengths.
682
00:41:24 --> 00:41:26
So lambda has to be one.
683
00:41:26 --> 00:41:27
Right.
684
00:41:27 --> 00:41:27
OK.
685
00:41:27 --> 00:41:33
That's worth committing to
memory, that could show up
686
00:41:33 --> 00:41:33.99
again.
687
00:41:33.99 --> 00:41:34
OK.
688
00:41:34 --> 00:41:38
So what's the answer now to
this question,
689
00:41:38 --> 00:41:42
what can the eigenvalues be?
690
00:41:42 --> 00:41:48
There's only two possibilities,
and they are one and the other
691
00:41:48 --> 00:41:52
one, the other possibility is
negative one,
692
00:41:52 --> 00:41:56
right, because these have the
right magnitude,
693
00:41:56 --> 00:41:58
and they're real.
694
00:41:58 --> 00:41:58
OK.
695
00:41:58 --> 00:41:59
TK.
true -- OK.
696
00:41:59 --> 00:42:01
True or false?
697
00:42:01 --> 00:42:04
A is sure to be positive
definite.
698
00:42:04 --> 00:42:08
Well, this is a great matrix,
but
699
00:42:08 --> 00:42:11
is it sure to be positive
definite?
700
00:42:11 --> 00:42:11
No.
701
00:42:11 --> 00:42:16.12
If it could have an eigenvalue
minus one, it wouldn't be
702
00:42:16.12 --> 00:42:17.58
positive definite.
703
00:42:17.58 --> 00:42:21
True or false,
it has no repeated eigenvalues.
704
00:42:21 --> 00:42:22
That's false,
too.
705
00:42:22 --> 00:42:27
In fact, it's going to have
repeated eigenvalues if it's as
706
00:42:27 --> 00:42:29.84
big
as three by three,
707
00:42:29.84 --> 00:42:33.6
one of these c- one of these,
at least, will have to get
708
00:42:33.6 --> 00:42:34
repeated.
709
00:42:34 --> 00:42:34
Sure.
710
00:42:34 --> 00:42:37
So it's got repeated
eigenvalues, but,
711
00:42:37 --> 00:42:38
is it diagonalizable?
712
00:42:38 --> 00:42:41
It's got these many,
many, repeated eigenvalues.
713
00:42:41 --> 00:42:45
If it's fifty by fifty,
it's certainly got a lot of
714
00:42:45 --> 00:42:46
repetitions.
715
00:42:46 --> 00:42:48
Is it diagonalizable?
716
00:42:48 --> 00:42:48
Yes.
717
00:42:48 --> 00:42:53
All symmetric matrices,
all orthogonal matrices can be
718
00:42:53 --> 00:42:54
diagonalized.
719
00:42:54 --> 00:42:57
And, in fact,
the eigenvectors can even be
720
00:42:57 --> 00:42:58.67
chosen orthogonal.
721
00:42:58.67 --> 00:43:00
So it could be,
sort of, like,
722
00:43:00 --> 00:43:06
diagonalized the best way with
a Q, and not just any old S.
723
00:43:06 --> 00:43:06
OK.
724
00:43:06 --> 00:43:09
Is it non-singular?
725
00:43:09 --> 00:43:15.18
Is a symmetric orthogonal
matrix non-singular?
726
00:43:15.18 --> 00:43:15
Sure.
727
00:43:15 --> 00:43:21
Orthogonal matrices are always
non-singular.
728
00:43:21 --> 00:43:26.24
And, obviously,
we don't have any zero
729
00:43:26.24 --> 00:43:27
Eigenvalues.
730
00:43:27 --> 00:43:31
Is it sure to be
diagonalizable?
731
00:43:31 --> 00:43:34
Yes.
732
00:43:34 --> 00:43:42
Now, here's a final step --
show that one-half of A plus I
733
00:43:42 --> 00:43:48
is A -- that is,
prove one-half of A plus I is a
734
00:43:48 --> 00:43:51
projection matrix.
735
00:43:51 --> 00:43:51
OK?
736
00:43:51 --> 00:43:53.41
Let's see.
737
00:43:53.41 --> 00:43:55
What do I do?
738
00:43:55 --> 00:43:59
I could see two ways to do
this.
739
00:43:59 --> 00:44:07
I could check the properties of
a projection matrix,
740
00:44:07 --> 00:44:11
which are what?
741
00:44:11 --> 00:44:13
A projection matrix is
symmetric.
742
00:44:13 --> 00:44:18
Well, that's certainly
symmetric, because A is.
743
00:44:18 --> 00:44:20
And what's the other property?
744
00:44:20 --> 00:44:24
I should square it,
and hopefully get the same
745
00:44:24 --> 00:44:25
thing back.
746
00:44:25 --> 00:44:29
So can I do that,
square and see if I get the
747
00:44:29 --> 00:44:32
same thing back?
748
00:44:32 --> 00:44:37
So if I square it,
I'll get one-quarter of A
749
00:44:37 --> 00:44:41
squared plus two A plus I,
right?
750
00:44:41 --> 00:44:47.43
And the question is,
does that agree with p- the
751
00:44:47.43 --> 00:44:49
thing itself?
752
00:44:49 --> 00:44:51
One-half A plus I.
753
00:44:51 --> 00:44:51
Hm.
754
00:44:51 --> 00:44:57
I guess I'd like to know
something about A squared.
755
00:44:57 --> 00:45:00
What is A squared?
756
00:45:00 --> 00:45:04
That's our problem.
757
00:45:04 --> 00:45:05
What is A squared?
758
00:45:05 --> 00:45:11
If A is symmetric and
orthogonal, A is symmetric and
759
00:45:11 --> 00:45:12
orthogonal.
760
00:45:12 --> 00:45:15
This is what we're given,
right?
761
00:45:15 --> 00:45:19
It's symmetric,
and it's orthogonal.
762
00:45:19 --> 00:45:21
So what's A squared?
763
00:45:21 --> 00:45:21
I.
764
00:45:21 --> 00:45:26.13
A squared is I,
because A times A -- if A
765
00:45:26.13 --> 00:45:31
equals its own inverse,
so A times A is the same as A
766
00:45:31 --> 00:45:36
times A inverse,
which is I.
767
00:45:36 --> 00:45:39
So this A squared here is I.
768
00:45:39 --> 00:45:42
And now we've got it.
769
00:45:42 --> 00:45:48
We've got two identities over
four, that's good,
770
00:45:48 --> 00:45:54
and we've got two As over four,
that's good.
771
00:45:54 --> 00:45:54
OK.
772
00:45:54 --> 00:46:02
So it turned out to be a
projection matrix safely.
773
00:46:02 --> 00:46:07
And we could also have said,
well, what are the eigenvalues
774
00:46:07 --> 00:46:08
of this thing?
775
00:46:08 --> 00:46:12
What are the eigenvalues of a
half A plus I?
776
00:46:12 --> 00:46:16
If the eigenvalues of A are one
and minus one,
777
00:46:16 --> 00:46:19
what are the eigenvalues of A
plus I?
778
00:46:19 --> 00:46:24
Just stay with it these last
thirty seconds here.
779
00:46:24 --> 00:46:28
What if I know these
eigenvalues of A,
780
00:46:28 --> 00:46:32
and I add the identity,
the eigenvalues of A plus I are
781
00:46:32 --> 00:46:34
zero and two.
782
00:46:34 --> 00:46:39
And then when I divide by two,
the eigenvalues are zero and
783
00:46:39 --> 00:46:39
one.
784
00:46:39 --> 00:46:43
So it's symmetric,
it's got the right eigenvalues,
785
00:46:43 --> 00:46:47
it's a projection matrix.
786
00:46:47 --> 00:46:51
OK, you're seeing a lot of
stuff about eigenvalues,
787
00:46:51 --> 00:46:55
and special matrices,
and that's what the quiz is
788
00:46:55 --> 00:46:56
about.
789
00:46:56 --> 00:46:59
OK, so good luck on the quiz. |
Basic Plane Geometry Pdf
Plane Geometry - University of Virginias College at Wise chapter 1 introduction the opening lines in the subject of geometry were written around 300 b c by the greek mathematician euclid in 13 short books gathered into a Plane Geometry - University of Virginias College at Wise
Notes on basic algebraic geometry - Purdue University these are my notes for an introductory course in algebraic geometry i have trodden lightly through the theory and concentrated more on examples Notes on basic algebraic geometry - Purdue University
Contents Basic Definitions - University of Chicago projective geometry 3 3 linear algebra with geometries many of the concepts and theorems from linear algebra can be applied to the structures of geometries which Contents Basic Definitions - University of Chicago
Basic Lighting Laws and Some Notes on Pole Spacing Geometry 2 lighting luminaire above the ground the horizontal spacing of one pole to the next and the cutoff angle of the luminaire are all important issues in outdoor Basic Lighting Laws and Some Notes on Pole Spacing Geometry |
Geometry Workbook For Dummies
9780471799405
ISBN:
0471799408
Pub Date: 2006 Publisher: Wiley & Sons, Incorporated, John
Summary: From proofs to polygons -- solve geometry problems with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear -- this hands-on guide focuses on helping you solve the many types of geometry problems you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll shar...pen your skills and improve your performance. You'll see how to work with proofs, theorems, triangles, circles, formulas, 3-D figures, and more! 100s of Problems! Step-by-step answer sets clearly identify where you went wrong (or right) with a problem Get the inside scoop on geometry shortcuts and strategies Know where to begin and how to solve the most common problems Use geometry in practical applications with confidence
Ryan, Mark is the author of Geometry Workbook For Dummies, published 2006 under ISBN 9780471799405 and 0471799408. Two hundred eighty nine Geometry Workbook For Dummies textbooks are available for sale on ValoreBooks.com, one hundred thirty four used from the cheapest price of $0.75, or buy new starting at $9.95 |
The Basic Math DVD Series helps students build confidence in their mathematical knowledge, skills, and ability.
In this episode, students will learn how to solve basic equations. This lesson introduces a method for solving relatively simple linear equations of one variable, starting with simple equations such as 3x = 15 and x + 5 = 11 and eventually moving on to equations such as 3(x + 5) - 6x = 11 - 2x + 8. The importance of checking answers using the original equation is emphasized throughout the lesson. Grades 3-7. 30 minutes on DVD. |
Concise Introduction to MATLAB
9780073385839
ISBN:
0073385832
Pub Date: 2007 Publisher: McGraw-Hill Companies, The
Summary: A Concise Introduction to Matlab is a simple, concise book designed to cover all the major capabilities of MATLAB that are useful for beginning students. Thorough coverage of Function handles, Anonymous functions, and Subfunctions. In addition, key applications including plotting, programming, statistics and model building are also all covered. MATLAB is presently a globally available standard computational tool for ...engineers and scientists. The terminology, syntax, and the use of the programming language are well defined and the organization of the material makes it easy to locate information and navigate through the textbook.
Palm, William J., III is the author of Concise Introduction to MATLAB, published 2007 under ISBN 9780073385839 and 0073385832. Five hundred eighteen Concise Introduction to MATLAB textbooks are available for sale on ValoreBooks.com, fifty used from the cheapest price of $21.93, or buy new starting at $77.55Writing programs to do tedious computations quickly and making a graphical representation of that data. One example is the differential equation solver. Very versatile built in functions to do that task. |
Elementary Algebra
9780321577290
ISBN:
0321577299
Pub Date: 2010 Publisher: Pearson Education
Summary: Carson, Tom is the author of Elementary Algebra, published 2010 under ISBN 9780321577290 and 0321577299. One hundred thirty seven Elementary Algebra textbooks are available for sale on ValoreBooks.com, one hundred nine used from the cheapest price of $7.13Comments: ALTERNATE EDITION: no cd or access cards-ANNOTATED TEACHER EDITION-same content as the student ed- with teaching tips and answers- May havehilighting-writing-Has Used Book stickers on the cover.
There wasn't a part of the book that was least helpful to me because all the steps to the equation or story problem were explained well and I did not need to go and waste extra time to try and figure it out on my own.
The material that was needed for the class I had to take and it had all the answers I needed to learn the material for that class. It gave great examples for problems in each lesson of the chapters!! |
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More About
This Book
basics will find that egghead's Guide to Algebra offers just what they need to be able to score high in school, on the GED, and on college entrance exams.
egghead's tips and advice for improving algebra skills
Ideal subject review for students taking the SAT, ACT, and GED exams
Foundational learning for students who need basic and remedial instruction
Dozens of sample exercises and solutions with helpful tips and strategies
Easy-to-follow lessons with fun graphics that provide essential information to help those who learn visually |
From Arithmetic to Algebra
Teachers can help students make the transition by developing their algebraic thinking early on.
Increasingly, algebra is the focus of mathematics discussions in schools and districts across the United States. Policymakers, professional organizations, and researchers emphasize the importance of developing algebraic reasoning at increasingly earlier ages. The National Mathematics Advisory Panel (2007) has issued initial reports stating that students need to develop understanding of concepts, problem-solving skills, and computational skills related to algebra in grades preK–8. In 2006, the National Council of Teachers of Mathematics published the Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, which emphasizes connections to algebra as early as kindergarten and promotes the development of algebraic reasoning across the elementary and middle school grades. Finally, mathematicians and mathematics educators are speaking up about the need to increase teachers' awareness and abilities for teaching algebra across the grades (Wu, 1999).
Multiple factors are driving the increased emphasis on algebra proficiency. For many educators, the primary concern is the poor performance of U.S. students on national and international assessments of mathematics ability. On the 2005 National Assessment of Educational Progress (NAEP), only 6.9 percent of 17-year-olds scored at or above proficiency on multistep problem solving and algebra (National Center for Education Statistics, 2005). On the algebra subtest of the 2003 Trends in International Mathematics and Science Survey (TIMSS), U.S. 8th graders scored below many economic competitors, such as Japan, the Russian Federation, Korea, Singapore, and China. These results suggest that a majority of U.S. students are not proficient in algebra by the time they exit middle school or high school.
Although the academic performance of U.S. youth as a whole is important, No Child Left Behind (NCLB) emphasizes the need to monitor the progress of subgroup populations that have traditionally performed below expectations. On the 2005 NAEP, 59 percent of black students, 50 percent of Hispanic students, and 45 percent of American Indian students did not meet proficiency at the 8th grade level. Similarly, 69 percent of students with disabilities and 71 percent of English language learners did not reach this benchmark (National Center for Education Statistics, 2005). These results highlight the crucial need to develop algebraic thinking across the grades and focus on providing the best instructional practices for all students.
Why Algebra?
Employers often expect their employees to translate work-related problems into general mathematical models, from calculating discounts for merchandise to operating technology-based equipment and machinery. Many careers in the fields of science and technology demand high levels of mathematics competence to solve complex problems, such as chemical equations involved in the study of drug interactions. Algebra is also helpful in daily life, from applying formulas for calculating miles per gallon of gasoline to using functions to determine the profit of a business venture.
Research suggests that students who pass Algebra II in high school are 4.15 times more likely to graduate from college than other students are (Adelman, 1999). This has led many state education agencies to raise graduation requirements to include courses in Algebra II. Currently, 13 states require students to take Algebra II to graduate from high school, up from just two states in 2005 (Achieve, 2007). Many states and school districts are considering implementing higher mathematics standards to promote college readiness and future success for their graduates.
What Is Algebra?
When we think about algebra in the curriculum, we often think of a separate area of mathematics concerned with symbols and equations, such as 3x + 7y - 2 = 30. Mathematics curriculums often reinforce the notion of separateness by identifying algebra as a distinct strand with such subtopics as patterning, data analysis, simple functions, and coordinate systems. However, arithmetic and algebra are not mutually exclusive areas of mathematical study.
Basic algebra, as opposed to modern or abstract algebra, extends learners' understanding of arithmetic and enables them to express arithmetical understandings as generalizations using variable notation. Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole-number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems. Teachers may introduce number properties as "truths" or axioms without developing students' deep conceptual understanding or providing multiple experiences applying these properties.
When teachers introduce integers and rational numbers in later elementary grades, many of these "truths" about numbers and operations don't generalize to addition and subtraction of positive and negative numbers or multiplication and division of fractions. By the time algebra is introduced in middle school, many students view mathematical principles as subjective and arbitrary and rely on memorization in lieu of conceptual understanding.
The National Council of Teachers of Mathematics has attempted to bridge the gap between arithmetic and algebra by embedding algebraic reasoning standards in elementary school mathematics. From grades 3 to 5, algebra is embedded with number and operations as one of the three main focal points; beginning in grade 6, algebra is the predominant topic. However, it is not always clear how to develop students' algebraic thinking as they learn about numbers, operations, properties of numbers, data display and analysis, and problem solving. Teachers need support in learning how to integrate these topics and provide rich and explicit instruction to their students in early algebraic thinking.
Teaching Algebra for Transfer
Teachers' understanding of mathematics influences the quality of their instruction. Many elementary school teachers have limited experience with mathematics and lack the knowledge and skills to teach mathematics effectively (Ball, Hill, & Bass, 2005). Moreover, most credentialing programs for elementary school teachers require minimal college-level mathematics courses despite calls for considerably more extensive requirements (Conference Board of the Mathematical Sciences, 2001). Aside from developing their content knowledge in mathematics, these teachers can benefit from some general instructional practices that can help them teach arithmetic for transfer to algebra.
Whenever possible, teachers should model precisely what they want students to be able to do, using multiple examples that illustrate the range of problem types that students must solve on their own. Demonstration models should include careful verbal explanations that explicitly detail for students how to perform each step of the problem. As students develop expertise, teachers can make fewer verbal explanations and focus less on each individual step.
Teachers often have difficulty modeling for students how to think about mathematics problems conceptually. Rather than initially using numeric symbols to solve a problem, teachers might use concrete objects or semi-concrete representations (such as pictures) to help represent the underlying concepts behind specific problems. Teachers will find that explaining the concept of 2/3 ÷ 1/3 is more complex than explaining how to use the "invert and multiply" algorithm. To develop deep conceptual understanding, teachers should draw on different types of examples that represent problems.
For example, teachers can use concrete objects to visually represent that the problem 2/3 ÷ 1/3 = □ means the same thing as "how many 1/3s are there in 2/3?" Presenting the problem this way helps students understand what it means to divide any number by a fraction and "see" that in this example, there are "2 1/3s in 2/3." However, using visual models to help students understand how to solve problems involving division by fractions breaks down quickly when the numerical values in the problem are not artificially constrained, such as in the problem 9/23 ÷ 11/15 = □. Without using the "invert and multiply" algorithm, this problem becomes difficult to solve. After students understand the meaning of division of fractions, instruction should focus on applying the algorithm in a step-by-step fashion. With clear verbal explanations and explicit modeling, students can understand
why the algorithm works and
what it means to divide by fractions.
In addition to hearing teachers' verbal explanations, students should share their verbal explanations to further develop conceptual understanding. Here again, carefully chosen examples can provide a rich source of discussion as students explain why 2 × 54 = 2 × 50 + 2 × 4 (an application of the distributive property); why 72 - 6 ≠ 72 (an application of the identity property of subtraction); or why 5 + 2 = 2 + 5 (an application of the commutative property of addition). Students should be able to describe the properties of numbers in their own words—such as through telling a story or describing what is happening in a picture that has an obvious numerical focus—as well as in symbolic notation, and they should be able to apply these principles in multiple contexts.
For example, young students might demonstrate the commutative property of addition by using concrete objects, such as groups of marbles. Students might explain the commutative property by showing that reordering the groups of marbles does not change the sum of the marbles when the groups are added together. Once they understand the concept, the teacher might ask the students to provide multiple representations of the commutative property using symbolic notation.
Students also need to demonstrate their own understanding and skills. Teachers can gauge how well students solve problems in relatively straightforward ways. Students can work different types of problems and apply algorithms to solve them. Teachers can set proficiency goals for students and monitor student progress toward these goals.
Algebra-Specific Instructional Strategies
Algebraic reasoning builds on students' deep understanding of numbers and their relationships. Some mathematics researchers have identified areas of arithmetic that provide the foundations for algebra. These include
Numbers and number relationships (quantities and magnitudes).
Operations (functional relationships between numbers).
Field axioms or number properties (commutative, associative, distributive, identity, inverse, and so on).
Other topics linked to algebra include geometry, data analysis, proportional reasoning, and measurement. These topics provide rich opportunities for developing early algebraic reasoning as students learn about functional relationships in these areas (Van de Walle, 2004).
To develop algebraic reasoning, students must understand the following four key components (Milgram, 2005).
Variables and Constants
As students progress through elementary school, they learn about number systems—from counting, to whole numbers, to integers, to rationals, to real numbers. Studying number systems builds students' understanding that each new system is an extension of the previous system and that all number systems are embedded in the real-number system. As such, each system satisfies the basic rules of associativity, commutativity, and distributivity.
As we introduce students to variables, a key insight for students to grasp is that algebraic expressions, in which variables replace real numbers, will also satisfy the properties with which they are familiar. For example, when teachers introduce the distributive property, they can extend instruction from the context of whole numbers and integers to expressions with variables. They can follow a discussion of the problem
6 × (2 + 9) = 6 × 2 + 6 × 9
with a discussion of
6 × (t + 9) = 6 × t + 6 × 9.
Representing and Decomposing Word Problems Algebraically
Key to abstract reasoning and using algebra to solve problems is using algebraic expressions to describe problems. For example, students who think in algebraic terms easily translate the phrase "if you add 3 to a number times itself" into n2 + 3. Students need to apply this conversion of phrases to solve word problems. Teachers can help students master this skill by modeling and using language that identifies the "unknown" in a problem and then translates the process of finding the unknown into mathematical statements and equations.
Consider the following word problem:
Maria needs to find the weight of a box of cereal using a balancing scale. Maria puts 6 identical boxes of cereal on one side of the scale. To balance the scale, Maria puts 2 more identical boxes of the same cereal and 3 4-pound blocks on the other side of the scale. How much does each box of cereal weigh?
Teachers can model how to solve this problem by first identifying the unknown component (the weight of each box of cereal, labeled y) and the known components (the number of boxes of cereal and the weight and number of the blocks). Next, teachers can help students understand how to translate these elements into a mathematical statement to solve for the unknown (6y = 2y + 12). Students can check answers by inserting various numerical values into equations to verify solutions. This last step is about more than just getting the correct answer; it is an important step in problem solving because it encourages students to reflect on the original problem and determine whether the answer is reasonable.
For many students, improving skills at translating or converting problems to algebraic expressions will pose challenges. Students need to learn to break the problem into separate parts and then convert each part to an expression or equation that acknowledges the restrictions that the problem places on it (for example, the phrase "times itself"). Students will also need to recognize when a problem contains irrelevant information.
Symbol Manipulation
Many adults associate symbol manipulation with algebra because their memories of basic algebra are with the struggles of moving abstract symbols about the page "to solve for x." Although isolating the variable is still the goal for symbol manipulation, students need to understand that manipulating symbols in an equation merely simplifies the equation in a manner that enables us to get the answer we are seeking. Lawful manipulation of the symbols results in an equation that has the same solutions as the original equation.
Related to this topic is a common misconception about the equality rule and the equal sign. Many students in the early grades view a number sentence or mathematical formula as something "to do," most often with input on the left and output on the right. Consider the number sentence 5 + 3 = □. Students interpret this as adding the quantities 5 and 3 to find the specific answer of 8. Students may not view the following as possible solutions to the same problem:
5 + 3 = 3 + 5
5 + x = 8
8 = 5 + 3
5 + 3 = 2 + 6
Teaching equality and the meaning of the equal sign as a symbol that indicates both sides are balanced (as symbolized, for example, by a balance scale) provides opportunities for students to see equations as more than something to act on or a problem for which they must seek a single solution. Encouraging students to generate multiple solutions to 5 + 3 prepares them for working with variables, understanding and applying the commutative property and the inverse property of addition and subtraction.
Functions
Students should begin to learn elements of functions early in their school careers. Teachers need to strategically teach students to build patterns in which each input has only one output. Milgram (2005) provides an example of how kindergarten teachers can help their students understand simple functions. By sorting and classifying objects on the basis of unique properties, students can understand the association between objects in one set and unique objects (or features of the object) in another set. For example, students can sort objects by color. If each object has a specific color, the object is the input and the color is the output. Sorting the objects by color is an example of a function. As students progress in their understanding, teachers can explicitly model symbolic representations of functions.
Later students will learn to graph the Cartesian coordinates of the members of the input and output sets (domain and range). Next, they'll develop an understanding of how the domain and range represent a "rule of correspondence" that can be described using function notation, a convention in mathematics. Ultimately, these early insights into functions assist learners in understanding linear algebra and, later, curvilinear and quadratic functions and the role they play in mathematical relationships.
Finally, to help students develop algebraic reasoning in problem solving, students must develop a degree of certainty about the properties of number systems that allow us to manipulate and operate on numbers. Teachers can build this certainty in students by teaching the process of mathematical induction so students understand that their actions must be verifiable mathematically to be lawful and useful (Milgram, 2005). Teachers often teach mathematical induction as a procedure without sufficiently understanding why induction is so crucial for students' cognitive development in mathematics.
Starting Early
Because the goal of teaching algebra is to help students develop abstract reasoning in problem solving, schools should begin to develop these skills in students at the elementary level. By systematically and explicitly incorporating concepts of algebra in elementary school mathematics, schools can help students avoid developing many misconceptions about number and number relationships, operations, and application of number properties. Teaching mathematics in the elementary grades to transfer to algebraic concepts may promote success for all students engaged in mathematical reasoning.
My "Aha!" Moment
Mathematics suddenly interested me when I encountered calculus at age 16. Before then, I never saw much point in the subject beyond basic arithmetic, and looking back I now realize why. Other than basic number skills and a bit of trigonometry, no subject generally taught before calculus shows how mathematics makes a difference in the world.
Logical thinking is important in earlier math classes, but not mathematical thinking.
The enormous power of mathematics—and its beauty—lies in the vast range of the subject beyond high school mathematics. The mathematics taught in school is what I call abstracted math—and it really amounts to little more than formalized common sense. You can call it math, but it really isn't.
What our modern world depends on—big time—is what I call constructed math. This is the rule-based, abstract reasoning system that forms the basis of all science and engineering, and a lot else besides. It isn't really abstracted from the world; rather, we humans create it to apply to the world. By and large, this kind of mathematics cannot be learned before the upper levels of high school; it requires too much mental sophistication. But there is no reason why we can't teach such mathematics descriptively, where the goal is awareness and understanding, not the ability to do it. I am sure that if I had been taught that way, I would have been interested in math long before I was.
Leanne R. Ketterlin-Geller
(lketterl@uoregon.edu) is Assistant Professor and Kathleen Jungjohann
(kjj@uoregon.edu) is Senior Instructor and Research Assistant at the University of Oregon, Eugene. David J. Chard
(dchard@smu.edu) is Dean of the School of Education and Human Development at Southern Methodist University, Dallas, Texas. Scott Baker (sbaker@uoregon.edu) is Director of Pacific Institutes for Research, Eugene, Oregon. |
want to learn the backbone of calculus, then you should check out (real) analysis. All of the material's importance is easily motivated by your calculus experience. After all that calculus though, I'd imagine you're tired of it. Abstract algebra is a nice deviation and will be hugely important with your linear algebra and as your mathematical sophistication advances. Another option might be combinatorics, which is different and more an application of set theory and discrete math.
If discrete math, set theory, and proofs haven't been integrated into your curriculum thus far, those should actually be priority numero uno for you. Can't get past calc or LA without them.
Basically this. If you have gone up to LA without taking a discrete, you'll need to read up on that before going any further. Real Analysis and Abstract Algebra (though not so much) are huge jumps in logic. It's basically like not knowing algebra, then taking a calculus class. The more you learn about sets, proof, and general logic, the better off you will be.
I should add, after this subjects, math branches out into many categories that have nothing to do with the typical "math" that people are used to seeing. Topology is really interesting, and if you love shapes and geometry, that is one goal, but it takes heavily from the core ideas in Real Variables, and you might not want to take that until after Linear II.
Leave a comment with things you had in mind, or general areas you thought "hey, that was cool, how did they get to that?", and I can give you a better answer.
I really do like things with applications. The actual thing motivating this question was looking at the general relativity fields equation and being amazed that I still understood absolutely nothing about it. So I guess topology ties in with the whole differential geometry field.
Topology is actually more on the real math side of things. It's not exactly what you'd think it is from it's name.
I'm more versed in pure math at the moment, so i haven't looked much into general relativity.
Best bet is to find a physics professor who teaches general relativity and ask where you can start in order to understand the material. My bet is that you are only missing a few key concepts that would bridge the gap pretty quickly. |
Math Resources and Portals
Cycloid Optical Illusion Will Boggle Your Mind. Optical illusions typically prey on your mind's ability to fill in blanks in the visual stimulus it is receiving.
Sometimes this can be so strong, your mind continues to be tricked even after it has been explained. Case in point: this video from brusspup, purveyor of mind-melting illusions.
Matematicas Visuales | Home. In MatematicasVisuales you will find visual expositions of mathematical concepts.
MatematicasVisuales intends to complement the work initiated by artiludios , a site with games, puzzles and mathematical curiosities . Reading Miguel de Guzmán I found a demonstration of the line of Simpson and the Steiner Deltoid . It serves as an introduction to the geometry section. The concept of function and its graphical representation are a key concept and we dedicate special attention to it in the analysis section. Geometric representation of the complex numbers facilitates its visualization.
CR Algebra Review. Prealgebra Review A review of the concepts in Prealgebra as preparation to enter Elementary Algebra (Math 380).
Elementary Algebra Review A review of the concepts in Elementary Algebra as preparation to enter Intermediate Algebra (Math 120). Intermediate Algebra Review A review of the concepts in Intermediate Algebra as preparation to enter a Transfer Level math class (Math 5, 15, 25, or 30). Important Information Procedure: Each review course is broken up into 6 modules. Each module has 2 or 3 skills (labeled A, B, ...) to be reviewed.
This gives the same information as the pink-covered booklet and the yellow-covered booklet available in exams. Currently the tables are available in and pdf format. These can be downloaded and/or viewed with Adobe Acrobat. You can access the individual sections or the tables as a whole.
Making Math and Science Connections. Fill-It-In Outline Mathematics. Free Mathematics Resources OnLine. What about free graphing and algebra on your mobile phone?
XCalc is a free Java Applet which offers a graphing calculator for any mobile phone which runs Java (which is nearly all of them now!) Think about the possibilities for classes that do not yet have access to graphing calculators! While limited (it does not yet have a trace facility) it DOES offer 2d and 3d graphing which is quite fast and powerful.
The goal of this project was to collect and organize available INTERNET resources on Authentic Mathematical Assessment applicable to Secondary Schools. What is it?
MacTutor History of Mathematics. Sage: Open Source Mathematics Software. Jim Loy's Mathematics Page. Go to my home page Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com.
"He must be a 'practical' man who can see no poetry in mathematics.
" - W. |
material presented in the following pages are for middle school students, high school students, college students, and all...
see more
The material presented in the following pages are for middle school students, high school students, college students, and all who are interested in mathematics. You will find interactive programs that you can manipulate and a lot of animation that helps you to grasp the meaning of mathematical ideas. |
97808218478796813 Analysis is designed to bridge the gap between the intuitive calculus usually offered at the undergraduate level and the sophisticated analysis courses the student encounters at the graduate level. In this book the student is given the vocabulary and facts necessary for further study in analysis. The course for which it is designed is usually offered at the junior level, and it is assumed that the student has little or no previous experience with proofs in analysis. A considerable amount of time is spent motivating the theorems and proofs and developing the reader's intuition. Of course, that intuition must be tempered with the realization that rigorous proofs are required for theorems. The topics are quite standard: convergence of sequences, limits of functions, continuity, differentiation, the Riemann integral, infinite series, power series, and convergence of sequences of functions. Many examples are given to illustrate the theory, and exercises at the end of each chapter are keyed to each section. Also, at the end of each section, one finds several Projects. The purpose of a Project is to give the reader a substantial mathematical problem and the necessary guidance to solve that problem. A Project is distinguished from an exercise in that the solution of a Project is a multi-step process requiring assistance for the beginner student. |
In this problem set, learners will analyze an image of carbon dioxide emissions in the continental US in a given year to answer a series of questions. Answer key is provided. This is part of Earth Math: A Brief Mathematical Guide to Earth Science...(View More) and Climate ChangeThis is a book containing over 200 problems spanning over 70 specific topic areas covered in a typical Algebra II course. Learners can encounter a selection of application problems featuring astronomy, earth science and space exploration, often with...(View More) more than one example in a specific category. Learners will use mathematics to explore science topics related to a wide variety of NASA science and space exploration endeavors. Each problem or problem set is introduced with a brief paragraph about the underlying science, written in a simplified, non-technical jargon where possible. Problems are often presented as a multi-step or multi-part activities. This book can be found on the Space Math@NASA website.(View Less) |
Resources for Algebra 1 and Algebra 2
Using the internet as a teaching tool
Algebra 1
Algebra 2
General HELP
1-
HOTMATH will help you with many step-by-step solutions to the odd problems of almost every textbook
2-
PURPLE MATH Offers a variety of complete lessons with quizzes and extra tutorials. These are just some
topics your students can use. There are many more topics.
3- Fee is required
4-
5- Search for the lesson that you want. Copy the hyperlink and then import that
video to your computer by using a free software from
For the first time only:
a- Go to
b- Download the Free Ipod Video Converter 2.92
c- Install or run the program and create an icon on your
desktop (Just to find it)
d- Open the Free IPOD converter
e-
f- Click on the Youtube icon
g- Paste the link of the video that you want to show your
students
h- Click that you want to automatically convert your video to
an Ipod format (mp4)
i- Make sure to select the folder for your file (Go to
parameters)
j- Click download now
k-
6- Use your textbook websites
7- Test for Juniors and Seniors: KEMPT. To get there type or go |
Algebra 1 is the first year course of Algebra that is required for a high school diploma. Topics include equations, functions, graphing, statistics, inequalities, exponents, quadratics, polynomials, and rational functions. All Algebra I students will be required to take the State End of Instruction test at the end of the school year.
Rules
1.Bring proper material to class.
2.Respect your teacher, classmates, and classroom belongings.
3.Be in your seat when the bell rings!
4.Do not take calculators out of the classroom.
Consequences
1.First Offense: Warning
2.Second Offense: Parent Contact & Detention (time will be set by teacher after school!)
3.Third Offense: Office Referral
Classroom Procedures
1.Be in your assigned seat when the bell rings. Start on the warm-up problems from the board or overhead.
2.Take notes during lectures and discussions.
3.All homework assignments will be due in class the following day at the beginning of thehour unless otherwise told.
4.The bell does not dismiss the class; I do.
5.Building tardy policy will be enforced. Must be in assigned seat to not be considered tardy.
6.You will be required to have your agenda to leave the classroom.
Assignments
1.Each day's assignments should be in pencil on a separate sheet of regular (not spiral) notebook paper.
2.Each sheet of the assignment should be labeled with your name, date, and class period.
3.Assignments should be neat, showing all required work, and with answers circled or boxed.
4.Assignments lacking in required work will be returned to redo the assignment showing proper work.
5.Assignments are graded for accuracy and competition. Students will grade their own assignments in class and record the number missed on their paper. Grades will be recorded randomly but usually about 2 to 3 assignment grades per week.
6.Modeling of showing proper work on assignments is done every day in class.
7.Assignments are designed to start in class where most work can be completed in 10-30 minutes.
Teacher Website
Make sure you visit my teacher website. The weekly agenda is always posted along with any handouts from class. If you are absent, please check this site out for materials that were missed.
Getting Help
There is always a way to get help with your Algebra!
1.Tuesday and Thursday tutorial period. New this year!!
2.Teacher Website - check this website for links, suggested, etc. Lots of information posted on this site for homework help.
3.Online Textbook Resources: Video to every example problem in the book, practice problems and practice test. This is a great resource!
4.Before or after school in my room. Make sure you ask permission to see if I am busy that day!!!
1.Extra Credit – no more than 5% of total grade will be given on any extra credit assignment.
2.Test - Tests and/or Quizzes are generally given after a unit has been taught. Students are given opportunities for test corrections to better their grade. Test Corrections are limited to a total of 10 points maximum on any test. Students will have the option of retaking one test per nine week period for a maximum grade of 75% (Semester test not included).
3.Time Limits - All tests are timed. Students will be expected to turn in what is completed at the end of the class period.
Grading
Grades are taken on assignments, test, quizzes, projects, etc. Semester test are comprehensive and will be given at the end of each semester. All major tests will be reviewed in class with a test review worksheet. The following guidelines are used with grades:
·Assignments, daily work, project, etc. - 40 % of total grade
·Test & quizzes - 60% of total grade
Grading scale
90-100 A
80-89 B
70-79 C
60-69 D
59 and Below F
Textbook
All students will check out a textbook. Students will also be given login information to use textbookresources online. Included in the resources is an online textbook, videos of each section with examples, practice problems, practice test, etc. I have a classroom set of books in my classroom, so textbook is NOT needed at school.
State Testing
All Algebra I students will be required to take the State EOI Algebra I test in the spring.This is the first of three math test required by the State Department of Education in high school.All 8th Grade Algebra1 students will be required to take the 8th Math OCCT test in the spring also.Note: State testing is expected to change next school year with the new Common Core Standards being testing for the first time |
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How to Count: An Introduction to Combinatorics, Second Edition (Discrete Mathematics and Its Applications)
by:R.B.J.T. Allenby, Alan Slomson...
Show More the authors present most of the topics through concrete problems, they also emphasize the importance of proofs in mathematics. New to the Second EditionThis second edition incorporates 50 percent more material. It includes seven new chapters that cover occupancy problems, Stirling and Catalan numbers, graph theory, trees, Dirichlet's pigeonhole principle, Ramsey theory, and rook polynomials. This edition also contains more than 450 exercises. Ideal for both classroom teaching and self-study, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusion-exclusion principle, generating functions, recurrence relations, and Pólya's counting theorem |
More About
This Textbook
Overview
Combining standard Volumes I and II into one soft cover edition, this helpful book explains how to solve mathematical problems in a clear, step-by-step progression. It shows how to think about a problem, how to look at special cases, and how to devise an effective strategy to attack and solve the problem. Covers arithemetic, algebra, geometry, and some elementary combinatorics. Includes an updated bibliography and newly expanded |
wow... good one. I would go with B as well, but was not sure why to eliminate A. nice explanation.
zaarathelab wrote:
1. The graphical illustrations mathematics teachers use enable students to learn geometry more easily by providing them with an intuitive understanding of geometric concepts, which makes it easier to acquire the ability to manipulate symbols for the purpose of calculation. Illustrating algebraic concepts graphically would be equally effective pedagogically, even though the deepest mathematical understanding is abstract, not imagistic. The statements above provide some support for each of the following EXCEPT: (A) Pictorial understanding is not the final stage of mathematical understanding. (B) People who are very good at manipulating symbols do not necessarily have any mathematical understanding. (C) Illustrating geometric concepts graphically is an effective teaching method. (D) Acquiring the ability to manipulate symbols is part of the process of learning geometry. (E) There are strategies that can be effectively employed in the teaching both of algebra and of geometry.
IMO B
It is mentioned in the argument that "deepest mathematical understanding is abstract, not imagistic". Therefore, the pictorial understanding cannot provide the deepest understanding and hence cannot be the final stage. Hence, there is support for A |
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Classroom Voting Questions: Linear Algebra
This is a reasonable set of multiple choice clicker questions that can be used in both building and testing concepts in linear algebra. The questions are organized in a couple of different formats on the site, which is also nice. |
For Grades 7-10, this series emphasizes the development of understanding mathematical concepts and their applications, as well as proficiency in problem solving, mathematical reasoning and higher order thinking. Students will work on investigative skills, communication in mathematics, appropriate computation and estimation skills and mental calculation through graded exercises. Exercises are provided for different levels, while chapter reviews, challenging and problem-solving questions, revision exercises for review, the history of math and other elements are also included. Many questions require students to apply knowledge to new situations rather than following a procedure. Textbook 2 is recommended for Grade 8; 444 pages, hardcover. Teacher involvement is generally required.
Do your kid a favor and get this book
Date:May 16, 2011
Mark
Location:Los Angeles, CA
Age:45-54
Gender:male
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
Excellent book! My wife and I are both engineers, so math is important to us. Our son attends a great school but needed some other approaches to understanding the same material. This book, and the workbook with extra problems, fulfill that role very well. |
...
0764563742 no time, you'll be ready to tackle other concepts in this book such as
Factors and prime numbers
Integers, exponents, and scientific notation
Measurements, the metric system, and graphs
Variables and algebraic equations
CliffsQuickReview Basic Math and Pre-Algebra acts as a supplement to your textbook and to classroom lectures. Use this reference in any way that fits your personal style for study and review — you decide what works best with your needsJerry Bobrow, PhD, is an award-winning teacher and educator. He is a national authority in the field of test preparation. As executive directory of Bobrow Test Preparation Services, Dr. Bobrow has been administering the test preparation programs for most California State Universities for the past 27 years. Dr. Bobrow has authored more than 30 national best-selling test preparation books including Cliffs Preparation Guides for the GRE, GMAT, MSAT, SAT I, CBEST, NTE, ACT, and PPST. Each year he personally lectures to thousands of students on preparing for these important basic review
a good review book that goes from basic math through pre-algebra without skipping parts or dragging. The explanations are clear, and detailed and there are enough practice questions to make sure you get the point. I would highly suggest this!The book if most helpful for anyone who needs help with math.
The book also helps those who need to review math.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. |
More About
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Overview
This is a lucid introduction to some of the mathematical ideas which are useful to biologists. Professor Maynard Smith introduces the reader to the ways in which biological problems can be expressed mathematically, and shows how the mathematical equations which arise in biological work can be solved. Each chapter has a number of examples which present further points of biological and mathematical interest. interest. Professor Maynard Smith's book is written for all biologists, from undergraduate level upwards, who need mathematical tools. Only an elementary knowledge of mathematics is assumed. Since there are already a number of books dealing with statistics for biologists, this book is particularly concerned with non-statistical |
Algebra Unplugged - 95 edition
Summary: A lighthearted, sometimes irreverent introduction to the concepts, vocabulary and strategies of first year algebra. Designed for people who learn best by reading, it includes no exercises. Beginning with some pre algebra concepts, like working with fractions, it explains linear equations, quadratic equations and graphing in easy to understand, non frightening language. Ideal for people who think they hate math to read before they take the class or as a supplement during it. It provides an additi...show moreonal explanation to the confused and comfort for the fearful. For nearly twenty years this book has been helping math-phobes survive their most dreaded class |
Thinking Mathematically - 85 edition
Summary: Three easy steps for solving problems are presented in this useful book. Capitalizing on how students think and learn, the authors present brief hints and suggestions to help you teach mathematics effectively.
11/01/1982 Paperback Used-Good Book in good or better condition. Dispatched same day from US or UK warehouse.
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0201102382 Good title in good condition. Pages are clean and tight. Covers have light shelf wear. Satisfaction guaranteed. If item not as described, return for refund of purchase price.
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An authoritative, full-year course on both group theory and ordinary character theory—essential tools for mathematics and the physical sciences
One of the few treatments available combining both group theory and character theory, Groups and Characters is an effective general textbook on these two fundamentally connected subjects. Presuming only a basic knowledge of abstract algebra as in a first-year graduate course, the text opens with a review of background material and then guides readers carefully through several of the most important aspects of groups and characters, concentrating mainly on finite groups.
Challenging yet accessible, Groups and Characters features:
* An extensive collection of examples surveying many different types of groups, including Sylow subgroups of symmetric groups, affine groups of fields, the Mathieu groups, and symplectic groups
* A thorough, easy-to-follow discussion of Polya-Redfield enumeration, with applications to combinatorics
* Inclusive explorations of the transfer function and normal complements, induction and restriction of characters, Clifford theory, characters of symmetric and alternating groups, Frobenius groups, and the Schur index
* Illuminating accounts of several computational aspects of group theory, such as the Schreier-Sims algorithm, Todd-Coxeter coset enumeration, and algorithms for generating character tables
As valuable as Groups and Characters will prove as a textbook for mathematicians, it has broader applications. With chapters suitable for use as independent review units, along with a full bibliography and index, it will be a dependable general reference for chemists, physicists, and crystallographers.
Editorial Reviews
Booknews
A textbook for a graduate mathematics course of one or two semesters. Combines group and character theory, with a focus on finite groups, and includes some computational aspects. Readers are assumed to have completed a graduate-level course in abstract algebra and have a reasonable facility with linear algebra. Also suitable as a reference for chemists, physicists, and crystallographers |
Synopses & Reviews
Publisher Comments:
Suitable for advanced undergraduates and graduate students, this text offers a complete introduction to the basic concepts of linear algebra. Interesting and inspiring in its approach, it imparts an understanding of the subject's logical structure as well as the ways in which linear algebra provides solutions to problems in many branches of mathematics.
The authors define general vector spaces and linear mappings at the outset and base all subsequent developments on these concepts. This approach provides a ready-made context, motivation, and geometric interpretation for each new computational technique. Proofs and abstract problem-solving are introduced from the start, offering students an immediate opportunity to practice applying what they've learned. Each chapter contains an introduction, summary, and supplementary exercises. The text concludes with a pair of helpful appendixes and solutions to selected exercises.
Synopsis:
Suitable for advanced undergraduates and graduate students, this text introduces basic concepts of linear algebra. Each chapter features multiple examples, proofs, and exercises. Includes solutions to selected problems. 1988 edition.
Synopsis:Suitable for advanced undergraduates and graduate students, this text introduces basic concepts of linear algebra. Each chapter features multiple examples, proofs, and exercises. Includes solutions to selected problems. 1988 edition.
"Synopsis"
by Firebrand, |
This course develops the ability to utilize mathematical skills and technology to solve problems at a level found in non-mathematicsintensive programs. Topics include applications to percent, ratio and proportion, formulas, statistics, function notation, linear functions, probability, sampling techniques, scatter plots, and modeling. Upon completion, students should be able to solve practical problems, reason and communicate with mathematics, and work confidently, collaboratively, and independently.
2013FA - New State Prereq: (DMA 010 and DMA 020 and DMA 030 and DMA 040 and DMA 050) |
Mathematical Analysis covers a wide variety of topics that will be of great interest to students of pure mathematics or mathematics and philosophy. Aimed principally at graduate and advanced undergraduate students, its primary goal is to discuss the fundamental number systems, N, Z, Q, R, and C, in the context of the branches of mathematics for which they form a starting point; for example, a study of the natural numbers leads on to logic (via Gödel's theorems), and of the real numbers (as constructed by Cauchy) to metric spaces and topology. The author offers a refreshingly original and accessible approach, presenting standard material in new ways and incorporating less mainstream topics such as long real and rational lines and the p-adic numbers. With a discussion of constructivism and independence questions, including Suslin's problem and the continuum hypothesis, the author completes a wide-ranging consideration of the development of mathematics from the very beginning, concentrating on the foundational issues particularly related to analysis. |
McDougal Littell Algebra1 *Destination Math does not align to all standards. Those standards are not shown on this document. This document is a correlation of Destination Math, to McDougal Littell Algebra1. 2
Student Help Sheet for Getting HoltAlgebra Online 1) Go to website: ... The online version of HoltAlgebra I features not only your textbook, but also interactive features that will enhance your learning experience. Additional Practice Welcome to HoltMcDougal On"net o}gal line My Subscriptions ...
HoltMcDougal Physics balances a conceptual study of physics with quantitative problem solving that is demonstrated in consistent, four-step method. ... textbook • There are Section Study Guides found at point of use online at HMDScience.com
HOLTALGEBRA1HOLT GEOMETRY HOLTALGEBRA 2. Dr. Edward Burger is an award-winning teacher. ... programs, including a 1990 Geometry textbook and a high school series in 1997. Freddie Renfro Former Director of Mathematics Instruction, K–12 |
Essentials of Trigonometry - With CD - 4th edition
Summary: Intended for the freshman market, this book is known for its student-friendly approach. It starts with the right angle definition, and applications involving the solution of right triangles, to help students investigate and understand the trigonometric functions, their graphs, their relationships to one another, and ways in which they can be used in a variety of real-world applications. The book is not dependent upon a graphing calculator.
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Intermediate Algebra
9780321358356
ISBN:
032135835X
Edition: 2 Pub Date: 2006 Publisher: Addison-Wesley
Summary: This student-focused text addresses individual learning styles through the use of a complete study system that starts with a learning styles inventory and presents targeted learning strategies designed to guide students toward success in this and future college-level courses. Students who approach math with trepidation will find that Intermediate Algebra, Second Edition, builds competence and confidence. The study sy...stem, introduced at the outset and used consistently throughout the text, transforms the student experience by applying time-tested strategies to the study of mathematics. Learning strategies dovetail nicely into the overall system and build on individual learning styles by addressing students unique strengths. The authors talk to students in their own language and walk them through the concepts, showing students both how to do the math and the reasoning behind it. Tying it all together, the use of the Algebra Pyramid as an overarching theme relates specific chapter topics to the big picture of algebra.
Carson, Tom is the author of Intermediate Algebra, published 2006 under ISBN 9780321358356 and 032135835X. Thirty four Intermediate Algebra textbooks are available for sale on ValoreBooks.com, thirteen used from the cheapest price of $5.69, or buy new starting at $24Book is in good condition; cover shows moderate signs of wear. Pages are unmarked by pen or highl... [more]Book is in good condition; cover shows moderate signs of wear. Pages are unmarked by pen or highlighter ALL ANSWERS and/or TEACHING NOTES in margins. DOES NOT include supplements such as CDs or acces [more]
ALTERNATE EDITION: INSTRUCTORS EDITION. Exact same content as the student version only may include ALL ANSWERS and/or TEACHING NOTES in margins. DOES NOT include supplements such as CDs or access codes. Orders ship within 1-2 business days w/ free tracking[less] |
This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives students the information as to how these ...Show synopsisThis manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives students the information as to how these problems are solved Student Solutions Manual for Gustafson/Frisk's Beginning...Good. Student Solutions Manual for Gustafson/Frisk's Beginning and Intermediate Algebra: An Integrated App |
I purchased this book roughly 6-7 weeks before the actual mathematics exam, thinking that it was too late to revise the majority of the year 9 curriculm. But I found it simple, fun and easy to understand because of its unique teaching methods. For instance it had useful pictures, strange jokes and questions at the end of each chapter to test your knowledge. All of which proved to be extremely helpful, due to the amount of time I had to revise.
Somehow CGP manage to produce guides which contain all the hard facts kids need for their exams, but which present them in a friendly, digestible format accessible to both children and adults. This comprehensive guide includes sections on numbers, algebra, shapes and statistics, presented with the usual CGP humour and cartoons. It is useful both as a reference book (when are shapes congruent and similar? What are the 8 simple rules of geometry?) and as a work-your-way through-it revision guide. Conclusion: everything you need for revising (and learning) KS3 maths.
My son was having problems in maths due to lessons missed after illness. I had very little knowledge of modern secondary level maths so this book was perfect to help us both to tackle some tricky new topics.
my 13 year old daughter was falling behind in her maths so i approached her maths teacher who recommended this book. we bought it for her and 4 weeks later she scored a 7C and secured her place in the top maths set for year 8. well worth every penny. i highly recommend this book.
A really good book that is worth reading it helps with your maths and covers all the catagories studied through Key Stage Three. With funny jokes and pictures it really makes Maths come to life, a brilliant read!!!
Whilst revising for my sats i have used this book and found it very informative.The revision guide written by Paddy Gannon is stress free.Whilst revising, it can be difficult to persue the revision but with this revision guide you are but happy to read it.The book gives information in a friendly,simple manor.I recommend this book to those who wish to do well in their sats or any other major exam.Thank you for reading my reveiw i am but happy to reveiw this product.
I'd give this zero stars if I could, everything about it is awful. The opening message is that maths is boring so they've added some fun. Nice one. If the design was done by a professional they should be ashamed, it's horrible to look at but worse it's messy and difficult to work out where to read first. Poorly conceived and executed. |
Description of Exploring the World of Mathematics by John HudsonMath doesn't have to be difficult, and John Tiner shows that it can actually be fun. Students of different ages and skill levels can use this fascinating book.
Intended as a supplement to a homeschool curriculum, Exploring the World of Mathematics is more than just a math book. Tracing the history of mathematics principles and theory, it includes stories and tips showing math to be practical for everyday use. It also uses many examples of mathematics from the Bible and explains the timekeeping methods used in biblical times.
Included are the following: basic mathematical principles including some simple algebra, geometry, and scientific math; Egyptian and Greek contributions to mathematics; Math involving time; the seasons; and measurements. |
(with CD-ROM, BCA/iLrn™ Tutorial, and InfoTrac)
College Algebra, 5th Edition
Outlines and Highlights for College Algebra by James Stewart, Lothar Redlin, Saleem Watson, Isbn : 9780495565215
Outlines and Highlights for College Algebra by James Stewart, Lothar Redlin, Saleem Watson, Isbn : 9780495565215
Summary
Learn to think mathematically and develop genuine problem-solving skills with Stewart, Redlin, and Watson's COLLEGE ALGEBRA, Sixth Edition. This straightforward and easy-to-use algebra book will help you learn the fundamentals of algebra in a variety of practical ways. The book features new tools to help you succeed, such as learning objectives before each section to prepare you for what you're about to learn, and a list of formulas and key concepts after each section that help reinforce what you've learned. In addition, the book includes many real-world examples that show you how mathematics is used to model in fields like engineering, business, physics, chemistry, and biology. |
great explanations of the three major Calculus concepts: limits, derivatives, and integrals/antiderivatives. I explain the ideas of calculus in such a way that you actually understand them. I highlight the key points (most teachers just lecture and then you are stuck looking at your note... |
Stochastic geometry deals with models for random geometric
structures. Its early beginnings are found in playful
geometric probability questions, and it has vigorously developed
during recent decades, when an increasing number of real-world
applications in various sciences required solid mathematical
foundations. Integral geometry studies geometric mean values with
respect to invariant measures and is, therefore, the appropriate... more...
Now more that a quarter of a century old, intersection homology
theory has proven to be a powerful tool in the study of the
topology of singular spaces, with deep links to many other areas of
mathematics, including combinatorics, differential equations, group
representations, and number theory.
Like its predecessor, An Introduction to Intersection Homology
Theory, Second Edition introduces the power and beauty of
intersection homology, explaining the... more...
An easily accessible introduction to over three
centuries of innovations in geometry
Praise for the First Edition
". . . a welcome alternative to compartmentalized treatments
bound to the old thinking. This clearly written, well-illustrated
book supplies sufficient background to be self-contained."
—CHOICE
This fully revised new edition offers the most comprehensive
coverage of modern geometry currently available at an... more...
Presents an in-depth analysis of geometry of part surfaces and
provides the tools for solving complex engineering problems
Geometry of Surfaces: A Practical Guide for Mechanical
Engineers is a comprehensive guide to applied geometry of
surfaces with focus on practical applications in various areas of
mechanical engineering. The book is divided into three parts on
Part Surfaces, Geometry of Contact of Part Surfaces and Mapping
of the... more...
Essentials of the integral geometry in a homogenous space are
presented and the focus is on the basic results and applications.
This book provides the readers with new findings, some being
published for the first time.
Describing a striking connection between topology and algebra,
rather than only proving the theorem, this study demonstrates how
the result fits into a more general pattern. Throughout the text
emphasis is on the interplay between algebra and topology, with
graphical interpretation of algebraic operations, and topological
structures described algebraically in terms of generators and
relations. Includes numerous exercises and examples.
Presenting a classical approach to the foundations and development
of the geometry of vector fields, this volume space, three
orthogonal systems, and applications in mechanics. Other topics,
including vector fields, Pfaff forms and systems in n-dimensional
space, foliations and Godbillon-Vey invariant, are also considered.
There is much interest in the study of geometrical objects in
n-dimensional Euclidean space, and this volume provides a useful
and... more...
This is an introduction to diophantine geometry at the advanced
graduate level. The book contains a proof of the Mordell
conjecture which will make it quite attractive to graduate
students and professional mathematicians. In each part of the
book, the reader will find numerous exercises.
This book provides a comprehensive introduction to modern global
variational theory on fibred spaces. It is based on differentiation
and integration theory of differential forms on smooth manifolds,
and on the concepts of global analysis and geometry such as jet
prolongations of manifolds, mappings, and Lie groups.
The book will be invaluable for researchers and PhD students in
differential geometry, global analysis, differential equations on |
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Brand new. We distribute directly for the publisher. Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we...show more learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas.With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces.The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces.The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces.The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added. ...show less
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New
El Pinarillo Books Morden,
2006 Paperback New Book New and in stock. 8/10/2006112 |
Geometry Second Semester FinalExamReview 1. Mr. Jones has taken a survey of college students and found that 1 out of 6 students are liberal arts majors. If a college has 7000 students, what is the best estimate of the number of students who are liberal arts majors? a. 1167
GeometryFinalExamReview - This review covers the major topics in the material that will be tested on the finalexam. It is not necessarily all inclusive and additional study and problem solving practice may be required to fully
1 Geometry Goncepts I: FinalExamReview Multiple Choice Identify the choice that best completes the statement or answers the question. In the diagram, the dashed figure is the image of the solid figure.
Released T est Questions Geometry Introduction - Geometry The following released test questions are taken from the Geometry Standards Test. ... The following table lists each reporting cluster, the number of items that appear on the exam, ...
GEOMETRYFINALEXAMREVIEW - PRESENTATIONS We are nearing the end of the year, and the time for review is upon us once again. During the course of this class, you have been exposed to a variety of different lessons.
FinalExam scores will be worth a minimum of 20% of the student's final grade. ... review the results from the teacher panels and to develop weight distributions across the domains for ... The NC Final Exams for Geometry and Algebra II will consist of four-response-option MC items.
FinalExamReview MATD 0330 Austin Community College Page 1 of 12 Review for Basic Math Skills FinalExam In order to be prepared for the finalexam, students should be able to do all of the following problems and related problems as well. Students ...
Geometry A Credit By Exam Study Guide ... The CBE/EAs are a global exam grounded in the TEKS and are not designed to be a finalexam ... In order to be successful on the exam, you must study and review all of the concepts listed above.
Honors GeometryFinalREVIEW Short Answer 1. Find the lateral area of a cone if the height is 17 centimeters and the slant height is 19 centimeters. Use 3.14 for !. Round to the nearest tenth if necessary. 2. There are 84 boys in a freshman class of 146 students. |
Editorial Reviews
Review
"The book certainly contains a wealth of information...a rewarding pleasure to leaf through its pages." Mathematical Reviews
"...Not merely readable but entertaining and also easy to browse, this book should help lift undergraduate mathematics to a higher level of mathematical sophistication and maturity. Highly recommended." Choice
Book Description
This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and non-commutative. Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner that is
This is a delightful book that covers broad areas of theoretical and practical mathematics from the standpoint of Group Theory and Fourier Analysis. Although Fourier Analysis is first in the title, I think Finite Groups should come first. The author takes the discrete perspective of traditionally continous functions of classical physics and other applications - which not suprisingly brings finite groups into the picture. This includes the applications of quadratic residues and primitive roots to areas not usally touched by number theory - such as the ceilings of concert halls. This is a fun book, and it doesn't pretend to be comprehensive or complete with regards to its choosen subjects. I'd call the book "A Grand Tour of Finite Groups and Fourier Analysis with Applications" if I were the publisher. The price is suprisingly affordable given the subject where new books usually cost O($100).
For students and users who need the facts! Perhaps they have come accross a group problem in programming, in physics, in some course or other, or in a research assignment. Over the years, when teaching, I am often asked by students for directions to the facts of groups(usually finite) and their harmonic analysis. And too often, I have had to send the poor student to a multi-volume book set on the general theory. Sure much of it can be specialized to what the case demands. But the user[in e.g., programming, algorithms, fast transforms, error-correction codes, crystals, symmetry, quantum theory, engineering...] typically isn't ready for the big picture yet, and more often than not, she will need to first look for a gentle introduction--just the facts!-- and written in a delightful style! The one that this author has perfected in her other books. Thanks! Now I do feel good about sending my students to this lovely little book. Great book! -- And gentle on my student's budget too.
The book presents a wealth of facts regarding finite structures related with the Fourier transform and its many applications - not only the usual stuff on fast and discrete Fourier transforms (FFT/DFT), but also on Cayley graphs, error-correcting codes, group representation, and the Selberg trace formula, among other things, e.g., an introduction to the idea of random walks on groups. However, while I liked the choice of subjects and the book 'hits the point' many times, sometimes you get lost in the somewhat sketchy presentation, as if it had been assembled directly from slide presentations and lecture notes. The book also has some typos like non-matching parentheses, spacing errors, and dislocated super- and subscripts, most of them harmless.
Dr. Terras is a good math writer, though, and succeeds in making an otherwise potentially dull subject entertaining, either by demonstrating erudition - she cross-references ideas from different fields with apparent ease - or by interspersing her book with wit. For example, in chapter 15 (p. 243) we read that "A chapter of a mathematics book often begins with a long parade of definitions. Sorry, but it appears that we are at the beginning of a chapter." Fine.
I recommend the excellent and highly entertaining "Number Theory in Science and Communication," by Manfred R. Schroeder, as companion reading. In Schroeder's book you will find a more structured (perhaps more elementary) presentation of a large subset of the subjects presented by Terras.
My rate: 5 stars for the innovative choice of topics and quality of the math and -1 star for the sketchy presentation. This book deserves a better editorial and typographical treatment (e.g., the type chosen to represent the Cyrillic letter shah is a shame, some figures seem to have been extracted from shabby PowerPoint slides, etc.) as well as a hardcover in a future, ideally updated, edition. |
Use Advanced Algebra II
Summary: This module contains a table of every module within the three books of Kenny Felder's course on "Algebra II", with links to the modules.
Over a period of time, I have developed a set of in-class assignments, homeworks, and lesson plans, that work for me and for other people who have tried them. The complete set comprises three separate books that work together:
The Concepts Book provides conceptual explanations, and is intended as a reference or review guide for students; it is not used when teaching the class.
The Teacher's Guide provides lesson plans; it is your guide to how I envisioned these materials being used when I created them (and how I use them myself).
Instructors should note that this book probably contains more information than you will be able to cover in a single school year. I myself do not teach from every chapter in my own classes, but have chosen to include these additional materials to assist you in meeting your own needs. As you will likely need to cut some sections from the book, I strongly recommend that you spend time early on to determine which modules are most important for your state requirements and personal teaching style.
One more warning is important: these materials were designed for an Advanced Algebra II course. For such a course, I hope this will provide you with ready-to-use textbook and lesson plans. If you are teaching a Standard or Remedial-level course, these materials will still be useful, but you will probably have to cut or reduce some of the most conceptual material, and supplement it with more drill-and-practice than I provide.
The following table of contents provides a list of topics covered in this course with links to each module. You can use these links to move between the books or to jump ahead to any topic |
Algebra and Trigonometry
This best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling ...Show synopsisThis best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking. Comprehensive and evenly paced, this book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas. The authors' attention to detail and clarity, the same as found in James Stewart's market-leading "Calculus text," is what makes this text the market leader |
Elementary Plane Geometry
This updated Second Edition was designed primarily for a quick review of high school level geometry in a one-semester course. Techniques of geometric ...Show synopsisThis updated Second Edition was designed primarily for a quick review of high school level geometry in a one-semester course. Techniques of geometric construction are still introduced early in the course. Some major changes in this new edition include numbered sections and exercises, many more examples, more numerical problems accompanying exercise sets, a new chapter on numerical trigonometry, additional theorems that strengthen the logical development of the material, and reasons provided for all statements |
Elementary Linear Algebra
9780471669609
ISBN:
0471669601
Edition: 9 Pub Date: 2004 Publisher: Wiley
Summary: This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation. Clear, accessible, step-by-step explanati...ons make the material crystal clear. The authors spotlight the relationships between concepts to give a unified and complete picture. Established the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues.
Howard Anton is the author of Elementary Linear Algebra, published 2004 under ISBN 9780471669609 and 0471669601. Four hundred sixty five Elementary Linear Algebra textbooks are available for sale on ValoreBooks.com, one hundred thirty seven used from the cheapest price of $3.56, or buy new starting at $50.00 |
An interactive box plot applet that allows users to put in their own data that is part of a large collection of platform...
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An interactive box plot applet that allows users to put in their own data that is part of a large collection of platform independent, interactive, java applets and activities for K-12 mathematics and teacher education
An interactive scatterplot applet that allows users to put in their own data that is part of a large collection of platform...
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An interactive scatterplot applet that allows users to put in their own data that is part of a large collection of platform independent, interactive, java applets and activities for K-12 mathematics and teacher education.
This site can be used equally well for demonstration materials in lectures and for tutorial purposes. Topics range from basic...
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This site can be used equally well for demonstration materials in lectures and for tutorial purposes. Topics range from basic algebra through first-year calculus. Solved exercises throughout make the site especially appropriate for individual study. Interactive elements are added using LiveMath and Macromedia Flash.
This applet is a web based lab that explores the properties of linear functions. It is one in a series of other precalculus...
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This applet is a web based lab that explores the properties of linear functions. It is one in a series of other precalculus labs by the same author. The directions for using Graph Explorer are contained in the Cartesian Coordinates applet.
OpenAlgebra.com is a free online algebra study guide and problem solver designed to supplement any algebra course. There are...
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OpenAlgebra.com is a free online algebra study guide and problem solver designed to supplement any algebra course. There are hundreds of solved problems, video solutions, sample test questions, worksheets, and interactives. |
Calculus : Single and Multivariable - 5th edition
Summary: Calculus teachers recognize Calculus as the leading resource among the ''reform'' projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The fifth edition uses all strands of the ''Rule of Four'' - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are ...show morenot fundamentally unique. Readers will also gain access to WileyPLUS, an online tool that allows for extensive drills and practice. Calculus teachers will build on their understanding in the field and discover new ways to present concepts to their students14868 +$3.99 s/h
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Is mathematics a creative discipline?Do you need to be creative to be good at math?Does creativity help you or hurt you when you are workingon math problems?What is the difference between an
exercise
and a
problem
?Spend a few minutes answering these questions onpaper. Put your response in the journal section of your notebook. It might make for a good introduction for your January journal entries.
Transformation Geometry
A transformation of the plane can be viewed as afunction.
Definition
: A one-to-one function with the set of allpoints in the plane as the domain and the range is calleda
transformation
.What is a one-to-one function?
A function is one-to-one if the elements of the range arenot repeated. Each element of the range corresponds toonly on element in the domain.For transformations, this means that no two points can getmapped to the same point.The inverse of a One-to-One function is also a function. |
Miramar, FL ACT Mathinite math is very similar to Math modelling which is the act of creating functions or equations that describe a given application or situation. Finite Math also has Matrix Algebra, Probability, Statistics, and logic. I have taught or tutored these disciplines for 15 years |
Introduction to graph theory
In recent years graph theory has emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as ...Show synopsisIn recent years graph theory has emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology and genetics. Robin Wilson???s book has been widely used as a text for undergraduate courses in mathematics, computer science and economics, and as a readable introduction to the subject for non-mathematicians. The opening chapters provide a basic foundation course, containing definitions and examples, connectedness, Eulerian and Hamiltonian paths and cycles, and trees, with a range of applications. This is followed by two chapters on planar graphs and colouring, with special reference to the four-colour theorem. The next chapter deals with transversal theory and connectivity, with applications to network flows. A final chapter on matroid theory ties together material from earlier chapters, and an appendix discusses algorithms and their efficiency.Hide synopsis
Description:New. Graph Theory has recently emerged as a subject in its own...New. Graph Theory has recently emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology, and genetics. This book provides a comprehensive introduction to the subject. Printed Pages: 184 |
Mathematical Sciences Institute - Andrew Talmadge
Technology-driven professional development for middle and secondary level mathematics teachers. Week-long summer courses led by Nils Ahbel, Karen Bryant, Doug Kuhlmann, Ron Lancaster, Ira Nirenberg, and others have included Using the iPad to Enrich and
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Mathematics
This blog of instructional posts, which dates back to January, 2011, has included articles that introduce concepts of scale factor, logarithms, calculus, surds, and more.
...more>>
(SparkNotes.com) - WebCT.com & iTurf Inc.
Over 100 guides for mathematics ranging from pre-algebra topics to advanced work in calculus, written by students and recent graduates of Harvard University. The site also includes message boards for beginner, high school, and advanced math, calculus,
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Math Everywhere, Inc. (MEI)
Online Internet mathematics courses that are written in interactive notebooks using Mathematica. Take a whole course, or part of a course in calculus, pre-calculus, differential equations, or the geometry of n-space with matrices. This program offers
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Math Homework Help
Email contact for homework help in pre-algebra, algebra I and II, college algebra, geometry, trigonometry, pre-calculus, and calculus. Site also contains a math history timeline; math dictionary; some basic differentiation and integration rules; trigonometric
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Mathie x Pensive - Gregory Taylor
Musings on mathematics, teaching, and more by an Ottawa (Canada) high school math teacher who majored in computer science and graduated from the mathematics program at the University of Waterloo before receiving his B. Ed from Queen's University. Posts,
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Math Index - George Mason University
A glossary of math terms with illustrated examples of problems and tips for understanding concepts, from addition of positive and negative numbers to the definition of the derivative. Part of the DAU (Defense Acquisition University) Math Refresher course.
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MathMagic on the Web - Alan A. Hodson; The Math Forum
MathMagic was a K-12 telecommunications project developed in El Paso, Texas. The intent of the project was to provide motivation for students to use computer technology while increasing problem-solving strategies and communications skills. MathMagic posted
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Math Mistakes - Michael Pershan, editor
Compilation, analysis, and discussion of the mathematical errors that students make. Pershan, who teaches high school math in New York City, has posted a new error on this blog every day since June, 2012.
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Math Movie Network - Shawn Avery
This collection of student-made videos stars kids explaining math procedures and concepts. Formerly known as "the Student Made Math Movie" wiki, the self-proclaimed "Home of Student Made Math Movies" features a section labeled "practice" embedded with
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Math Mutation - Erik Seligman
These "show notes" accompany a short podcast of the same name that explores "fun, interesting, or just plain weird corners of mathematics that you probably didn't hear in school." Posts, which date back to December, 2011, have included "Flat People Give
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MathNotations - David Marain
Math investigations, challenges, problems of the day, and standardized test practice that emphasize the development of conceptual understanding in mathematics. Marain's blog, which dates back December of 2006, also features dialogue on issues in mathematics
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Math Right Now
An interactive mathematics website which features real-time multimedia solutions and companion lessons for each question in a large database. Create your own individualized problem sets and practice exams from among topics in the areas of algebra, geometry,
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Introduction to Mathcad 15 Mathcad 15, 3/eis ideal for Freshman or Introductory courses in Engineering and Computer Science. Introduces Mathcad's basic mathematical and data analysis functions (e.g., trigonometric, regression, and interpolation functions) using easy-to-follow examples, then applies the functions to examples drawn from emerging or rapidly developing fields in engineering. ESourcePrentice Hall's Engineering Sourceprovides a complete, flexible introductory engineering and computing program. ESource allows professors to fully customize their textbooks through the ESource website. Professors are not only able to pick and choose modules, but also sections of modules, incorporate their own materials, and re-paginate and re-index the complete project.prenhall.com/esource
Ronald Larsen, Ph.D., P.E., is the Department Head of the Chemical and Biological Engineering Department at Montana State University. |
Essential Math With Application - 8th edition
Summary: The latest book from Cengage Learning on Essential Mathematics As in previous editions, the focus in ESSENTIAL MATHEMATICS with APPLICATIONS remains on the Aufmann Interactive Method (AIM). Users are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. The role of ''active participant'' is crucial to success. Presenting students with worked examples, and then providing ...show morethem with the opportunity to immediately work similar problems, helps them build their confidence and eventually master the46.44 +$3.99 s/h
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Student Solutions Manual for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus, 6th:Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. |
Maimi, OK SAT MathFinite math is very similar to Math modelling which is the act of creating functions or equations that describe a given application or situation. Finite Math also has Matrix Algebra, Probability, Statistics, and logic. I have taught or tutored these disciplines for 15 years |
Concept, notation, order, equality, types of matrices, zero matrix, transpose of
a matrix, symmetric and skew symmetric matrices. Addition, multiplication and
scalar multiplication of matrices, simple properties of addition, multiplication
and scalar multiplication. Non-commutativity of multiplication of matrices and
existence of non-zero matrices whose product is the zero matrix (restrict to
square matrices of order 2). Concept of elementary row and column operations.
Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here
all matrices will have real entries).
2. Determinants: (20) Periods
Determinant of a square matrix (up to 3 x 3 matrices), properties of
determinants, minors, cofactors and applications of determinants in finding the
area of a triangle. Adjoint and inverse of a square matrix. Consistency,
inconsistency and number of solutions of system of linear equations by examples,
solving system of linear equations in two or three variables (having unique
solution) using inverse of a matrix.
UNIT-III: CALCULUS
Applications of derivatives: rate of change, increasing/decreasing functions,
tangents & normals, approximation, maxima and minima (first derivative test
motivated geometrically and second derivative test given as a provable tool).
Simple problems (that illustrate basic principles and understanding of the
subject as well as real-life situations).
3. Integrals: (20) Periods
Integration as inverse process of differentiation. Integration of a variety of
functions by substitution, by partial fractions and by parts, only simple
integrals of the type as below to be evaluated.
Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without
proof). Basic properties of definite integrals and evaluation of definite
integrals.
4. Applications of the Integrals: (10) Periods
Applications in finding the area under simple curves, especially lines, areas of
circles/ parabolas/ellipses (in standard form only), area between the two above
said curves (the region should be clearly identifiable).
5. Differential Equations: (10) Periods
Definition, order and degree, general and particular solutions of a differential
equation. Formation of differential equation whose general solution is given.
Solution of differential equations by method of separation of variables,
homogeneous differential equations of first order and first degree. Solutions of
linear differential equation of the type:
where p and q are functions of x.
UNIT-IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY
1. Vectors: (12) Periods
Vectors and scalars, magnitude and direction of a vector. Direction
cosines/ratios of vectors. Types of vectors (equal, unit, zero, parallel and
collinear vectors), position vector of a point, negative of a vector, components
of a vector, addition of vectors, multiplication of a vector by a scalar,
position vector of a point dividing a line segment in a given ratio. Scalar
(dot) product of vectors, projection of a vector on a line. Vector (cross)
product of vectors.
2. Three - dimensional Geometry: (12) Periods
Direction cosines/ratios of a line joining two points. Cartesian and vector
equation of a line, coplanar and skew lines, shortest distance between two
lines. Cartesian and vector equation of a plane. Angle between (i) two lines,
(ii) two planes. (iii) a line and a plane. Distance of a point from a plane.
UNIT-V: LINEAR PROGRAMMING |
Elementary Algebra
Elementary Algebra, 6/e is part of the latest offerings in the successful Dugopolski series in mathematics. Given the importance of examples within a ...Show synopsisElementary Algebra, 6/e is part of the latest offerings in the successful Dugopolski series in mathematics. Given the importance of examples within a math book, the author has paid close attention to the most important details for solving the given topic. Dugopolski includes a double cross-referencing system between the examples and exercise sets, so no matter which one the students start with, they will see the connection to the other.Hide synopsis
1.
Softcover,
McGraw Hill Learning Solutions,
2009224790224790.
Description:Good. Hardcover. Missing components. May include moderately...Good. Hardcover. Missing components. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780073533506Very good. Hardcover. Instructor Edition: Same as student...Very good. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Has minor wear and/or markings. SKU: 9780073340760-3-0-1Fine. 0073533505 ANNOTATED INSTRUCTOR'S EDITION contains the...Fine. 0073533505 ANNOTATED INSTRUCTOR'S EDITION contains the COMPLETE STUDENT TEXT with some instructor comments or answers. May not include student CD or access code. |
Critical Assignment 1 - Designing A Technology-Rich Lesson
Teacher: Mr. Nixon Subject: Trigonometry Grade Level: 10th – 11th
Lesson Topic: Trigonometric Functions Time Required: 4 weeks
Sunshine State Standards:
Low Cognitive Complexity
MA.912.T.1.3: State and use exact values of trigonometric functions for special angles: multiples of
and (degree and radian measures).
MA.912.T.1.4: Find approximate values of trigonometric and inverse trigonometric functions using
appropriate technology.
Moderate Cognitive Complexity
MA.912.T.1.1 Convert between degree and radian measures.
MA.912.T.1.2 Define and determine sine and cosine using the unit circle.
MA.912.T.1.5 Make connections between right triangle ratios, trigonometric functions, and circular
functions.
MA.912.T.1.7 Define and graph inverse trigonometric relations and functions.
MA.912.T.2.1 Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant,
cosecant) in terms of angles of right triangles.
MA.912.T.2.4 Use the area of triangles given two sides and an angle or three sides to solve real-world
problems.
MA.912.T.3.4 Solve trigonometric equations and real-world problems involving applications of
trigonometric equations using technology when appropriate.
High Cognitive Complexity
MA.912.T.1.6 Define and graph trigonometric functions using domain, range, intercepts, period,
amplitude, phase shift, vertical shift, and asymptotes with and without the use of graphing technology.
MA.912.T.1.8 Solve real-world problems involving applications of trigonometric functions using graphing
technology when appropriate.
MA.912.T.2.2 Solve real-world problems involving right triangles using technology when appropriate.
MA.912.T.3.4 Solve trigonometric equations and real-world problems involving applications of
trigonometric equations using technology when appropriate.
1
Access Points:
MA.912.T.2.In.a Compare the length of the straight sides in a right triangle with the length of the side
opposite the right angle (hypotenuse) by measuring the sides.
MA.912.T.2.In.b Identify and construct right triangles to solve real-world problems.
MA.912.T.2.Pa.a Recognize a right triangle in objects, pictures, or signs in real-world situations.
MA.912.T.2.Su.a Measure the sides of a right triangle to determine which side is the longest.
MA.912.T.2.Su.b Use right triangles to solve real-world problems.
Instructional Analysis - Prior Knowledge
Declarative Knowledge - Students should already know that…
1. Functions are setup with an independent variable x and a depend variable y; and should always be
setup in the form y=x or x=y
2. The horizontal axis of a graph is the independent variable x of a function and domain of the graph
3. The vertical axis of a graph is the dependent variable y and the range of the graph.
4. A right triangle has at least one angle equal to 90 degrees and the sum of the other two angle equal
90 degrees.
Students should already know how to…
1. Use a calculator to do basic arithmetic
2. Setup function properly to be entered and graphed in a graphing calculator.
3. Maneuver through websites and programs; and log into websites to access homework and blogs.
Prior Knowledge Assessment Plan
As students enter the classroom, they will be handed a piece of paper with 10 problems that
they will need to start as soon as they are seated. The problems will vary according to the
knowledge needed to start this lesson. After the class has started, 6 students at a time will put
answers on the board. In a class of 30 and 6 students per a problem, each student writes up at
least 2 answers. This will allow me to access the student's prior knowledge; as well as, the
student's confidence level in their abilities. The problems will be worked out on a separate
sheet of paper and handed in after any necessary review of prior knowledge.
2
Instructional Analysis – Declarative Knowledge
In this lesson, students will learn that….
1. Radian is a measurement of an arc length
2. The evaluation of trigonometric functions through a measurement of degrees or radians is a
ration of the sides of a right triangle created by that measurement of degrees or radians.
3. A unit circle is a circle with a radius of one and an arc length of 2π.
4. Relationship between the Pythagorean Theorem, unit circle, and trigonometric functions.
Instructional Analysis – Procedural Knowledge
In this lesson, students will learn how to….
1. Enter trigonometric functions into a graphing calculator
2. Setup the graphing calculator to view period length of the function.
3. Translate trigonometric functions along the x and y axis, increase or decrease the frequency,
and increase or decrease the amplitude.
4. Evaluate without a calculator all six trigonometric functions of given angles in increments of
n(π/2), n(π/3), n(π/4), and n(π/6) from 0 to 2π.
Interdisciplinary connections:
Students will be able to relate trigonometric concepts to applications in physics, such as, lunar and
solar trajectories, ocean tidal shifts, alternating current wave propagation, and applied forces to planes
in the form of right triangles.
Common Misunderstandings or Misconceptions:
1. Inputting trigonometric functions into the calculator with the wrong angle of measurement to
that of which the calculator is set for. Example is degrees when the calculator is set for radians.
2. Order of operations of trigonometric functions.
Plan to address these:
1. To alleviate entering wrong angle of measurements into the calculator, during the lesson of
conversion between degrees and radians explain that the calculator will give trigonometric
responses based on the mode in which it is set for. Also during examples, remind the students to
determine what mode the calculator is set in.
2. After the lesson on the description of each trigonometric function is established, a short review
of functions within functions. Explaining that trigonometric functions are functions, such as, f(x)
and should be treated as such. If there is an equation within the trigonometric function, then it
needs to be solved first to establish the angle of the function. The same as, f o g(x); f(x) needs to
be solved before g(x) can be solved.
Learning Objectives:
3
Knowledge –
1. Students will know how to convert degrees to radians and radians to degrees.
2. Students will be able to define all six trigonometric functions
3. Students will know the relation all six trigonometric functions have to the unit circle and right
triangle
Comprehension –
1. Students will be able to show the values for all six trigonometric functions that correspond to
increment values along the unit circle of n (π/2), n (π/3), n (π/4), and n (π/6) from 0 to 2π.
2. Students will be able to fully describe the unit circle and label all the important radian measures
along the unit circle
Application -
1. Students will be able to demonstrate the use of trigonometric functions to find the postion of
objects that travel on circular arcs or distance between two points on a circle 80% of the time
2. Students will be able to demonstrate the use of trigonometric functions to find the height of an
object given their distance and angle of attack at least 80% of the time
3. Students will be able to demonstrate the use of a right triangle to find the distance between two
objects by using right triangles at least 80% of the time.
Analysis –
1. Students will be able to identify a right triangle
2. Students will be able to distinguish the hypotenuse from the other legs of a triangle and how all
three legs are associated to all six trigonometric functions
3. Will be able to distinguish sinusoidal graphs from co-sinusoidal graphs
4. Students will be able to identify the parts of a trigonometric function of the constanstce a,b,c,
and d; a*f(bx+c)+d.
Synthesis -
1. Student will be able to compose a function based a given graph of a standard trigonometric
function af(bx+c)+d
2. Students will be able to formulate the necessary information to evaluate the distance between
two points.
Evaluation -
1. Students will be able to assess heights of objects using right triangles
2. Students will be able to choose correct formulas that relate to situational conditions.
Learner Analysis:
1. Age Range & Gender – Girls and boys ranging from 15-18 years of age
2. Race – White, Hispanic, and African American
3. Socio-economic background – Middle class to upper class
4. Learning Style Preferences – Visual, auditory, and kinesthetic
5. Disabilities & Exceptionalities – ESE students will be allotted all their extra provision that are
outlined according to their IEP.
4
6. Motivation – All students will be allowed to retake any test, but may have accompanying
knowledge of current material and past material. Before they are allowed to retake the test they
must show all homework that has been assigned and problems in the blog. The better grade will
replace the test. Because future material builds on previous material, this will serve as an
incentive to go back over previous material.
7. Cognitive Skills (concrete, abstract, etc.) – The student will need to rely on their concrete
knowledge and back ground of geometry and algebra to understand how to derive the abstract
concepts of trigonometric functions and evaluate the functions at various intervals along the
unit circle. The units cognitive skills will range from low to high, but will predominantly be in the
moderate to high level of cognitive skill.
Assessment Plan
Formative:
1. Students will be given a "Bell Ringer" at the beginning of class to find out how much they have
retained from previous lessons.
2. Homework will be reviewed and students will write and answer problems on the board.
3. The students' blogs will be reviewed to make sure that they are keeping up with the work and
performing up to standards of the class.
Summative:
1. Students will be given a unit test that will cover all the material presented in class and in the
blog. The unit test will evaluate that students knowledge and understand of the material and
ability to apply it to real world problems
2. All real world situational problems that were in the blog will be pulled together and submitted
as a quiz grade.
3. The Web Quest project will show that the student can analyze and evaluate a real world
problem given the knowledge base of the current material and will create a final project to be
submitted as a test grade.
Instructional Strategies:
How will you introduce the lesson?
The introduction of the unit will be presented in a power point. The power point will give a true relation
between what the student is learning and how they will use it every day.
How will you teach concepts during the lesson?
New material will begin with a 15-20 minutes lecture that will present the new concept and a few
examples. Then 15-20 minutes will be used for the students to present random problems in the book on
the board and present them to the class. The following day, class will start with a bell ringer; and a
review of the material that was presented the day before, and go over the home work the night before.
5
The rest of the class time will be used for collaborative team work on the current blog questions; which
will be 6 real life questions composed from the current and previous material.
How will you conclude the lesson?
The lesson will be concluded with a web quest that will be a culmination of the real life blog questions.
The details of the web quest will be introduced two weeks before the web quest is to be started.
Classroom & Technology Management Strategies:
While students are using the blog within the classroom, the teacher will roam among the computer
stations to provide help, keep students on task, and make sure the work is being completed in the time
allotted.
Learning Activities:
During the introduction of the lesson, students will:
Students will listen and ask question during the power point presentation
The power point presentation is to facilitate an overall understanding what the lesson is about
The student will gain a purpose of what the lesson is for, and future uses and what
During the lesson, students will (Mention Web Quest):
First day of new material
Have a 15-20 minute lecture that will present new concepts and include examples on the use of
the new material.
Spend 15-20 minutes working problems on the board and presenting their answers to the class.
Second day of new material
The class will begin with a bell ringer and 10 minutes review the homework assigned the night
before.
The rest of class the students will divide into groups of 4 members that will not change during
this lesson and will work on the collaborative class blog site.
The blog will have questions relative to the current lesson and will have an overall encompassing
facilitation toward the web quest that will be do at the end of the lesson.
As groups complete the blog assignments they will use the website to navigate to web quest and
continue review or completing necessary work for the web quest.
At the conclusion of the lesson, students will:
Understand the basic principles of trigonometric functions
How trigonometric functions relate to right triangles
Combine all blog questions into a portfolio and turn in for a quiz grade
Complete the web quest in which the students will use current knowledge to analyze and apply
to real life situations and create a solution.
6
Materials & Resources: (List everything you will need)
Supplies:
Outline of the lesson
Dry erase markers
Computers
Graphing calculator
Technology tools (Software and hardware):
Teacher created website that will direct students to the power point, blog, and web quest
Blog site to post helpful information for other groups to access
URL's you will use:
(Will be determined as the blog and web quest are created)
Self-Evaluation:
Teacher NETS addressed:
I plan on meeting NETS by teaching the appropriate use of technology and integrating it into the lesson
in meaningful ways. I will model appropriate technology use throughout all the lessons. The lessons
allow for students to incorporate digital resources in creative ways, allowing for student growth in both
mathematics and technology. Using the blog and webquest as assessment tools, I am able to expose the
students to non-standard ways of expressing mastery of a concept. I plan to provide this lesson to
others who are teaching this subject area so that they may share this learning experience with their
students or use the lesson as a springboard for developing their own unique lesson.
Student NETS addressed:
Throughout the course of this lesson, all of the NETS for students will be addressed. The students will be
actively using technology, participating in a digital community, conducting research, collaborating, and
problem solving. By the end of the lesson, students will not only be proficient in trigonometric functions,
but also in the many forms of technology integrated into this lesson.
Lesson Strengths:
The lesson draws its strength from it being student driven with a small amount of time used for teacher
provided lecture and examples. This will allow students to gain confidence in understanding and
applying the material to the homework problems. The student driven process will help foster or
continue the student's active learning experience by allowing them to seek the answers to the blog
questions and web quest.
Lesson Weaknesses:
Because the lesson is student driven, then the momentum of the lesson is primarily depicted by the
students. This can cause the learning time to slow or even stale cause the instructional time to increase.
7
Possible Solutions or Action Plan for Improvement:
If the teacher is not use to positively motivating students, then I would recommend a work shop that
teaches how to change body language and syntax to help create a positively motivating atmosphere.
Also, it is important to keep on top of student's questions as they arise. If a student is waiting for a
question to be answered, then chances are the student is not moving along with the lesson and will also
cause the momentum to stale |
Sketchpad® Dynamic Geometry® software gives students a tangible, visual way to learn mathematics that increases their engagement, understanding, and achievement. And it's not just for geometry. Use it for elementary and middle school math, algebra, precalculus, and calculus. More Info »
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Link to official The Geometer's Sketchpad site
The Geometer's Sketchpad was added by Reprotected in
Oct 2010 and the lastest update was made in
Oct 2010. There is a history of activites on
The Geometer's Sketchpad in our Change Log and Activity Log.
It's possible to update the information on The Geometer's SketchpadThe Geometer's SketchpadGraph is an open source application used to draw mathematical graphs in a coordinate system. Anyone who wants to draw graphs of functions will find this program useful. The program makes it very easy to...
Geometry Pad is a dynamic geometry application for iPad and Android tablets. Geometry Pad is your personal assistant in teaching and learning geometry through practice. With the Geometry Pad you can create...
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Alumni of our math program have been very successful. Our alumni profiles feature some of their positions, including Actuary, Medical Doctor, Lawyer, High School Teacher and Vice President of Information Management.
The five professors in the our math department have a wide variety of mathematical interests, including game theory, mathematical modeling, statistics, chaos theory, geometry, knot theory, graph theory, and differential equations. They are also interested in interdisciplinary applications of math in fields such as political science, economics, education, biology, chemistry and physics |
Saxon's Math in Focus: The Singapore Approach program is the American edition of the highly popular and successful Singapore Approach to mathematics. This approach places strong emphasis on problem-solving and model-drawing skills, with definitions of core math concepts explained in extremely simple terms.
Instructional advantages to the Math in Focus program include paced instruction for optimum mastery of concepts, consistent use of visual models and manipulatives for concrete/abstract awareness, and problem-solving methods that help students visualize and understand math concepts.
Math in Focus is available in semester homeschool kits as well as student packs. The semester kits include a Student Textbook, Student Workbook, and Teacher's Edition, with Assessments available separately. Student packs include Student Textbooks and Student Workbooks for both semesters as well as Assessments.
Volume A is designed for the first half of the year, and Volume B is designed for the second half. Teacher's Editions contain complete program support, including Chapter Overviews with math background, cross-curricular connections, and a planning guide. Student Books allow for age-appropriate and mathematically sound practice, assessment, and development of problem-solving and thinking skills. Assessments provide a clear picture of student progress. |
New. Considered a classic by many, A First Course in Abstract...New. Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. Features " This classical approach to abstract algebra focuses on applications. " The text is geared toward high-level courses at schools with strong mathematics programs. " Accessible pedagogy includes historical notes written by Victor Katz, an authority on the history of math. " By opening with a study of group theory, this text provides students with an easy transition to axiomatic mathematics Contents Chapter 0. Sets and Relations Chapter 1. Groups and Subgroups Chapter 2. Permutations, Cosets, and Direct Products Chapter 3. Homomorphisms and Factor Groups Chapter 4. Rings and Fields Chapter 5. Ideals and Factor Rings Chapter 6. Ectension Fields Chapter 7. Advanced Group Theory Chapter 8. Factorization Chapter 9. Automorphisms and Galois Theory Appendix: Matrix Algebra Printed Pages: 456..24968 Brand new book. International Edition. Ship...New. 12920249 |
introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.
Most helpful customer reviews
The content of this book is fairly complete. But the treatment of each subject is not very detailed. So you end up with a 288 pages book which deals with the same amount of content that the Zienkiewicz & Taylor. And I really don't like the typesetting
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
9 reviews
15 of 15 people found the following review helpful
Best book on mathematics of FEMJune 20 2010
By
A. I. Haque
- Published on Amazon.com
Format: Paperback
This book provides an excellent mathematical introduction to the Finite Element Method. All of the necessary functional analysis concepts are provided. This book is aimed at mathematicians and not engineers!
Cautionary notes: 1. A strong background in mathematical analysis is required to understand the proofs. 2. Few implementation details provided. 3. Focus is primarily on elliptic PDEs. Only 1 chapter each for Parabolic and Hyperbolic PDEs.
This was the textbook I had for a grad math course in FEM. I had previously studied FEM on my own through an engineering text. Engineering FEM books explain FEM through a bottom-up approach: formulating the elements and then assembling them. Mathematical FEM books explain FEM through a top-down approach: formulating the solution space and then specifying the elements. This book follows the mathematical approach, so you will not find it useful in terms of implementation or applications. However, the proofs are fairly easy to follow (for a mathematician).
6 of 6 people found the following review helpful
Among the best introduction to FEMOct. 7 2010
By
Luciano Teresi
- Published on Amazon.com
Format: Paperback
I am dealing with FEM since many years, and in many ways. I use FEM as a computational tool, I teach FEM at graduate level, and I am involved in developing new computational tools related to FEM.
Among the many books devoted to the subject, that by C. Johnson is definitely one of the best; my opinion is that no other book can introduce you to the method as seamlessly, yet accurately, as this book does. I strongly suggest the book to anyone interested in the subject. Despite its age (first published in 1987), it is still extremely useful.
5 of 5 people found the following review helpful
Very goodNov. 7 2010
By
Vedran Jagodnik
- Published on Amazon.com
Format: Paperback
Verified Purchase
I bought this book for my phd course. Although the book is written on a pure mathematical way (and I'm not an mathematician) I did understand almost everything. Nevertheless, before starting to read this you should have in your mind that you NEED to have some basic of advanced mathematics. I would recommend this book to engineers who are starting to use FEM analysis, and graduate students who are learning the basics of FEM.
3 of 3 people found the following review helpful
chapters 1-5 are a good introduction to FEMJune 2 2013
By
L. de Vito
- Published on Amazon.com
Format: Paperback
At least chapters 1-5 make it a good introductory book. Chapters 6-13 are not so well-written, not so clear: The author goes way too fast and included advanced materials. It would have been better to stay at an introductory level and to further trim it down. It cannot claim 5 stars because of numerous weaknesses: - as already observed by another reviewer, it is typed in something like Word instead of Latex, hence reading lots of equations is a painful experience ! - it is light on discontinuous Galerkin method and heavy on the streamline diffusion method since the latter was the most promising method at the time this little book was written; since then, the streamline diffusion method is dead and the discontinuous Galerkin method is the standard FE extension to deal with hyperbolic system of equations. If it does not suffice: - no Matlab / Octave source code to illustrate - no errata - no solution to the exercises either in the book or on the author's home page. Nevertheless, at this very low price, I would recommend it for a concise introduction.
Great OverviewJune 12 2014
By
Michael Harmon
- Published on Amazon.com
Format: Paperback
This book is a gem. Slightly out of date, but really informative. It covers a lot of material in a more of an overview way, lightly touching on each topic. It is not a book about how to do FEM from an engineering or computational way, more in the mathematical light. In that sense its definitely a great 2nd book on FEM or as a reference on FEM. Even though it is now almost 30 years old a lot of it is still relevant today and its interesting to see a time capsule of where FEM stood and was going at that time. My only complaint is the font, however you get used to that after a little while. |
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Mathematica is revolutionizing mathematics. What the electronic calculator did for arithmetic in the '70s, a powerful new computer language, Mathematica, is doing for all mathematical calculations today. In this task-specific guide to Mathematica for mathematically sophisticated applications, Stan Wagon's goal is to develop the user's mathematical imagination and to unveil the full capacity of the program to achieve impressive results-from standard numerical computations to complex programs at the frontiers of mathematics.
Hardcover, ISBN 0716722291 Publisher: W.H. Freeman & Company, 1991 Hardcover. Used - Good Good . Book shows a small amount of wear to cover and binding. Some pages show signs of use. Sail the Seas of Value.
Hardcover, ISBN 0716722291 Publisher: W.H. Freeman & Company, 1991 Used - Good, Usually ships in 1-2 business days, Book shows a small amount of wear to cover and binding. Some pages show signs of use. Sail the Seas of Value.
Hardcover, ISBN 0716722291 Publisher: W.H.Freeman & Co Ltd, 1991 Good. US Edition. Former Library book. Shows some signs of wear, and may have some markings on the inside. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!. |
Synopses & Reviews
Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments.
Publisher Comments:
The fifth edition of Calculus brings together the best of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus. The author team's extensive experience teaching from both traditional and innovative books and their expertise in developing innovative problems put them in an unique position to make this new curriculum meaningful to students going into mathematics and those going into the sciences and engineering. Calculus: Single Variable, 5e exhibits the same strengths from earlier editions including the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. The conceptual and modeling problems, praised for their creativity and variety, continue to motivate and challenge students. The fifth edition includes even more problems and additional skill-building exercises Ingram Wiley |
This site is part of the NCTM's Student i-Math Investigations website. It uses algebra and discrete mathematics to analyze...
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This site is part of the NCTM's Student i-Math Investigations website. It uses algebra and discrete mathematics to analyze population changes in a trout pond. Included are applets for numerical and graphical analysis.
Flash program that tests students' graphical understanding of addition or subtraction of functions. Uses two...
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Flash program that tests students' graphical understanding of addition or subtraction of functions. Uses two randomly generated trig graphs that students then need to add and subtract to find the new graph. Appropriate for use when discussing special cases of graphing trig functions or for use when discussing algebra of functions in a precalculus class even if students are not yet familiar with trig.trigonometry, college algebra, math, precalculus, algebra of functions, addition of ordinates, subtraction of ordinates |
Teenage son liked it as he nows sees the point of maths. Says this book makes it relevant in a way his teachers fail to. He is happily revising for his GCSE maths now without complaint. You can buy various versions tailored to different industries and I recommend them. |
Essential Mathematical Methods for Physicists
9780120598779
ISBN:
0120598779
Pub Date: 2003 Publisher: Academic Pr
Summary: This new adaptation of Arfken and Weber's bestselling Mathematical Methods for Physicists, Fifth Edition, is the most modern collection of mathematical principles for solving physics problems. Additional explanations and examples provide models and context for methods applicable to a wide range of physics theories and applications. Features: · Many detailed, worked-out examples illustrate how to use and apply mathema...tical techniques to solve physics problems · Frequent and thorough explanations help readers understand, recall, and apply the theory · Introductions and review material provide context and emphasis on key ideas · Many routine exercises reinforce basic, foundational concepts and computations "True to the title, this new text achieves a comprehensive coverage of the "essential" topics in mathematical physics at the undergraduate level. This new version is filled with enlightening examples, which is the key to undergraduate teaching . More importantly, many examples are real problems from various fields of physics. Illustration of the mathematical principles behind the solution of these problems further enhances the connection between this course and other courses in a physics curriculum. " David Hwang, University of California at Davis
Weber, Hans J. is the author of Essential Mathematical Methods for Physicists, published 2003 under ISBN 9780120598779 and 0120598779. Five hundred seventy five Essential Mathematical Methods for Physicists textbooks are available for sale on ValoreBooks.com, one hundred twenty three used from the cheapest price of $20.48, or buy new starting at $47Corners/spine lightly worn/bent, covers lightly scuffed, copyright page lightly stained, front fly/first few pages lightly stained/creased at right top corner, text pages clean/tight/bright |
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Overview
Loaded with helpful hints and easy-to-understand examples — and now with 20 percent more content — this book covers the type of mathematical problems that golf course superintendents face every day. You'll learn how to handle budgeting estimates and figure how much fertilizer or pesticide you should buy. You'll also find guidance on the ordering of topdressing, irrigation volume and coverage, area and volume calculations, and many other aspects of golf course operation that require a thorough understanding of basic mathematical principles. Whatever the management calculation may be, this book has the solution you need! Example problems and answers help you get the job done right.
Editorial Reviews
Booknews
Christians (horticulture, Iowa State U.-Ames) and Agnew, with a North Carolina turf company, explain how many of the tasks of a golf course superintendent involve mathematics, and show how to set up the problems and solve them. They include additional practice problems in each chapter, with answers at the back, to allow readers to get some experience. Among the problems are calculating area, pesticides, irrigation, and seeding rate. Chapters also look at the metric system, and applying small amounts of materials to greenhouse pots. No dates are noted for earlier editions. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Product Details
Meet the Author
Nick E. Christians, PhD, is a professor of horticulture at Iowa State University, Ames, Iowa, and is the recipient of numerous honors, includingthe Outstanding Educator Award from the American Society of Horticultural Sciences and the Iowa Legislative Teaching Excellence Award. Michael L. Agnew, PhD, is a senior field technical manager at Syngenta Professional Products, Greensboro, North Carolina, and former associate professor and turfgrass extension specialist at Iowa |
Elementary Linear Algebra - 9th edition
Summary: Presents the fundamentals of linear algebra in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. This substantial revision includes greater focus on relationships between concepts, smoother transition to abstraction, early exposure to linear transformations and eigenvalues, more emphasize on visualization, new material on least squares and QR-decomposition and a greater number of proofs. Exercise sets ...show morebegin with routine drill problems, progress to problems with more substance and conclude with theoretical problems2004047166960112 |
Princeton Junction PrecalI show students how to use this information to know what to do when solving math/science problems and develop smarter, more efficient problem solving strategies to put them ahead of their peers.
3.) TARGETING & ELIMINATING WEAKNESSES - By targeting their specific weaknesses and focusing on impro... |
Essential Matlab for Engineers and Scientists for an Amazon Gift Card of up to £6.35, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book Description
Publication Date: 1 Nov 2012Whether you are learning Matlab on your own, or using it as an accompaniment, this book guides you through simple exercises first. It takes you through more difficult problems as you progress. I would recommend that you read through the worked examples in the book, then try and do it on your own a few times until you eventually can do it without looking at the book, which really builds your confidence. For this book to be really useful, you need to practise as you go along. This isn't a book just to read. Very well presented, and better than previous editions.
Depends what you need - it clarified a few very very basic things, but doesn't explain how to implement Euler / Runga-Kutta / Du-Fort Frankel nor Crank-Nicolson methods which I feel are essential for engineers.
5.0 out of 5 starsshould have started with this book first24 July 2013
By Ernest Bylander - Published on Amazon.com
Format:Paperback|Verified Purchase
After trying several "how-to" books I tried this one. It is a winner and is written by an excellent teachers. I have MatLab because it is required by the book: "Tutorial on Neural Systems Modeling". And MatLab seems to have a very steep learning curve, and this book seems to scale the methods of ML successfully. I have used various such programs over the years: Basic, from DOS thru 3 HP versions and Visual Basic, Mathcad, 2 versions of Linux, C, C+, Mathematica etc. Also MatLab is based on Matrices, and I have had undergrad vectors, graduatecourse in Matrices, and a few Matrix Quantum Mechanics courses. However I found Matlab almost insurmountable. Hahn and Valentine overcome these drawbacks and even show where tutorials can also be found on the MatLab site. As an aside, it took from last October through this February to lease some part of MatLab. And several books later did help-some until this summer when I bought this book. I find this book to be the answer to my learning difficulties.
2 of 2 people found the following review helpful
5.0 out of 5 starsGood Book for learning MatLab23 Jan 2014
By Robert Kettelkamp - Published on Amazon.com
Format:Paperback|Verified Purchase
This is a good starter book for getting familiar with using MatLab. I would recommend this book to anyone not fluent in MatLab
2 of 3 people found the following review helpful
5.0 out of 5 starsExcellent introductionary book29 Aug 2013
By Oleg Agafonov - Published on Amazon.com
Format:Paperback
It is an excellent introductionary book, which I often use as a reference. It is well written; it has good structure and useful examples. It would be very useful if you just started to use Matlab and it provides good overview of Matlab's major functions.
0 of 1 people found the following review helpful
3.0 out of 5 starsNot excelente!13 Mar 2014
By Joćo Manuel Ferreira Verde - Published on Amazon.com
Format:Paperback|Verified Purchase
There are some errors in the book. Excellent book for beginners - it has solutions and many problems solved. It is not bad choice but if you want to evolve more...you have to buy another book. Maybe one month after start working on Matlab.
0 of 1 people found the following review helpful
4.0 out of 5 starsCollege Text Choice24 Jan 2014
By lindy - Published on Amazon.com
Format:Paperback
We had to include MatLab into an introductory engineering class, and this book fits that bill very nicely. It will be used in the Fall semester this year. |
Multivariable Calculus - 4th edition
Summary: CALCULUS 4/e brings together the best of both new and traditional curricula to meet the needs of even more instructors teaching calculus. The author team's extensive experience teaching from both traditional and innovative books and their expertise in developing innovative problems put them in an unique position to make this new curriculum meaningful to students going into mathematics and those going into the sciences and engineering. This edition will work well for ...show morethose departments who are looking for a calculus book that offers a middle ground for their calculus instructors.
CALCULUS 4/e exhibits the same strengths from earlier editions including the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. The conceptual and modeling problems, praised for their creativity and variety, continue to motivate and challenge students. ...show less
Functions of Several Variables A Fundamental Tool: Vectors Differentiating Functions of Many Variables Optimization: Local and Global Extrema Integrating Functions of Many Variables Parameterized Curves and Vector Fields Line Integrals Flux Integrals Calculus of Vector Fields05 +$3.99 s/h
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northeastbooks FL Miami, FL
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Introduction to Problem Solving Grades 3-5
9780325009704
ISBN:
0325009708
Edition: 2 Pub Date: 2007 Publisher: Heinemann
Summary: Susan O'Connell is the editor of Heinemann's Math Process Standards series, as well as the author its volumes Introduction to Problem Solving (grades PreK - 2, 3 - 5, and 6 - 8) and Introduction to Communication (grades PreK - 2, 3 - 5, and 6 - 8). She also wrote the popular Now I Get It (Heinemann, 2005). Sue has a varied background, including years as a classroom teacher, a school-based instructional specialist, a ...testing coordinator, a talented-and-gifted teacher, a district school-improvement specialist, and a university professional-development schools coordinator. Currently she is a project consultant for a federal teacher-quality grant in the College of Education at the University of Maryland. Additionally, she is an educational consultant, conducts mathematics seminars for teachers throughout the country, and a Heinemann Professional Development Provider.
O'Connell, Susan is the author of Introduction to Problem Solving Grades 3-5, published 2007 under ISBN 9780325009704 and 0325009708. Three hundred twenty five Introduction to Problem Solving Grades 3-5 textbooks are available for sale on ValoreBooks.com, one hundred three used from the cheapest price of $20.41, or buy new starting at $30.42 |
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About the book:
This applied introduction to the mathematics of probability and statistics emphasizes the existence of variation in almost every process, and how the study of probability and statistics helps us understand this variability. Designed for students with a background in calculus, it reinforces basic mathematical concepts with numerous real-world examples and applications to illustrate the relevance of key concepts.
Hardcover, ISBN 0023557303 Publisher: Macmillan Library Reference23557303 Publisher: Collier Macmillan Macmillan :: Collier Macmillan, 1983 2nd ed.. Used - Good. Shows some signs of wear, and may have some markings on the inside. Find out why millions of customers rave about Better World Books. Experience the best customer care and a 100% satisfaction guarantee. 2nd ed.
Hardcover, ISBN 0023557303 Publisher: Macmillan Library Reference, 1983 Good. US Edition. Shows some signs of wear, and may have some markings on the inside. Find out why millions of customers rave about Better World Books. Experience the best customer care and a 100% satisfaction guarantee.. |
Elementary Algebra-Student Solution Manual - 5th edition
Summary: Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
PAPERBACK Good 1111989028$23 |
Featured Research
from universities, journals, and other organizations
Math goes viral in the classroom
Date:
December 14, 2009
Source:
University of Alberta
Summary:
At least a dozen Alberta high-school calculus classrooms were exposed to the West Nile virus recently. Luckily, it wasn't literally the illness. Educators used the virus as a theoretical tool when they designed materials for use in an advanced high-school math course.
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At least a dozen Alberta high-school calculus classrooms were exposed to the West Nile virus recently.
Luckily, however, it wasn't literally the illness. University of Alberta education professor Stephen Norris and mathematics professor Gerda de Vries used the virus as a theoretical tool when they designed materials for use in an advanced high-school math course. The materials allow students to use mathematical concepts learned in their curriculum to determine the disease's reproductive number, which determines the likelihood of a disease spreading.
The approach is a marriage of science and math, subjects the researchers say seem to exist in separate worlds at a secondary-school level, but that when brought together can effectively bring real-world scenarios into the classroom to enhance learning and understanding.
Not to mention answering that ages old high-school student question: "why do I need to know this?"
"This piece was designed to satisfy an optional unit in Math 31 (Calculus), for which there are no materials, so we said, 'let's fill the gap,'" said Norris. "These materials show a real application of mathematics in the biology curriculum for high-school students."
Norris and de Vries chose a published academic math paper on the transmission of the West Nile virus and modified it -keeping the science intact, but making it readable and practical for high-school calculus students.
The information and equations in the original paper dealing with disease transmission were then used as the basis for calculus math problems to be solved by the students. Students were presented with a variety of materials that covered topics and concepts such as rate of change, exponential growth-decay models, and models for the carriers of the virus, including mosquitoes and infectious and susceptible birds. The students' mathematical skills were then put to use in determining the spread of the disease using various parameters, which included variables such as biting rate and the probability of infection.
Norris underlines that the project challenged the students to see and understand science in a different fashion from what they learn inside the science curricula. He points out that high-school classroom scientific experiments are "proven" science and have been around for at least 300 years, in many cases. For the students to discover that real scientists often work with some assumptions that they know to be false in order to reach their conclusions was certainly an eye-opening realization for them, he says.
"There's no way out of the fact that the knowledge you gain from science is imperfect; it's tentative and subject to change," said Norris. "I think that's what struck the students between the eyes."
Both researchers agree that this form of collaborative, interdisciplinary learning can take place across all subject areas. De Vries and Norris are currently working on another project that focuses on population genetics that will fit into Grade 12 biology and math courses.
"It's mathematics in the real world. Kids are always asking, 'why am I learning this,'" she said. "All of a sudden the mathematics that kids have learned comes together in a project like |
complete curriculum for the second semester of pre algebra. It is designed to be used with the first semester book as well as the student workbook with keys. All these books are available... More > from Simplified Solutions for Math on LULU. Contact Simplified at ss4math@gmail.com for more information and a complete set of PowerPoint presentations for each lesson, free with purchase. Completely self-contained, ideal for home schooling as well as traditional classrooms.< Less
This book reviews the necessary concepts that appear on the Pennsylvania Algebra I Keystone Exam - Module 1. The fifteen Also included are two 20-question practice exams that include an answer key and scoring guidelines to gauge a student's ability level on the exam. Answers for all questions are provided to check the student's work and understanding.< Less
This book reviews the necessary concepts that appear on the Pennsylvania Algebra I Keystone Exam - Module 2. The twelve
Also included are two 20-question practice exams that include an answer key and scoring guidelines to gauge a student's ability level on the exam. Answers for all questions are provided to check the student's work and understanding.< Less
This book provides two free practice exams based upon the Algebra I Keystone Exams in Pennsylvania. The book is free to educators to use and reproduce. If you are interested in purchasing the... More > complete program, please feel free to visit: Less
This series of 26 warm-ups reviews eligible Algebra I Keystone Exam content for the Pennsylvania Department of Education's Algebra I Keystone Exam. Each review section includes 5-6 multiple-choice... More > questions and 1 two-part constructed response question. Answers for all questions are included as well as two complete practice exams per module.< Less
This collection of cards provides students with the essential information about key topics in high school mathematics and algebra. Each card provides a visual representation of the concept to assist... More > in remembering the formulas, definitions, and procedures needed to succeed in math class. From square roots to measurements and graphing linear equations to test-taking strategies, these cards use visual cues so students can relate the images to the concepts.
To better prepare your student for math class, visit to see other research-based books to organize, motivate, and prepare your students to be successful on math assessments, homework, and problem solving. For students transitioning to high school, we offer a self-guided book to ease the transition and improve grades.< Less |
Prealgebra instruction includes a review of the basics of mathematics and a thorough introduction to integers, basic equations and word problems. This course is designed to develop the skills and understanding to perform the fundamental operations on whole numbers, fractions and decimals. Topic |
Basic College Mathematics - 4th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics, Fourth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success Binding solid. Very Good copy. Covers clean with minimal wear. Text contains no apparent markings No CD or access code included. All items shipped to US include delivery confirmation....show more Thanks for looking! ...show less
$111649409-5-0
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GICW Books Hillsboro, OR
Reading copy. May have notes, underlining or highlighting. Dust jacket may be missing.10.74 +$3.99 s/h
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newrecycleabook centerville, OH
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Quick Overview
MASTER MATH: ESSENTIAL PHYSICS presents, teaches, and explains the fundamental topics of algebra-based physics. It includes engaging, fun examples and applications, as well as challenging practice problems with explanatory answers. Master Math: Essential Physics was written for you, the student, parent, teacher, tutor, or curious thinker. It covers the essentials of high school and college curricula. It can serve as a supplement to your textbook, a handy reference, or a tutor for lifetime learners. Topics encompass motion, force, momentum, Newton's Laws, equilibrium, friction, forces in nature, energy, work, elasticity, harmonic motion, static and moving fluids, heat, temperature, gas, electric fields, electromagnetism, direct and alternating current, waves, sound, radiation, light and optics, and an introduction to relativity, quanta, the atom, dark matter, and dark energy |
Tagged QuestionsIn physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself had the joy of discovering AMS' Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect ...
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I really liked Teach Yourself Logic: A Study Guide by the user Peter Smith. It is a thorough guide how to teach yourself logic and set-theory from scratch up to any level with book recommendations for ...A little background: I am currently a sophomore (studying mathematics) at an unknown university in the Middle East. My mother is European so it does not make sense to study mathematics in the Middle ...
I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
I'm finishing my undergraduate degree in 6 weeks and I'm pretty happy with how my education is coming along so far. I can write proofs, solve many different problems, and I even have some idea as to ...
I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...
I have a strong interest in maths, and I feel that advanced physics is cool too (although I've only studied classical mechanics at high school, which is kind of boring). So I'm not sure about which ...
Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
I'm a high school student who just finished elementary school.Though since I was into math I started going through advanced math while I was in elementary school and I pretty much finished most of the ...
I was just wondering if someone would be kind enough to tell me in what order (I know that there is no real "best order") one would most profitably study these subjects/books:
(edited to conform with ...
There are many books, written by highly decorated academics, which feature proofs that I can hardly comprehend in an acceptable amount of time. Roughly each week, it happens that I find myself having ...
I have taken some courses in Convex optimization.I want to work on the bi-level optimization. Now I would like to know a little bit more about the pure mathematical side. Is there any good books in ...
I'm struggling a bit with functions(limits, squeeze theorem, etc).
I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would ...
The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear.
Being human, ...
I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to ...
Suppose that a student is doing self study from some book: "Introduction to subject $X$" because he wants to learn the first things about the subject $X$. Now, I know that the first rule is "do lots ...
The popularity of books like Visual Group Theory and Visual Complex Analysis validates the importance of pictures and visualization for complex subjects. Unfortunately, I'm not aware of similar books ...
I currently have an engineering-style education in mathematics. We covered quite a lot of material (e.g. real and complex analysis, some probability theory and graph theory), but more often than notI'm mostly self-taught and while I know (and use) many advanced mathematical topics, I often enough find holes in my understanding of lower level math. Is there an exam (or series of exams) I could ...
Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ... |
Book Description
These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs.
These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs. |
An Introduction to Computational Combinatorics
This book describes algorithms of mathematical methods and illustrates their application with examples. The mathematical background needed is ...Show synopsisThis book describes algorithms of mathematical methods and illustrates their application with examples. The mathematical background needed is elementary algebra and calculus6.
Cambridge University Press,
1979Description:Good. 0521294924 USED BOOK in good condition| No supplements|...Good. 0521294924 describes...New. This item is printed on demand. This book describes algorithms of mathematical methods and illustrates their application with examples. The mathematical background needed is elementary algebra and calculus |
books.google.com - Ever since the groundbreaking work of J.J. Kohn in the early 1960s, there has been a significant interaction between the theory of partial differential equations and the function theory of several complex variables. Partial Differential Equations and Complex Analysis explores the background and plumbs... Differential Equations and Complex Analysis |
algebra
Bibliography
General history
B.L. van der Waerden, A History of Algebra: From al-Khwārizmī to Emmy Noether (1985), is a highly respected classic. Two works that contain selections from original mathematical texts, including many that are directly relevant to the history of algebra, are David Eugene Smith, A Source Book in Mathematics (1929, reissued 1959); and John Fauvel and Jeremy Gray (eds.), The History of Mathematics: A Reader (1987, reissued 1990).
Ancient and Greek algebra
Among the books on ancient mathematics, including sections on algebra, the reader may consult O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed. (1969, reissued 1993); Richard J. Gillings, Mathematics in the Time of the Pharaohs (1972, reprinted 1982); and Jens Hoyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (2002).
The degree to which algebraic ideas do or do not appear in Greek geometric texts has been widely discussed by historians. A comprehensive summary of work on this controversial question appears in Michael N. Fried and Sabetai Unguru, Apollonius of Perga's Conica: Text, Context, Subtext (2001).
Indian and Chinese algebra
Among the few English-language books on the history of algebra in India and China, the following are recommended: C.N. Srinivasiengar, The History of Ancient Indian Mathematics (1967); and Ulrich Libbrecht, Chinese Mathematics in the Thirteenth Century: The Shu-Shu Chiu-Chang of Ch'in, Chiu-Shao (1973).
Islamic algebra
Research on Islamic mathematics has vigorously developed in recent years. Two important works are Roshdi Rashed (Rushdi Rashid), The Development of Arabic Mathematics: Between Arithmetic and Algebra (1994; originally published in French, 1984); and J.L. Berggren, Episodes in the Mathematics of Medieval Islam (1986).
Algebra in Renaissance Europe
Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, trans. from German (1968, reprinted 1992), is one of the most important accounts of the evolution of the concept of number from the ancient Greeks to the 17th century. Paul Lawrence Rose, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo (1975), is a highly respected history of Renaissance mathematics.
Modern algebra
Leo Corry, Modern Algebra and the Rise of Mathematical Structures (1996), traces the emergence of the structural approach, as well as efforts to develop a metatheory of structures. |
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Math A
The them entirely. Barron's brand new Regents Exams and Answers book for this subject covers all topics on the new Math A exam. It also presents all seven of the actual New York State Regents Math A exams that have been issued to date. All exams appear in this book with answers to all questions. These ever popular guides include study tips, test-taking strategies, score analysis charts, and other valuable features. They are an ideal source of practice and test preparation. The detailed answer explanations make each exam a practical learning experience.
About the author (2004)
Lawrence S. Leff has helped make mathematics and tests more accessible to students through his work and his writing. Leff is assistant principal and math supervisor at Franklin D. Roosevelt High School in Brooklyn, New York. His writing is centered primarily on examination preparation guides. His books, which target high school students, have included, Let's Review: Sequential Mathematics, a teacher planning and student review and math test preparation guide, and Preparation for the CLEP--College-Level Examination Program. Some of Leff's math review guides originally were intended for New York area high school students participating in the three-year mathematics curriculum there. Lawrence Leff has also published mathematics materials not strictly for test-taking, including Geometry the Easy Way. |
ethi...
...This class usually introduces trigonometry, a detailed study of circles. This course also goes into the relationship between parabolas and circles, and the equations and graphs of all conic sections. Whereas algebra is the study of variables, numbers that change, calculus is a study of how variables change -- their rate of change, to be specific. |
Summary: Focusing on the important ideas of geometry, this book shows how to investigate two- and three-dimensional shapes with very young students. It introduces methods to describe location and position, explores simple transformations, and addresses visualization, spatial reasoning, and the building and drawing of constructions. Activities in each chapter pose questions that stimulate students to think more deeply about mathematical ideas. The CD-ROM features fourteen arti...show morecles from NCTM publications. The supplemental CD-ROM also features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. ...show less
Edition/Copyright:01 Cover: Paperback Publisher:National Council of Teachers of Mathematics Published: 01/28/2001 International: NoShows definite wear, and perhaps considerable marking on inside. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!
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Let's Graph
Text and photographs introduce the concept of graphing and present examples of two different kinds of graphs.Text and photographs introduce the concept of graphing and present examples of two different kinds of graphs Let's Graph (Yellow Umbrella Books) This book is in Good...Good. Let's Graph (Yellow Umbrella |
Slay the calculus monster with this user-friendly guide Calculus For Dummies, 2 nd Edition makes calculus manageable—even if you're one of the many students who sweat at the thought of it. By breaking down differentiation and integration into digestible concepts, this guide helps you build a stronger foundation with a solid understanding... more...
From differentiation to integration - solve problems with ease Got... more...
Everyday math skills can be painlessly learned and easily mastered, transforming readers from a person who doesn't know the meaning of APR into someone who understands credit card rates. Ryan's guide is broken into sections which review basic arithmetic from fractions to percents. more...
Many colleges and universities require students to take at least one math course, and Calculus I is often the chosen option. Calculus Essentials For Dummies provides explanations of key concepts for students who may have taken calculus in high school and want to review the most important concepts as they gear up for a faster-paced college course.... more...
One of the best women tennis players of all time, Justine Henin?s dazzling array of shots, combined with her immense personal courage, have earned her enormous respect among current players and former champions alike. Justine?s success is all the more remarkable considering that off-court she has had to weather a series of family tragedies, a debilitating... more...
Just the critical concepts you need to score high in geometry This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming... more...
Immortalised in the film Chariots of Fire, Harold Abrahams remains one of the all-time great British Olympians, but in many ways his true story is even more dramatic and moving than as portrayed on the big screen. Now, in this powerful and painstakingly researched biography, Mark Ryan vividly recounts the sprinter?s struggles and triumphs, his private... more... |
Best Fit Least-Squares (BFLS) is a required technique for many STEM subjects. It is a method to compute a linear model for a set of data points. Due to its utility, BFSL is frequently taught to STEM students before they have sufficient mathematical experience to follow the mechanics of a derivation. Not only does this fail to produce procedural and conceptual understandings, but also it encourages students to view formulae and algorithms as things to be looked up, rather than derived. This is discourages students from developing productive dispositions. In this paper, we describe the "Close Fit Rod-Spring" (CFRS) approach to the problem. This approach computes the resting position of a rigid rod, which is connected by vertically oriented springs to a set of data points. This method of derivation results in two linear equations that may be solved for the slope and intercept of the best fit line. The result is equivalent to BFSL. However, it is achieved in an intuitively understandable and mathematically accessible way for high school students. |
Wondering about weather? Learn Mother Nature's secrets
Meteorology Demystified presents a complete explanation of essential physical and scientific concepts before delving into a more detailed look at various weather phenomena. more...
turned off by the dry, complicated presentation? Problem solved! Astronomy Demystified is a totally... more...
Best-selling Demystified author and electronics expert Stan Gibilisco has penned the perfect introductory book for consumers, hobbyists, and students alike. Coverage includes essential topics such as current and power supplies, wireless, digital principles, measurement and monitoring, transducers and sensors, location and navigation, and more.The fast and easy way to get up-to-speed on alternative energy
Because of current events, geopolitics, and natural disasters, the cost of fuel is front and center in our lives. This book provides a concise look at all forms of energy, including fossil fuels, electric, solar, biodiesel, nuclear, hydroelectric, wind, and renewable fuel cells.... more...
Almost every student has to study some sort of mathematical proofs, whether it be in geometry, trigonometry, or with higher-level topics. In addition, mathematical theorems have become an interesting course for many students outside of the mathematical arena, purely for the reasoning and logic that is needed to complete them. Therefore, it is not... more...
Master calculus from the comfort of home!
Want to "know it ALL" when it comes to calculus? This book gives you the expert, one-on-one instruction you need, whether you're new to calculus or you're looking to ramp up your skills. Providing easy-to-understand concepts and thoroughly explained exercises, math whiz Stan Gibilisco serves as your own... more...
Master pre-calculus from the comfort of home! Want to "know it ALL" when it comes to pre-calculus? This book gives you the expert, one-on-one instruction you need, whether you're new to pre-calculus or you're looking to ramp up your skills. Providing easy-to-understand concepts and thoroughly explained exercises, math whiz Stan Gibilisco serves... more... |
This eBook reviews some advanced topics in algebra, including exploring the nature of polynomials, functions, equations and identity's, examining the mathematical nomenclature used in multiplication and division. We consider multiplying out brackets, taking out common factors, manipulating algebraic fractions and simplifying expressions. We include an extensive selection of questions.
The 2 in 1 Times Table Tales is a compilation of Times Table Tales and the Times Table Tales Workbook that goes with it. Get both books for a super price! If you follow the steps in this simple, yet powerful book, your child will easily learn their facts. This game-changing book on learning the multiplication table uses brain research to super-charge your child's learning. Get started today!
Times Table Tales Workbook is a companion to the popular Times Table Tales book! If you follow the steps in this simple, yet powerful book, your child will easily learn their facts. This game-changing book on learning the multiplication table uses brain research to super-charge your child's learning! This book gives you step by step instructions and lists. Try it today!Are you taking calculus right now and it's kicking your butt? You're not alone; when I was teaching calculus, I realized that textbooks suck!
I wrote the Practically Cheating Calculus Handbook so that you don't have to struggle any more. This handbook contains hundreds of step-by-step explanations for calculus problems from differentiation to differential equations -- in plain English!
Practice division with third secondEnhance essential elementary algebra skills with these twenty-one practice problems. Each problem has an expression with mixed operators to reinforce the concept of operator precedence and use of parentheses. Choose a problem from the problem list, and then confirm your answer by easily navigating the link to the complete instructive solution. Most appropriate for 4th and 5th grade students second first thirdMathematics V10 Home StudyPractice and hone important addition skills with this second first |
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but now comes the hard part: Understanding the structure and function of
proteins. The 100,000 proteins in our bodies initiate, control, or perform every
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Proteins assemble and re-assemble in an infinitesimal space and, most often,
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New: Edgewood College Undergraduate Math Journal
For submission guidelines and other information, please contact Steven Post.
WHAT CAN I DO WITH A MATHEMATICS MAJOR?
While the main motivation to choose math as a major should stem from a combination of keen interest and high ability in math, students are naturally concerned about the opportunities available to a mathematics major or a mathematics teaching major after graduation. At this time, the math major appears to be in a better position than many other majors for employment in business, industry, government agencies, and teaching. The prospects are also good for well-qualified students to obtain support for graduate studies in either mathematics or mathematics education. Also a major in mathematics is excellent preparation for further study in many other fields.
In order to help you clarify your thoughts on what you want to get out of your collegiate experience as a math major, here are some questions to ask yourself:
Why do I like mathematics? What is it about math that attracts me to majoring in mathematics?
What type of mathematics do I like? Do I like the computational aspect? The rigor and logic? The problem solving experience? The theoretical aspect? Which content areas interest me?
What do I want to do for a career? Do I want to teach, or do I want pursue other avenues? If you want to teach, then:
what age group(s) do you want to teach? PreK, 1-6, 6-9, 9-12, college?
do you want to teach just mathematics, or do you want to have the flexibility to teach other fields as well?
If you are not interested in a career in teaching, then: are you interested in a career in business, industry, government, nonprofits, other alternatives?
How much education do I want to complete? Bachelors, Masters, or Ph.D.? Do I want to enter the work force right after graduation with the option to pursue graduate work later?
What do I need to do in order to further my career prospects?
What should I be doing academically to further my goals? Should I pick up a minor in another area? Should I try to double major?
What extracurriculars should I become involved in to further my goals? For example, should I get involved with the math club? Should I participate in the MCM Modeling Competition?
What types of work experience should I try to get to further my goals? Should I consider volunteer work experiences such as tutoring? Should I consider internships?
What organizations should I become involved in? What conferences or meetings might it be helpful to attend? |
...
More About
This Book
calculus textbook, it could also be used by college students or adult learners to refresh long-forgotten concepts and skills.
Homework Helpers: Calculus is a straightforward and understandable introduction to differential calculus and its applications. It covers all of the topics in a typical Calculus class, including:
Limits
Continuity
The product, quotient and chain rules
Implicit differentiation
Related rates
Graphical analysis
Optimization |
Mathematics (MAT)
Courses
This course is designed to offer a fast-paced review of basic arithmetic skills for students who have previously mastered these skills. Topics include all arithmetic operations on whole numbers, fractions, decimals and percents. Upon completion, students should be able to demonstrate mastery of basic computational skills, as well as their application to relevant mathematical problems.
This course is designed to offer a fast-paced, intensive review of skills taught in MAT 060. Emphasis is placed on working with signed numbers, problems involving proportions and per cents, as well as simplifying expressions and solving equations in algebra. Upon completion, students should be able to demonstrate mastery of pre-algebra computations and to solve relevant, multi-step problems.
This course establishes a foundation in algebraic concepts and problem solving. Topics include signed numbers, exponents, order of operations, simplifying expressions, solving linear equations and inequalities, graphing, formulas, polynomials, factoring, and elements of geometry. Upon completion, students should be able to apply the above concepts in problem solving using appropriate technology.
Prerequisites: Take MAT 060 with a minimum grade of C
Corequisites: Take One: RED 080 or ENG 085
This course is designed to offer a fast-paced, intensive review of skills taught in MAT 070. Emphasis is placed on working with exponents, order of operations, simplifying algebraic expressions, solving linear equations and inequalities, graphing, formulas, polynomials, and factoring. Upon completion, students should be able to demonstrate mastery of introductory algebra concepts and apply these principles in solving problems.
This course continues the study of algebraic concepts with emphasis on applications. Topics include factoring; rational expressions; rational exponents; rational, radical, and quadratic equations; systems of equations; inequalities; graphing; functions; variations; complex numbers; and elements of geometry. Upon completion, students should be able to apply the above concepts in problem solving using appropriate technology.
This course is designed to offer a fast-paced review of skills taught in Intermediate Algebra, MAT 080. Topics include factoring; graphing; functions; geometry; solving systems of equations and inequalities; and evaluating, simplifying, and solving rational, radical, and polynomial expressions and equations. Upon completion, students should be able to demonstrate mastery of intermediate algebra concepts and apply these principles in solving problems.
This course covers algebraic concepts with emphasis on applications. Topics include those covered in MAT 070 and MAT 080. Upon completion, students should be able to apply algebraic concepts in problem solving using appropriate technology. |
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