Lecture Notes (free download for
educational/non-profit purposes only)
the following Singapore Math textbooks:
Primary Mathematics Textbook 5A (U.S. Edition) (You may
have this book already from your Math 130 class; do not
buy this book if you don't have it; copies of the few pages we need will
be provided for free in class, or you can download them below!)
Homework will be assigned on Thursday at the beginning of lecture and due the
following Thursday at the beginning of class.
There will be no credit for late homework except
in case of illness or family emergency.
HW # 1 DUE 1/30: Primary 5A, p. 25, # 1, 4, 7, 10 (using
bar diagrams) and p. 60, # 2, 4, 6, 8 (in two ways, using both methods on
page 58); Lecture Notes, p. 4, # 1.1
HW # 2 DUE 2/6: Lecture Notes, p. 8, # 1.3, 1.4;
NEM 1, p. 25, # 5-7 and p. 27, # 1, 5; Primary 6A, p. 29, # 3, 6 and
p. 33, # 2, 6 and p. 38, # 4, 6 and p. 60, # 7, 10 and p. 68, # 8, 10
(all using bar diagrams!)
HW # 3 DUE 2/13: Primary 6A, p. 82, # 2, 4, 7 (using
pictures as in text); NEM 1, p. 141, # 2dfh, 4dfh and p. 147,
# 2dfh, 4dfh (In the last two exercises, explain each step carefully
by a rule or a definition!)
HW # 4 DUE 2/20: Lecture Notes, p. 23, # 2.1; NEM 1,
p. 139, # 4dg, 5, 7, 10, 13 and p. 174, # 10, 13 and p. 181, # 17, 21;
NEM 2, p. 117, #4, 6, 8, 9 (GST = general sales tax)
HW # 5 DUE 2/27: Lecture Notes, p. 33, #2.2 and p. 35,
#2.4 and p. 41, # 2.5; NEM 1, p. 158, # 3dhl and p. 161, # 12, 24 and
p. 166, # 4, 10, 17, 20 (with the method of this section, i.e., using only
one variable!); NEM 3A, p. 77, # 1df, 3abc, 4bd, 6bd, 8
HW # 6 DUE 3/13: NEM 2, p. 53, # 4, 6, 10 (literal =
linear) and p. 78, # 17, 37 and p. 141, # 6, 12 (by elimination method)
and p. 148, # 10, 20 and p. 151, # 11, 18, 20, 21
HW # 7 DUE 3/27: Lecture Notes, p. 61, # 3.1-3.2;
NEM 2, p. 159, # 1 and p. 163, # 3dfhj and p. 164, # 4dhl, 5-7
HW # 8 DUE 4/3: NEM 3A, p. 47, # 4, 18, 22 and
p. 50, # 5, 10, 12, 14 and p. 55, #3-5 (Typo for # 22 on p. 49:
Second inequality should read "2x+1 < x+2")
HW # 9 DUE 4/10 NEM 2, p. 122, # 3, 5, 9, 10;
NEM 3A, p. 102, # 2, 10, 12 and p. 106, # 1bfj, 2 and
p. 109, # 2, 6, 8, 12, 14
HW # 10 DUE 4/17: NEM 2, p. 28, # 3dhl, 4bdh and
p. 31, # 2bdf and p. 33, # 6dhlpt; NEM 3A, p. 114, # 6, 10, 16, 17;
also solve:
|.5x+4| > 2.5
|4(x-5)| ≤ 3
|x-1| < 2x+5
HW # 11 DUE 4/24: NEM 2, p. 65, # 1hlpt and
p. 66, # 3, 7, 11, 14; NEM 3A, p. 31, # 1dh, 2dh and
p. 39, # 12, 15, 17, 20; Lecture Notes p. 98, # 5.1 and p. 99, # 5.2
(Note: Solve all NEM 2 quadratic equations by factorization!)
HW # 12 DUE 5/1: NEM 2, p. 133, # 3 and
p. 135, # 1, 2, 4, 5; NEM 3A, p. 34, # 1bf, 2lptz, 4bd and
p. 41, # 2, 4 and p. 129, # 5 and p. 138, # 1, 5; also solve:
2x^{2} < 5x-1
-x^{2}+5 ≥ 2x
HW # 13 DUE 5/8: Lecture Notes, p. 125, # 6.1,
Add'l Math, p. 78, # 1d, 2df, 4df and p. 81, # 1dg, 3bd, 5 and p. 83,
#1c, 2f, 3c (note that in the Singapore math books, "index" = "exponent",
and "surd" = "algebraic expression involving roots")
HW # 14 DUE 5/15: Add'l Math, p. 87, # 1e, 2b, 3di and
p. 91, # 2cd, 4cf and p. 97, # 2dgh, 6, 8, 10 and p. 100, # 2, 5, 7 and
p. 103, # 2ace, 4, 9, 13
Each project requires a group of 3 to 4 students to compare how the topic is
introduced in one of our Singapore math schoolbooks and one American math
schoolbook of your choice. Each student should pick a group and a joint
topic. (Pick an American schoolbook of your choice from CIMC, e.g.,
the one you yourself used in your own school,
or one that is currently used in a school you are familiar with.
The more different that book is from our Singapore math schoolbooks, the
more interesting the project will be.)
The comparison I am looking for is how the mathematical concepts
are introduced. I'm not necessarily looking for a value judgment from you,
which one you like better, but mainly for an investigation of how the
presentation of the topic differs from a mathematical point of view.
E.g., is the concept defined differently? Are the examples by which it is
introduced very different? Etc.?
Your project will consist of a short (joint) paper (of 2-3 pages), with copies
of the relevant "other" schoolbook pages attached, and a short presentation
(10-15 minutes) in class about this comparison of topics. You can use an
overhead projector or copied handouts if you prefer. (I'll help you get
an overhead projector if you need one.) Presentations will start in mid-April.
I suggest that each group meet with me briefly a few
days before your presentation for feedback.
Here is a list of possible topics (with reference to treatment in the Lecture Notes):
How are the properties of arithmetic (commutative law, etc.) first
presented? (Cf. Proposition 1.2.)
How is the slope of a line defined? (How is it motivated that one can
define the slope as one single number? (Cf. section 2.4.)
How is solving two linear equations in two unknowns presented? (Cf.
section 2.5.)
How is estimation presented? (Cf. section 3.4.)
How is the concept of a function first introduced?
How are the concepts of a function being 1-1 and onto, and of an inverse
function, introduced? (Cf. sections 4.1 and 4.2.)
How is absolute value, and solving equations involving absolute value,
presented? (Cf. section 4.3.3.)
How is the concept of a quadratic function first motivated?
(Cf. section 5.1.)
What are the steps by which solving quadratic equations is introduced?
(Cf. section 5.2.)
How is sketching the graph of a quadratic function presented?
(Cf. section 5.4.)
COPIED PAGES FROM OTHER SINGAPORE PRIMARY MATH BOOKS:
For copyright reasons, the links below are password protected.
Please get your user name and password from me by
@math.wisc.edu">email if you have forgotten it.
Use of these links will be monitored, and unauthorized use is prohibited.
Makeup exams will be scheduled only with the instructor's
consent, and only in case of illness or family emergency or conflict with
another required class. In the latter case, please let me know as soon as
possible!
Calculators of any kind are discouraged for any part of this course and
will not be allowed during exams.
Midterm Exam 1: Tuesday, March 4, in class
Midterm Exam 2: Tuesday, April 15, in class
Final Exam: Thursday, May 15, from 2:45 pm to 4:45 pm, location TBA
This course is the first of three courses of the math component of the
Mathematics-Science Dual Minor for all Elementary Education
and Special Education majors wishing to enhance their content preparation in
mathematics and science. This minor is particularly suitable for those
Elementary Education majors seeking Middle Childhood-Early Adolescence
certification and intending to teach mathematics and science in middle school.
The other two math courses for this minor, Math 136 and Math 138, will focus on
precalculus and early calculus; and on probability, statistics and
combinatorics, respectively.
Math 135 focuses on the mathematical content needed to teach pre-algebra
and algebra in upper-level elementary and middle school.
The core instructional goals of Math 135 are:
problem-solving;
making mathematically grounded arguments about the strengths and weaknesses
of a range of solution strategies (including standard techniques)
examining the rationale behind middle-school students' mathematical work
and how it connects to prior mathematical understanding and future
mathematical concepts
flexible use of multiple representations such as graphs, tables, and
equations (including different forms)
using functions to model real-world phenomena
modeling real-world problems ("word problems") as mathematical problems and
then interpreting the mathematical solution in the real-world context
symbolic proficiency (solving equations and inequalities, simplifying
expressions, factoring, etc.)
The mathematical content topics of Math 135 include:
Review: basic properties of the real numbers
Linear Functions: proportional relationships, linear equations and systems
of linear equations, linear inequalities
Quadratic Functions: different forms of quadratic equations, factoring
quadratic polynomials, completing the square and quadratic formula,
graphing quadratic functions, quadratic inequalities, brief discussion of
polynomial functions
Exponential Functions: understanding the difference between additive and
multiplicative growth, exponential rules, exponential growth and decay,
brief discussion of logarithmic functions
The outcomes of a sample space are called equally likely if all of them have the same chance of occurrence. It is difficult to decide whether or not the outcomes are equally likely.
Flipping two coins:
{two heads, one head and one tail, two tails} is a sample space. However, it is NOT the sample space with equally likely outcomes.
{two heads, two tails} is made of some equally likely outcomes, but it is NOT a sample space because it is not a list of all possible outcomes.
Please download and try these. GeoGebra (free
mathematics software for teaching geometry, algebra and calculus in
middle school and high school) UCB_LOGO (free
programming software for studying programming language and geometry) LOGO (LOGO applet on the web)
Objective Use GeoGebra to create a tessellation with quadrilaterals. Doing the construction on the computer allows you to change the shape of your original quadrilateral and see how the tessellation changes.
Study EGT section 4.5 by yourself.
Go to GeoGebra Web to load a web-based applet version of GeoGebra.
Go to the View menu and unselect the "Axes" and "Algebra View" options. You should now have just a blank window to work with.
Use the "Polygon" tool to draw a quadrilateral.
Construct the midpoints of all four sides of the quadrilateral. To do this, select the "Midpoint or Center" tool and then click on the side of the quadrilateral you want to bisect.
Rotate the quadrilateral 180° around each midpoint. To do this, first select the "Rotate Object around Point by Angle" tool then click on the interior of the quadrilateral and then the midpoint you just created. Change the value the number to 180 leaving the degree unit alone. Color all four new copies with one color. To do this hold the CTRL key and click (right click on a PC) on each of the new quadrilaterals to bring up a menu. The bottom option is "Object Properties." Click on it and then select the color tab.
Your picture should look something like this:
Use translations to finish the tessellation. To do a translation, you first need to specify the length and direction, which is called a "Vector". To create a Vector, select the "Vector between Two Points" tool and click on two of the points you used to create your original quadrilateral. You should now see an arrow along that segment. Select the "Translate Object by Vector" tool and then use the mouse to select your quadrilateral and then click on the Vector you created.
Move the corners of the original quadrilateral. Examine what happens when it is not convex. Do you still have a tessellation? What symmetries does the picture have?
Add your name to the sketch using the Text palette. Save it to the desktop and email it to @math.wisc.edu">kwonmath.wisc.edu.
Use this tessellation to answer #8 above and #8 in EGT p. 106 and turn them in separately by May 8.
Here's a different way to create a quadrilateral tessellation just for your own interest and fun. Do not follow this.
EXAMS:
Makeup exams will be scheduled only with the instructor's
consent, and only in case of illness or family emergency or conflict with
another required class. In the latter case, please let me know as soon as
possible!
Calculators of any kind are discouraged for any part of this course and
will not be allowed during exams. A compass set (a compass, a setsquare, a ruler, a protractor) is required to take the exams.
Midterm Exam 1: Thursday, Feb. 23,
Midterm Exam 2: Thursday, Mar. 29,
Final exam: Wednesday, May 16,
10:05-12:05 location TBA
This course is NOT about how to calculate 25 divided by 5 NOR how to
teach a dividion algorithm. It is about what kinds of serious
mathematics are behind the calculations, algorithms, symbols, and
diagrams usually assumed to be simple and trivial to future elementary
and middle school teachers. Making future teachers seriously engage in
mathematics that they will teach in their near future is the main goal
of this course.
HW 10 due on Nov. 20 6.3 p.149 #7, #11 6.4 p.154 #2, #6, 6.5 p.158
#1, #4, #6, #8
HW 11 due on Dec. 4 6.6 p.165 #2(b)(d)(f), #6(b)(c)(i), #11, 7.1 p. 172 #6,
#8
HW 12 Due on Dec. 11 7.2 p.177 #4, #8(PRACTICE4C p.60 #3, teacher's solution for
5,9), 7.3 p.181 #3, #6 8.1 p. 190 #1, #3, 8.2 p.196 #1, #6
Problems that you need to work on for the final 8.3 p. 200 #4, #7, #10, 9.1 p. 207 #1, #5,
#11, 9.2 p. 214 #4(e)(f)(j), #6, 9.3 p. 221 #1, #4, #5
Math 113 Trigonometry
C&I 636 Mathematical Knowledge for Teaching: Number Development and
Generalization (co-instructor)
C&I 637 Mathematical Knowledge for Teaching: Rational Number and
Proportional Reasoning (co-instructor)
C&I 639 Mathematical Knowledge for Teaching: Algebra and Functions (co-instructor)
C&I 640 Mathematical Knowledge for Teaching: Experimentation, Conjecture, and Reasoning (co-instructor)
C&I 675 Advanced Placement Summer Institute: Calculus AB and BC
(General Seminar)
Math 130, 131, and 132 are the three math content courses required
for all elementary
education and special education majors, together with the math methods course
C&I 370 (see the
UW course guide) taught by the
Department of
Curriculum and Instruction (course supervisor:
Prof. Anita Wager).
Math 130, 131, and 132 are usually taken in sequence; contact me (preferably
by email:
@math.wisc.edu">kwonmath.wisc.edu)
to be granted an exemption from this requirement.
(C&I 370 has Math 130 and 131 as prerequisites and
can be taken concurrently with Math 132.)
The three courses Math 130-131-132 have a prerequisite of
Math 101 (which is offered by UW-Madison
in both the fall and spring semester), an equivalent course
elsewhere, or (most commonly) placement intoMath 112 (see
general placement test information,
sample math placement tests, and
placement score evaluation).
Note, however, that students do not have to
take Math 112 to take Math 130-131-132.
Also note that Math 141 does not give you the placement
into Math 112 required to enroll in Math 130-131-132!
Please note that effective fall semester 2012,
any student wanting to register for a Math 13x course must have a grade of at
least C in all prerequisite Math 13x courses (unless exempted from these
courses).
Under certain circumstances, students can be exempt from Math 130 and/or 131
(see here for the precise
rules), but not from Math 132. Similarly, some courses from
other universities may transfer toward Math 13x credit, see the
UW
Transfer Information System for the most common courses transferable
from other UW campuses; for all other questions about possible course
credit transfer for Math 13x courses, contact me
(preferably by email:
@math.wisc.edu">
kwonmath.wisc.edu).
Note that Math 130 also meets the Quantitative Reasoning Requirement Part A of the
UW-Madison School of Education, and that Math 131 and 132
together, or Math 135, or any calculus course, meet the Quantitative Reasoning Requirement Part B of the
UW-Madison School of Education. (Education students not in
elementary or special education are advised, however, to meet the Quantitative
Reasoning requirements via other courses (check for the lists of courses for
QR-A and QR-B in the
UW course guide).
Math 135, Math 136 and Math 138 are the three math content courses of the new
Mathematics-Science Dual Minor intended for all Elementary
Education and Special Education majors wishing to enhance their content
preparation in mathematics and science. (Students taking this minor are
exempt from Math 132!) This minor is particularly suitable
for those Elementary Education majors seeking Middle Childhood-Early
Adolescence
certification and intending to teach mathematics and science in middle school.
(The last math content course for this minor, Math 138, is being offered for
the first time in spring 2011; Math 132 can no longer be taken as its
substitute unless you have taken it before spring 2011.)
This minor was supported by a $2,000 scholarship of the Brookhill Foundation
for all students who have completed Math 135 by the end of spring 2012.
Once you have completed the requirements for the $500 level,
download this form,
fill it out, and give it to me (or put it into my mailbox on the 2nd floor of Van Vleck Hall).
Once you have also completed the requirements for the remaining $1,500 level,
download this form,
fill it out, have EAS (139 Education Bldg.) sign it, and give it to me
(or put it into my mailboxes on the 2nd floor of Van Vleck Hall).
In either case, I'll handle the rest once I have your form(s), and your
scholarship will show up in your MyUW in 4-6 weeks.
Math 135: Algebraic Reasoning for Teaching Mathematics:
taught in one section in spring semester only, using
Lecture Notes
(free download for educational/non-profit purposes only)
the following Singapore Math textbooks:
Primary Mathematics Textbook 5A (U.S. Edition)
(You may have this book already from your Math 130 class;
do not buy this book if you don't have it;
copies of the few pages we need will be provided for free in class)
Math 136: Pre-calculus and Calculus for Middle School Teachers:
taught in one section in fall semester only, consisting of a large lecture
of Math 171 ("Calculus with Algebra and Trigonometry I")
and a special 3-hour discussion section, using
Math 130-131-132 are the math content courses preparing students to become
elementary or middle school teachers. The students and especially the content
in these courses are very different from those found in other mathematics
classes in that they focus on a "profound understanding of elementary
mathematics".
Special interest in how teachers are prepared, and some familiarity with
current developments in how mathematics is taught in schools, are essential
for a TA in these courses.
Normally, a TA teaching these courses would be someone with a minor or a
special interest in mathematics education since it is desirable to have some
background in educational psychology and how someone learns mathematics. Since
these courses are also taught by faculty, appointments to teach them are made
by the TA coordinator of the department in consultation with me and Prof. Lempp.
TA's interested in teaching one of these courses should contact me
by email:
@math.wisc.edu">
kwonmath.wisc.edu.
Each single section of Math 130, 131 and 132 corresponds to a 50% appointment
level.
This includes some required special TA training during the last few weeks of
the previous semester, some required meetings with other Math 13x TA's as well
as me and other faculty during the semester, and a fair amount of grading
homework.
"Mathematics presented with rigor is a systematic deductive science
but mathematics in the making is an experimental inductive science."
"Heuristic reasoning is good in itself.
What is bad is to mix up heuristic reasoning with rigorous proof.
What is worse is to sell heuristic reasoning for rigorous proof."
Brown Center 2006 Report on American Education,
Brookings Institute (see Part II: "The Happiness Factor in Student Learning"
on the strong negative correlation between math scores and self-confidence
in international comparison studies)