Math 107Y
Fall 2000
M107Y COURSE GUIDE—FALL 2000
TEXTBOOK
Mathematical Reasoning for Elementary Teachers, second edition.
By Long and DeTemple.
THE NATURE OF THE MATH 107 SERIES
These courses are mathematics courses, and so the primary goal of these courses is to improve,
broaden, and deepen student facility with, appreciation for, and understanding of mathematics.
The content of the course has been chosen specifically to be of most benefit to those who aspire
to be K-8 teachers. The topics of the courses support and extend the expectations set forth by the
California Content Standards in Mathematics, K-7 (February 1998). Students in the course
may be tacitly familiar with many of the topics in the courses, but will be challenged throughout
to move their knowledge of these areas toward a pedagogical level (see page 4 of this course
guide).
The focus of instruction is on your learning, and therefore several different instructional
techniques are used throughout the course sequence, including making use of concrete learning
tools for hands-on explorations. Where possible, such techniques will be tied to the California
Mathematics Framework (1999).
The Math 107YZ courses are not courses on "methods for teaching elementary school
mathematics", or on "learning the mathematics elementary students must know.” However, the
student-centered focus, the variety of instructional techniques, and the ties to the CA standards
and Framework will help you begin building an advanced perspective on the teaching of
elementary mathematics.
Numeracy, the ability to represent, communicate, and interpret quantitative information, has
become an integral part of our social definition of literacy. Thus, another goal for the course is to
strengthen your spoken and written communication within mathematics. Writing well in
mathematics requires fluency in the precise descriptive language of mathematics and careful
logical organization. The writing portions of assignments will also serve to start you on your
way to making the transition from learner to teacher in mathematics.
COURSE EXPECTATIONS
In this course you will engage in a wide variety of situations and contexts which give rise to the
mathematical concepts essential for K-8 teaching. These concepts are significant in the
mathematical world, so you should expect to find this a challenging college-level mathematics
course.
In class, we will often work in collaborative groups of 3 or 4 to stimulate dialogue between
"hands-on" activity (physical activity with blocks, sticks, dice, paper, chips, etc.) and systematic
inquiry (using physical and linguistic tools to communicate understanding, justify your thinking,
and pose new problems). In your work outside of class, you may collaborate in groups of any
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Math 107Y
Fall 2000
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size that you find productive, but you must turn in a personal report for each of the
assignments.
Other expectations:



The best way to insure your successful completion of this course is to come to class and
keep up with the assignments!
Be reminded that 2 student hours devoted to assignments and preparation for every hour
of classroom time is a reasonable expectation for an average student.
Barring unforeseen medical or other serious conditions, I expect you to be in class on time
every day. If you must miss a class, please leave a message on my answering machine
(826-4921) prior to class time and be sure to contact a classmate to find out what you missed.
COURSE CONTENT (Tentative schedule):
Problem-solving skills will be emphasized and practiced throughout the course. A topical index
is given below to indicate the general outline for the course. Be aware that I will supplement
our textbook substantially. Make sure you receive a copy of all supporting material
throughout the semester.
I.
Mathematical Thinking (Textbook reference – Chapter 1)
Week 1, 2
Week 3
II.
Writing/Problem-Solving
Patterns and Functions
Number Sense in the Counting Numbers (Textbook reference, Chapters 2-3)
Week 4
Sets and Counting/Numeration Systems
Week 5-6
Week 7
Algorithms for the four operations
Mental Mathematics
Test #1: September 25
III.
Number Theory (Textbook reference, Chapter 4)
Week 8
Week 9
Factors and Multiples
Clock Arithmetic/Diophantine Equations
Test #2: October 23
IV.
Extending Number Sense to Integers and Rationals (Textbook Reference, Chapters 5,6)
Week 10
Week 11
Week 12
Arithmetic in the Integers and Rationals
Arithmetic in the Rationals
Decimals and percents
Test #3: November 15
V.
Algebra and Algebraic Thinking (Textbook Reference, Chapter 7)
Math 107Y
Week 13
Week 14-15
Fall 2000
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Ratio and Proportion
Algebraic Thinking
Final Examination (Cummulative): December 22, 10:20 AM – 12:10 PM
EVALUATION AND ASSESSMENT:
40% Weekly Assignments.
Assignments may include exercises from the text, summaries of problem-solving work done in
class, explanations to the “Why?” questions of elementary mathematics, and reflections on the
mathematics we are studying relative to the CA Academic Content Standards for grades K – 7.
Each assignment will have its own point value. Each will be graded with a ratio of point earned
to total point value. The smallest ratio of your eleven assignments will be dropped before
calculating your composite grade for these assignments.
REVISION POLICY: If you turn in an assignment on the designated due date, you may resubmit
it (once) for re-grading. All re-submissions must include what you originally submitted for the
assignment (with my initial comments) and the revised portions of the assignment (on new
paper – please don’t just write something new on your original paper). Re-submissions are
due 1 week after you receive the original graded assignment back from me.
Typically, Weekly assignments are due on Mondays.
60% Tests and Final Examination.
The tests will be patterned after the work you have done on the assignments and draw heavily
from the content of the class sessions. You will be allowed an 8.5 x 11-in. helpful-hints page
(which you create) for each of the quizzes. We will schedule two hours for each quiz, although
each will be written so that a typical student can complete the test in about an hour. The final
examination will heavily emphasize the last three weeks of class, but will also include selected
topics from the entire course.
Math 107Y
Fall 2000
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A Few Tips on Writing Mathematics
Modified from a document prepared by Tracy Rusch, Wright State University
The purpose of assignments in this course is to develop skills in understanding and
communicating mathematics. It is not to give you busy work or drill. Hence, most of the
exercises and problems in your assignments will require responses in sentence or paragraph
forms. Very few will only require a numerical response.
In your writing of mathematics I offer the following tips on
Incorrect mathematics
Mathematics is the most precise of subjects. Every statement you make must be correct. A
solution will be ruined by a single false step such as
a 3 a
 1
b3 b
or
x  y  x
2
2
 y2
or
x   x
5
2
25
23  5
or
If you are not sure about a statement, check it for some special cases. For example, if you
2
n
2
5
suspect that n  2 , try a few values of n. You will soon see that 5  2 , so the statement is
false.
English
Good communication requires good English. A correct solution garbled by bad English may be
worthless to the reader. The rules of grammar (spelling, punctuation, sentence structure) apply to
mathematics.
For clear communication, present one idea at a time. Since an idea is expressed by a complete
sentence, write in complete sentences.
Non-sentence
Sentence
a b
a  b  15
since x is positive
Since x is positive, x  2 .
Take a number m. Where m is odd.
Take a number m, where m is odd.
x  y 
x  y  x
2
x 2  2xy  y 2
2
2
 2xy  y 2
Math 107Y
Fall 2000
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In general, be short and eliminate unnecessary words.
Long
Short (and better)
An even number multiplied by an even number
results in an even number.
The product of two even numbers is even.
x signifies the value of the solution
x is the solution
the result you get when you add up all the scores
the sum of the scores
If you have a right triangle and if you construct
squares on all three sides of the triangle, then
when you add up the areas of the squares on the
two sides, what you get is the area of the square
on the hypotenuse.
In a right triangle, the square of the hypotenuse
equals the sum of the squares of the two legs.
Avoid pronouns (it, this, that, which).
Bad
To solve it, take away 3 from both sides then
take it and divide by 4.
Good
To solve 4x  3  0 , subtract 3 from each side.
3
This gives 4x  3 , thus x 
.
4
To avoid pronouns, give names to quantities and use those names. Don't solve "it," solve
4x  3  0 .
Write mathematics: Strike a balance between words and symbols.
The notation of mathematics is clean and precise.
appropriate rather than lengthy prose.
Use mathematical symbols where it is
Words
Mathematics
If you start with any number and subtract one
from the square of the number, then the answer
you will get is the same as the answer you will
get if you multiply one more than the number by
one less than it. [Ugh! 43 words]
Let x be any number. Then
x 2 1  x 1x 1.
If you multiply an odd number by 2, it becomes
even. [A number cannot "become" another
number.]
Twice an odd number is even.
the answer that you get when you add a and b
ab
together
Honor the equal sign.
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Oranges do not equal apples; numbers do not equal sets. The equal sign has a precise meaning; it
is not a punctuation mark such as a dash –. This symbol is very abused, even by some
mathematicians!
Bad
Good
n = even = 2n
[Note: n does not equal 2n.]
If n is even, the n = 2k for some k.
n 2  16  n  4
[16 does not equal ±4.]
n 2  16 ; hence, n  4 .
An n-gon = n  2180
[A polygon does not equal a number of degrees.]
The sum of the interior angles of an n-gon is
n  2180 .
Answer the question.
When you finish a problem, go back and read it again. Be sure you have given a clear answer to
the question asked.
Example: I drive from West Lafayette to Chicago, 125 miles, at 50 mph, then go from Chicago
to Minneapolis, 550 miles, at 55 mph. How long does it take me to drive from here to
Minneapolis?
Nonanswer
125 + 550 = 675
Example:
Answer
The time to travel from West Lafayette to
Chicago is 125 miles/50 mph = 2.5 hours. The
time from Chicago to Minneapolis is 550
miles/55 mph = 10 hours. Thus, the total time is
12.5 hours.
Explain why the sum of two consecutive integers is not divisible by 4.
Nonanswer
1 + 2 = 3, not divisible by 4
2 + 3 = 5, not divisible by 4
[Two special case do not prove a general
statement.]
Answer
Let n and n+1 be two consecutive integers.
Their sum is
n  n 1  2n 1
which is odd, hence not divisible by 4.
Math 107Y
Fall 2000
The building blocks of mathematical language
Number, Variable, Expression, Relation, and Statement
Number
A number is a concept.
A numeral is a representation of that concept.
Number Sets
Counting Numbers
Whole Numbers
Integers
Rationals
Reals
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Variable
A variable is simply a name we introduce into a mathematical setting or
discussion with the intention of assigning it some value.
Perhaps the fundamental difficulty for many students making the transition from
arithmetic to algebra is their failure to recognize that the symbol x stands for a
number. For example, the equation 3(2x-5)+4(xs-2)=12 simply means that a
certain number x has the property that when the arithmetic operations 3(2x5)+4(x-2) are performed on it as indicated, the result is 12. The problem is to
find that number (solution).
Mathematics Framework for California Public Schools, K-12, 1999, Page 158
At different times we may assign it different values. Or we may simply talk (write) about
assigning it a value, but never actually do it. The value of a variable could be any mathematical
object: a number, a vector, a matrix, and expression, a function, or a character string. A variable
in mathematics is the analog of a pronoun in English.
It is important to realize that developing students' understanding of variable is complicated,
precisely because the concept of variable itself is multifaceted. Consider the following equations
and their variables:
30 = 5x
A = LW
y = 3x.
1 n
1
n
Unknown In the first equation, x stands for a specific, non varying number that can be
determined by solving the equation.
Measured Quantity The second equation is usually considered a formula with A, L, and
W standing for the quantities area, length, and width. These quantities actually feel more
like knowns than unknowns.
Independent and Dependent Variables The third equation shows x independently taking
on a variety of values, and the corresponding y varying depending on the value of x.
Universal variables The fourth equation generalizes an arithmetic pattern where n can
take all real numbers (but not zero).
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Math 107Y
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Expression
An expression is a description of how to calculate something.
Expressions are sometimes called rules or formulas. When you evaluate an expression you
actually follow the description and carry out the prescribed sequence of steps in the calculation,
and the resulting object is called the value of the expression.
Expressions are made up of numbers, variables, operations, and known functions. Parenthesis
and the accepted order of operations govern the interpretation of the description of calculation.
For example,
2  7
3
is an expression whose value is 3.
If we evaluate the expression
x  5 x  4
3
when x assumes the value 2, then the value of the expression is  .
2
The expressions
x  5 x  4 and
x  5 x
4
have the same value when x = 2 (there are other ways to reorganize parenthesis and operations to
produce the same value). These two expressions are equivalent, in that they will produce the
same value for any given value of the variable x. One common task in algebra is to simplify
expressions. A “simplified expression” is a description of how to calculate something
“efficiently”. However, what two different people think of as being efficient can vary greatly –
so the performing a “simplification” may produce more than one acceptable result.
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Math 107Y
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Relation
A relation is a mathematical way to describe a relationship between elements of
two different sets of objects or elements within the same set of objects.
(Notice that I did not say “a functional relationship” – this is special kind of relationship that we
will discuss under functions.)
Let’s consider an example. Suppose that we want to describe the relationship, “is a multiple of,”
that exists among the elements in the set of whole numbers {0,1,2,3,4,…}. First, let’s name the
relationship M. There are several ways to describe counting numbers that are “related” by M.
For example:
15 is a multiple of 3
15M3
(15,3)  M
I could also graph the point (15,3) on a coordinate plane, where the first coordinate is a
multiple of the second coordinate.
Note that each of these descriptions is ORDER SPECIFIC (that is why we often call objects like
(15,3) a coordinate pair) So order matters—for example, (3,15)M.
The more common relations that we use in mathematics are, “is equal to”, “is less than”, “is
greater than”, and “is not equal to”.
Math 107Y
Fall 2000
Statement
A mathematical statement is a sentence to which a truth-value
(True or False) may be assigned.
Two statements are called equivalent if they have the same truth value under all possible
conditions.
The “verbs” of most mathematical statements – at least those that we write symbolically — are
descriptions of relations. The most common statements in school mathematics are equations and
inequalities (corresponding to the relations of equality and inequality, respectively).
Solving equations: What most of us think of when we picture the process of solving an equation
is a process of rewriting the equation into a form that is equivalent. (Two equations are
equivalent if they have the same truth values for set of solutions for a given variable. A “solution
set for a given variable” is the set of all values for that variable that maintain the equality
relationship. Another way to say this: two equations are equivalent if they have the same truth
value given any values for the variables in the equation.)
The first basic skills that must be learned in Algebra I are those that related to
understanding linear equations and solving systems of linear equations. In
Algebra I the students are expected to solve only two linear equations in two
unknowns, but this is a basic skill. …Moreover, modern applications of
mathematics rely on solving systems of linear equations more than on any other
single technique that students will learn in kindergarten through grade twelve
mathematics.
Mathematics Framework for California Public Schools, K-12, 1999, Page 159
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