Math 103 Lecture 4 notes page 1
Math 103 Lecture 4 notes
A Simple Problem: What is 2 more than 3 times 4?
1. How would you solve this problem?
2. How might students solve this problem?
Student Solutions to A Simple Problem:
Student A
Student B
Student C
Student D
(3 + 2) x 4 = 20
3 x 4 = 12
3 x 4 = 12 + 2 = 14
(3 x 4) + 2 = 14
+ 2
14
1. Classify each solution as correct or incorrect.
2. What does each solution indicate about the student’s understanding?
3. What feedback would you give to each student?
Order of Operations: “Please Excuse My Dear Aunt Sally”
P = Parentheses first
E = Exponents second
MD = Multipy or Divide in order of appearance from left to right
AS = Add or Subtract in order of appearance from left to right
Challenges:
#1: Express each of the numbers 1 –10 using 4’s at least one negative sign, and any operations.
Examples: 1 = –44/–44
1 = (4/4)^(–44)
1 = –4 + 4 + (4/4)
#2: Write the digits 1 through 9 in order. How many ways can you add plus/minus signs between the
numbers or use no operation symbol at all to obtain a total of 100?
Example: 1 + 2 + 3 – 4 + 5 + 6 + 78 + 9 = 100
Negative Numbers have been known since ancient times. The Chinese had red and black computing rods
to distinguish between positive and negative. (Accountants today speak about being "in the red" or "in
the black.") The Hindus used negative numbers to represent debt and denoted a negative by enclosing
the number in a circle. Mathematicians of the 16th century were reluctant to accept negative numbers as
"true" roots of equations and referred to them as "absurd." By the 18th century, negative numbers were
universally accepted.
Children know that a temperature of 5 degrees below zero is referred to as "minus five" and written –
5°. Young children encounter negative numbers when they use calculators to find answers for expression
such as 2–5, or continue to "count down" past zero and see –1, –2, –3 appear on the display. Other real
world examples are profit and loss, above and below sea level, winning and losing points, golf scores
above and below par, a hockey player's "plus-minus" statistic, and positive and negative electrical
charges.
Number Theory is the study of integers, their properties, and relations among them.
NCTM Standards emphasize the importance of studying concepts from number theory in middle grades.
Integers are both positive and negative "whole numbers." (Whole numbers, by definition, are only
positive)
J = I = {. . . , –4, –3, –2, –1, 0, 1, 2, 3, 4, . . . }
Natural numbers are positive counting numbers starting with 1. Zero is not a natural number.
N = { 1, 2, 3, 4, . . . }
Math 103 Lecture 4 notes page 2
Negative numbers should be introduced at grades 3-5 through use of familiar models such as
temperature or owing money. The number line is a helpful model, and students should recognize that
points to the left of 0 can be represented by numbers less than 0.
In grades 5-7, students should extend these initial understandings of negative numbers as useful for
noting relative changes or values.
Teachers can introduce negative numbers as opposites of counting numbers.
The opposite of 3 is –3 (read "negative 3").
Zero is neither positive nor negative.
Use the number line to establish order relations.
–4
–3
–2
–1
0
1
2
3
4
5
Any negative number < zero.
Any negative number < any positive number.
Ex: –7 < –4 because –7 is further to the left.
Definition of "Less Than" for Integers:
For any integers a and b, a is less than b,
written a < b, if and only if there exists a positive integer k such that a + k = b.
. . . in other words: a < b if b – a = positive integer.
Ex:
–7
+2
–8
–3
–0.45
–0.5
Distance is always a positive number or zero. The distance between the points corresponding to an
integer and 0 is called the Absolute Value of the integer. Thus the absolute value of both 4 and –4 is 4,
written |4| = 4 and |–4| = 4.
Formal definition:
|x| = x when x ≥ 0
|x| = –x when x < 0
Numbers on a Number Line Problem:
The numbers 0, 1, x, y, and –z are marked on the number line below.
Y
0
x
1
–z
For each pair of numbers, circle the number that has the greater value. IF the two numbers are equal,
circle both of the numbers.
A.
y
–y
D.
x–y y–x
B.
z
–(–z)
E.
|y|
–y
C.
2y
y
F.
x+y x–y
Meaningful Analogies for Operations on Integers
Credit-Debit Analogy: Creation of Debt
Note: A transactions involving a debit would be represented by the addition of a negative number
Math 103 Lecture 4 notes page 3
Jaesook had just $12 left after paying the rent, buying food, and paying some bills. She was happy
about having some extra money until the mail arrived with a $15 bill for the dentist. So what’s Jaesook’s
money condition now?
Credit-Debit Analogy: Accumulation of Debt
Note: A transactions involving a debit would be represented by the addition of a negative number
Jaesook was not happy about owing $3. If her situation was not unhappy enough, Jaesook found another
bill of $7 for a book. So what’s her money condition now?
Credit-Debit Analogy: Reduction of Debt
Note: A transactions involving a debit would be represented by the addition of a negative number
Jaesook was very unhappy about owing $10. She opened another letter and found a refund check for $5.
“This $5 could not have come at a better time,” she exclaimed. So what’s her money condition now?
Credit-Debit Analogy: Elimination of Debt
Note: A transactions involving a debit would be represented by the addition of a negative number
Now Jaesook thought, I only owe $5. She opened a card. Inside was a birthday gift of $10.
“Wonderful! ” thought Jaesook. So what’s her money condition now?
Football Analogy
With the football analogy, midfield is considered the origin, the Righties Team tries to advance the
football as afar as possible to the right, and the Lefties, Team tries to advance the ball as far as
possible to the left. The operation of addition indicates a play by the Righties Team. Positive integers
indicate a gain of yardage; negative integers indicate a loss.
Charged-Particles Analogy
This concrete analogy builds on children’s familiar change-add-to understanding of addition, and can be
most helpful in understanding the addition of negative integers in particular. A positive integer can be
thought of as a collection of positive charges; a negative integer as a collection of negative charges.
Equalize Analogy
Because subtraction can have an equalize as well as a take-away meaning, the subtraction of integers can
be viewed as equalizing two values.
Time-Line Analogy
This analogy can combine the advantage of the familiar experience of determining age, using a number
line in the form of a time line, and connecting mathematical ideas to other content area such as social
studies. Many children recognize that a person’s age can be computed by subtracting his or her birth
year from the current year. In effect, they are computing the difference between an earlier date
(smaller number) and a later date (larger number). This can be a useful skill when studying history.
Ex: Rome’s first emperor Augustus ruled from 27 B.C. to 14 A.D. How long was he emperor?
Multiplication:
Repeated-Addition or Repeated Subtraction Analogy
(–3) x 5 and 5 x (–3) are equivalent
(because multiplication is commutative)
can be thought of repeated addition of (–3) five times,
(–3) + (–3) + (–3) + (–3) + (–3)
Math 103 Lecture 4 notes page 4
(–3) x (–5)
can be thought of repeated subtraction,
ie, subtracting (–5) three times.
A real-world example: a 5¢ off coupon where the store redeems triple the value, net result = +15¢
Rate Analogy for multiplying two negative integers:
We can talk about a situation where a student will earn $5 a day (+5) or lose $5 a day. If we talk about
the future, 2 days from now, we will represent this situation with a +2. In contrast, 2 days ago will be –2.
Mary earns $5 every day. Two days from now, how will Mary’s money compare to the amount she has
now?
Mary lost $5 a day. Two days from now she will have how much?
Mary has been receiving $5 per day to reach the amount she has now. Two days ago, she had how much
less?
Mary has been losing $5 a day to get to the amount she has now. Two days ago, how did Mary’s money
compare to the amount she has now?
Verbalization of such situations is sufficient for many 12-14-year-olds. However, for some, it may be
necessary to use physical objects (play money) to visualize.
Charged Particles Analogy for Multiplication
Can use two-color counters to model: (–2) x (–3)
Start with “zeros” in the bin.
Division:
Division of integers can be viewed as doing the opposite of multiplying integers: as a missing-factor
problem; or can be viewed as a divvy-up or measure out.
Divisibility:
If a and b are any integers, then b divides a, written b|a, if and only if there is a unique integer c such
that a = bc. Examples: 2 | 12
3 | 12
12 does not divide 2
If b|a, then b is a factor, or a divisor, of a, a is a multiple of b.
Do not confuse the relation b|a (b divides a), with the operation b|a which means b ÷ a
True or False?
1.
0 | 12
2.
0 is even
3.
If 3 | a, then 3 | na
4.
1|a
5.
a|1
6.
d | a implies d | na
7.
d | a, d | b implies d | (a + b)
8.
d | (a + b) implies d | a and d | b
9.
d | a, d | b implies d | (a–b)
Math 103 Lecture 4 notes page 5
Divisibility Rules
A number
A number
A number
A number
A number
A number
A number
A number
is
is
is
is
is
is
is
is
divisible
divisible
divisible
divisible
divisible
divisible
divisible
divisible
by
by
by
by
by
by
by
by
2 if the last digit is even.
3 if the sum of digits is divisible by 3.
4 if the last two digits are divisible by 4.
5 if the last digit is 0 or 5.
6 if it is divisible by 3 and 2.
8 if the last three digits are divisible by 8.
9 if the sum of the digits is divisible by 9.
10 if the last digit is 0.
All integers >1 can be classified as Prime or Composite.
Sieve of Eratosthenes is used to find Primes. (handout)
Mathematicians have looked for a formula that produces only primes, but none has been found. One
result was n2 – n + 41, for n from 0 to 40.
In 1998, Roland Clarkson, a 19-yr-old student at Cal State, showed that 23021377 – 1 is prime. This is
an example of a Mersenne prime (form 2n–1). There are only 38 known Mersenne primes.
Anther type of prime is a Sophie Germain prime, which is an odd prime p for which 2p + 1 is also prime.
Ex: 3, 5, 11, 23 are Sophie Germain primes.
Fundamental Theorem of Arithmetic:
Each composite number can be written as a product of primes in one, and only one way except for the
order of the prime factors in the product.
Ex: the prime factorization of 260 is 25•5•13
The least common multiple (LCM) of two integers a and b is the least positive integer that is
simultaneously a multiple of both a and b.
To find the LCM, find the product of the each of the primes that are factors of either number in their
prime factorization.
Ex:
180 = 2•2•3•3•5
and
LCM = 23•32•5•7 = 2520
168 = 2•2•2•3•7
The greatest common divisor (GCD or GCF) of two integers a and b is the greatest integer that divides
both a and b.
To find the GCD, find the product of the common factors in the prime factorization.
Ex:
180 = 2•2•3•3•5
and
GCD = 2•2•3 = 12
168 = 2•2•2•3•7
Use Venn diagrams to find LCM and GCD (UpClose notes)
Math 103 Lecture 4 notes page 6
Which of the following problems involve finding common factors; which, common multiples?
A: Lester had an 18x24 inch piece of construction paper. He wanted to edge the paper with equalsized squares. How large could he make the squares?
B: Landon was planning a party and wanted everyone at the party to have the same number of candies.
He was planning on having six people at the party. However, the Leopold twins had developed spots
that morning, and it was unclear whether they had a contagious disease and would be able to attend
the party. How many candies should Landon buy so that four or six children would get the same
number?
C: Lyndon and Lydia started running around their house at the same time and from the same place.
Lydia, who was the younger but in better shape, ran around the house in 3 minutes. Lyndon required
5. Assuming they could keep up this pace for 60 minutes, how many times would lyndon and Lydia
arrive at the starting point together?
D: Mr Jung wanted to enclose his backyard with a fence. The yard measured 65 feet x 104 feet. He
wanted to cut the top of the fence so that it formed a series of arcs. Mr. Jung was very finicky: He
did not want any partial arcs. What is the largest length the arcs would have to be cut so that each
side was entirely composed of whole arcs?
E: Miss Please was having a difficult first year of teaching and was a nervous wreck. Her doctor
prescribed some red pills she was to take every 4 hours and some blue pills she was to take every 6
hours. Miss Please felt especially calm whenever she took both pills simultaneously. If she started
taking the teacher’s little helpers at the same time, how often would Miss Please feel particularly
mellow?
F. The Math Book from Hell had the following rather unrealistic fraction question:
What is the sum of 1/54 and 1/72 of a mile?
Rodney sensed that 54x72 (3888) was not the lowest common denominator and that using it would
mean a lot of extra work. How could the lowest common denominator for 1/54 and 1/72 be found?