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Introduction
Prepared by Sa’diyya Hendrickson
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Package Summary
• Definitions and Properties of Integers
• Investigation I: Adding Integers
• Verifying Results: Investigation I
• Investigation II: Multiplying integers
• Summarizing Strategies
• Let’s Play! (Exercises)
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Definitions and Properties
Level: Z
1. Integers (Z): The set of integers is the set Z = {. . . . . . , −3, −2, −1, 0, 1, 2, 3, . . . . . .}.
The integers 1, 2, 3, . . . are positive and the integers −1, −2, −3, . . . are negative.
Zero is neither positive nor negative. Below is an integer number line.
2. absolute value: The absolute value of an integer a (denoted by |a|) is its distance
from zero. e.g. | − 4| =
, since −4 is a distance of
units from zero.
3. opposite: The opposite of an integer a is the integer −a that is the mirror image
(reflection) about zero on the integer number line. e.g. 3 and −3 are opposites.
4. subtraction: Subtracting two integers a and b (i.e. a − b) is the same as taking a
and adding the opposite of b. In other words, a − b = a + (−b).
5. multiplying by −1: To multiply an integer a by −1 is the same as reflecting over
zero on the integer number line. Therefore, multiplying a by −1 creates the opposite
of a! i.e. (−1)a = −a. e.g. (−1)(−2) = −(−2) = 2 since 2 is the opposite of −2.
Let a, b, and c be integers.
1. Adding opposites: a + (−a) = 0 since, by the Definition 4, a + (−a) = a − a = 0.
2. Commutative Property of addition: a+b = b+a
e.g. 7+(−2) = (−2)+7
3. Commutative Property of multiplication: a · b = b · a
e.g. 7 · (−2) = (−2) · 7
4. Associative Property of multiplication: a · b · c = (a · b)c = a(b · c)
can multiply by pairing or grouping numbers in the product!
5. Distributive Property: a(b ± c) = a(b) ± a(c)
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i.e. we
e.g. 3(7 − 2)) = 3(7) − 3(2)
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Investigation I
Level: Z
Let’s begin by establishing a few conventions. In the expression a + b :
• The “a” tells us to move a units on the number line (starting from zero).
Then, the “b” tells us to move b units from our current location.
For example: Consider 2 − 5. When trying to determine a and b, remember that (by
definition of subtraction) 2 − 5 = 2 + (−5). Therefore, a = 2 and b = −5.
• If the number (a or b) is positive, we move to the right.
From our example, we must first move 2 units right (from zero), which causes us to
land on the number 2.
• If the number (a or b) is negative, we move to the left.
Then, from our example, we move 5 unit to the left of 2, causing us to land at −3.
Use a number line to solve the exercises below. Then, identify any similarities between
parts (a) and (b) and try to develop general strategies for questions in each case.
Case 1: “Like” Signs
Adding when the integers have the same sign. i.e. both positive or both negative.
1. (a) 5 + 4
(b) −5 − 4
2. (a) 2 + 6
(b) −2 − 6
3. (a) 3 + 3
(b) −3 − 3
Case 2: “Opposite” Signs
Adding when the integers have opposite signs. i.e. one positive and one negative.
1. (a) 5 − 4
(b) −5 + 4
2. (a) 2 − 6
(b) −2 + 6
3. (a) −7 + 1
(b) 7 − 1
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Verifying Results
Level: Z
Case 1: “Like” Signs (integers are either both positive or both negative)
• We want to prove that for ”like” signs, we simply add the absolute values
of the numbers and the answer will keep their sign.
a) For a + b (integers both positive), simply add and the answer remains positive!
b) For −a − b (integers both negative), we must show that −a − b = −(a + b).
By our rules we have: − a − b = (−1)a − b
by
= (−1)a + (−b)
by
= (−1)a + (−1)b
by
= (−1)(a + b)
by
= −(a + b)
by
Case 2: “Opposite” Signs (one integer is positive and the other is negative)
• Prove that for ”opposite” signs, we can simply subtract from the larger
absolute value and the integer that coincides determines the sign.
a) For a − b (where a is larger than b), subtract as usual to get a positive answer!
b) For a − b (where b is larger than a), we must show that a − b = −(b − a).
By our rules we have: a − b = a + (−b)
by
= (−b) + a
by
= (−b) + (−(−a))
by
= (−1)b + (−1)(−a)
by
= (−1)(b + (−a))
by
= −(b + (−a))
by
= −(b − a)
by
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Multiplying Integers
Level: Z
Use the integer number line (below) and the definition of multiplying by −1
to calculate the following:
1. (−1)(3) =
2. (−1)(−1)(3) =
3. (−1)(−1)(−1)(3) =
4. (−1)(−1)(−1)(−1)(3) =
5. (−1)(−1)(−1)(−1)(−1)(3) =
Consider the following product: (−2)(5)(−4).
Question: Based on the pattern, should the answer be positive or negative? Why?
Let’s use the rules that we’ve learned to verify our prediction:
We have : (−2)(5)(−4) = (−1)(2)(5)(−1)(4)
by
= (−1)(−1)(2)(5)(4)
by
= (1)(2)(5)(4)
by
= (1 · 2)(5)(4)
by
= (2 · 5)(4)
by
= (10)(4)
= 40
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Summarizing Strategies
Level: Z
Why? Because we’ve seen that the strategies are different depending on whether we
are adding integers or multiplying integers.
S1: Like signs or opposite signs?
S2: Add or subtract?
• If “like” signs ⇒ add/combine the absolute value of each integer
• If “opposite” signs ⇒ subtract the absolute values (larger − smaller)
S3: Who (which “group”) determines the sign?
• If “like” signs ⇒ the answer keeps that sign!
• If “opposite” signs ⇒ the “bigger group” determines the sign (i.e. the integer with
the larger absolute value)
S1: How many negative numbers are in this product?
S2: Is this number even or odd?
S3: Should the answer be positive or negative?
• If even ⇒ multiply the absolute values and record a positive result!
• If odd ⇒ multiply the absolute values and record a negative result!
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Let’s Play!
Level: Z
1. Calculate: (Be sure to use your strategies!)
Example: 5 − 8 + 3 − 4
There are two popular approaches: (1) Combine the positives and negatives seperately, then combine the two resulting integers or (2) Use Order of Operations.
For this example, I will use approach (1):
5 − 8 + 3 − 4 = (5 + 3) + (−8 − 4)
by the Commutative Property of Additon
= 8 − 12
“opposite” signs ⇒ subtract
= −(12 − 8)
negatives are ”winning” ⇒ answer is negative
= −4
(a) −7 − 2
(b) −13 + 15
(c) 25 − 30
(d) −5 + 7 − 10
(e) −8 − 3 + 12 − 4
(f) 6 + 8 − 13 + 5
2. For each exercise: (i) Express in expanded form (ii) Determine if the
solution will be positive or negative (iii) Calculate
Example: (−3)5
i. (−3)5 = (−3)(−3)(−3)(−3)(−3)
ii. Negative answer because there is an odd number of negatives in the product.
iii. (−3)(−3)(−3)(−3)(−3) = −(3 · 3 · 3 · 3 · 3) = −243
(a) (−5)2
(b) (−3)3
(c) (−10)4
(d) (−2)5
3. For each exercise:
(i) Determine if the solution will be positive or negative (ii) Calculate
(a) (−7)(−2)
(b) (−8)(−(−2))
(c) (−6)(2)(−(−4))
(d) (−1)(−5)(−3)(2)(−1)
(e) −(−(−2))(−3)(7)
(f) (3)(−2)(−(−(3)))(−1)
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Let’s Play!
Level: Z
4. Write the expression in expanded form and calculate:
Example: 2(3 − 5) = 2(3) − 2(5)
by the Distributive Property
= 6 − 10
= −4
(a) 6(−1 + 7)
(b) 4(8 − 10)
(c) 3(−2 − 5)
(d) 5(4 − 3)
5. Calculate the following: (Remember to use Order of Operations!)
(a) (5(−2) + 32 ) + ((−4) − 52 ) + (−3)(−2)3 ÷ 6
(b) 3 − 2((−2)4 − 11) + (−9)(−4) ÷ 6 + (1 − 32 )
Now that we know integers, let’s expand our definitions of divides and factor
so that they include all integers.
i. divides (verb):
If a and b are integers, we say that “b divides a” if we can find some integer c, such
that a = b · c (i.e. there is no remainder when a is divided by b). If c exists, then it
also divides a. We will refer to b and c as a “pair” that makes a.
ii. factor (noun):
If a and b are integers, we say that “b is a factor of a” if b divides a. Therefore,
by the definition of “divides,” there exists some integer c (that is also a factor), such
that a = b · c.
So, −2 is a factor of 6 since there exists the integer −3 such that 6 = (−2)(−3).
In general, the factors of 6 include the positive factors (discussed in the previous lesson), and their opposites (discussed in this lesson)!
i.e. Factors of 6 include: ±1, ±2, ±3, ±6
6. Find all pairs, from the factors between −12 and 12, using divisibility rules.
e.g. (−2 and 4) or (2 and −4) are pairs that make −8!
(a) 180
(b) −216
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(c) 759
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(d) −1320
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