Download Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and
Dividing Integers
Introduction
In this unit, students will multiply and divide integers, and multiply positive and negative fractions
by integers. Students will understand the rules for multiplication of integers by extending the
properties of operations to negative numbers. Students will interpret products and quotients of
integers by describing real-world contexts. Students will apply the order of operations to
expressions involving brackets, positive and negative numbers, whole-number exponents, and
all four operations. Students will substitute negative numbers into equivalent expressions to
verify equivalence.
A note about models. There are many ways to remember the rules for determining when the
product of two integers is positive or negative. Many models can help students remember the
rules, but the Common Core State Standards emphasize the importance of students using the
properties of operations (e.g., the distributive property) to understand why the rules were made
in the first place. If you teach any models such as the patterns we introduce in NS7-26 and
NS7-27 or other frequently used models (e.g., number lines or integer tiles), be sure to explain
that the purpose of these models is to help students remember the rules, not to help them
understand why the rules were made.
Fraction notation. We show fractions in two ways in our lesson plans:
1
Stacked:
Not stacked: 1/2
2
If you show your students the non-stacked form, remember to introduce it as new notation.
In addition to the BLMs provided at the end of this unit, the following Generic BLM, found in
section J, is used in Unit 7:
BLM 1 cm Grid Paper (p. J-1)
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-1
NS7-25
Multiplying Integers (Introduction)
Pages 176–177
Standards: 7.NS.A.2a, 7.NS.A.2c
Goals:
Students will multiply two integers, one positive and the other negative.
Prior Knowledge Required:
Can add negative integers
Can add negative integers on a number line
Can extend increasing and decreasing sequences of integers that have constant gap
Can recognize opposite integers
Knows the commutative property of multiplication
Knows the bracket notation for adding integers
Knows the gains and losses notation for adding integers, such as −3 − 4
Vocabulary: commutative property, integer, negative, opposite, positive
Review adding negative integers. Remind students that to add negative numbers, you can
add them as though they are all positive then put a negative sign in front. Write on the board:
(−3) + (−4) = −3 − 4 = −7
SAY: To add −3 and −4, you can add 3 and 4 then make the answer negative. 3 + 4 is 7,
so −3 + (−4) is −7. Remind students that they can also write the addition of negative numbers
without brackets, the same as they do for gains and losses. A loss of $3 followed by a loss of $4
gets you a loss of $7.
Exercises: Add the integers.
a) 3 + 1
b) −3 − 1
c) 5 + 1
e) 13 + 7
f) −13 − 7
g) 120 + 500
Answers: a) 4, b) −4, c) 6, d) −6, e) 20, f) −20, g) 620, h) −620
d) −5 − 1
h) −120 − 500
Multiplication is a short form for repeated addition. Write on the board:
2 + 2 + 2 + 2 + 2 = 10
_____ × 2 = 10
ASK: How can we write the addition as a multiplication? (5 × 2) Draw on the board:
0
H-2
1
2
3
4
5
6
7
8
9
10
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Remind students that when adding on a number line, you start at zero. ASK: How would you
write (−2) + (−2) + (−2) + (−2) + (−2) as a multiplication? (5 × (−2)) Draw on the board:
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
Ask a volunteer to show this addition on the number line. (see answer below)
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
Remind students again that you can write addition of negative numbers without brackets, the
same as you do for the notation for gains and losses. Write an example on the board:
5 × (−2) = (−2) + (−2) + (−2) + (−2) + (−2) = −2 − 2 − 2 − 2 − 2
Exercises: Write each product as repeated addition. Then find the answer.
a) 5 × (−3)
b) 3 × (−2)
c) 4 × (−3)
d) 2 × (−4)
e) 3 × (−1)
f) 2 × (−7)
Selected solution: a) (−3) + (−3) + (−3) + (−3) + (−3) = −3 − 3 − 3 − 3 − 3 = −15
Answers: b) −6, c) −12, d) −8, e) −3, f) −14
(MP.8) Compare a × (−b) to a × b. Write on the board:
a) 5 × (−3)
a) 5 × 3
b) 3 × (−2)
b) 3 × 2
c) 4 × (−3)
c) 4 × 3
d) 2 × (−4)
d) 2 × 4
e) 3 × (−1)
e) 3 × 1
f) 2 × (−7)
f) 2 × 7
To encourage students to compare the two sets of problems and see their association, have
volunteers say the answers to the problems in the first row (−15, −6, −12, −8, −3, −14), then the
answers to the problems in the second row. (15, 6, 12, 8, 3, 14). Emphasize how the answers to
the first row of problems have the opposite sign to the answers to the second row of problems.
For example, 5 × (−3) is going to be negative because you are adding five −3s. When you
multiply 5 × 3, you are adding five +3s.
(MP.7) Exercises: Multiply the positive numbers first, then find the answer to the other
multiplication.
a) 4 × 5 = ____, so 4 × (−5) = ____
b) 3 × 6 = ____, so 3 × (−6) = ____
c) 7 × 5 = ____, so 7 × (−5) = ____
d) 8 × 2 = ____, so 8 × (−2) = ____
e) 1 × (−2) = ____
f) 2 × (−2) = ____
g) 3 × (−4) = ____
h) 5 × (−2) = ____
Bonus:
i) 5 × (−100) = ____
j) 30 × (−1,000) = ____
k) 300 × (−400,000) = ____
Answers: a) 20, −20; b) 18, −18; c) 35, −35; d) 16, −16; e) −2; f) −4; g) −12; h) −10;
Bonus: i) −500; j) −30,000; k) −120,000,000
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-3
Using patterns to see the rule another way. Write on the board:
5 × 4 = _____
5 × 3 = _____
5 × 2 = _____
5 × 1 = _____
5 × 0 = _____
5 × (−1) = _____
5 × (−2) = _____
Have volunteers dictate the answers to the first five multiplications (20, 15, 10, 5, 0), then guide
students to look at the pattern in the answers. ASK: What is the next term in the pattern? (−5)
And the next? (−10) SAY: When the number you multiply 5 by decreases by 1, the product
decreases by 5 each time. So because 5 × 0 is 0, we know that 5 × (−1) = −5, which is
5 less than 0. Fill in the last two blanks on the board. (−5, −10) Continue the pattern on the
board:
5 × ( _____ ) = ______
ASK: If we continue this pattern, what is the next number we multiply 5 by? (−3) How did you
get that? (because it is 1 less than −2) And what is the next number in the product column?
(−15) How did you get that? (because it is 5 less than −10)
Exercises: Complete the pattern.
10 × 4 = _____
10 × 3 = _____
10 × 2 = _____
10 × 1 = _____
10 × 0 = _____
10 × _____ = _____
10 × _____ = _____
10 × _____ = _____
Answers: 40, 30, 20, 10, 0, 10 × (−1) = −10, 10 × (−2) = −20, 10 × (−3) = −30
Multiplying (−a) × b. Write on the board:
2 × (−7) = ______
−7 × 2 = ______
SAY: When you multiply the same two numbers but in different orders, you always get the same
answer. ASK: What property is that called? (the commutative property) Have a volunteer tell you
the answer to the first problem. (−14) ASK: What is the answer to the second problem? (−14)
SAY: When mathematicians were deciding on the rules for multiplying integers, they wanted to
make sure that the commutative property would continue to hold. So if a positive number
multiplied by a negative number is negative, then so is a negative number multiplied by a
positive number.
H-4
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Exercises: Use the commutative property to multiply.
a) 6 × (−2) = ____, so −2 × 6 = ____
b) 4 × (−8) = ____, so −8 × 4 = ____
c) 5 × (−10) = ____, so −10 × 5 = ____
d) 25 × (−2) = ____, so −2 × 25 = ____
Answers: a) −12, −12; b) −32, −32; c) −50, −50; d) −50, −50
Emphasize that 2 × (−7) and −7 × 2 are both opposite of 2 × 7 or 7 × 2. Point out that it is easy
to multiply a positive number and a negative number. Just multiply as though they are both
positive, then make the answer negative.
Exercises:
1. Multiply mentally.
a) −8 × 3
b) 9 × (−1)
c) −1 × 9
d) −2 × 12
e) 15 × (−2)
f) −12 × 2
g) −13 × 2
h) 25 × (−3)
Answers: a) −24, b) −9, c) −9, d) −24, e) −30, f) −24, g) −26, h) −75
2. Multiply.
a) If 31 × 12 = 372, what is 31 × (−12)?
Answers: a) −372, b) −684
b) If 19 × 36 = 684, what is −19 × 36?
Extension
a) How does −7 × 4 compare to 7 × 4?
b) Predict how −7 × (−4) compares to 7 × (−4).
c) If −34 × (34) = −1,156, what is −34 × (−34)?
Answers: a) −28 and 28. Both multiplications provide a number with the same magnitude but
opposite signs (i.e., one is positive and one is negative); b) Changing the sign of one of the
numbers in part a) changed the sign of the answer, and that is what we are doing in part b) as
well. So since 7 × (−4) = −28, then −7 × (−4) = 28; c) 1,156
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-5
NS7-26
Multiplying Integers
Pages 178–179
Standards: 7.NS.A.2a, 7.NS.A.2c
Goals:
Students will multiply integers and fractions, determining when the answer is positive or
negative.
Prior Knowledge Required:
Can multiply a positive integer by a negative integer
Can multiply a whole number by a fraction
Vocabulary: commutative property, integer, negative, opposite, positive
Review multiplying a positive and a negative integer. Write on the board:
2×3=6
2 × (−3) = _____
−2 × 3 = _______
Have volunteers fill in the blanks. (−6, −6) Remind students that if they can multiply two positive
numbers, then they can multiply a positive and a negative number too.
Using patterns to find the rule for multiplying any number, positive or negative, by −1.
Write on the board:
−1 × 4 = ____
−1 × 3 = ____
−1 × 2 = ____
−1 × 1 = ____
−1 × 0 = ____
−1 × (−1) = ____
−1 × (−2) = ____
−1 × (___) = ____
ASK: Since 1 × 4 is 4, what is −1 × 4? (−4) Have volunteers provide the other answers, up to
−1 × 0. (−4, −3, −2, −1, 0) Have students look at the pattern. ASK: Are the answers getting
bigger or smaller? (bigger) By how much? (1 bigger each time) What is the next answer in the
pattern? (1) Write “1” in the blank beside −1 × (−1). Have a volunteer fill in the next answer (2).
Now point students to the pattern of the numbers being multiplied by −1. ASK: How are those
numbers changing? (they are getting smaller) Have a volunteer finish the last equation.
(−1 × (−3) = 3)
H-6
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
(MP.8) Ask students to look at the pattern and see if they can find a shortcut for multiplying by −1.
(change the sign of the number you are multiplying by or take the opposite number) Remind
students that multiplication is commutative, so −1 can be first or second and the same rule
applies.
Exercises: Multiply.
a) −1 × 2
b) 9 × (−1)
c) −8 × (−1)
d) −1 × (−13)
e) −1 × 8
f) 15 × (−1)
g) −1 × (−10)
h) −9 × (−1)
Bonus:
i) −1 × 13,000
j) −8,000,000 × (−1) k) 9 × 3 × (−1)
l) 7 × 8 × (−1)
Answers: a) −2; b) −9; c) 8; d) 13; e) −8; f) −15; g) 10; h) 9; Bonus: i) −13,000; j) 8,000,000;
k) −27; l)−56
Using patterns to determine the rule for multiplying any number, positive or negative, by
any negative number. SAY: The rule for multiplying by −1 is to change the sign. Let’s see if we
can find the rule for multiplying by −3.
Exercises: Copy the chart and complete the pattern.
−3 × 3
−3 × 2
−3 × 1
−3 × 0
−3 × _____
−3 × _____
−3 × _____
Answers: −9, −6, −3, 0, −3 × (−1) = 3, −3 × (−2) = 6, −3 × (−3) = 9
ASK: How does multiplying by −3 compare to multiplying by 3? (the answers are opposite)
SAY: So to multiply by −3, first multiply by 3 then change the sign. Tell students that this is what
they did when multiplying by −1, too. It’s just that the first step of multiplying by 1 wasn’t visible
because you don’t need to change the number at all when you multiply by 1. SAY: You can
multiply by any negative number this way: multiply by the positive number, then change the
sign.
Exercises: Multiply.
a) 3 × (−2) = _____, so −3 × (−2) = _____
b) 4 × 5 = _______, so −4 × 5 = ______
c) 2 × (−8) = _____, so −2 × (−8) = _____
d) 3 × 7 = _______, so −3 × 7 = ______
Answers: a) −6, 6; b) 20, −20; c) −16, 16; d) 21, −21
Developing and using the rule to multiply two negative numbers. Write on the board:
3 × (−4) = −12, so −3 × (−4) = 12
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-7
SAY: Here’s the pattern. A positive number (point to 3) multiplied by a negative number (point to
−4) is negative (point to −12), so a negative number (point to −3) multiplied by a negative
number (point to −4) is positive (point to 12). Tell students that in the next lesson they will learn
more about why two negative numbers multiply to a positive number.
Exercises: Multiply mentally.
a) −1 × (−1)
b) −3 × (−9)
c) −4 × (−10)
e) −7 × (−8)
f) −7 × (−12)
g) −15 × (−3)
Answers: a) 1; b) 27; c) 40; d) 100; e) 56; f) 84; g) 45; h) 1,000
d) −10 × (−10)
h) −100 × (−10)
(MP.8) Summarizing all the rules for multiplying integers. Write on the board:
−3 × (−2)
4 × (−6)
−7 × 8
−5 × (−5)
4×2
3 × (−6)
Have students signal whether the answer is positive (thumbs up) or negative (thumbs down).
Write the correct sign under each product after students signal the answer. (+, −, −, +, +, −)
Point to each product in turn and ASK: Do the two numbers being multiplied have the same sign
or different sign? Students can signal thumbs up for the same sign and thumbs down for a
different sign. Write “same” or “different” on the board as students signal their answers. The final
picture should look like this:
−3 × (−2)
4 × (−6)
−7 × 8
+
−
−
+
different
same
same
different
−5 × (−5)
4×2
+
same
3 × (−6)
−
different
ASK: If two numbers have the same sign, is their product positive or negative? (positive) If two
numbers have different signs is their product positive or negative? (negative) SAY: When both
numbers being multiplied are positive or both are negative, their product is positive. When one
number is positive and the other is negative, their product is negative.
(MP.7) Exercises: Multiply mentally.
a) 2 × (−8)
b) −3 × 0
c) 4 × (−2)
e) −10 × 9
f) −8 × (−8)
g) −19 × (−2)
Answers: a) −16, b) 0, c) −8, d) −60, e) −90, f) 64, g) 38, h) −66
d) 12 × (−5)
h) 11 × (−6)
In the Bonus exercises below, fill in the first two squares for part a) as −12 and 24.
SAY: −3 × 4 is −12. So to fill in the next square, I multiply −12 by −2, which is +24 or just 24.
You can keep going to find the entire product.
Bonus: Keep track as you go along to multiply many numbers.
a)
−3 × 4 × (−2) × (−1) × 10 =
H-8
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
b)
2 × (−2) × (−11) × 2 × (−1) =
Answers: Bonus: a) −12, 24, −24, −240; b) −4, 44, 88, −88
Multiplying a negative fraction and a whole number. Tell students that you can multiply
negative fractions by whole numbers the same way you multiply positive fractions by whole
numbers. Show the relationship by drawing a number line on the board, as shown below:
−2
−1
0
æ 2ö
6
3 ´çç- ÷÷÷ = çè 5 ø
5
3×2
1
2 6
3´ =
5 5
2
3×2
SAY: If you know how to multiply a whole number by a positive fraction, then multiplying the
whole number by the opposite negative fraction gets the opposite answer.
Exercises: Multiply.
æ 3ö
æ 5ö
b) 3 ´çç- ÷÷÷
a) 2 ´çç- ÷÷÷
çè 5 ø
çè 8 ø
6
15
8
Answers: a) - , b) - , c) 5
8
7
æ 2ö
c) 4 ´çç- ÷÷÷
çè 7 ø
SAY: Remember, if you can multiply positive integers, then you can multiply any integers. The
same is true for fractions: If you can multiply positive numbers, then you can multiply any
numbers. Write on the board:
æ 2ö
3 ´çç- ÷÷÷ =
çè 5 ø
2
-3 ´ =
5
æ 2ö
-3 ´çç- ÷÷÷ =
çè 5 ø
æ 6
6 6ö
Have volunteers provide the answers. çç- , - , ÷÷÷
çè 5
5 5ø
4 7
(MP.3) Exercises: Multiply. Use the same rules you use for multiplying integers to multiply any
positive and negative numbers.
æ 4ö
æ 4ö
æ 2ö
1
b) −1 × çç- ÷÷÷
c) −5 × çç- ÷÷÷
d) - × 9
a) 3 × çç- ÷÷÷
çè 3 ø
çè 7 ø
çè 10 ø
2
æ
ö
5
3
2
5
e) −3 ×
f) - × (−8)
g) −12 × çç- ÷÷÷
h) - × 100
ç
è
ø
8
3
2
3
9
15
Answers: a) −4, b) , c) 1, d) - , e) - , f) 8, g) 8, h) −250
2
8
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-9
SAY: Some people write brackets around both numbers when multiplying integers or positive
and negative fractions. Then they usually write the + sign in front of a positive number. It looks a
bit different but you still multiply the same way.
Exercises: Multiply.
a) (−3) × (+2)
b) (+4) × (−3)
æ 1 ö÷
æ 4ö
e) (-3)´çç+ ÷÷÷
d) çç+ ÷÷´(-5)
çè 2 ø
çè 3 ø
c) (−8) × (−5)
æ 3ö
f) (-2)´çç- ÷÷÷
çè 5 ø
6
5
Answers: a) −6, b) −12, c) 40, d) - , e) −4, f)
2
5
Extensions
(MP.3) 1. The product of 1 and 9 is less than 1 + 9. Can the product of two negative numbers be
1
less than their sum? Explain. Hint: Start with some examples, such as −2 × (−3) or −6 × ( - ).
2
Answer: No, because the product will be positive and the sum will be negative.
(MP.3) 2. Tell students that once you know the rule for multiplying a positive and a negative
number and that −1 × (−1) is 1, you can multiply any two negative numbers. Have students
explain each step in the following example:
−5 × (−2) = −1 × 5 × (−1) × 2
= −1 × (−1) × 5 × 2
= 1 × 10
= 10
Answer:
−5 × (−2) = −1 × 5 × (−1) × 2
= −1 × (−1) × 5 × 2
= 1 × 10
= 10
because −5 = −1 × 5 and −2 = −1 × 2
because 5 × (−1) = (−1) × 5
because −1 × (−1) = 1 and 5 × 2 = 10
because 1 × any number is itself
(MP.7) 3. Tell students that they can think of 3/4 × 2 as the point that is 3/4 of the distance from
0 to 2, as shown below:
0
3
3
´2 =
4
2
1
2
Ask students how they would write a multiplication for the point that is 3/4 of the distance from 0
to −2. (3/4 × (−2)) Have students use this method to show the following on a number line:
1
1
3
a) ´(-3)
b) ´(-5)
c) ´(-2)
2
4
5
H-10
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
3
5
6
Answers: a) - , b) - , c) 2
4
5
(MP.1, MP.5) 4. Puzzles with adding and multiplying integers. Tell students that you want to
find two integers that add to −5 and multiply to +6. Together, list the following pairs of integers
that add to −5 (pairs with two negative numbers have been left out on purpose):
0, −5
1, −6
2, −7
3, −8
4, −9
Have students find the products of the pairs listed so far: 0, −6, −14, −24, −36. ASK: Do you
think the next pair will have an answer that is closer to or farther from +6? Should we continue
what we are doing? (no, we are getting farther from the product we want) Suggest that instead
of going down the list, we should go up the list because as we go up the list, we are getting
closer to the answer we want. Have students tell you what goes next, above the pair
0, −5. (−1, −4) Find the product. (+4) Continue until you get the product +6. (−2, −3)
Point out that we needed to find two integers that both add to −5 and multiply to +6.
ASK: Instead of finding pairs that add to −5, what else could we look for? (pairs that multiply
to +6) Together, list all the pairs of numbers that multiply to +6:
+1, +6
+2, +3
−1, −6
−2, −3
Have students add the numbers in each pair (7, 5, −7, −5) The pair that adds to −5 is
−2, −3. Discuss which method students like better. Emphasize that there are likely to be a lot
fewer possibilities to check if students first find all pairs that multiply to the desired result rather
than all pairs that add to the desired result.
Have students find two numbers that …
a) multiply to −20 and add to +8.
c) multiply to −20 and add to −8.
Answers: a) 10, −2; b) 10, 2; c) −10, 2
5. Complete the multiplication chart.
a)
b)
×
+2
−3
×
+4
+5
−5
−3
b) multiply to 20 and add to 12.
c)
+2
−3
×
+
−
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
+
−
H-11
Answers:
a)
×
+2
+4
8
−5
−10
b)
−3
−12
15
c)
×
+5
−3
+2
10
−6
−3
−15
9
×
+
−
+
+
−
−
−
+
6. Complete the chart by deciding if the product will be positive or negative.
a
b
c
a×b×c
+
+
+
−
+
+
+
−
+
+
+
−
−
−
+
−
+
−
+
−
−
−
−
−
Answers: +, −, −, −, +, +, +, −
(MP.3) 7. Tell students that subtracting is the same thing as multiplying by −1, then adding. For
example:
5 − (−3) = 5 + (−1) × (−3)
Have students explain why this is the case.
Sample answer: Multiplying by −1 gets the opposite of the number and subtracting a number is
the same thing as adding its opposite.
H-12
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
NS7-27
Using the Distributive Property to Multiply Integers
(Advanced)
Pages 180–181
Standards: 7.NS.A.2a, 7.NS.A.2c
Goals:
Students will develop and apply the formula for multiplying integers.
Prior Knowledge Required:
Can add and subtract integers
Knows and can apply the distributive property
Can use repeated addition to multiply whole numbers
Can use the order of operations with brackets for these operations: +, −, ×, ÷
Vocabulary: commutative property, distributive property, equivalent expression, integer,
negative, positive, sign
Review the distributive property of multiplication over addition. Write on the board:
2+2+2+2+2+2+2
3×2
+
4×2
= 7×2
SAY: If you start with three 2s and add four more 2s, you end up with seven 2s, because
7 is 3 + 4. Write on the board:
3 × 2 + 4 × 2 = (3 + 4) × 2
SAY: These two expressions are equivalent because multiplication distributes over addition. In
fact, the distributive property of multiplication says that you could replace 2, 3, and 4 with any
other numbers and the two expressions would still be equivalent. Demonstrate by replacing
3 with 6, 2 with 5, and 4 with 7, as shown below:
6 × 5 + 7 × 5 = (6 + 7) × 5
Have students verify the expressions are still equivalent. (30 + 35 = 65 and 13 x 5 = 65)
Exercises:
(MP.7) 1. Write an equivalent expression that uses the same numbers.
a) 3 × 6 + 4 × 6
b) 7 × 4 + 7 × 2
c) 5 × 4 + 4 × 4
d) 4 × 4 + 4 × 3
e) (2 + 7) × 3
f) (3 + 3) × 5
g) 2 × (4 + 3)
Answers: a) (3 + 4) × 6, b) 7 × (4 + 2), c) (5 + 4) × 4, d) 4 × (4 + 3), e) 2 × 3 + 7 × 3,
f) 3 × 5 + 3 × 5, g) 2 × 4 + 2 × 3
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-13
(MP.1) 2. Evaluate the expressions in Exercise 1 to verify that both expressions are equivalent.
Answers: a) 18 + 24 = 42 and 7 × 6 = 42, b) 28 + 14 = 42 and 7 × 6 = 42, c) 20 + 16 = 36 and
9 × 4 = 36, d) 16 + 12 = 28 and 4 × 7 = 28, e) 9 × 3 = 27 and 6 + 21 = 27, f) 6 × 5 = 30 and
15 + 15 = 30, g) 2 × 7 = 14 and 8 + 6 = 14
Review the distributive property of multiplication over subtraction. Tell students that
multiplication distributes over subtraction too. Write on the board:
2+2+2+2+2+2+2
7×2−3×2=4×2
SAY: If you start with seven 2s and take away three of them, you end up with four 2s, because
7 − 3 is 4. Write on the board:
7 × 2 − 3 × 2 = (7 − 3) × 2
Exercises: Use the distributive property to write an equivalent expression that uses all the
same numbers.
a) (5 − 3) × 7
b) 8 × 4 − 5 × 4
c) (3 − 2) × 6
d) (5 − 5) × 6
e) 7 × 5 − 5 × 5
Answers: a) 5 × 7 − 3 × 7, b) (8 − 5) × 4, c) 3 × 6 − 2 × 6, d) 5 × 6 − 5 × 6, e) (7 − 5) × 5
(MP.3) Investigate the distributive law when the subtraction results in a negative integer.
Write on the board:
2 × (3 − 7) = 2 × 3 − 2 × 7
ASK: What is 3 − 7? (−4) Write on the board:
2 × (−4) = _____ − ______
Have volunteers fill in the blanks. (6 − 14) ASK: What is 6 − 14? (−8) Remind students that the
answer to 6 − 14 is the opposite of 14 − 6. SAY: We’ve just shown that 2 × (−4) is −8 and we
only had to multiply positive numbers and subtract them to do so. Tell students that the rules for
multiplying integers were chosen as they were so that the distributive property and all the other
properties of multiplication continue to hold for negative numbers.
(MP.1, MP.7) Exercises: Use the distributive property to multiply 2 × (−4) in different ways.
Make sure you always get −8.
a) 2 × (1 − 5)
b) 2 × (0 − 4)
c) 2 × (6 − 10)
Answers: a) 2 − 10 = −8, b) 0 − 8 = −8, c) 12 − 20 = −8
ASK: Using this method of subtracting positive products, what is the easiest way to multiply
2 × (−4)? (subtract 0 − 8)
H-14
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Exercises: Multiply by replacing the negative number with zero minus its opposite, then using
the distributive property.
a) 7 × (−4) = 7 × (0 − 4)
b) 8 × (−3) = 8 × (0 − 3)
c) 6 × (−4)
d) 3 × (−3)
Answers: a) 7 × 0 − 7 × 4 = 0 − 28 = −28, b) −24, c) −24, d) −9
SAY: Once you know a positive times a negative is negative and you decide to make the
commutative property hold, then a negative times a positive must also be negative. Write on the
board:
−4 × 7 = 7 × (−4) = −28
SAY: But you can also use the distributive property directly on −4 × 7 = (0 − 4) × 7.
Exercises: Use the distributive property to multiply.
a) −4 × 7
b) −3 × 5
c) −2 × 8
Answers: a) 0 − 28 = −28, b) 0 − 15 = −15, c) 0 − 16 = −16
(MP.5) Using the distributive property to multiply two negative integers. Write on the
board:
(−3) × (−2)
Remind students that we want all the properties of operations to hold for negative numbers.
SAY: I want to use properties of operations to show that multiplying two negatives gets a
positive. ASK: Can I use the commutative property? (no) PROMPT: Is (−2) × (−3) any easier
than (−3) × (−2)? (no) SAY: Let’s try using the distributive property. Write on the board:
(−3) × (−2) = (−3) × (0 − 2)
= (−3) × 0 − (−3) × 2
(MP.1) SAY: I want the distributive property to hold for negative numbers too, so that we can
use it to evaluate (−3) × (−2). Pointing to the last expression on the board, ASK: How can we
evaluate this equivalent expression? PROMPT: Do we know how to multiply by 0? (yes) Do we
know how to multiply a negative number and a positive number? (yes) Do we know how to
subtract a negative number? (yes) SAY: So we know how to do all the components of this
problem. We’ve broken down one difficult problem that we didn’t know how to do into three
easier problems that we know how to do.
ASK: What is (−3) × 0? (0) SAY: Anything times 0 is 0. ASK: What is (−3) × 2? (−6) Continue
writing on the board:
= 0 − (−6)
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-15
Remind students that subtracting a negative integer is the same as adding a positive integer.
Continue writing on the board:
=0+6
=6
SAY: So −3 × (−2) is +6.
Exercises: Use the distributive property to multiply.
a) −2 × (−5) = 0 − 2 × (−5) = _____
b) −1 × (−15) = 0 − __________ = _____
c) −5 × (−7) = 0 − __________ = _____
d) −8 × (−5) = 0 − __________ = _____
e) −7 × (−9) = 0 − __________ = _____
f) −3 × (−10) = 0 − __________ = _____
Selected solutions: a) 0 − (−10) = 10, b) 0 − (−15) = 15
Answers: c) 35, d) 40, e) 63, f) 30
After students finish, point to part a) and ASK: How does −2 × (−5) compare to 2 × 5? (they
have the same answer) Repeat for parts b), c), and d). SAY: You can multiply two negative
numbers as though both numbers are positive. You don’t even need to change the sign. Write
on the board:
negative × negative = positive
Exercises: Multiply without using the distributive property.
a) −3 × (−4)
b) −2 × (−5)
c) −8 × (−4)
d) −6 × (−2)
e) −7 × (−3)
f) −4 × (−4)
Answers: a) 12, b) 10, c) 32, d) 12, e) 21, f) 16
(MP.3) Moving negative signs around in a product. Write on the board:
−3 × 4 = _____
3 × (−4) = ______
Have volunteers dictate the answers. (−12, −12) SAY: You can move a negative sign from the
first term to the second and still get the same answer. Write on the board:
−3 × 4 = 3 × −4
SAY: These are equivalent expressions because they have the same answer.
Exercises: Write an equivalent expression by moving the negative sign to the other number.
a) −2 × 7
b) −8 × 9
c) −5 × 4
Bonus: Make several equivalent expressions by moving the negative sign to any other number.
−2 × 3 × 4 × 5
Answers: a) 2 × (−7); b) 8 × (−9); c) 5 × (−4); Bonus: 2 × (−3) × 4 × 5, 2 × 3 × (−4) × 5,
2 × 3 × 4 × (−5)
H-16
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Extensions
(MP.8) 1. Investigate multiplying sequences of integers to determine the rule for how to find the
sign of the answer. Complete the table below as a class. (answers are in italics)
Number of Negative
Multiplication
Answer
Sign of Answer
Integers in Product
−3 × 2 × (−5) × 10
2
300
+
−3 × (−2) × (−5) × 10
3
−300
−
−3 × (−2) × (−5) × (−10)
4
300
+
−3 × (−2) × (−5) × (−10) × (−2)
5
−600
−
−3 × (−2) × 5 × (−10) × (−2)
4
600
+
−3 × (−2) × 5 × 10 × (−2)
3
−600
−
ASK: When is the answer positive? (when the number of minus signs is even) When is the
answer negative? (when the number of minus signs is odd)
Have students use their discovery to multiply 2 × (−5) × 2 × (−5) × (−2) × 5 × (−7) × (−3).
(−21,000)
(MP.7) 2. Use 0 = −2 + 2 and 0 × (−5) = (−2 + 2) × (−5) to show that if we want the distributive
property to hold, then −2 × (−5) must be +10.
Answer:
0 = −2 + 2
0 × (−5) = (−2 + 2) × (−5)
0 = −2 × (−5) + 2 × (−5)
0 = −2 × (−5) + (−10)
So −2 × (−5) is the opposite integer to −10 and so must be +10.
3. Multiply two ways. In the first method replace the first integer with 0 minus a positive integer,
and in the second method replace the second integer with 0 minus a positive integer. Make sure
you get the same answer both ways.
a) −4 × (−5)
b) −3 × (−7)
c) −5 × (−6)
d) −2 × (−8)
Selected solution: a) −4 × (−5) = (0 − 4) × (−5) = 0 × (−5) − 4 × (−5) = 0 − (−20) = 20 and
(−4) × (0 − 5) = −4 × 0 − (−4) × 5 = 0 − (−20) = 20
Answers: b) 21, c) 30, d) 16
(MP.1) 4. Use the distributive property to do −3 × (−2) six ways. Make sure you always get the
same answer.
a) −3 × (1 − 3)
b) −3 × (2 − 4)
c) −3 × (3 − 5)
d) (0 − 3) × (−2)
e) (1 − 4) × (−2)
f) (2 − 5) × (−2)
Bonus: −3 × (98 − 100)
Selected solution: a) −3 × 1 − (−3) × 3 = −3 − (−9) = −3 + 9 = +6
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-17
5. What properties are being used?
a) −3 × 5 = (3 × (−1)) × 5 = 3 × (−1 × 5) _________________________
= 3 × (5 × (−1)) _________________________
= 3 × (−5)
b) 2 × (−4) = 2 × ((−1) × 4) = (2 × (−1)) × 4 ______________________
= (−1 × 2) × 4 ______________________
= −1 × (2 × 4) ______________________
= −1 × 8
= −8
Answers: a) associative, commutative; b) associative, commutative, associative
6. Have students complete the table.
p
q
−p × q = ____
p × (−q) = _____
2
−2
2
−2
3
−2
−3
−2
Are the Expressions Equal?
5
5
−5
−5
−4
−1
2
−5
When students are done, conclude that products of variables have the same answer when you
move a negative sign from one term to another. So these expressions are equivalent, no matter
what p and q are: p and q might be both positive, both negative, or one positive and one
negative.
Answers:
p
q
−p × q = ____
p × (−q) = _____
Are the Expressions Equal?
2
−2
2
−2
3
−2
−3
−2
H-18
5
5
−5
−5
−4
−1
2
−5
−2 × 5 = −10
2 × 5 = 10
−2 × (−5) = 10
2 × (−5) = −10
−3 × (−4) = 12
2 × (−1) = −2
3×2=6
2 × (−5) = −10
2 × (−5) = −10
−2 × (−5) = 10
2 × 5 = 10
−2 × 5 = −10
3 × 4 = 12
−2 × 1 = −2
−3 × (−2) = 6
−2 × 5 = −10
yes
yes
yes
yes
yes
yes
yes
yes
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
NS7-28
Multiplying Integers in the Real World
Pages 182–184
Standards: 7.NS.A.2a, 7.NS.A.3
Goals:
Students will represent real-world situations with integer equations, including those involving
multiplication and addition.
Prior Knowledge Required:
Knows the order of operations, without exponents
Can add, subtract, and multiply integers
Knows when situations can be represented by integers
Vocabulary: integer, negative, positive, sign
Review real-world situations described by integers. Have students brainstorm things in the
real world that they can describe by integers. (sample answers: temperature, electric charge,
golf scores, +/− ratings in basketball or hockey, bank balances, gains and losses, elevation
above or below sea level)
Describing change as positive or negative. Tell students that in most of the situations they
suggested above the integer is describing a situation, such as the current temperature or the
elevation at a particular spot or the strength of an electric charge. But integers can also
represent a change in the situation. Point out that students have already seen this with gains
and losses or debits and credits in the way they represent a change in the bank balance.
Explain that increases are considered positive and decreases are considered negative. Have
volunteers demonstrate the first two exercises below before assigning the rest to all students.
Exercises: Describe the change as a positive or negative number.
a) A baby gained 14 pounds in his first year.
b) A baby lost 0.6 pounds in her first week.
c) Ed earned $60 from working.
d) Lynn lost $40 on the sidewalk.
e) Max climbed 1,500 feet.
f) The temperature decreased 4.5°F.
Bonus:
g) The temperature changed from −3°F to +2°F.
h) Jane started the day with $25 and has $55 at the end of the day.
i) Roy started on the 12th floor of his building and walked down to the 8th floor.
Answers: a) +14; b) −0.6; c) +60; d) −40; e) +1,500; f) −4.5; Bonus: g) +5; h) +30; i) −4
Constant change as repeated addition and multiplication. Tell students that sometimes the
same change happens repeatedly. Write on the board:
(+10) + (+10) + (+10) = 3 × (+10)
= +30 or 30
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-19
SAY: If I earned $10 each hour for 3 hours, then my change in money can be described as
+30 dollars.
Exercises: Write a multiplication equation to show the amount of change.
a) Ted gained $10 every hour for 5 hours.
b) The temperature dropped by 2°F every hour for 6 hours.
c) Vicky’s bank account balance decreased $5 every week for 8 weeks.
d) A hiker climbed 12 feet every hour for 3 hours.
Answers: a) 5 × (+10) = +50, b) 6 × (−2) = −12, c) 8 × (−5) = −40, d) 3 × (+12) = +36
Representing time using integers by putting 0 as right now. Tell students that they can
think of future time as positive and past time as negative. SAY: We’ll use zero to represent right
now. Write on the board:
3 weeks from now ______
3 weeks ago _____
Have volunteers tell you what integer can be used to represent 3 weeks from now and
3 weeks ago (+3, −3). SAY: When a problem involves two different units, you need to carefully
say what the units are. But when there is no confusion you can ignore the units until you are
done. So the same integer can represent many different quantities.
Exercises: Describe the time as an integer.
a) 2 years ago
b) 1 minute ago
c) 30 seconds from now
Answers: a) −2, b) −1, c) +30, d) +30
d) 30 days from now
Writing integer multiplication equations to show real-life situations. Tell students to
pretend that somebody is climbing a mountain and their elevation is increasing at a rate of
20 feet per minute. Tell students that their current elevation is at sea level. ASK: What integer
do we use to represent sea level? (0) Write on the board:
current elevation = 0 ft
5 minutes from now the elevation will be _______
ASK: What will the elevation be 5 minutes from now? (+100 ft) Write that in the blank.
ASK: How did the volunteer get the answer? (multiplied 5 × 20) Tell students that you want to
be even more specific and write the multiplication of integers. Write on the board:
(+5) × (+20) = +100
SAY: The 5 minutes are in the future so we use +5. The 20 feet each minute is an increase in
elevation so we use +20. Continue writing on the board:
5 minutes ago the elevation was _____ × ______ = _______
H-20
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Tell students that the person has been climbing for a while and that they started below sea
level. Point out that that doesn’t mean that they were underwater; many places on Earth are
below sea level but are still on dry land. ASK: What was the elevation 5 minutes ago?
(−100 ft) How did the volunteer get the answer? (multiplied −5 × 20) Fill in the blanks on the
board:
5 minutes ago the elevation was: _(−5)_ × _(+20)_ = _−100_
Pointing to −5, ASK: Why is this negative? (because the 5 minutes is in the past) Pointing to
+20, ASK: Why is this positive? (because the elevation is increasing) SAY: So 5 minutes ago,
the climber’s elevation was −100 ft.
(MP.2, MP.6) Emphasize the importance of including the units in the answer even though the
units are not used when doing the calculation.
Exercises: Write an integer equation to show your answer. Zack’s elevation is currently 0 m.
a) Zack increases his elevation 5 m every hour.
i) What will his elevation be 3 hours from now?
ii) What was his elevation 3 hours ago?
b) Zack decreases his elevation 5 m every hour.
i) What will his elevation be 3 hours from now?
ii) What was his elevation 3 hours ago?
Answers: a) i) (+5) × (+3) = +15, so +15 m; ii) (+5) × (−3) = −15, so −15 m;
b) i) (−5) × (+3) = −15, so −15 m; ii) (−5) × (−3) = +15, so +15 m
Review adding integers. SAY: When two integers have the same sign, you can add them by
adding their absolute values, which are the number parts without the signs. The answer has the
same sign as both numbers. Write on the board:
3+4=7
(−3) + (−4) = −7
Remind students that you can write adding integers as if you are adding gains and losses. Write
on the board:
−3 − 4 = −7
SAY: When two integers have opposite signs you can add them by subtracting their absolute
values. The answer has the same sign as the number with the larger absolute value. Write on
the board:
(−3) + (+4) = +1
−3 + 4 = +1
Exercises: Add.
a) 3 + 9
b) −4 − 1
e) 13 + 7
f) 9 − 6
(+3) + (−4) = −1
+3 − 4 = −1
c) −3 + 8
g) −7 − 14
d) −8 + 6
h) −25 − 5
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-21
Answers: a) 12, b) −5, c) 5, d) −2, e) 20, f) 3, g) −21, h) −30
Review subtracting integers. SAY: You can subtract a number by adding the opposite
number. Write on the board:
2 − (+5) = 2 − 5 = −3
2 − (−5) = 2 + 5 = 7
Exercises: Subtract.
a) −1 − (−3)
b) −4 − (−4)
c) 3 − (−5)
e) −20 − (+15)
f) 31 − (−2)
g) −11 − (+7)
Answers: a) 2, b) 0, c) 8, d) 10, e) −35, f) 33, g) −18, h) 22
d) 8 − (−2)
h) 10 − (−12)
Applying the order of operations to multiplication of integers. Remind students that
multiplication and division are done before addition and subtraction unless brackets tell you
otherwise. Write on the board:
4−1×3=4−3=1
but
(4 − 1) × 3 = 3 × 3 = 9
Exercises:
1. Evaluate.
a) 9 − 4 × 3
b) (9 − 4) × 3
c) (11 − 3) × 5
e) (−5 − 3) × 2
f) −5 − 3 × 2
g) 6 − 6 × 3
Answers: a) −3, b) 15, c) 40, d) −4, e) −16, f) −11, g) −12, h) 0
d) 11 − 3 × 5
h) (6 − 6) × 3
2. Write and evaluate an expression to show the amount that Anna ends up with.
a) Anna started with $4. She gained $2 each hour for 8 hours.
b) Anna started with a debt of $35. She gained $8 each hour for 4 hours.
c) Anna started with $150. She lost $10 each day for 10 days.
d) Anna started with a debt of $200. She lost $150 each month for 6 months.
Answers: a) 4 + 8 × 2 = 4 + 16 = $20, b) −35 + 8 × 4 = −35 + 32 = −$3,
c) 150 −10 × 10 = 150 − 100 = $50, d) −200 −150 × 6 = −200 − 900 = −$1,100
Word problems practice.
(MP.4) Exercises: Write an integer equation to find the answer.
a) Mark owes his dad $80. He earns $6 per hour for 11 hours of work. Is he still in debt?
b) Tina owes her mother $50. She earns $8 per hour for 9 hours of work. Does she have
enough money left over to buy a $20 book?
c) Pedro started with a debt of $60. He earned $10 per hour for 9 hours of work. Then he
bought 6 items that cost $4 each. Is he still in debt?
d) Helen started with a debt of $70. She earned $6 per hour for 20 hours work. How many
$4 phone apps can she buy?
Answers: a) −80 + 6 × 11, = −14, he is still $14 in debt; b) −50 + 8 × 9 = 22, she has $22 left
over, which is enough money to buy a $20 book; c) −60 + 10 × 9 − 6 × 4 = 6, he is not in debt,
he has $6 left over; d) −70 + 6 × 20 = 50, and 50 ÷ 4 = 12 R 2, so she can buy 12 apps and
have $2 left over
H-22
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
3 5
5
2
Extensions
1. Write an equation involving positive and negative numbers to show your answer.
2
a) The temperature is 0°F. The temperature increases
of a degree every hour. What will the
3
temperature be in 6 hours?
3
of a degree every day. What was the
b) The temperature is 0°F. The temperature decreases
5
temperature 5 days ago?
3
c) The temperature is 32°F. The temperature decreases 1 degrees every hour. What will the
5
temperature be 4 hours from now?
9
degrees every minute. What was the
d) The temperature is −8°F. The temperature increases
4
temperature 5 minutes ago?
æ 2ö
æ 3ö
Answers: a) çç+ ÷÷÷´ (+6) = +4 , so +4°F; b) çç- ÷÷÷´ (-5) = +3 , so +3°F;
çè 3 ø
çè 5 ø
æ 8ö
32
2
3
°F or 25.6°F;
c) ( 32 + çç- ÷÷÷´ (+4) = 32 = 32 - 6 = 25 ), so
çè 5 ø
5
5
5
æ 9ö
45
1
1
d) -8 + çç+ ÷÷÷´ (-5) = -8 °F or −19.25°F
= -8 - 11 = -19 , so 
çè 4 ø
4
4
4
1 4
9
1
2. Translate the description into an expression and evaluate the expression.
a) Add 3 and 9. Then divide by −2. Then add 5.
b) Multiply by −3 and −4. Then subtract 8. Then add 5.
c) Subtract 12 from −3. Then add 7. Then divide by 2.
d) Add 4 and −5. Subtract 8 from the result. Then divide by 7.
Answers: a) (3 + 9) ÷ (−2) + 5 = −1, b) (−3) × (−4) − 8 + 5 = 9, c) (−3 − 12 + 7) ÷ 2 = −4,
9
d) (4 + (−5) − 8) ÷ 7 = 7
3. Write the expression in words.
a) 5 + (−7) × 3
b) (5 + (−7)) × 3
c) (5 − 3) × 3 + (−2)
d) 5 − 3 × 3 + (−2)
e) 4 × (3 + (−2) × (−5))
f) 4 × 3 + (−2) × (−5)
g) (4 × 3 + (−2)) × (−5)
h) 4 × (3 + (−2)) × (−5)
Answers:
a) Multiply −7 and 3. Then add 5.
b) Add 5 and −7. Then multiply by 3.
c) Subtract 3 from 5, then multiply by 3, then add −2.
d) Multiply 3 by 3, then subtract from 5, then add −2.
e) Multiply −2 and −5, then add 3, then multiply by 4.
f) Multiply −2 and −5. Multiply 4 and 3. Then add the results.
g) Multiply 4 and 3. Then add −2. Then multiply by −5.
h) Add 3 and −2, then multiply by 4, then multiply by −5.
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-23
NS7-29
Dividing Integers
Pages 185–186
Standards: 7.NS.A.2b, 7.NS.A.2c, 7.NS.A.3
Goals:
Students will develop and apply the formula for dividing integers.
Prior Knowledge Required:
Can multiply integers
Knows the relationship between multiplication and division
Vocabulary: integer, negative, positive, sign
(MP.3) Apply the relationship between multiplication and division to negative numbers.
SAY: If you can find the missing number in a multiplication, then you can divide. That’s true for
negative numbers the same as it is for positive numbers. Write on the board:
−3 × ______ = 12, so 12 ÷ (−3) = ______
SAY: A negative number times something is a positive number. ASK: Is the something positive
or negative? (negative) SAY: Once you determine the sign, you just have to look at the
numbers. ASK: 3 times what is 12? (4) Write −4 in both blanks. Repeat the questions for
____ × (−4) = (−20) and ____ × 3 = −18. (5, −6)
Exercises: Divide by finding the missing number in the product.
a) 2 × ____ = 14, so 14 ÷ 2 = _____
b) 3 × ____ = 24, so 24 ÷ 3 = _____
c) −3 × ____ = −27, so −27 ÷ (−3) = _____
d) 11 × ____ = −88, so −88 ÷ 11 = _____
e) −3 × ____ = −60, so −60 ÷ (−3) = _____
f) −100 × ____ = 200, so 200 ÷ (−100) = _____
Answers: a) 7, 7; b) 8, 8; c) 9, 9; d) −8, −8; e) 20, 20; f) −2, −2
(MP.1, MP.7) Developing a rule for dividing integers. Write on the board:
To find −12 ÷ 3 = _____, ask: 3 × _____ = −12
To find (−) ÷ (+) = _____, ask: (+) × _____ = (−)
SAY: If you can find the missing number in a product, you can answer the division. So if you can
find the missing sign in a product, then you can find the correct sign of the division answer.
Exercises:
1. Use multiplication to finish writing the signs.
a) (+) ÷ (+) =
b) (+) ÷ (−) =
c) (−) ÷ (+) =
H-24
d) (−) ÷ (−) =
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Answers: a) (+) × ____ = (+), so (+); b) (−) × ____ = (+), so (−); c) (+) × ____ = (−), so (−);
d) (−) × _____ = (−), so (+)
2. Use the answer to the first division to do the other three divisions.
a) 49 ÷ 7 = 7
b) 55 ÷ 11 = 5
c) 36 ÷ 2 = 18
49 ÷ (−7) = _____
55 ÷ (−11) = _____
36 ÷ (−2) = _____
−49 ÷ 7 = _____
−55 ÷ 11 = _____
−36 ÷ 2 = _____
−49 ÷ (−7) = _____
−55 ÷ (−11) = _____
−36 ÷ (−2) = _____
Answers: a) −7, −7, 7; b) −5, −5, 5; c) −18, −18, 18
SAY: The rules for dividing are the same as for multiplying: If both numbers have the same sign,
the answer is positive. If the numbers have different signs, the answer is negative.
Exercises: Divide mentally.
a) 60 ÷ (−12)
b) −40 ÷ (−5)
c) −26 ÷ 2
d) (−65) ÷ (−5)
e) −120 ÷ 20
f) 1,600 ÷ (−40)
Answers: a) −5, b) 8, c) −13, d) 13, e) −6, f) −40
Writing the answer to a division question as a decimal. Remind students that the answer to
a division question can be written as a fraction. Write on the board:
7÷5=
7
5
2 ÷ 25 =
2
25
Remind students that to turn a fraction into a decimal we need a denominator that is a power of
10 (10, 100, 1,000, etc.). For each fraction above, ASK: What is the smallest denominator we
can use? (10, 100) Write on the board:
7
=
5 10
2
=
25 100
Have volunteers fill in each numerator. (14, 8) SAY: When a fraction has a denominator that is a
power of 10, it is easy to change it to a decimal. Write the numerator and put the decimal point
so that there is the same number of digits after the decimal point as there are zeros in the
denominator. Show this on the board:
7 14
=
= 1.4
5 10
2
8
=
= 0.08
25 100
Exercises: Use your answer to the first division to do the other three divisions.
a) 49 ÷ 10 = ______
b) 8 ÷ 5 = _____
49 ÷ (−10) = _____
8 ÷ (−5) = _____
−49 ÷ 10 = _____
−8 ÷ 5 = _____
−49 ÷ (−10) = _____
−8 ÷ (−5) = _____
Answers: a) 4.9, −4.9, −4.9, 4.9; b) 1.6, −1.6, −1.6, 1.6
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-25
SAY: Some people write brackets around both numbers when dividing integers. Then they put
positive signs in front of positive numbers. It looks a bit different but you still divide the same
way.
Exercises:
1. Fill in the blanks.
(+7) ÷ (+20) = _____
(+7) ÷ (−20) = _____
(−7) ÷ (+20) = _____
(−7) ÷ (−20) = _____
Answers: 0.35, −0.35, −0.35, 0.35
2. Divide. Write your answer as a decimal.
a) 25 ÷ (−100)
b) (+35) ÷ (−70)
c) −3 ÷ (20)
d) −47 ÷ (−100)
e) −15 ÷ (−60)
f) (−19) ÷ (+5)
Answers: a) −0.25, b) −0.5, c) −0.15, d) 0.47, e) 0.25, f) −3.8
(MP.3) Writing the division answer as a fraction of integers. SAY: You can write the answer
to an integer division as a fraction of integers. Write on the board:
7¸5 =
7
5
−7 ÷ 5 =
-7
5
7 ÷ −5 =
7
-5
−7 ÷ (−5) =
-7
-5
Exercises: Write each division answer as a fraction of integers.
a) 3 ÷ (−4)
b) −8 ÷ 9
c) −2 ÷ (−15)
d) 1 ÷ (−12)
3
-8
-2
1
Answers: a)
, b)
, c)
, d)
-4
-12
9
-15
SAY: If the numerator and denominator have opposite signs, then the value is negative. If they
have the same sign, the value is positive.
Equivalent fractions of integers. Write on the board:
−7 ÷ 5 =
7
5
−7 ÷ (−5) =
7
5
7 ÷ (−5) =
7
5
For each question, ASK: Is the answer positive or negative? (negative, positive, negative) Write
the correct signs in the boxes. Summarize by writing on the board:
-7
7
=5
5
H-26
-7
7 7
=+ =
5 5
-5
7
7
=-5
5
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Exercises:
5
a) Circle the expressions that have the same value as - .
3
-5
5
5
-5
3
3
-3
-3
5
b) Circle the expressions that have the same value as .
8
-5
5
5
-5
-8
8
8
-8
-2
.
-3
-2
3
2
-2
, and
3
-3
Bonus: Circle the expressions that have the same value as -
-2
2
-3
-3
-5
5
-5
2
Answers: a)
and
; b)
; Bonus: - ,
-8
3
3
-3
2
3
-
2
3
Average change. Tell students that on December 14, 1924, the temperature at Fairfield, MT,
dropped from 63°F to −21°F in 12 hours (from noon to midnight). ASK: How much did the
temperature change in total? (84°F) Is the change positive or negative? (negative) How do you
know? (because the temperature decreased) Write on the board:
−84°F in 12 hours
Average change per hour = −84° ÷ 12
Remind students that if the total temperature change was −84°F in 12 hours, then the average
change in each hour is obtained by division. Ask a volunteer to calculate the average change
per hour. (−7°F)
Exercises: Fill in the blanks. Write the average change as an integer. Include the units.
a) An elevator rose from 0 ft to 60 ft in 30 seconds.
b) Fred’s bank account balance changed from $34 to −$16 in 5 days.
c) Milly’s bank account balance changed from −$7 to $17 in 3 days.
d) In Spearfish, SD, on January 22, 1943, the temperature went from −4°F to 45°F in 2 minutes.
e) The value of a car decreased from its initial value of $23,000 to a value of $19,500 after
5 months.
f) A skydiver fell from an elevation of 60,000 ft to 45,000 ft in 3 minutes.
Answers: a) +2 ft per second; b) −$10 per day; c) +$8 per day; d) +24.5°F per minute;
e) −$700 per month; f) −5,000 ft per minute
Some students might think that the change from −7 to +17 is 20 because, when subtracting
positive numbers, the difference between two numbers with the same ones digit is always a
multiple of 10. Emphasize that this is not the case when subtracting a negative number from a
positive number.
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-27
Extensions
2
1. Use multiplication to check that −2 ÷ 5 = - .
5
æ 2 ö÷
10
Answer: 5 ´çç- ÷÷ = = -2
çè 5 ø
5
2. a) Predict which divisions will have an answer greater than 1. Explain how you made your
prediction. Hint: Write the divisions as fractions.
A. −9 ÷ (−10)
B. −11 ÷ (−10)
C. −3 ÷ (−4)
D. −6 ÷ (−5)
b) Check your predictions from part a) by writing the answer as a decimal.
c) Estimate where −97 ÷ (−101) is on a number line.
Answers:
a) The fractions are all positive and a positive fraction is greater than 1 when its numerator is
greater than its denominator, so I predict A and C will have answers less than 1, and B and D
will have answers greater than 1.
b) A has answer 0.9 < 1, B has answer 1.1 > 1, C has answer 0.75 < 1, and D has answer 1.2 > 1
c) The quotient should be less than, but very close to, 1.
0
H-28
1
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
NS7-30
Powers
Pages 187–188
Standards: 7.NS.A.2a, 7.NS.A.2c, 7.NS.A.3
Goals:
Students will evaluate powers by using repeated multiplication and will investigate the properties
of powers.
Prior Knowledge Required:
Can multiply negative numbers
Understands multiplication as repeated addition
Vocabulary: base, exponent, negative, opposite, positive, power, squared
Introduce powers as repeated multiplication. Remind students that multiplication is repeated
addition. Write on the board:
5×3=3+3+3+3+3
SAY: Just like multiplication is a short form for repeated addition, powers are a short form for
repeated multiplication. Write on the board:
35 = 3 × 3 × 3 × 3 × 3
SAY: 5 times 3 means add five 3s together, and 3 with a raised 5 means multiply five 3s
together. Write on the board:
exponent
base
35
Tell students that 3 is called the base, 5 is called the exponent, and 35 is called a power of 3.
SAY: The base is the number you are multiplying and the exponent is how many of them you
are multiplying. Point out that in math a base is the bottom part of something, the same as in
English. Write on the board:
42
34
25
Have students signal the base, then the exponent for each expression. (4, 3, 2; 2, 4, 5) Write on
the board:
2×2×2
4×4×4
5×5
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-29
Have students signal the base, then the exponent for each expression. (2, 4, 5; 3, 3, 2) Ask a
volunteer to write 94 as a product. (9 × 9 × 9 × 9)
Ask a volunteer to write 3 × 3 × 3 × 3 × 3 × 3 × 3 as a power. (37)
Exercises: Write the product as a power.
a) 2 × 2 × 2 × 2
b) 8 × 8
c) 1 × 1 × 1
d) 10 × 10
e) 12 × 12 × 12
f) 100 × 100
Bonus:
g) 1,500 × 1,500 × 1,500
h) 100,000 × 100,000
Answers: a) 24; b) 82; c) 13; d) 102; e) 123; f) 1002; Bonus: g) 1,5003; h) 100,0002
Evaluating powers. Write on the board:
23 = 2 × 2 × 2
=
_______
SAY: You can find repeated multiplications by keeping track as you go along. First find the
result of 2 × 2 (write 4 in the first square), then multiply your answer by 2 (write 8 in the next
square). So multiplying three 2s gets 8 (write that in the blank). Write on the board:
26 = 2
×
2 ×
2
×
2
×
2
×
2
=
________
Ask a volunteer to evaluate the power using the same method. (4, 8, 16, 32, 64)
Exercises:
1. Evaluate the power. Keep track of the power as you go along.
a) 25 = 2
×
2 ×
2
×
b) 53 = 5
×
5 ×
5
=
2
×
2
=
________
________
c) 104 = 10 × 10 × 10 × 10
Answers: a) 32, b) 125, c) 10,000
=
________
2. Evaluate.
a) 24
b) 42
c) 15
d) 33
e) 62
f) 43
g) 02
h) 301
10
11
Bonus: If 2 = 1,024, what is 2 ?
Answers: a) 16; b) 16; c) 1; d) 27; e) 36; f) 64; g) 0; h) 30; Bonus: 2,048
H-30
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Order matters in powers. ASK: Which two answers in the exercises above are the same?
(parts a) and b)) SAY: I know that order doesn’t matter in multiplication. ASK: Do you think order
matters in powers? Have students signal thumbs up for yes and thumbs down for no. Write on
the board:
23 =
32 =
Have students evaluate both powers, then ask volunteers to say the answer. (8, 9) SAY: These
answers are different so the power doesn’t always stay the same when the base and exponent
trade places.
Exercises: Find another two numbers that show that order matters in powers.
Bonus: Is an even number to an odd exponent even or odd? Is an odd number to an even
exponent even or odd?
Sample answers: 25 and 52; Bonus: an even number to any exponent is always even, an odd
number to any exponent is always odd.
Patterns in the powers of 10. Write on the board:
102 = 10 × 10 = _____
103 = 10 × 10 × 10 = _________
104 = 10 × 10 × 10 × 10 = ___________
105 = 10 × 10 × 10 × 10 × 10 = _______________
(MP.8) ASK: What is 10 × 10? (100) Write “100” in the first blank. Cover the last 10 in 103 and
SAY: 10 × 10 is 100, so multiplying one more 10 is 1,000 because 100 × 10 = 1,000. Write
“1,000” in the next blank. Have volunteers fill in the next two blanks. (10,000 and 100,000)
ASK: How can you get the answer from the exponent? (write 1, then write the number of 0s that
is equal to the exponent)
Exercises: Evaluate the powers of 10.
a) 108
b) 109
c) 1015
Bonus: Which is larger, 108 × 109 or 1015?
Answers: a) 100,000,000; b) 1,000,000,000; c) 1,000,000,000,000,000; Bonus: 108 × 109
Powers with exponent 1. Write on the board:
34 = 3 × 3 × 3 × 3
31 = 3
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-31
Remind students that 34 means multiply four 3s. Tell students that they can think of 31 as just
one 3, so there is no multiplication to do and the answer is just 3. SAY: This is similar to the way
we think of multiplication as repeated addition. Write on the board:
4×3=3+3+3+3
1×3=3
SAY: When there is only one 3 to add there is no addition to do, so the answer is just 3.
Exercises: Evaluate the power.
a) 11
b) 51
c) 91
1
1
d) 12
e) 20
f) 501
Bonus: 123,456,7891
Answers: a) 1; b) 5; c) 9; d) 12; e) 20; f) 50; Bonus: 123,456,789
Powers with exponent 2.
Exercises: What is the area of a square with the given side length? Write your answer as a
power.
a) 3 cm
b) 4 cm
c) 7 cm
d) 8 cm
2
2
2
2
2
2
Answers: a) 3 cm , b) 4 cm , c) 7 cm , d) 82 cm2
Tell students that 3 to the exponent 2 is often referred to as “3 squared” because it is the area of
a square with side length 3.
Powers with base 1. Write on the board:
11 = 1
12 = 1 × 1 = ____
13 = 1 × 1 × 1 = ______
14 = 1 × 1 × 1 × 1 = _____
Have volunteers tell you the answers. (1 every time) SAY: No matter how many 1s you multiply,
you always get 1.
Exercises: Evaluate the power.
b) 15
c) 19
a) 11
d) 112
e) 120
f) 150
123,456,789
Bonus: 1
Answers: a) 1, b) 1, c) 1, d) 1, e) 1, f) 1, Bonus: 1
H-32
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Powers with base −1. Write on the board:
(−1)1 = _____
(−1)2 = _____
(−1)3 = _____
(−1)4 = _____
(−1)5 = _____
(−1)6 = _____
(−1)7 = _____
(−1)8 = _____
Remind students that to find each successive power of −1, they can just multiply the last one
by −1. ASK: How do you multiply a number by −1? (take the opposite) ASK: What is −1 to the
exponent 1? (−1) Write “−1” in the first blank. ASK: What is the opposite of −1? (+1 or 1)
SAY: So (−1) × (−1) is +1. Write “+1” in the next blank. Have volunteers continue to fill in the
blanks. (−1, +1, −1, +1, −1, +1)
(MP.8) ASK: When is the answer +1? (when the exponent is even) PROMPT: Look at the
exponents. ASK: When is the answer −1? (when the exponent is odd) Explain that multiplying
an even number of negative numbers together gets a positive answer and multiplying an odd
number of negative numbers together gets a negative answer.
Exercises: Evaluate the power.
a) (−1)11
b) (−1)16
c) (−1)20
d) (−1)21
e) (−1)25
f) (−1)36
Answers: a) −1, b) +1, c) +1, d) −1, e) −1, f) +1
Powers with any negative base. Write on the board:
31 = ______
(−3)1 = _____
32 = ______
(−3)2 = _____
33 = ______
(−3)3 = _____
34 = ______
(−3)4 = _____
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-33
Remind students that to find each successive power of 3, they can just multiply the last one by 3.
Have volunteers fill in the first column. (3, 9, 27, 81) Then have students dictate the first two
answers in the second column. (−3, +9). Remind students that to find each successive power
of −3, they can just multiply the last one by −3. ASK: How do you multiply a number by −3?
(multiply by 3 and take the opposite) Have volunteers complete the second column (−27, +81)
Tell students to look at the exponents. ASK: When is a power of −3 negative? (when the
exponent is odd) Write on the board:
217 = 131,072
ASK: What is (−2)17? SAY: I know the number part is 131,072 but is the answer positive or
negative? (negative) How do you know? (because 17 is odd)
Exercises: Evaluate the exponents.
b) If 233 = 12,167, what is (−23)3?
a) If 312 = 531,441, what is (−3)12?
c) If 119 = 2,357,947,691, what is (−11)9?
d) If 144 = 38,416, what is (−14)4?
Answers: a) 531,441; b) −12,167; c) −2,357,947,691; d) 38,416
Ordering powers. Remind students that they can compare two negative numbers by
comparing their positive opposites first. Write on the board:
2 < 5, so −2 > −5
ASK: How do you compare a positive number and a negative number? PROMPT: Which
number will always be greater? (the positive number)
Exercises:
a) Order the numbers from least to greatest. Use the answers calculated before or calculate
them again.
(−3)1, (−3)2, (−3)3, (−3)4.
b) Without calculating (−3)5 and (−3)6, predict where they go in the order from part a).
Answers: a) (−3)3 < (−3)1 < (−3)2 < (−3)4, b) (−3)5 < (−3)3 < (−3)1 < (−3)2 < (−3)4 < (−3)6
Extensions
1. Write on the board:
6+3
63
6−3
exponent
addends
minuend subtrahend
base
Point out that when order matters, the two words are different. When order doesn’t matter the
two words are the same. Ask if anyone knows what the words are for multiplication where order
doesn’t matter (factor × factor) and for division where order does matter (dividend ÷ divisor).
H-34
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
2. a) What is the greatest common factor of 56 and 57?
b) What is the lowest common multiple of 56 and 57?
Answers: a) 56, b) 57
(MP.8) 3. Continue the patterns in the exponents and in the values of the powers.
b) 23 = 8
c) 33 = 27
a) 103 = 1,000
2
2
10 = 100
2 =4
32 = 9
101 = 10
21 = 2
31 = 3
?
?
2 = ??
3? = ??
10 = ??
Answers: a) 100 = 1, b) 20 = 1, c) 30 = 1
NOTE: Students will learn in Grade 8 that the any non-zero base to the exponent zero is equal to 1.
4. a) Computer codes are written as sequences of 0s and 1s. There are two possible sequences
of length 1 and four possible sequences of length 2.
0
1
00
10
01
11
Have students write all the 0-1 sequences of length 3 to determine how many there are. (there
are eight such sequences: 000, 100, 010, 110, 001, 101, 011, 111) Have students predict how
many sequences of length 4 there are. (16, because the number of sequences of a given length
is always multiplied by 2 to get the number of sequences of the next length) Show students how
to quickly change the sequences of length 2 to get all the sequences of length 3, and how doing
so demonstrates that the number of sequences of length 3 is double the number of length 2:
Add a 0 to the end of all the sequences of length 2, then add a 1 to the end instead.
Have students continue the pattern to determine how many 0-1 sequences there are of lengths
5, 6, 7, 8, 9, and 10. (32, 64, 128, 256, 512, 1,024) Guide students to think where they have
seen some of these numbers in relation to computers. (example: 256 megabytes)
b) Which power of 2 is closest to 1,000? When students finish, tell them that a kilobyte is really
1,024 bytes, not 1,000 bytes.
c) A megabyte is about 1,000,000 bytes. Exactly how many bytes are in a megabyte?
Hint: 1,000,000 = 1,000 × 1,000.
Answers: b) 1,024; c) 1,024 × 1,024 = 1,048,576
5. Patterns in ones digits of powers.
a) Complete the chart.
2n
21
22
23
24
25
26
27
28
29
2
4
8
16
=
2
4
8
6
Ones Digit
10
b) Predict the ones digits of 2 , 211, and 212, and explain the basis for your predictions.
c) Predict the ones digit for 222. Explain how you made your prediction.
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-35
(MP.1) d) What is the ones digit of 27,051? Hint: Add another row to the table for the remainder of
the exponent when divided by 4. For example, 1 ÷ 4 = 0 R 1, 5 ÷ 4 = 1 R 1, and 9 ÷ 4 = 2 R 1.
2n
21
22
23
24
25
26
27
28
29
2
4
8
16
=
2
4
8
6
Ones Digit
Remainder of Exponent When
1
1
1
Divided by 4
Use that pattern to predict where 7,051 will fit in the ones digit category.
NOTE: In part c), students might work backward to find where the exponent 22 is in the onesdigit category. (the same as 18, 14, 10, 6, and 2, so 222 will have ones digit 4, just like 22)
However in part d), they will not be able to use this method so they will need to look for a better
strategy. The hint provides that strategy.
When students finish, point out that it is quite surprising that we can find the ones digit of 27,051
when we know so little about the number itself. We don’t know any of its other digits or even
how many digits there are.
Answers:
a)
2n
21
22
23
24
25
26
27
28
29
2
4
8
16 32 64 128 256 512
=
2
4
8
6
Ones Digit
2
4
8
6
2
b) 4, 8, 6
c) 4
d) 8
Remainder of Exponent When
1
2
3
0
1
2
3
0
1
Divided by 4
(MP.3) 6. a) Mathematicians have proven that if a and b have GCF = 1 and a is a prime
number, then a is a factor of ba − 1 − 1. This is called Fermat’s Little Theorem.
Check this for:
a = 2 and b = 3
a = 2 and b = 5
a = 3 and b = 2
a = 3 and b = 4
a = 3 and b = 5
a = 3 and b = 10
a = 5 and b = 2
a = 5 and b = 3
a = 5 and b = 4
a = 7 and b = 10
your own example: a = ____ and b = ____
(make sure a is a prime number and the GCF of a and b is 1)
H-36
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
b) For a = 4 and b = 7, check whether a is a factor of ba − 1 − 1. Is this a counterexample to the
statement in part a)? Explain.
c) For a = 5 and b = 10, check whether a is a factor of ba − 1 − 1. Is this a counterexample to the
statement in part a)? Explain.
Answers: b) 73 − 1 = 342, and 4 is not a factor of 342. This is not a counterexample though,
because 4 is not prime; c) 104 − 1 = 9,999, and 5 is not a factor of 9,999. This is not a
counterexample though, because 5 and 10 do not have GCF = 1.
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-37
NS7-31
The Order of Operations with Powers and
Negative Numbers
Pages 189–191
Standards: 7.NS.A.2a, 7.NS.A.2c, 7.NS.A.3
Goals:
Students will use the correct order of operations when evaluating expressions that involve
powers, all four operations, and brackets.
Students will substitute negative numbers into simple equivalent expressions to verify their
equivalence.
Prior Knowledge Required:
Can multiply two positive and/or negative numbers when at least one of the numbers is an
integer
Can add and subtract negative numbers
Knows the order of operations with brackets and all four operations
Can expand expressions
Can add expressions
Can substitute positive numbers into expressions
Vocabulary: base, equivalent expression, exponent, integer, negative, order of operations,
positive, power, sign
Materials:
BLM Magic Cards (pp. H-46–47, see Extension 1)
grid paper or BLM 1 cm Grid Paper (p. J-1, see Extension 3)
(MP.3) Extending the order of operations to negative numbers. Tell students that the order
of operations applies to negative numbers the same way it does to positive numbers. Write on
the board:
1. Do operations in brackets first.
2. Then do multiplication and division from left to right.
3. Then do addition and subtraction from left to right.
Exercises: Use the correct order of operations to calculate.
a) 5 × (−4) × 3
b) 6 ÷ (−2) × 3
c) 5 × (−2 + 8)
d) 2 × 4 − (−6) ÷ 2
e) 7 − (4 − 8)
f) (5 − 15) ÷ (−5)
g) −5 × (−3) − (6 + 7)
h) (24 + 2 × (−6)) ÷ 3
Selected solution: h) (24 + (−12))  3 = 12  3 = 4
Answers: a) −60, b) −9, c) 30, d) 11, e) 11, f) 2, g) 2
H-38
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Bonus: Add brackets to make the equation true.
a) 5 − 7 × (−2) + 4 = −9
b) 5 − 7 × (−2) + 4 = 8
c) 5 − 7 × (−2) + 4 = −4
Answers: a) 5 − 7 × ((−2) + 4) = −9, b) (5 − 7) × (−2) + 4 = 8, c) (5 − 7) × ((−2) + 4) = −4
The order of operations with powers. Tell students that powers are just numbers so you can
add, subtract, multiply, and divide them. Write on the board:
32 + 42 = _____ + _____
ASK: What is 32? (9) What is 42? (16) Fill in the blanks on the board, then write the total. The
final equation should look like this:
32 + 42 = 9 + 16 = 25
SAY: You have to start by evaluating the powers, then you can add, subtract, multiply, or divide
them.
Exercises: Evaluate the powers first. Then multiply, divide, add, or subtract.
a) 3 × 52
b) 1000 ÷ 53
c) 32 − (−2)3
d) −43 ÷ (−2)2
Bonus:
f) 22 × 34
g) 103 ÷ 23
Selected solution: d) −64 ÷ 4 = −16
Answers: a) 75, b) 8, c) 17, e) −1, Bonus: f) 324, g) 125
e) 23 − 32
The order of operations with powers, but not brackets. Remind students that there is an
agreed-upon order of doing operations. ASK: Which operation do you perform first, addition or
multiplication? (multiplication) SAY: You also evaluate powers before doing multiplication. One
way to remember this is to think of multiplication as repeated addition and powers as repeated
multiplication. Just like repeated addition is done before addition, repeated multiplication is done
before multiplication. Write the new order of operations on the board and keep it on the board:
1. Evaluate powers.
2. Do all multiplication and division in order from left to right.
3. Do all addition and subtraction in order from left to right.
Exercises:
1. Which operation is done first? Do the operation, then rewrite the expression.
b) 5 + 32
c) 55 ÷ 51
a) 5 × 32
2
2
e) 125 − 5
f) 15 + 3 × 5
d) 5 − 2
Answers: a) 5 × 9, b) 5 + 9, c) 55 ÷ 5, d) 5 − 4, e) 125 − 25, f) 15 + 15
2. Do the operations one at a time, in the correct order.
a) 27 ÷ 32
b) 25 × 10 ÷ 2
c) 3 × 32 − 9
d) 3 + 42 ÷ (−2)
e) 1,000 ÷ (−2)3
f) 7 − 32 × (−2)
Answers: a) 27 ÷ 9 = 3; b) 250 ÷ 2 = 125; c) 3 × 9 − 9 = 27 − 9 = 18;
d) 3 + 16 ÷ (−2) = 3 + (−8) = −5; e) 1,000 ÷ (−8) = −125; f) 7 − 9 × (−2) = 7 − (−18) = 25
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-39
Evaluate expressions within brackets before evaluating powers. SAY: When there are
brackets, always evaluate the expression in brackets first. Write on the board:
(1 − 3)2 = ______
1 − 32 = ______
Refer students to the first expression on the board. SAY: 1 minus 3 is in brackets so do that
first. ASK: What is 1 − 3? (−2) Continue writing on the board:
(1 − 3)2 = (−2)2 = ______
Have a volunteer dictate the answer. (+4 or 4) Repeat for the second expression on the board.
SAY: This time there are no brackets so evaluate the power first, then subtract. Have volunteers
dictate the answers to each stage. (1 − 9 = −8)
Exercises: Evaluate the expression.
a) (7 − 5)3
b) 73 − 53
c) (5 − 8)2
d) 52 − 82
f) 13 − 33
g) (2 × 3)2
h) 22 × 32
e) (1 − 3)3
Answers: a) 8, b) 218, c) 9, d) −39, e) −8, f) −26, g) 36, h) 36
The order of operations with powers and brackets. Summarize the correct order of
operations on the board:
1. Do operations in brackets.
2. Evaluate powers.
3. Do multiplication and division, from left to right.
4. Do addition and subtraction, from left to right.
For the Bonus exercises below, SAY: Sometimes an exponent has an expression that needs to
be evaluated first.
Exercises: Evaluate the expression.
b) (100 ÷ 5)2 × 2
c) 100 ÷ (52 × 2)
d) 5 + 34 − 2 × 5
a) 100 ÷ 52 × 2
3
3
2
e) 64 ÷ (−2) × 4
f) 64 ÷ ((−2) × 4)
g) (−8 ÷ (−2)) × 3
h) −8 ÷ (−2)2 × 3
Bonus:
i) 38 − 6
j) (−2)6 ÷ 2
k) 58 − 3 × 2
l) (7 − 10)3 − 2 + 1
m) (−2)(10 − 2) ÷ 2
n) (−2)4 + 2 ÷ 2
o) (−8)2 ÷ (−2)4 × (3 + 2)20 ÷ 5
Answers: a) 8, b) 800, c) 2, d) 76, e) −32, f) −2, g) 48, h) −6, Bonus: i) 9, j) −8, k) 25, l) 9,
m) 16, n) −32, o) 2,500
Adding brackets to make an expression true. Write on the board:
4 − (2 × 3)2 =
(4 − 2) × 32 =
4 − 2 × 32 =
(4 − 2 × 3)2 =
H-40
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Have volunteers dictate each step, using the correct order of operations. (see answers below)
4 − (2 × 3)2 = 4 − 62 = 4 − 36 = −32
(4 − 2) × 32 = 2 × 32 = 2 × 9 = 18
4 − 2 × 32 = 4 − 2 × 9 = 4 − 18 = −14
(4 − 2 × 3)2 = (4 − 6)2 = (−2)2 = 4
Exercises: Add brackets where necessary to make the equation true.
a) 6 − 5 × 3 = 3
b) 6 − 5 × 3 = −9
c) 22 ÷ 11 − 2 = 0
e) 4 − 12 + 9 − 6 = −12
f) 4 − 12 + 9 − 6 = 0
d) 4 − 12 + 9 − 6 = 12
Answers: a) (6 − 5) × 3 = 3, b) 6 − (5 × 3) = −9, c) (22 ÷ 11) − 2 = 0, d) (4 − 1)2 + 9 − 6 = 12,
e) 4 − (12 + 9) − 6 = −12, f) 4 − (12 + 9 − 6) = 0
Bonus: Evaluate the expression: 25 − 32 + 2 × 22. Then find as many ways as you can to add
brackets to change the value of the expression.
Sample answers: 25 − 32 + 2 × 22 = 24, 25 − 32 + (2 × 2)2 = 32, (25 − 3)2 + 2 × 22 = 492,
(25 − 3)2 + (2 × 2)2 = 500, 25 − (32 + 2 × 22) = 8, 25 − (32 + (2 × 2)2) = 0,
25 − (32 + 2) × 22 = −19, 25 − (32 + 2 × 2)2 = −144
(MP.3) Write on the board:
25 − 3(2 + 2) × 22
ASK: Why doesn’t this make sense? (the exponent doesn’t mean anything without the base)
Point out, however, that a closing bracket can be put between a base and its exponent, as in
many cases above, because the expression in brackets becomes the base for the exponent.
Review the pattern for multiplying many integers. Write on the board:
(+) × (+) = (+)
(+) × (−) = (−)
(−) × (+) = (−)
(−) × (−) = (+)
SAY: The sign doesn’t change when you multiply by a positive number, but the sign does
change when you multiply by a negative integer. Write on the board:
(−1) × (−2) × (−3) × (−1) × (−5)
SAY: As I go along multiplying, I want to keep track of just the sign of the answer, not the
values. The first one is negative so I’ll write “−” in the first box. Next I multiply by a negative
number, so the sign will change to be positive. Write “+” in the second box and have volunteers
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-41
continue writing the signs in the next three boxes. (−, +, −) Repeat with the following product:
(−1) × (+2) × (+3) × (−3) × (−4) × (+3) × (−3) × (−4)
(−, −, −, +, −, −, +, −) SAY: Sometimes we had to change the sign and sometimes we didn’t.
Exercises: Write the sequence of signs for these products.
a) (−2) × (−3) × 4
b) (−3) × 5 × (−2) × (−4)
c) (−10) × (−2) × (−3) × (−5) × (−1) × (−7)
d) (−3) × 2 × (−4) × (−5) × 6 × (−2) × (−3) × 3 × (−8)
Answers: a) −, +, +; b) −, −, +, −; c) −, +, −, +, −, + ; d) −, −, + −, −, +, −, −, +
SAY: For each negative number you had to change the sign. When there are an odd number of
negative signs, the answer will be negative. When there are an even number of negative signs,
the answer will be positive.
Exercises: Determine if the value is positive or negative.
a) −2 × (−3) × (−4)
b) 2 × (−1) × 3 × (−4)
c) −8 × (−5) × (−6) × (−2) × (−1) × (−9) × (−7)
d) 5 × 2 × (−4) × (−5) × (−6) × (−2) × 3 × 4 × 4
Answers: a) −, b) +, c) −, d) +
Equivalent algebraic expressions. Remind students that 2(x + 3) and 2x + 6 are equivalent
expressions. Write on the board:
weighs x kg
x+3
weighs 1 kg
2(x + 3) = 2x + 6
SAY: A triangle and 3 circles weigh x + 3 kg, so 2 triangles and 6 circles weigh twice as much.
So the expressions 2(x + 3) and 2x + 6 are equivalent. In one expression (point to 2(x + 3)) you
are adding first then multiplying. In the other expression (point to 2x + 6) you are multiplying first
then adding. If these expressions really are equivalent, they should be equal for any value of x,
even negative numbers.
Exercises: Check that the expressions 2(x + 3) and 2x + 6 are equal for each value.
a) 0
b) −1
c) −5
d) −3
Answers: a) 2 × 3 = 6 and 0 + 6 = 6, b) 2 × 2 = 4 and −2 + 6 = 4,
c) 2 × (−2) = −4 and −10 + 6 = −4, d) 2 × 0 = 0 and −6 + 6 = 0
H-42
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Review multiplying a fraction by a whole number. Write on the board:
2×3
2
6
´3 =
5
5
Exercises: Check that the expressions 2(x + 3) and 2x + 6 are equal for each value.
1
1
5
3
c) d)
b) a)
2
2
2
4
Answers: a) 2 × 7/2= 7 and 1 + 6 = 7, b) 2 × (5/2) = 5 and −1 + 6 = 5, c) 2 × 1/2 = 1
and −5 + 6 = 1, d) 2 × 15/4 = 15/2 and 3/2 + 6 = 15/2
Review multiplying a decimal by a whole number. Write on the board:
1.7 × 2 = 3.4
SAY: To multiply 1.7 × 2, first multiply 17 × 2 = 34, then put the decimal point so that there is
one digit after the decimal point because that is what there is in the question.
Exercises: Check that the expressions 2(x + 3) and 2x + 6 are equal for each value.
a) 0.3
b) −0.3
c) 1.5
d) −1.5
Answers: a) 2 × 3.3 = 6.6 and 0.6 + 6 = 6.6, b) 2 × 2.7 = 5.4 and −0.6 + 6 = 5.4,
c) 2 × 4.5 = 9 and 3 + 6 = 9, d) 2 × 1.5 = 3 and −3 + 6 = 3
Extensions
1. Have students complete BLM Magic Cards. Tell them that they can learn a magic trick by
following the instructions on the BLM. The cards on the BLM are an application of the fact that
every number is a sum of powers of 2 or one more than a sum of powers of 2.
To create their own magic cards for the Bonus question, students write 1, 2, 4, 8, and so on
(1, then the powers of 2) in sequence in the top left of each card. They then extend the table in
Question 4 and use the table to fill in the rest of the numbers on the cards. To extend the table
in Question 4, students will need to write the numbers from 32 to 63 as sums of powers of 2
(or one more than a sum of powers of 2).
For example: 41 = 32 + 8 + 1, so students will know to put 41 on cards 1 (for the 1 in the sum),
4 (for the 8 in the sum), and 6 (for the 32 in the sum).
Card 1
Card 2
Card 3
Card 4
Card 5
Card 6
1
2
4
8
16
32
NOTE: In Grade 8, students will learn that 1 is in fact 20. In Grade 7 you do not need to teach
that 1 is a power of 2.
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-43
2. a) How many 8s must I add together to get a sum equal to 83?
b) How many 8 by 8 squares do I need to have a total area of 83?
Answers: a) 64, because 83 = 82 × 8 = 64 × 8; b) 8, because 83 = 8 × 82
3. Connecting algebra to geometry. Have students copy the following picture of two squares
onto grid paper or BLM 1 cm Grid Paper:
a) Have students find the area of the shaded part in two ways, keeping the numbers 12 and 17
in the expressions.
i) Subtract the area of the small square from the area of the big square.
ii) Cut out the shaded region, and cut and rearrange it into a rectangle. ASK: What is the
length and the width of the rectangle in terms of 12 and 17?
b) Have students write an equation based on the two ways of finding the area of the shaded
region.
c) Have students make their own picture to prove that 192 − 162 = (19 + 16)(19 − 16).
d) Tell students that the dimensions of the squares can be 12 and 17, or 16 and 19, or any two
numbers. Because the numbers can be any numbers, we can use variables to represent them.
Show students how to change the equation in c) to an equation with variables:
a2 − b2 = (a + b)(a − b). Challenge students to substitute various numbers into this equation to
see if it is true and to fill in the table below (students should make up their own examples for the
last two rows):
H-44
a
b
a+b
a−b
(a + b)(a − b)
a2
b2
a 2 − b2
5
3
8
2
16
25
9
16
7
4
9
2
10
9
8
3
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
Selected answers: a) i) area = 172 − 122, ii) length = 17 + 12, width = 17 − 12,
area = (17 + 12)(17 − 12); b) 172 − 122 = (17 + 12)(17 − 12)
Some students might notice that even when a − b is negative, the equation still holds. This is
interesting because the picture of the two squares doesn’t apply to that case, but it was the
picture that helped us guess the general formula in the first place.
4. A rule for the sum of powers of 2. Find a rule for the sum of the first n powers of 2 by
completing the following chart.
Sum of First n
n
2n
2n+1
Powers of 2
1
2
2
4
2
4
2+4=6
8
3
8
2 + 4 + 8 = 14
4
16
5
6
Write a rule for the sum of the first n powers of 2. Predict the sum of the first 10 powers of 2.
Answer: 2n+1 − 2, so the sum of the first 10 powers of 2 is 211 − 2 = 2,048 − 2 = 2,046.
5. Have students complete the chart by filling in the answers and continuing the patterns in the
last two rows.
1=1
1=1
1+2=3
13 + 23 = _____
1+2+3=6
13 + 23 + 33 = _____
1 + 2 + 3 + 4 = ____
13 + 23 + 33 + 43 = ____
1 + 2 + 3 + 4 + 5 = ____
13 + 23 + 33 + 43 + 53 = ____
1 + 2 + 3 + 4 + 5 + 6 = ____
13 + 23 + 33 + 43 + 53 + 63 = ____
Have students think about and describe how you can get the numbers in the second column
from the numbers in the first column.
Answer: The numbers in the second column are the square of the sums in the first column. For
example, 13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2.
Teacher’s Guide for AP Book 7.1 — Unit 7 The Number System
H-45