Download Unit 5 Number Sense

Unit 5 Number Sense
In this unit, students will study exponents, including powers of negative
integers, and the order of operations with exponents and integers. Students
will also study negative fractions and decimals.
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Meeting Your Curriculum
Ontario students should cover all topics in this unit. For WNCP students,
this unit is optional.
Teacher’s Guide for Workbook 8.2
Q-1
NS8-104 Introduction to Powers
Pages 137–138
Ontario: 8m1, 8m2, 8m4,
8m7, 8m11
WNCP: optional, [R, T, C];
9N1, 9N2
Vocabulary
power
exponent
base
first power, second power,
and so on
Process Expectation
Revisiting conjectures that
were true in one context.
Process assessment
8m1, 8m4, [R, T]
Workbook p 138 Question 11
Q-2
Goals
Students will evaluate powers by using repeated multiplication and
investigate properties of powers.
PRIOR KNOWLEDGE REQUIRED
Can multiply a sequence of more than two whole numbers
(e.g., 2 × 3 × 5)
Can use a calculator to multiply numbers
Can add, subtract, multiply, and divide two whole numbers
Knows that multiplication commutes (e.g., 2 × 4 = 4 × 2)
Introduce powers as repeated multiplication. First, remind students that
multiplication is a short form for repeated addition. Then point out that
just as we can add the same number repeatedly, we can multiply the same
number repeatedly. Introduce the concepts power, base, and exponent
as on Workbook page 137. Emphasize that in math, just as in English, a
base is the bottom part of something. Assign questions according to the
progression on Workbook page 137 Questions 1–5.
Investigate the commutativity of powers. Remind students that
multiplication is commutative—the order we write the numbers in doesn’t
matter. For example, 2 × 5 = 5 × 2. ASK: Do you think the same is true
for powers—will it matter which number is written as the base and which
number is written as the exponent? Have students explain the basis for their
prediction. (Sample answers: I think it won’t matter because of the example
I found in Question 5; I think it won’t matter because it doesn’t matter
for both addition or multiplication and just as multiplication is repeated
addition, powers are repeated multiplication; I can see in a simple example
like 23 and 32 that order does matter—23 is even and 32 is odd, or 23 = 8
and 32 = 9; I think it will matter because an odd base will give an odd result
and an even base will give an even result, so interchanging an even number
and an odd number will make a difference.) Then have students do the
Investigation on Workbook page 137 and discuss the results. Emphasize
that sometimes predictions turn out not to be true even when we have good
reasons for making the predictions.
Introduce the terms first power, second power, and so on. See
Workbook page 138 Questions 7 and 8.
Add, subtract, multiply, and divide powers. Remind students of the
order of operations. ASK: Which operation do you perform first, addition or
multiplication? (multiplication) Why? (because multiplication is a short form
for repeated addition and 3 + 4 × 5 actually means 3 + 5 + 5 + 5 + 5, so
you have to add the 5 four times—that is, multiply 4 × 5—and then add the
result to the 3) Emphasize that powers are repeated multiplication. When
you see a power such as 23, you are really multiplying 2 three times, so
Teacher’s Guide for Workbook 8.2
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Curriculum
Expectations
5 × 23 really means 5 × 2 × 2 × 2 = 5 × 8 = 40. This means that you have
to evaluate the powers before multiplying. Tell students to evaluate powers
before doing multiplication or division, just as they do multiplication or
division before addition or subtraction. So the new order of operations is:
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.
(NOTE: Do not introduce brackets in the same expression as powers at this
point.) Have students do Workbook page 138 Question 12.
ACTIVITY
The cards on BLM Magic Cards (pp Q-24–Q-25) 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” (BLM Question
10), students write 1, 2, 4, 8, and so on (1 and then the powers of 2)
in sequence in the top left of each card, extend the table in BLM
Question 4, then use the table to fill in the rest of the numbers on the
cards. To extend the table in BLM 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
1
Card #2
2
Card #3
4
Card #4
8
Card #5
16
Card #6
32
Extensions
1. The greatest common factor of 56 and 57 is ______. (56)
2. The lowest common multiple of 56 and 57 is ______. (57)
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Process Expectation
Modelling, Visualizing
3. Connecting algebra to geometry. This extension is good preparation
for NS8-105–106 Extension 3. Have students copy the following picture
of two squares onto grid paper (not in their notebooks, as students will
need to cut the squares out).
12
17
Number Sense 8-104
Q-3
Have students find the area of the shaded part in two ways:
a) Subtract the area of the small square from the area of the big
square. (172 − 122)
b) Cut out the shaded region, and cut and rearrange it into a
rectangle. What is the length of the rectangle in terms of 12 and 17?
(17 + 12) What is the width of the rectangle in terms of 12 and 17?
(17 − 12)
c) Have students write an equation using only the numbers 12 and 17
and the exponent 2, based on the two ways of finding the area of
the shaded region. ANSWER: 172 − 122 = (17 + 12)(17 − 12).
d) Have students make their own picture to prove that 192 − 162 =
(19 + 16)(19 − 16).
e) 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):
Algebra
a
b
5
3
7
4
9
2
10
9
8
3
a + b a − b (a + b)(a − b)
8
2
16
a2
b2
a2 − b2
25
9
16
Some students might notice that even when a − b is negative, the
equation still holds. This is interesting, because the picture doesn’t
apply to that case, but it was the picture that helped us guess the
general formula in the first place.
4. Recall that taking a number to the exponent 2 is called squaring the
number. Explain that taking a number to the exponent 3 is called cubing
the number; and just as 72 can be read “seven squared,” 73 can be read
“seven cubed.” Why is that? (because the volume of a cube of side
length 7 is 7 × 7 × 7 = 73)
5. Comparing powers without computing them directly.
a) Write the smallest power of 2 that will make the statement true.
Q-4
5 = 51 < 2
52 < 2
53 < 2
Teacher’s Guide for Workbook 8.2
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connection
ANSWERS: 3, 5, and 7.
b) Continue the pattern above. Which power is bigger now?
54 ____ 2
ANSWER: 54 > 29
Challenge students to explain why the pattern doesn’t continue
to hold. (Explanation: You are continually multiplying the powers
of 2 by 22 = 4 and the powers of 5 by 5, so eventually the power
of 5 will become larger.)
c) Determine how many digits the number 212 has without calculating
it, as follows.
i) Use 24 > 10 to explain why 212 > 103.
ii) Use 23 < 10 to explain why 212 < 104.
iii) Use your answers to i) and ii) to determine how many digits
212 has.
ANSWERS:
i) 212 = 24 × 24 × 24 > 10 × 10 × 10 = 103
ii) 212 = 23 × 23 × 23 × 23 < 10 × 10 × 10 × 10 = 104
iii) Any number between 103 = 1000 and 104 = 10 000 has 4 digits,
so 212 has 4 digits.
d) i) Calculate the powers to determine the correct sign (> or <).
53 _____ 102
54 _____ 103
ii) Without calculating it, determine how many digits the number
512 has.
ANSWERS:
i) 53 = 125 > 100 = 102 and 54 = 625 < 1000 = 103
ii) 512 = 53 × 53 × 53 × 53 > 102 × 102 × 102 × 102 = 108 and
512 = 54 × 54 × 54 < 103 × 103 × 103 = 109.
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So 512 is between 108 and 109. This means that it has 9 digits.
Encourage students who don’t see this immediately to use a T-table
to see that if a number is between 108 and 109 (as 512 is), then it
has 9 digits (some students might think it has 8 digits). The T-table
could look like this:
number is between…
number has ___ digits
10 and 102
2
10 and 10
3
3
103 and 104
4
2
Number Sense 8-104
Q-5
rom the same pattern, a number between 108 and 109 will have 9 digits,
F
since the number of digits is equal to the higher power of 10.
Process assessment
7m1, [R]
6. a)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:
n
2n
sum of n powers of 2
2n + 1
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 = 2048 − 2 = 2046.
b) How big are powers of 2? You have a choice between 2 prizes:
Prize A: monthly payments of $1000
Prize B: monthly payments of powers of $2 ($2, $4, $8, $16,
and so on)
If you won the prize for 10 months, which prize would you prefer?
(Prize A: $1000 × 10 = $10 000 but 211 − 2 = $2046) Which would
you prefer if you won for 20 months? (Prize B: $1000 × 20 =
$20 000 but 221 − 2 = $2 097 150) If you start receiving these prizes
in January, what would be the first month where the total payout
from the Prize B would be more than the total payout from Prize A?
(the 13th month, which is January of the 2nd year)
7. a) Computer codes are written as sequences of 0s and 1s. SAY:
There are two possible sequences of length 1 and four possible
sequences of length 2. Show them on the board:
0
1
00
10
01
11
Have students write all the 0-1 sequences of length 3. ASK: How
many are there? (8) Write them on the board to summarize. ASK:
How many sequences of length 4 do you think there will be? Show
students how you can quickly change the sequences of length 3 to
get all the sequences of length 4, and how doing so demonstrates
that the number of sequences of length 4 is double the number of
length 3: add a 0 to the end of all the sequences of length 3 and
then add a 1 to the end instead.
Q-6
Teacher’s Guide for Workbook 8.2
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Write on the board:
2 sequences of length 1
4 sequences of length 2
8 sequences of length 3
16 sequences of length 4
Have students continue the pattern to determine how many 0-1
sequences there are of lengths 5, 6, 7, 8, 9, and 10. ASK: Do these
numbers remind you of anything? (they are powers of 2) Where
have you seen any of these numbers in relation to computers?
(EXAMPLE: 256 Megabytes)
Process assessment
8m5, [CN]
b) ASK: Which power of 2 is closest to 1000? (the 10th) Tell students
that a kilobyte is really 1024 bytes, not actually 1000 bytes.
c) A megabyte is about 1 000 000 bytes. Exactly how many bytes
are in a megabyte? HINT: 1 000 000 = 1000 × 1000. Answer:
1024 × 1024 = 1 048 576.
8. Students will need to be very familiar with writing numbers as sums
of powers of 2, or one more than a sum of powers of 2. Doing BLM
Magic Cards will provide this familiarity. EXAMPLE:
217 = 128 + 64 + 16 + 8 +1 = 27 + 26 + 24 + 23 + 1.
The ancient Egyptians used this observation to multiply any number by
any other number! For example, to calculate 37 × 217, start with 37 and
double it continually:
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37 × 1 37 × 2 37 × 4 37 × 8 37 × 16 37 × 32 37 × 64 37 × 128 Since 217 = 128 + 64 + 16 + 8 +1,
then 37 × 217 = 37 × (128 + 64 + 16 + 8 +1)
= 37 × 128 + 37 × 64 + 37 × 16 + 37 × 8 + 37 × 1
= 4736 + 2368 + 592 + 296 + 37
= 8029
a) Have students check this answer using the standard algorithm
for multiplication.
b) Use the ancient method of multiplying to do the following questions
and check your answer by using the standard algorithm:
Number Sense 8-104
= 37
= 74
= 148
= 296
= 592
= 1184
= 2368
= 4736
15
× 19
13
× 27
37
× 29
Q-7
c) When you do these questions, you have a choice in which number
to expand. For example, to multiply 15 × 19, you could either write
15 = 1 + 2 + 4 + 8 and find 19 × 1 + 19 × 2 + 19 × 4 + 19 × 8
or you could write 19 = 1 + 2 + 16 and find 15 × 1 + 15 × 2 +
15 × 16. Do each of the above questions using the other way and
decide which way you like better for each question and why.
d) State one thing you like about the ancient Egyptian method of
multiplying. (Sample answer: As long as I can add, I only have to
multiply by 2 or double numbers; I don’t have to remember the
times tables.)
e) State one thing you don’t like about the ancient Egyptian method of
multiplying. (Sample answer: It’s a lot more work than using the
standard algorithm!)
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Q-8
Teacher’s Guide for Workbook 8.2
NS8-105 Investigating Powers
NS8-106 Powers and Expanded Form
Pages 139–141
Curriculum
Expectations
Ontario: 8m1, 8m2, 8m7,
8m11, 8m12
WNCP: optional, [R, C];
9N1, 9N2
Goals
Students will evaluate, compare, and order simple powers and
investigate properties of powers. Students will solve equations
involving powers. Students will write numbers in expanded form
using power notation.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
power
exponent
base
expanded form
Can evaluate a power by using repeated multiplication
Can write a number in expanded form
Review power, base, and exponent. Have students identify the base and
exponent in several powers. Students can signal their answers by holding
up the correct number of fingers.
Patterns in powers of 2. See Workbook page 139 Question 1. Point
out that if you know a power of 2, you can find the next power of 2 by
multiplying by 2, i.e., doubling the number.
Patterns in powers of 10. Have students make a chart like the one on
Workbook page 139, Question 2a), for powers of 10 instead of powers of 2.
Then have students do these questions:
a) 103 × 102 = ______ × ______
b) 105 × 104 = ______ × ______
= ______
= ______
= 10
= 10
Continue with these problems:
Process Expectation
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Looking for a pattern
c) 104 × 103 = 10
d) 105 × 101 = 10
e) 103 × 103 = 10
f) 106 × 102 = 10
Draw students’ attention to the exponents. ASK: How can you get the
exponent in the answer from the other two exponents? (add the two
exponents) PROMPT: What operation can you use? ASK: Do you think
this will be true for any base, or just base 10? Demonstrate why the same
property holds for any base by using the teaching box on Workbook
page 139. Then have students do Workbook page 139 Questions 2–3.
Bonus
Process Expectation
Looking for a pattern,
Using logical reasoning
Number Sense 8-105, 106
3
2 × 33 × 34 = 3 7 × 72 × 73 × 74 × 75 × 76 × 77 × 78 = 7
Relating powers of 9 to powers of 3. Have students do Workbook
page 140 Question 4. When students finish, challenge students to explain
why the pattern holds. Emphasize that because each 9 in the product is
being replaced by a product of two 3s, there are twice as many 3s in the
product as there were 9s.
Q-9
After students finish Workbook page 140 Question 4, provide the following
bonus problem.
Bonus
Write each product as a power of 3:
a) 34 × 93 = 34 × 36 = 310
b) 33 × 92
c) 92 × 27
d) 94 × 34
Comparing and ordering powers with the same base or exponent.
Have students recall how easy it was to compare and order perfect squares:
432 < 472 because 43 < 47; there is no need to compute the squares to
check which one is larger. Have students articulate why this was the case.
(Sample answer: multiplying two smaller numbers together will get a
smaller result than multiplying two larger numbers together.) Then write
on the board: 433 < 473. Ask students if they think this is true and why. To
help them, rewrite the powers as repeated multiplication: 43 × 43 × 43 <
47 × 47 × 47. Emphasize that multiplying three smaller numbers together
will get a smaller result than multiplying three larger numbers together.
ASK: Which is larger: 4319 or 4719? (4719) How do you know? (because 47
is larger than 43 and multiplying 19 larger numbers together will result in a
larger number than multiplying 19 smaller numbers together) Emphasize
that as long as the exponent is the same, the power with the larger base is
larger. Then repeat the line of questioning for powers with the same base,
but different exponents. Emphasize that, for example, multiplying more 5s
together will get a larger result than multiplying fewer 5s together, so when
the base is the same, the power with the greater exponent is greater.
After students finish Workbook page 140 Question 6, provide the following
bonus problem.
Bonus
85 = 32
(Answer: 3, because both sides of the
equation are equivalent to 215)
Solving equations with powers. Show students how to use an organized
list to solve for the variable. By trying each possibility for x—when it is
either the base or the exponent—and evaluating the power, students can
determine which value for x solves the equation.
8m1,8m2, [R]
Workbook p 140 Question 8j)
NOTE: Workbook page 140 Question 9 is preparation for finding the bases
in Question 10. After students finish Workbook Question 10, provide the
following bonus problem.
Bonus
Write each expression as a power of a prime number.
a) 25 × 82
e) 25 × 55 ANSWERS: a) 211
b) 343 × 72
f) 43 × 27
b) 75
c) 3125 × 53 g) 2432
c) 58
d) 29
e) 57
d) 43 × 8
f) 213
g) 310
Powers and expanded form. Review writing numbers in expanded form
and then have students write numbers in expanded form using powers
of 10; see Workbook page 141 Questions 1–4. Then have students write
numbers written in expanded form, using power notation, as whole
numbers; see Workbook page 141 Question 5. Finally, have students
compare numbers written in expanded form as in Workbook page 141
Q-10
Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
Process assessment
Question 6. Encourage students who struggle with Question 6 to circle
the part in the two expressions that is different. Alternatively, students could
write both expressions as whole numbers in standard form.
Extensions
1. How many 8s must I add together to get a sum equal to 83?
(64 because 83 = 82 × 8 = 64 × 8)
2. How many 8 × 8 squares do I need to have in order to have a total
area of 83? (8 because 83 = 8 × 82)
3. 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. (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.)
Process Expectation
Visualizing
Show students how to visualize the equations in the second column
for small numbers:
Since 13 + 23 = 1 × 12 + 2 × 22, draw one 1 × 1 square and two
2 × 2 squares. Then try to fit these three squares into a 3 × 3
square, since 3 = 1 + 2, and the equation says 13 + 23 = (1 + 2)2.
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+
+
=
This is almost a 3 × 3 square—just the bottom right corner is
missing. But the two striped squares overlap in the middle, and
if we take the overlapping part of one of the striped squares and
move it to the bottom right corner, we get exactly a 3 × 3 square
(see next page).
Number Sense 8-105, 106
Q-11
So 13 + 23 = (1 + 2)2.
The next case is 13 + 23 + 33 = 1 × 12 + 2 × 22 + 3 × 32, so draw
one 1 × 1 square, two 2 × 2 squares, and three 3 × 3 squares,
and try to fit them into a 6 × 6 square, since 6 = 1 + 2 + 3 and the
equation says 13 + 23 + 33 = (1 + 2 + 3)2.
+
+
+
+
+
So 13 + 23 + 33 = (1 + 2 + 3)2.
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Challenge students to show the next case: 13 + 23 + 33 + 43 =
(1 + 2 + 3 + 4)2. Give students 1-cm grid paper (see page U-25)
and scissors. Students could colour the squares of different sizes
different colours, and make a poster to show their work.
Q-12
Teacher’s Guide for Workbook 8.2
NS8-107 Powers of Negative Numbers
NS8-108 Exponents, Integers, and Order
of Operations
Pages 142–143
Curriculum
Expectations
Ontario: 8m1, 8m2, 8m7,
8m23
WNCP: optional, [R, PS, C];
9N1, 9N2, 9N4
Goals
Students will evaluate powers of negative integers. Students will
evaluate expressions that involve integers, and that include brackets
and powers, using order of operations.
PRIOR KNOWLEDGE REQUIRED
Can add, subtract, multiply, and divide integers
Can use the order of operations without powers
Can write repeated multiplication using power notation
+ (+) = +
+ (−) = −
− (+) = −
− (−) = +
Review multiplying integers. Remind students that multiplying two positive
or two negative integers results in a positive integer, whereas multiplying a
positive integer and a negative integer results in a negative integer. Display
the chart shown in the margin. Have students practise:
a) 3 × (−4)
d) (+2) × (−7)
b) (−2) × (−5)
e) (−4) × (−5)
ANSWERS: a) −12
b) 10
c) −6
c) (−3) × 2
d) −14
e) 20
Have students do examples that require multiplying many integers.
Students can multiply two or more terms in each step.
a) (−2) × (−3) × (4)
b) (−3) × (5) × (−2) × (−4)
c) (−10) × (−2) × (−3) × (−5) × (−1) × (−7)
ANSWERS: a) 24
b) −120
c) 2100
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Work through Workbook page 142 Question 1a) together as a class,
then have students complete parts b)–e) on their own. When students are
finished, discuss the pattern: which powers are positive and which are
negative, and why. Emphasize that each time we multiply by a negative
number, we change the sign of the product, so having an even number
of negative terms makes the product positive, while an odd number of
negative terms makes the product negative.
Then have students complete Workbook page 142.
Review the correct order of operations with brackets but no powers.
1. Do operations in brackets.
2. Do multiplication and division, from left to right.
3. Do addition and subtraction, from left to right.
Emphasize, in particular, that multiplication stands for repeated addition,
so it makes sense to do multiplication before addition. In the expression
Number Sense 8-107, 108
Q-13
2 + 3 × 4, the 3 isn’t being added to the 2, it is telling how many 4s to add
together: 2 + 4 + 4 + 4. That’s why we do multiplication before addition.
Furthermore, multiplication and division are “opposite” operations that undo
each other, so it makes sense to do them at the same time, and not to do
one before the other. The same goes for addition and subtraction.
Extra Practice: Evaluate:
a) 3 − (7 + 2)
b) 3 − 7 + 2
e) 8 − (5 + 1) + 3 × (−2) − 8 ÷ (−2)
c) 8 × 3 − 5
d) 8 × (3 − 5)
f) 15 ÷ (−3) × (−4) − 3 × (2 − 4)
Solution:
e)Do operations in brackets: 8 − 6 + 3 × (−2) − 8 ÷ (−2). Then do
multiplication and division, from left to right: 8 − 6 + (−6) − (−4).
Then do addition and subtraction, from left to right:
2 + (−6) − (−4) = −4 − (−4)
= 0
f)Do operations in brackets: 15 ÷ (−3) × (−4) − 3 × (−2). Then do
multiplication and division, from left to right:
(−5) × (−4) − 3 × (−2) = 20 − (−6)
= 26
Process assessment
Review the correct order of operations with powers but no brackets.
8m2, [R]
Workbook p 142 Question 4
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.
−24= −(24)
(−2)4 = (−2) × (−2) × (−2) × (−2)
= (+4) × (−2) × (−2)
= −(2 × 2 × 2 × 2)
= (−8) × (−2) = −16
= +16
So −24 is the opposite of 24.
Extra Practice: Evaluate:
a) 3 × 52
e) 3 + 42 ÷ (−2)
b) 1000 ÷ 53 c) 32 − (−2)3
f) 1000 ÷ (−2)3 g) 7 − 32 × (−2)
ANSWERS: a) 75
b) 8
c) 17
d) −16
e) −5
d) −43 ÷ (−2)2
h) 12 ÷ (−4) × 5
f) −125
g) 25
h) −15
The order of operations with brackets and powers. Emphasize that
expressions inside brackets are evaluated first, even before powers.
Q-14
Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
Emphasize, in particular, that a power stands for repeated multiplication,
so it makes sense to evaluate the powers before doing any multiplication.
For example, look at 2 × 34. This expression really means 2 × 3 × 3 × 3 × 3,
because 34 means 3 × 3 × 3 × 3. Also, point out that the expressions
(−2)4 and −24 mean different things:
Extra Practice: Evaluate:
a) (7 − 5)3
d) 52 − 82
g) 3375 ÷ 52 × (−3)
b) 73 − 53
e) 5 + 34 − (2 × 5)
h) (7 − 3)2 × (−2)
c) (5 − 8)2
f) 3375 ÷ (52 × (−3))
ANSWERS: a) 8 b) 218 c) 9 d) −39 e) 76 f) −45 g) −405 h) −32
Include expressions in exponents. Tell students that sometimes there is
an expression in an exponent that needs to be calculated before you can
evaluate the power. All the numbers in the same exponent should be treated
as if they were in brackets and so evaluated first. EXAMPLE: 32 + 4 = 36.
Have students practise:
a) 38 − 6
e) (− 2)(4 + 2) ÷ 2
b) (−2)6 ÷ 2
f) (− 2)4 + 2 ÷ 2 c) 58 − 3 × 2
d) (7 −10)3 − 2 + 1
2
4
g) (−8) ÷ (−2) × (3 + 2)20 ÷ 5
c) 25
e) −8
ANSWERS:
a) 9
b) −8
d) 9
f) −32
g) 2500
Summarize the order of operations.
1.
2.
3.
4.
Do operations in brackets.
Calculate exponents and evaluate powers.
Do multiplication and division, from left to right.
Do addition and subtraction, from left to right.
Have students do Workbook page 143 Question 3 for practice.
Process Expectation
Problem Solving
Adding brackets changes the value of an expression. Write on the board
and evaluate: 25 − 32 + 2 × 22 (= 24).
Show students different ways of adding brackets to this expression and ask
them to evaluate the new expressions using the correct order of operations:
a) 25 − 32 + (2 × 2)2
d) 25 − (32 + 2 × 22)
g) 25 − (32 + 2 × 2)2
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
ANSWERS: a) 32
b) (25 − 3)2 + 2 × 22
e) 25 − (32 + (2 × 2)2)
b) 492
c) 500
d) 8
e) 0
c) (25 − 3)2 + (2 × 2)
f) 25 − (32 + 2) × 22
f) −19
g) −144
Then point out that some ways of adding brackets don’t make sense.
For example, it wouldn’t make sense to add an opening bracket between
a base and an exponent in a power, e.g., 25 − 3(2 + 2) × 22. Write this
expression on the board and 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 parts
a, b, c, e, and g above) because the expression in brackets becomes the
base for the exponent.
Process assessment
8m1, 8m2, [PS]
Workbook p 143 Question 6
Number Sense 8-107, 108
Have students add brackets to these expressions to get as many different
answers as they can.
a) 16 + 8 ÷ 2 + 6
b) (−5)6 ÷ 3 × 2
ANSWERS: a) 26, 3, 18, 17
c) (−2)5 + 2 × (−5)2 ÷ 5
b) 625 or −5
c) −22, −1500, 372.8
Q-15
NS8-109 Concepts in Powers
Page 144
Goals
Curriculum
Expectations
Students will investigate properties of powers.
Ontario: 8m1, 8m7, 8m13
WNCP: optional, [R, C];
9N1, 9N2
PRIOR KNOWLEDGE REQUIRED
Can write repeated multiplication as powers
Can use long division (or a calculator) to compute remainders
Can evaluate small powers
Vocabulary
Patterns in ones digits of powers. Have students complete this chart:
Process assessment
8m1, 8m7, [R, C]
Workbook p 144 Question 1
Process Expectation
2n
21
22
23
24
=
2
4
8
16
ones digit
2
4
8
6
25
26
27
28
After students complete the chart, have students predict the ones digit of
29, 210, 211, and 212, and explain the basis for their predictions. Ask students
to predict the ones digit for 222. (4, because every fourth term in the
sequence has the same ones digit, so 222 has the same ones digit as
218, 214, 210, 26, and 22)
Looking for a pattern
ASK: What is the ones digit of 27051? Start by listing the powers of 2, in
backwards order, that have the same ones digit: 27047, 27043, 27039. ASK:
Process Expectation
Is this a good method to use? Why not? Why did it work well for 222? Tell
students that although the method we used for 222 doesn’t work as well for
27051, we can still look at the easier problem for ideas. Write on the board:
Looking for a similar
problem for ideas,
Reflecting on what makes
a problem easy or hard
22, 26, 210, 214, 218, 222
Then write the sequence of exponents on the board: 2, 6, 10, 14, 18, 22.
ASK: How can we describe these numbers? (sample answers: they are
all of the form 4n + 2; they are all 4 more than the previous term) Then
ask students to think in terms of remainders: How can you describe these
numbers in terms of remainders? (They all have remainder 2 when dividing
by 4.) Emphasize that all these powers of 2 have the same ones digit as
22 (which is the first exponent in the sequence that has remainder 2 when
dividing by 4).
ASK: How does this help us with the harder problem of finding the ones
digit of 27051? PROMPT: What do we have to know about 7051 to help us
answer this question? (the remainder when dividing by 4) Have students
use long division to find this remainder, and then use this information to
determine the ones digit of 27051. (The remainder when dividing 7051 by 4
is 3, so 27051 has the same ones digit as 2 to the exponent 3. Since 23 is 8,
27051 has ones digit 8.)
Bonus
Q-16
W
hat is the ones digit of 251, 2151, 2251, 2351, 2451, 2551, 2651, 2238651,
and 2800951? (these all have remainder 3 when dividing by 4; in
Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
power
exponent
base
ones digit
remainder
fact the last 2 digits determine the remainder when dividing by 4;
students using the WNCP curriculum might remember this fact
from Grade 7)
When students finish, point out that it is quite surprising that we can find the
ones digit of 27051 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!
Process assessment
Extensions
8m1, 8m7, [R, C]
1. a) Write each power of 10 in standard form.
104 = _______
103 = _______
102 = _______
101 = _______
b) How do we get each number on the right side of the equations in
part a) from the number above it? (divide by 10) How do we get
each exponent in the left side of the equation from the exponent
above it? (subtract 1)
c) Continue the pattern in the exponents and in the numbers:
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
10 = _______
d)Repeat parts a) and b) for powers of 2, powers of 3, and powers of 5.
e) What is the zeroth power of any number? Explain. (Each power is
obtained from the previous one by dividing by the base. But the first
power of any base is the base itself, so the zeroth power is the base
divided by the base, which is always 1.)
f) Introduce negative powers. Have students continue the pattern:
102 = 100
101 = 10
100 = 1
10−1 = _______
10−2 = _______
ANSWERS: 1/10 or 0.1, and 1/100 or 0.01, because you divide by
10 to get the next term in the pattern.
2. Remind students that to find two or three consecutive numbers that add
to a number, we can use equations (e.g., x + x + 1 = 37).
a) Use an equation to find two consecutive numbers that add to:
Number Sense 8-109
i) 37
ii) 79
iii) 63
iv) 99
Q-17
b) Find half of:
i) 37
ii) 79
iii) 63
iv) 99
c) How are your answers to parts a) and b) related?
Explain to students that the two consecutive numbers are very close
together—they are only one apart. If they were exactly the same
and added to 37, they would be exactly half of 37, but because they
are only close to each other and not exactly the same, they are only
close to half of 37.
d) Use half of 85 to find two consecutive numbers that add to 85.
Tell students that you want to find three consecutive numbers that
add to 60. SAY: The three numbers are very close together—they
are only one apart. If the same number was added 3 times to get
60, what number would that be? (20) Tell students that they should
look for numbers close to 20 to find three consecutive numbers that
add to 60. (19 + 20 + 21 = 60)
e) Use 1/3 of 72 to find three consecutive numbers that add to 72.
(1/3 of 72 = 24, so 23 + 24 + 25 = 72)
f) Which whole number, when squared, is close to 182? Calculate
the square root of 182 on a calculator and use the result to find two
consecutive numbers that multiply to 182. (13 × 14 = 182)
g) Guess and check to find the whole number which, when cubed,
is closest to 1320. (103 = 1000 and 113 = 1331, so 11)
h) The product of three consecutive numbers is 1320. Find these
numbers. (10 × 11 × 12 = 1320)
i) The product of three consecutive numbers is 7980. Find these
numbers. (19 × 20 × 21 = 7980)
2. Prime factorizations, powers, squares, and square roots.
a) Write each square of a power as a power itself. EXAMPLE:
Q-18
(23)2 = 23 × 23 = (2 × 2 × 2) × (2 × 2 × 2) = 26
i) (32)2
Bonus
ii) (25)2
iii) (53)2
iv) (74)2
v) (85)2
(2324)2
Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
Tell students that you want to find two consecutive numbers that
multiply to 156. SAY: These two numbers are close together—they
are only one apart. If a number was multiplied by itself to get 156,
what would the number be? (the square root of 156) What two
consecutive numbers are closest to the square root of 156? (12 and
13 because 122 = 144 and 132 = 169) Tell students that if numbers
that are close together multiply to 156, they should be close to the
square root of 156, so a good guess for the two numbers is 12 and
13. Indeed, 12 × 13 = 156.
b)Teach students to write prime factorizations using power notation.
For example, 24 = 2 × 2 × 2 × 3 can be written as 24 = 23 × 3.
Have students write prime factorizations for these numbers using
power notation:
i) 12
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iv) 72
c)Why is 53 a prime factorization, but 82 is not a prime factorization?
(because 5 is prime, but 8 is composite and can be factorized
further) PROMPT: Draw the factor tree for each number—53 = 125
and 82 = 64.
d)If the prime factorization of a number is 23 × 75 × 118, what is the
prime factorization of its square? (26 × 710 × 1116)
e)How can you tell immediately from the exponents in a prime
factorization whether the number is a perfect square? (If all
exponents are even, the number is a perfect square; if any exponent
is odd, the number is not a perfect square.)
f) Write each square root of a power as a power itself. EXAMPLE:
2 6 = 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2 × 2)×(2 × 2 × 2) = 2 × 2 × 2 = 23
i)
ANSWERS: i) 24
Algebra
iii) 96
connection
ii) 75
28
ii)
34
iii)
310
ii) 32
iv)
iii) 35
516
iv) 58
g) If the prime factorization of a number is 28 × 310 × 516, what is the
prime factorization of its square root? (24 × 35 × 58)
4. 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:
= 2 and b = 3
a
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 prime and GCF of a and b is 1)
Number Sense 8-109
b) For a = 4 and b = 7, check whether a is a factor of ba − 1 − 1.
Is this a counter-example 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 counter-example to the statement in part a)? Explain.
Q-19
connection
Literature, Measurement
5. Read the book One Grain of Rice by Demi, and then work through
these questions individually, stopping after each one to check answers
and discuss.
a) If 1 grain of rice weighs 25 mg, how much rice (by mass) would
Rani get in a week? In 2 weeks? In 3 weeks? After the whole month
(30 days)? NOTE: Review the rule for finding the sum of n powers
from NS8-104 Extension 6: The sum of the first n powers of 2
is 2n + 1 – 2. For example, the sum of the first 6 powers of 2 is
2 + 4 + 8 + 16 + 32 + 64 = 27 – 2 = 126.
ANSWERS:
1 week: Number of grains = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127.
The mass of 127 grains is 127 × 25 mg = 3175 mg = 3.175 g.
2 weeks: The number of grains after 1 week is 27 − 1. Using the
same pattern, the number of grains after 2 weeks is 214 − 1 = 16 383
grains. The mass is 16 383 × 25 mg = 409 575 mg = 409.575 g.
3 weeks: 221 − 1 = 2 097 151 grains, for a total mass of 2 097 151
× 25 mg = 52 428 775 mg = 52 428.775 g = 52.428 775 kg.
30 days: There are 230 − 1 = 1 073 741 823 grains, for a total mass of
1 073 741 823 × 25 mg = 26 843 545 575 mg
= 26 843 545.575 g
= 26 843.545 575 kg
= 26.843 545 575 tonnes.
So Rani would get almost 27 tonnes of rice after 30 days.
b) If the volume of a grain of rice is 33.3 mm3, what would Rani need
in order to carry the rice home on the 7th day? The 14th day? The
21st day? After the whole month?
ANSWERS: Notice that on day 1, she needs to carry 1 grain; on
day 2, 2 grains; on day 3, 22 grains; on day 4, 23 grains; and on
day n, 2n − 1 grains.
14th day: 213 grains = 8192 grains for a volume of 8192 × 33.3 mm3
= 272793.6 mm3 = 272.7936 cm3. Rani could carry the rice in a bowl.
21st day: 220 grains = 1 048 576 grains for a volume of 1 048 576 ×
33.3 mm3 = 34 917 580.8 mm3 = 34 917.580 8 cm3 = 34.917 580 8
dm3. Rani needs a capacity of almost 35 L. You can demonstrate
this capacity if your school has 35 thousands cubes. Four big pails
should do.
Q-20
Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
7th day: 26 grains = 64 grains for a volume of 64 × 33.3 mm3 =
2131.2 mm3 = 2.1312 cm3. Rani could carry this much rice home
in the palm of her hand.
A whole month: 229 grains = 536 870 912 grains for a volume
of 536 870 912 × 33.3 mm3 = 17 877 801 369.6 mm3 =
17 877 801.369 cm3 = 17 877.801 369 dm3 = 17.877 801 369 m3.
This is almost 18 m3. Rani would need many sacks... and other
villagers to help her carry them!
c) After a whole month, would all the grains they have to store fit into
the classroom?
ANSWER: There would be 230 − 1 grains = 1 073 741 823 grains
for a volume of
1 073 741 823 × 33.3 mm3 = 35 755 602 705.9 mm3
≈ 35.8 m3
(This is likely to fit in most classrooms. Calculate the volume of your
classroom, roughly, by measuring the length and width of the floor
and the height of the walls.)
d) The residents of the village decide they won’t start eating the rice
until there is enough for everyone in the village to have 3 bowls of
rice a day. If a bowl of rice has 2000 grains, and the village has
250 people, on what day can the villagers start eating the rice?
ANSWER: They need to have 1 500 000 grains before they can start
eating. From part a), the answer is between 2 weeks and 3 weeks.
Check each day in turn:
15 days: 215 − 1 = 32 767 grains
16 days: 216 − 1 = 65 535 grains
17 days: 217 − 1 = 131 071 grains
18 days: 218 − 1 = 262 143 grains
19 days: 219 − 1 = 524 287 grains
20 days: 220 − 1 = 1 048 575 grains
21 days: 221 − 1 = 2 097 152 grains
So on day 21, the villagers can each have 3 bowls of rice a day.
Notice that on day 22, they would get 221 = 2 097 152 grains of rice,
so they have enough to eat 3 bowls each on that day too, and so on.
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
e) Assuming the villagers eat 3 bowls of rice a day as described in
part d), what is the volume of rice the villagers have to store at the
end of the month? How long can they eat from this supply of rice?
ANSWER: The villagers eat for 10 days (day 21 to day 30). So we
need to subtract 1 500 000 × 10 = 15 000 000 grains from the total
to be stored (see part c):
1 073 741 823 − 15 000 000 = 1 058 741 823 grains of rice.
To determine how many more days the villagers can eat for, divide
this total by the 1 500 000 grains they eat each day: 1 058 741 823 ÷
1 500 000 = 705.827 882. So the villagers can eat for 705 more
days, or almost 2 years.
Number Sense 8-109
Q-21
NS8-110 Negative Fractions and Decimals
Page 145
Curriculum
Expectations
Ontario: 8m1, 8m6, 8m7,
8m13
WNCP: optional, [V, R, C],
9N3
Vocabulary
integer
opposite integer
positive
negative
fraction
decimal
> and < symbols
Goals
Students will represent, compare, and order rational numbers
(i.e., positive and negative fractions and decimals to thousandths).
PRIOR KNOWLEDGE REQUIRED
Can compare and order fractions
Can compare and order decimals
Can compare and order integers
Can use long division to write fractions as decimals
Review comparing and ordering positive fractions and decimals.
Progress as follows:
1. Compare tenths, hundredths, and thousandths. EXAMPLE: Compare
0.132 and 0.105
2. Compare fractions with denominator 2, 4, 5, 8, or 20 to decimals by
changing the fraction to a fraction with denominator 10, 100, or 1000,
and then to a decimal. EXAMPLES: Compare
1
7
8
3
i)
and 0.23 ii)
and 0.431 iii)
and 1.584 iv)
and 0.36
4
20
5
8
3
3 ×125
375
= SAMPLE ANSWER: iv) = = 0.375 > 0.36.
8
8 ×125
1000
3. Compare any fraction to a decimal by changing the decimal to a
fraction and comparing the fractions. EXAMPLE: Compare 2/3 to 0.65
by comparing 200/300 to 65/100 = 195/300.
4. Compare any fraction to a decimal by changing the fraction to a
decimal by long division. EXAMPLE:
5. Order lists of fractions and decimals. EXAMPLE:
Order the list: 5 2/3, 5 3/5, 5.712, 5.615, 5.67.
Review comparing and ordering integers. First, recall that all negative
numbers are smaller than all positive numbers. Second, remind students
that an “opposite integer” is the number that is the same distance from 0
on a number line, but in the other direction; a number and its opposite add
to 0; to obtain the opposite number, keep the number part the same, but
change the sign (from + to − or from − to +).
Q-22
Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
Compare 2/3 to 0.65 by writing 2/3 as the repeating decimal 0.6 ,
which is greater than 0.65. (NOTE: If students are not comfortable
converting fractions to repeating decimals, students will need to
use the method of changing the decimal to a fraction instead of
changing the fraction to a decimal.)
Process Expectation
Using logical reasoning
Process Expectation
Visualizing
To order negative numbers, order the opposite positive numbers and put
the negative numbers in reverse order to their positive opposites. For
example, since 3 < 4 < 5, we know that −5 < −4 < −3. The fact that 4 is
less than 5 tells us that 5 is further from 0, but this means that −5 is further
from 0 than −4, which means it is more to the left, so −5 is less than −4.
Placing negative numbers on number lines. Have students mark the
positive integers 3, 7, and 8 with an X on a number line from −10 to 10 that
has only three points labelled: −10, 0, and 10. Then have students mark
their opposites, −3, −7, and −8, on the same number line. ASK: Do you
see any symmetry? (a vertical line through 0 is a mirror line)
Review placing positive fractions and decimals on number lines, and then
show students how to place negative fractions and decimals on number
lines by using the mirror image of the opposite positive numbers.
Process assessment
COPYRIGHT © 2011 JUMP MATH: NOT TO BE COPIED
8m6, 8m7, [V, C]
Workbook p 142 Question 2
Compare and order positive and negative fractions and decimals.
Combine the concepts above. EXAMPLE:
1 5
1
Order these numbers: 2, 1.3, −1.4, − 2 , , − 1 .
4 3
2
1 5
Step 1: Order the positive numbers: 2, 1.3, , .
4 3
1
5
So,
< 1.3 <
< 2.
4
3
1
Step 2: Order the opposite of the negative numbers: 1.4, 2, 1 .
2
1
So, 1.4 < 1 < 2.
2
Step 3: Order the negative numbers in reverse order from Step 2:
1
−2 < − 1 < −1.4
2
Step 4: Combine the lists from Steps 1 and 3, with all negative numbers
less than all positive numbers:
1
5
1
−2 < − 1 < −1.4 <
< 1.3 <
<2
2
4
3
Number Sense 8-110
Q-23