Download Activity 2.7 Answer Key - Campbell County Schools

Dividing Decimals
Getting Ready for the Dance
ACTIVITY
ACTIVITY 2.7 Investigative
2.7
Dividing Decimals
SUGGESTED LEARNING STRATEGIES: Close Reading, Use
Manipulatives, Look for a Pattern, Create Representations
Activity Focus
My Notes
• Dividing decimals by whole
numbers
• Dividing decimals by decimals
• Repeating decimals
The middle school students at Montgomery Middle School are
planning a dance. They will decorate the gym, hire a deejay, and
provide refreshments. The students have listed their expenses and
will share the cost of the dance.
Expense
Decorations
Deejay
Refreshments
Cost
$272.64
$168.64
$113.28
Materials
• BLM 2: Hundred grid
Chunking the Activity
Thirty-two students will attend the dance. What is each student’s
share of the total cost? You will have to divide to solve this problem,
so first explore some ideas about division.
Think about a simpler problem: 0.72 ÷ 9. Remember: 0.72
means 72 hundredths, or 72 out of 100.
#1–6
Ex 1
#7–10
#11
#12
100 or 1 whole. Shade ____
72 of the squares in
1. This grid represents ____
100
100
the grid.
#13–14
Ex 2
#15
Ex 3
#16–17
Ex 4
#18
Ex 5
Paragraph Close Reading
© 2010 College Board. All rights reserved.
1 Use Manipulatives, Create
Representations This model
will give students a concrete look
at dividing a decimal by a whole
number. Make sure that the students understand that the entire
chart represents one unit.
2. To divide by 9, you need to make 9 groups. Mark the shaded
sections into 9 groups. How many squares are in each group?
8 squares
W
We write 0.72 instead
o
of .72 for readability
purposes
purposes. Encourage students to
use the leading zero before the
decimal point when decimals have
a value less than one.
TEACHER TO
TEACHER
3. What is the value of one small square? 0.01
4. What is the value of each group of squares that you made? 0.08
5. Complete this number sentence.
0.72 divided by 9 = 0.08
2 Use Manipulatives Not all
students will mark the same
9 groups. Allow students to share
their representations.
Unit 2 • Operations with Numbers
© 2010 College Board. All rights reserved.
103-110_SB_MS1_2-7_SE.indd 103
103
Differentiating Instruction
12/16/09 5:55:33 PM
Some students may benefit by
using different colored pencils to
shade each group.
34 Look for a Pattern
Unit 2 • Operations with Numbers
103
ACTIVITY 2.7
continued
Dividing Decimals
Getting Ready for the Dance
6 Quickwrite Students have
studied long division in earlier
grades. The important concept
for them in this unit is the placement of the decimal point in the
quotient.
EXAMPLE 1 Close Reading,
Activating Prior Knowledge
Ask students to read through the
example on their own, and then
have a classroom discussion to
make sure that all students understand the process. You may want
to preclude this example with a
few review problems on dividing
with whole numbers.
SUGGESTED LEARNING STRATEGIES: Quickwrite,
Close Reading
My Notes
6. Look at 0.72 ÷ 9 set up as if dividing whole numbers.
0.08
____
9 0.72
MATH TERMS
An algorithm is a set of steps
or a procedure used to carry
out a computation or to solve
a problem.
a. Place the answer from Question 5 above the dividend.
b. How is the decimal point in the quotient placed in
relationship to the decimal point in the dividend?
Answers may vary. Sample answer: The decimal point in the
quotient is placed directly above the decimal point in the
dividend.
READING MATH
quotient
________
divisor dividend
A quotient is calculated by
dividing a dividend by a divisor.
EXAMPLE 1
The decorations for the dance will cost $272.64. What is each
student’s share of the cost of the decorations?
Step 1:
Step 2:
TRY THESE A
Think/Pair/Share
Determine the total number of students.
32 (given on the previous page)
Divide the cost of $272.64 by 32.
The algorithm is the one used for dividing
whole numbers.
8.52
______
32 272.64
- 256
166
- 160
64
- 64
0
Solution: Each student will pay $8.52 for dance decorations.
Notice that the decimal point is placed in the answer directly above
the decimal point in the dividend.
P
Page
105 may be the first
eexposure to estimation
com
using compatible
numbers for
some students. Have a class
discussion to help students
understand the difference
between using compatible
numbers and rounding. Help
students clarify when to use
each method.
TEACHER TO
TEACHER
© 2010 College Board. All rights reserved.
ACTIVITY 2.7 Continued
TRY THESE A
Find the quotient.
______
a. 25 168.75 6.75
_____
b. 7 339.5 48.5
104 SpringBoard® Mathematics with Meaning™ Level 1
MINI-LESSON: Compatible Numbers
Estimation using compatible numbers is often used with division of decimals.
Guide a discussion with students so that it leads to an agreement on a good
definition of compatible numbers.
Ask students to rewrite the following division problems using compatible
numbers and then estimate the quotient.
132 ÷ 54
878 ÷ 32
839.45 ÷ 11.6
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Dividing Decimals
ACTIVITY 2.7
Getting Ready for the Dance
ACTIVITY 2.7 Continued
continued
7 Create Representations
SUGGESTED LEARNING STRATEGIES: Guess and Check, Create
Representations, Think/Pair/Share, Quickwrite, Group Presentation
8 Think/Pair/Share
My Notes
It is helpful to estimate a quotient beforehand to see if your solution
makes sense. One method of estimating is to use compatible
numbers. For example, to divide 272.64 by 32, think of the closest
compatible numbers and find the quotient mentally.
9 Create Representations Ask
students if their estimates are
close to the actual quotient. Have
a discussion about why some
students make estimates that are
closer than others. The exact quotient for question 9 is 3.543125.
Students may have questions
about what to do with the
remainder as they work through
the problem.
MATH TERMS
272.64
270
32
30
270 ÷ 30 = 90
7. The deejay will cost $168.64. What is each student’s share of the
cost? Use compatible numbers to estimate, then solve.
Estimates will vary. Sample answer: 160 ÷ 40 = 4; $5.27
Compatible numbers are close
in value to the original numbers
in an arithmetic problem and
are used in place of the original
numbers to make it easier to
estimate an answer to the
problem.
8. Is your answer in Question 7 an exact number of dollars and
cents? If not, what should you do with the remainder?
It is exact; there is no remainder.
0 Think/Pair/Share Some
students may decide to “round
down” since the remainder after
carrying out the division process
10
through two decimal places is ___
32
5
___
or 16 , which is less than one-half.
9. Refreshments for the dance will total $113.38. Estimate first,
then find each student’s share of the cost.
Estimates may vary. Sample estimate: 120 ÷ 30 = 4.
Answers may vary. Two possible answers: $3.54 or $3.55.
10. Is your answer in Question 9 an exact number of dollars and
cents? If not, what should you do with the remainder?
When a division problem that
involves a decimal has a zero
remainder, we say that the
quotient is a terminating decimal.
© 2010 College Board. All rights reserved.
Answers may vary. Sample answer: No; round down or
round up.
a Quickwrite Have a guided
discussion about when you might
round up, down, or maybe not
round at all when dealing with
money. Ask whether the group of
students will have enough money
for refreshments if they round
down in Question 9.
11. When there is a remainder in a division problem involving
money, how should you round the quotient? Answers may
vary. Sample answer: In a case like Question 9, the answer
should be rounded up to make sure that the entire cost is
covered. ($3.54 × $32 = $113.28 and $3.55 × 32 = $113.60)
12. The students want to get the best buys on the refreshments.
They found three brands of pretzels at the grocery store.
Brand
Bob’s Pretzels
Crunchy Pretzels
Kettle Pretzels
Size
12 oz
14 oz
16 oz
Cost
$1.68
$1.89
$2.08
Cost of 1 Ounce
$0.14
b Group Presentation,
Debriefing This question is an
introduction to rates, which are
taught in unit 4. Debrief the class
on their responses to 12b.
$0.135
$0.13
a. For each brand, list the cost of an ounce of pretzels.
b. Which brand do you think will be the best buy? Explain.
Kettle Pretzels; they cost the least per ounce
Suggested Assignment
Unit 2 • Operations with Numbers
© 2010 College Board. All rights reserved.
PM
103-110_SB_MS1_2-7_SE.indd 105
105
CHECK YOUR UNDERSTANDING
p. 110, #1–3
12/16/09 5:55:48 PM
UNIT 2 PRACTICE
p. 136, #42–43
Ask students if they have
A
eever noticed the small
signs unde
underneath the products at
the grocery store that tell how
much one unit of the product
costs. Ask a grocery store manager for some of those signs to
use in your class discussion.
TEACHER TO
TEACHER
Unit 2 • Operations with Numbers
105
ACTIVITY 2.7
continued
Dividing Decimals
Getting Ready for the Dance
c Use Manipulatives
Guide students through this exercise. You may want to ask them
what they think the solution is
before marking the counter. Many
students may think that the solution is 3.5.
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Quickwrite, Close Reading
My Notes
13. The counter in the concession stand needs to be painted
before the dance. The counter is seven feet long and the students
want to alternate the colors blue and green in 0.5 ft sections.
0
Use inverse operations by asking
students what they would multiply
0.5 by to get 7.
2
3
4
5
6
7
a. How many 0.5 ft. sections will be painted? Mark the counter
to show the sections. 14 sections
d Quickwrite This question
b. Complete this number sentence: 7 ÷ 0.5 =
shows the students the value of
moving from making a model to
the using of an algorithm when
dividing by a decimal.
EXAMPLE 2 Close Reading
Guide students through
Example 2. Make sure that they
understand that you can always
change the divisor to a whole
number by multiplying by some
power of ten. Ask the students to
tell you the smallest power of 10
that they can use to make 1.2 a
whole number.
1
Counter
14
14. Suppose you wanted to divide 28.56 by 2.3. Why would it be
difficult to use a pictorial model to answer this question?
Answers may vary. Sample answer: It would be difficult to
draw a model exactly 28.56 units long and divide it into
sections 2.3 units long.
EXAMPLE 2
WRITING MATH
48 ÷ 1.2 can also be written as
___
48
___
or 1.2 48.
1.2
Divide 48 by 1.2.
Step 1:
Step 2:
Step 3:
Recall that multiplying a fraction
by 1 does not change its value
because of the Property of One. If
7
3
__
you multiply the fraction __
5 by 7 :
7 ___
3 __
21 , the resulting
__
×
=
5 7 35
3
fraction is equivalent to __
5.
Write the division problem as a fraction.
48
___
1.2
Rewrite the denominator so that it is a whole number.
The smallest number that we can use to do this is 10.
1.2 × 10 = 12
Since you multiplied the denominator by 10, you multiply the
numerator by 10 so that you have equivalent fractions.
© 2010 College Board. All rights reserved.
ACTIVITY 2.7 Continued
48 × ___
10 = ____
480
___
1.2
Step 4:
10
12
Divide 480 by 12.
40
____
12 480
- 48
00
Solution: The quotient is 40.
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106 SpringBoard® Mathematics with Meaning™ Level 1
Dividing Decimals
ACTIVITY 2.7
Getting Ready for the Dance
ACTIVITY 2.7 Continued
continued
TRY THESE B Guess and Check
(a), Think/Pair/Share (b, c)
Students should multiply by the
smallest power of ten possible in
order to write the denominator as
a whole number.
SUGGESTED LEARNING STRATEGIES: Guess and Check,
Think/Pair/Share, Create Representations, Close Reading
My Notes
TRY THESE B
5.8364 , find the smallest number by which you
a. For the fraction ______
2.173
can multiply the denominator to make it a whole number. Then
write an equivalent fraction with a whole-number denominator.
5836.4
a. 1000; ______
2173
e Create Representations,
Write each division problem as a fraction, rewrite it as an equivalent
fraction with a whole-number denominator, and then divide.
______
b. 2.7 13.041 4.83
EXAMPLE 3 Close Reading
Students are introduced to adding
zeros as placeholders in the dividend. Make sure that students are
moving the decimal point in the
divisor and dividend correctly. Ask
them to estimate their quotient
and check to see if their answer is
reasonable.
______
c. 0.52 6.5676 12.63
15. The decorating committee spent $22.95 on ribbon, which
costs $0.85 per meter. How many meters of ribbon did the
committee buy? 27 meters
Instead of multiplying the dividend and divisor by the same
power of 10 to make them both whole numbers, you can move the
decimal points.
EXAMPLE 3
____
Divide: 0.13 72.8
© 2010 College Board. All rights reserved.
Step 1:
_____
0.13. 72.8
Step 2:
First move the decimal point in this
divisor 2 places to the right to make
it a whole number.
Next move the decimal point in
this dividend the same number of
places. Notice that a zero must be
inserted in the dividend.
Step 3:
Then divide.
560
_____
13 7280
65
78
78
00
Solution:
• Multiplying by 10 moves the
decimal point one place to
the right.
• Multiplying by 100 moves the
decimal point two places to
the right.
If students are allowed to
u
use calculators, then you
may need to show them how to
enter fractions and long division
problems into the calculator
correctly. The dividend is always
entered first, followed by the
operation (÷), followed by the
divisor.
TEACHER TO
TEACHER
_______
0.13. 72.80.
560
Unit 2 • Operations with Numbers
107
12/16/09 5:55:56 PM
© 2010 College Board. All rights reserved.
PM
103-110_SB_MS1_2-7_SE.indd 107
Some students may benefit by
making “loops” as they move the
decimal point in the divisor and
dividend.
Recall the rules for multiplying by
powers of 10:
Unit 2 • Operations with Numbers
107
ACTIVITY 2.7 Continued
ACTIVITY 2.7
continued
Dividing Decimals
Getting Ready for the Dance
TRY THESE C Group
Presentation
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Create Representations, Marking the Text
My Notes
Suggested Assignment
TRY THESE C
CHECK YOUR UNDERSTANDING
p. 110, #4–5
Find each quotient.
_____
a. 0.45 103.5 230
____
b. 0.31 682
2200
UNIT 2 PRACTICE
p. 136, #44–46
16. Sharon wants to decorate each table with confetti. A bag of
confetti holds 546 grams. She will scatter 45.5 grams on each
table. How many tables will she be able to decorate? 12 tables
f Create Representations
g Create Representations
17. The deejay will play music for 2.5 hours. Each song lasts
approximately 3.75 minutes. What is the total number of songs
that can be played during the dance? 40 songs
Students will have to convert the
hours to minutes to work this
problem.
Sometimes the dividend is not evenly divided by the divisor.
EXAMPLE 4
___
Find the quotient 0.31 42 to the nearest tenth.
Step 1:
Move the decimal
point to create a
whole number. Then
divide. Notice the
remainder of 15.
Step 2:
Continue dividing by adding
two more zeros after the
decimal point.
Step 3:
There is still a remainder, but
now there are enough places
to round the quotient to the
nearest tenth.
0.3 = 0.30 = 0.300
because
30 ___
10 = ____
3 · ___
300
___
· 10 = _____
10 10 100 10 1000
______
0.31. 42.00.
Solution: The rounded quotient is 135.5.
135.
_____
31 4200.
- 31
110
- 93
170
- 153
15
© 2010 College Board. All rights reserved.
EXAMPLE 4 Marking the Text
Students will find quotients when
the remainder is not zero by adding zeros to the dividend so that
they have enough decimal places
(2) to round their answer to the
nearest tenth. Discuss how to
predict how many zeros to annex
to the right of the decimal point
when rounding to the nearest
tenth, hundredth, and so on.
135.48
_______
31 4200.00
- 31
110
- 93
170
- 153
150
- 124
260
- 248
12
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108 SpringBoard® Mathematics with Meaning™ Level 1
Dividing Decimals
ACTIVITY 2.7
Getting Ready for the Dance
ACTIVITY 2.7 Continued
continued
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Create Representations, Quickwrite, Close Reading, Look
for a Pattern
TRY THESE D Group Presentation Ask students to share their
answers on white boards. Make
sure that they are rewriting the
dividend and rounding correctly.
My Notes
TRY THESE D
Find each quotient to the nearest tenth.
___
a. 0.22 1.3
___
h Create Representations
b. 6.2 27
5.9
4.4
EXAMPLE 5 Close Reading
Help students understand that
we sometimes round repeating
decimals instead of writing them
with the bar over the repeating
digits. Ask them why they think it
might be more beneficial to round
the repeating decimal.
18. After spending two hours planning the dance, Mark, LaNita,
and Sam decide to go to the cafeteria and get a snack. The
cafeteria sells three containers of yogurt for $1.00.
a. How much will Mark, LaNita, and Sam each pay for one
container of yogurt?
Answers may vary. Sample answers: Two will pay $0.33 and
one will pay $0.34, or all three may have to pay 34¢ each.
b. What do you notice about the quotient?
Answers may vary. Sample answer: The 3 repeats in the
quotient.
MATH TERMS
A repeating decimal is a decimal that has one or more digits
to the right of the decimal point
that repeat forever.
SStudents may be familiar
22 as an approximaw
with ___
7
π If not, they will be
tion for π.
introduced to the concept of π in
Unit 5. In either case, you may
want to challenge students to
22 as a repeating decimal.
express ___
7
TEACHER TO
TEACHER
Numbers like these are called repeating decimals. Repeating
decimals can either be written with a bar over the digits that repeat
or can be rounded.
_______
© 2010 College Board. All rights reserved.
EXAMPLE 5
[3.142857]. Be sure students
Divide 7 by 11.
Step 1:
22 is an approxiunderstand that ___
7
22
mation for π and that, while ___
7
repeats, π is a nonrepeating,
non-terminating decimal.
Set up the problem and divide.
0.6363
______
11 7.0000
- 66
40
- 33
70
- 66
40
- 33
7
Step 2:
Notice the repeating digits in the quotient.
Solution: Th
_e digits 63 repeat in the quotient, so it can be written with
a bar as 0.63 or rounded to 0.64.
Unit 2 • Operations with Numbers
12/16/09 5:56:01 PM
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103-110_SB_MS1_2-7_SE.indd 109
PM
109
Unit 2 • Operations with Numbers
109
ACTIVITY 2.7 Continued
ACTIVITY 2.7
continued
Dividing Decimals
Getting Ready for the Dance
TRY THESE E Look for a Pattern
Suggested Assignment
My Notes
TRY THESE E
CHECK YOUR UNDERSTANDING
p. 110, #6–8
__
a. 3 5
−
1.6 or 1.67
__
b. 11 2
—
1.18 or 1.182
UNIT 2 PRACTICE
p. 136, #47–50
CHECK YOUR UNDERSTANDING
1. There are 5 groups of
two-tenths in 1.
Write your answers on notebook paper.
Show your work.
5. Sam paid $5.75 for 2.3 pounds of apples.
What was the cost for one pound?
1. Use a model to show how many groups of
0.2 are in 1.
6. Sam is saving $5.75 per week to buy a CD
player that costs $46. How many weeks
will he have to save before he can buy the
CD player?
5 and ___
14 .
7. Compare the quotients of __
4
11
a. How are they alike?
2. Estimate the quotient: 384.72 ÷ 19.475.
Explain your process.
3. Elliott is training for a 17.5 km race. His
track coach has separated the course into
2.5 km sections. Into how many sections is
the course separated?
8. MATHEMATICAL Why does moving the
R E F L E C T I O N decimal point in the
divisor and the dividend when dividing by
a decimal work?
© 2010 College Board. All rights reserved.
4. A jar contains 189 grams of mustard. How
many 4.5-gram portions can be made?
b. How are they different?
2. Answers may vary. Sample
answer: By using compatible
numbers, the estimate of the
quotient is 380 ÷ 20 = 19 or
380 ÷ 19 = 20.
3. 7 sections
4. 42 portions
5. $2.50
6. 8 weeks
a. Answers may vary. Sample
answer: Both are decimals.
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103-110_SB_MS1_2-7_SE.indd 110
b. Answers may vary. Sample
5
answer: __
is a terminating deci4
14 is a repeating decimal.
mal; ___
11
8. Answers may vary. Sample
answer: Moving the decimal
point the same number of
places in the divisor and
the dividend is the same as
multiplying them both by
the same number. This is the
Property of One at work in a
different way.
110 SpringBoard® Mathematics with Meaning™ Level 1
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© 2010 College Board. All rights reserved.
5
14 = 1.27
7. __
= 1.25; ___
4
11