Transcript Ratio and Proportion

Ratios, Proportions, and Percents
Equivalent Ratios vs. Equivalent Fractions
Cups Blue
2
4
6
Total Cups
3
6
9
2
3
4
6
6
9
Equivalent Fractions
More parts; smaller parts
Same whole amount
Same portion
2
3
4
6
6
9
Equivalent Ratios
Cups Blue
2
4
6
Total Cups
3
6
9
More parts; same size parts
More total paint
More blue pigment
Ratios
If you know that 2:3 is a part-to-part
relationship, when else can you deduce
from that ratio?
Tape Diagrams
• Best used when the two quantities have the
same units.
• Highlight the multiplicative relationship between
quantities.
yellow
blue
Tape Diagrams
yellow
blue
1. If you will use 10 quarts of blue paint, how many
quarts of yellow paint will you need?
2. If you will use 18 quarts of yellow paint, how
many quarts of blue paint will you need?
3. If you want to make 25 quarts of green paint,
how many quarts of yellow and blue will you
need?
Double Number Lines
• Best used when the two quantities have different
units.
• Help make visible that there are infinitely many
pairs in the same ratio, including those with
rational numbers
• Same ratios are the same distance from zero
Double Number Lines
Driving at a constant speed, you drove 14 miles in 20
minutes. On a “double number line”, show different
distances and times that would give you the same
speed. Identify equivalent rates below.
Distance
0 miles
7 miles
14 miles
0 minutes 10 minutes 20 minutes
Time
28 miles
40 minutes
Laundry Detergent Comparison
A box of Brand A laundry detergent washes 20 loads of
laundry and costs $6. A box of Brand B laundry
detergent washes 15 loads of laundry and costs $5.
What are some equivalent loads?
Brand A
Loads washed
20
Cost
$6
Brand B
Loads washed
15
Cost
$5
Unit Rates
Explain how to fill in the next tables with unit rates. Then
use the tables to make statements comparing the two
brands of laundry detergent.
Brand A
Brand B
Loads washed
20
3.33
Cost
$6
$1
Brand A
Loads washed
15
3
Cost
$5
$1
Loads washed
15
1
Cost
$5
$0.33
Brand B
Loads washed
20
1
Cost
$6
$0.30
Designing the Super Sandwich
Ratio Tables
It takes Paul 2 hours to bike 8 miles. How
long will it take him to bike 12 miles?
cc: Microsoft.com
Time
(hours)
Distance
(miles)
2
8
?
12
Ratio Tables
It takes Paul 2 hours to bike 8 miles. How
long will it take him to bike 12 miles?
cc: Microsoft.com
Time
(hours)
Distance
(miles)
1
4
2
8
?
12
Ratio Tables
It takes Paul 2 hours to bike 8 miles. How
long will it take him to bike 12 miles?
x3
cc: Microsoft.com
Time
(hours)
Distance
(miles)
1
4
2
8
3
12
x3
Susan and Tim save at constant rates.
On a certain day, Susan had $6 and
Tim had $14. How much money did
Susan have when Tim had $35?
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2
3
4
5
6
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9
10
2
4
6
8
10
12
14
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18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9
18
27
36
45
54
63
72
81
90
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60
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80
90
100
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9
10
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6
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10
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14
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18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9
18
27
36
45
54
63
72
81
90
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20
30
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80
90
100
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9
10
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18
20
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6
9
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15
18
21
24
27
30
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8
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16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
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12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
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16
24
32
40
48
56
64
72
80
9
18
27
36
45
54
63
72
81
90
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20
30
40
50
60
70
80
90
100
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2
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4
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20
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6
9
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18
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24
27
30
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8
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20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9
18
27
36
45
54
63
72
81
90
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20
30
40
50
60
70
80
90
100
1
2
3
4
5
6
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8
9
10
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9
18
27
36
45
54
63
72
81
90
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20
30
40
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60
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80
90
100
2
5
3
7
6
14
35
Factor Puzzles
6
14
35
Factor Puzzles
3
7
6
14
2
15
35
5
Ratio Tables
Three sweaters cost $18. What is the cost
of 7 sweaters?
Number
Cost
3
18
Ratio Tables
Three sweaters cost $18. What is the cost
of 7 sweaters?
Number
1
Cost
6
3
18
Ratio Tables
Three sweaters cost $18. What is the cost
of 7 sweaters?
Number
1
Cost
6
3
18
42
7
Your Turn
The ratio of Kate's stickers to Jenna's stickers
is 7:4. Kate has 21 stickers. How many
stickers does Jenna have?
Kate’s
Stickers
7
Jenna’s
Stickers
4
21
???
Solution Strategies
Strategy
Description
Build-up strategy
Students use the ratio to build up to the
unknown quantity.
Unit-rate strategy
Students identify the unit rate and then
use it to solve the problem.
Factor-of-change strategy
Students use a “times as many strategy.
Fraction strategy
Students use the concept of equivalent
fractions to find the missing part.
Ratio Tables
Students set up a table to compare the
quantities.
Cross multiplication
algorithm
Students set up a proportion
(equivalence of two ratios), find the cross
products, and solve by using division.
Cross Multiplication Algorithm
How does this work?
3
2
Step 1: Start with two equal fractions =
9
6
Step 2: Find a common denominator using each of the
two denominators.


2 by 9 , which is multiplying by 1
6
9
Multiply 3 by 6 , which is multiplying by 1
9
6
Source: IES Practice Guide:
Developing Effective Fraction Instruction for Kindergarten
Through 8 Grade

Multiply

th
Cross Multiplication Algorithm
Step 3: Calculate the result: (2 x 9)
(3 x 6)
=
(6 x 9)
(9 x 6)
Step 4: Note that the denominators are equal. If
two equal fractions have equal denominators, then
the numerators are also equal.
So, (2 x 9) = (3 x 6)
Source: IES Practice Guide:
Developing Effective Fraction Instruction for Kindergarten
Through 8th Grade
Comparing Mixtures
There are two containers, each containing a mixture of 1
cup red punch and 3 cups lemon lime soda. The first
container is left as it is, but somebody adds 2 cups red
punch and 2 cups lemon lime soda to the second
container.
•Will the two punch mixtures taste the same? Why or why
not?
Mixture 1
Mixture 2
PERCENTS
Percents
x3
0
20
0%
25%
40
50%
x3
60
80
75%
100%
Percents
÷10
0
4
8
16
0% 5% 10% 20%
24
32
40
48
56
64
72
30%
40%
50%
60%
70%
80%
90%
÷10
80
100%
Percents
0
4
8
16
0% 5% 10% 20%
24
32
40
48
56
64
72
30%
40%
50%
60%
70%
80%
90%
80
100%
Problem Strings
• Cathy Fosnot
• Problem string for a particular strategy are
meant to be done more than once
• Not intended to be used all at once,
handed out as worksheets or used as
independent work for the students
• Helps secondary students construct
mental numerical relationships
Percents – Start Unknown
• _____ is 100% of 40
• _____ is 5% of 40
• _____ is 200% of 40
• _____ is 1% of 40
• _____ is 50% of 40
• _____ is 6% of 40
• _____ is 25% of 40
• _____ is 0.5% of 40
• _____ is 10% of 40
• _____ is 13.5% of 40
Percents – Percent Unknown
• 10 is what percent of
20
• 5 is what percent of 50
• 5 is what percent of 20
• 15 is what percent of
50
• 15 is what percent of
20
• 2 is what percent of 50
• 2 is what percent of 20
• 17 is what percent of
50
• 3 is what percent of 20
• 39 is what percent of 40
Percents – Result Unknown
• 3 is 100% of _____
• 3 is 12% of _____
• 3 is 50% of _____
• 6 is 50%of _____
• 3 is 25% of _____
• 12 is 50% of _____
• 3 is 10% of _____
• 12 is 25% of _____
• 3 is 1% of _____
• 6 is 25% of _____
Percents
Jean has 60 text messages. Thirty-five
percent of them are from Susan. How
many text messages does she have from
Susan?
Percents
Your parents took your family out to
dinner. They wanted to give the waiter a
15% tip. If the total amount of the dinner
was $42.00, what should be paid to the
waiter as a tip?
Percents
x7
0
3
0%
6
x
5% 10%
35%
60
100%
x7
If 60 is 100% then 6 is 10% and 3 is 5%. Multiply 5%
by 7 to get to 35% and 3 by 7 to get 21.
Percents
0
0%
3
6
x
5% 10%
60
35%
100%
I know 10% is 6 and 5% is 3, so
10%
10%
10%
5%
35%
6
6
6
3
21
Percent of Decrease
• A coat selling for $120 is discounted 25%.
What is the sale price?
0
0%
100%
Percent of Decrease
• A coat selling for $120 is discounted 25%.
What is the sale price?
0
0%
120
100%
Percent of Decrease
• A coat selling for $120 is discounted 25%.
What is the sale price?
0
0%
x
75%
120
100%
Percent of Increase
• In a retail store the prices were increased
60% What would be the price of an item if
the original price was $20?
0
0%
100%
Percent of Increase
• In a retail store the prices were increased
60% What would be the price of an item if
the original price was $20?
0
20
0%
100%
x
160%
Percent of Increase
• In a retail store the prices were increased
60% What would be the price of an item if
the original price was $20?
0
20
0%
100%
x
160%
Percent of Increase
• A price of a pair of shoes is increased from
$24 to $80. What is the percent of
increase?
0
0%
100%
Percent of Increase
• A price of a pair of shoes is increased from
$24 to $80. What is the percent of
increase?
0
24
0%
100%
80
x
Resources
Developing Effective Fractions Instruction for
Kindergarten Though 8th Grade IES What Works
Clearinghouse
www.commoncoretools.wordpress.com
It’s All Connected: The Power of Proportional
Reasoning to Understand Mathematics Concepts
Carmen Whitman (Math Solutions)
3/26/2016 • page 54