Transcript Ratio and Proportional Relationships

CCSSM
National Professional
Development
Ratios and Proportions
Grades 6/7
Sara Anaya
Ayşe Şahin
Diana White
6th Grade
7th Grade
6.RP.1 - Definition of ratio
6.RP.2 - Unit rate(s)
associated to ratio
7.RP.1 – Unit Rates associated with
ratios (with fractions and decimals)
Whole numbers
(6.RP.3) Given a proportional relationship represent it a
variety of ways:
. A table
. Plotting points on a coordinate graph.
. Tape diagrams
7.RP.2.
Using the representations from 6th
grade
a. Decide whether two quantities are in
a proportional relationship.
b. Identify the constant of
proportionality
. Double number line diagrams.
6.RP.3 –Use ratios and rate reasoning to solve (single
step)real world mathematical problems using the
representations above and
Using the unit rate
7.RP.3 Use proportional
relationships to solve (multi-step)
ratios and percent problems.
c. See % as a rate
d. Ratio reasoning and manipulating units by
multiplying and dividing
7.RP.2
c. Represent proportional
6th graders are not asked to express proportional
relationships algebraically
relationships by equations.
d. Understanding the proportional
relationship on a graph most
importantly (0,0) and (1.r)
3
6.RP.1: The language of ratio and
proportion
• An example of how to introduce the notion of a ratio:
– Illustrative Mathematics 6.RP.1 Task example:
• Games at recess
Anaya, Şahin, White
4
6.RP.2 and 7.RP.1
• This recipe has a ratio of 3 cups of flour to 4 cups of
sugar.
• How many cups of flour would you use for 1 cup of
sugar?
– Answer this question using as many different methods as
possible.
Anaya, Şahin, White
Tape Diagram Representation
1 cup of flour
1 cup of flour
1 cup of flour
¾ cups of sugar
Author's Name IM&E CCSSM National PD
Table
# cups of flour
# cups of sugar
3
4
x2
x3
x4
x½
x¼
Author's Name IM&E CCSSM National PD
x2
6
8
9
12
x4
12
16
x½
3/2
2
3/4
1
x3
x¼
1
2
3
Cups
Cups
1
2
1
Cups of flour
3
2
4
3
Cups of sugar
1
2
3
4
Double number line representations don’t work as well
when the units are the same.
Author's Name IM&E CCSSM National PD
Changing the problem slightly, a double number line
becomes an appropriate representation:
This recipe calls for 3 tablespoons of butter for every 4 cups
of sugar. How many tablespoons of butter would you use
for 1 cup of sugar?
0 tbs
3 tbs
0c
4c
Tablespoons
Cups
Author's Name IM&E CCSSM
National PD
¸4
0 tbs ? tbs
3 tbs
0c
4c
Tablespoons
Cups
1c
¸4
• Using double number lines we directly translate the
problem into asking: what is 3 /4?
• (Standard 5.NF.3) ¾ tbs of butter for every cup of sugar.
Author's Name IM&E CCSSM National PD
10
• This recipe has a ratio of 3 cups of flour to 4 cups of
sugar.
– How many cups of sugar would you use for 1 cup of flour?
Solve this problem using a method you did not
use for the previous question.
Anaya, Şahin, White
11
7th Grade Unit Rate Problem
• This recipe calls for ¾ cup of flour for every ½ cup of
sugar. How many cups of flour would you use for every
1 cup of sugar?
Answer this question using as many different
methods as possible.
Anaya, Şahin, White
12
• This recipe calls for ¾ cup of flour for every ½ cup of
sugar. How many cups of flour would you use for every
1/4 cup of sugar?
Answer this question using as many different
methods as possible.
Anaya, Şahin, White
13
6.RP.3 and 7.RP.3
• Suppose Abby’s orange paint is made by mixing 1 cup
red paint for every 3 cups yellow paint and Zack’s
orange paint is made by mixing 3 cups red for every 5
cups yellow. Whose paint is yellower?
Anaya, Şahin, White
14
7th Grade Extension
• Suppose Abby makes orange paint by mixing 1 cup red
paint for every 3 cups yellow paint. She can only find
100 cups of the red paint, and as much of the yellow
paint as she likes. How many cups of orange paint can
she make?
Answer this question using as many different
methods as possible.
Anaya, Şahin, White