6.RP - Understand ratio concepts and use ratio reasoning to solve problems.

• 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

• 2. Understand the concept of a unit rate a/b associated with a ratio a : b with b ≠ 0, and use rate language in the context of a ratio relationship.

• 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.– a. Make tables of equivalent ratios relating quantities with whole number

measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

– b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

Highlighted Problem

• If 2 pounds of beans cost $5, how much will 15 pounds of beans cost?

Method 1

Pounds 2Dollars $5

How much would 4 pounds cost?

That’s just 2 more pounds and 5 more dollars.

+ 2

+ $5

Method 1

Pounds 2 4Dollars $5 $10

How much would 6 pounds cost?

That’s just 2 more pounds and 5 more dollars.

+ 2

+ $5

Method 1

Pounds 2 4 6Dollars $5 $10 $15

How much would 8 pounds cost?

That’s just 2 more pounds and 5 more dollars.

+ 2

+ $5

Method 1

Pounds 2 4 6 8Dollars $5 $10 $15 $20

How much would 10 pounds cost?

That’s just 2 more pounds and 5 more dollars.

+ 2

+ $5

Method 1

Pounds 2 4 6 8 10Dollars $5 $10 $15 $20 $25

How much would 12 pounds cost?

That’s just 2 more pounds and 5 more dollars.

+ 2

+ $5

Method 1

Pounds 2 4 6 8 10 12Dollars $5 $10 $15 $20 $25 $30

How much would 14 pounds cost?

That’s just 2 more pounds and 5 more dollars.

+ 2

+ $5

Method 1

Pounds 2 4 6 8 10 12 14Dollars $5 $10 $15 $20 $25 $30 $35

16 pounds is over the 15 we are looking for, so How much would 1 pounds cost?

+ 2

+ $5

Method 1

Pounds 2 4 6 8 10 12 14Dollars $5 $10 $15 $20 $25 $30 $35

16 pounds is over the 15 we are looking for, so How much would 1 pounds cost?

That’s half of our starting 2 pounds and $5.

2

2

Method 1

Pounds 2 4 6 8 10 12 14 1Dollars $5 $10 $15 $20 $25 $30 $35 $2.50

14 pounds costs $35 and then 1 more pound is another $2.50, so…

=

=

+

+

Method 1

Pounds 2 4 6 8 10 12 14 1 15Dollars $5 $10 $15 $20 $25 $30 $35 $2.50 $37.50

14 pounds costs $35 and then 1 more pound is another $2.50, so that makes $37.50 in all.

Method 2

Pounds 2 1Dollars $5 $2.50

You find 1 pound first.

2

2

Method 2

Pounds 2 1 15Dollars $5 $2.50 $37.50

If you know how much it costs for each pound then you can find any number of pounds by multiplying.

x 15

x 15

Method 3

Pounds 2 15Dollars $5 $37.50

With this perspective, the second column is seen as the first column times a number. To solve the proportion one first finds this number.

x 7.5

x 7.5

The previous method, done in one step.