What is a ratio?

A ratio is a comparison of two quantities or measures

What is the ratio of males to females in this room?

Part-to-part ratios

• Comparing part of a whole to another part of the same whole

Males : Females or Females : Males

• What is the whole in this example?

Part-to-Whole Ratios

• are a comparison of a part to the whole

• females : all teachers in the room

Using the diagram above, create as many ratios as you can and indicate if they are

part-to-part or part-to-whole ratios

Discuss the following statement

All fractions are ratios but not all ratios are fractions

Ratios that compare two measures are called rates

• km/L

• km/h

• $/item

• $/kg

• cm/km

To be proportional thinkers, students need to see ratios as multiplicative relationships

• What multiplicative relationships do you see in

a)9:4.5

b)2:10

• Sally measures two flowers to find they are 8 cm and 12 cm respectively. Two weeks later, they are 12 cm and 16 cm tall. Which flower grew more?

What is a proportion?

• A statement of equality between two ratios.

• The two ratios represent the same relationship

• Write two different ratios that tell what part of the rectangle is shadedo 2:3 and 4:6 represent the same relationship of shaded

parts to whole shape

• Write a correct proportion o 2:3 = 4:6 or

o Can be read as: 2 is to 3 as 4 is to 6

2 43 6

Using proportional reasoning

to

solve problems

• On which cards is the ratio of trucks to boxes the same?

• We describe this relationship as “within the ratio”

• On which cards is the ratio of trucks to trucks the same as the ratio of boxes to boxes ?

• We describe this relationship as “between the ratios”.

Solve this using a “within the ratio” relationship

• Since 5 is of 10, think 1.4 = of ?

• ? = 2.8

.?

5 1 4

10

12

12

Solve this using a “within the ratio” relationship

. ?.

2 1

18 9 3

Solve this using a “between the ratios” relationship

• Since 12 is 4 times 3, then ? is 4 times 2.

• ? = 8

OR

• Since 3 is of 12, then 2 is of ?

• ? = 8

• The 4 and the would be the “factor of change” from one ratio to another.

?

23 12

14

14

14

Solve this using a “between the ratios” relationship

?

715 30

Solve each one using a different method and explain

?

2 109

?

630 18

Solve each one using a different method and explain

. ?

. .

1 23 6 4 5

?

625 10

Would either relationship work to solve this problem? Explain.

• If 5 hectares of land is needed to grow 75 pine trees, how many hectares of land are need to grow 225 pine trees?

How would you solve each problem below? Look for 2 ways that are intuitive and be ready to

discuss your reasoning.

• Tammy bought 3 widgets for $2.40. At the same price what would 10 widgets cost?

• Tammy bought 4 widgets for $3.75. How much would a dozen widgets cost?

• Cross products should be used only if other methods are not more intuitive.

•

• Why does the cross product method work?

6 1113 x

• Within:

• is the unit price

5

$0.79

12

$ ?

5 120.79 ?

0.79 ?5 12

0.795

or

Solve with cross-products

• Why does this work?

Unit price (within ratio)

5 120.79 ?

12 apples $0.79? =

5 apples

• Between:

5

$0.79

12

$ ?

• is the factor of change

5 0.7912 ?

12 ?5 0.79

or

125

Solve with cross-products

• Why does this work?

Factor of change (between ratio)

5 120.79 ?

12 apples $0.79? =

5 apples

Proportional Thinkers

• Have a sense of covariation

• Distinguish proportional relationships from those that are not.

• Develop a wide variety of strategies for solving proportions and comparing ratios