Math 10Chapter 1 – Linear Measurement and Proportional

Reasoning

Lesson 1 – Ratio and Proportional Reasoning

Todays Objectives Solve problems that involve linear measurement using:

SI and imperial units of measure Estimation strategies Measurement strategies

Provide referents for linear measurements Justify the choice of units used for determining a

measurement in a problem Apply proportional reasoning to problems that involve

conversions between SI and imperial units of measure

Lesson 1.6 – Ratio and Proportional Reasoning We will soon use proportional reasoning to

convert linear units in one measurement system to units in the other.

Before doing this, it is important to review the concept of ratio and how it relates to proportion.

Ratio A ratio compares one number to another, showing a

relationship between the two. It is usually shown as a fraction where the first number is

the numerator and the second number is the denominator. Eg: ratio of 7 to 11

Ratio of 8 to 3

Ratio of 5 to 7

Sometimes ratios are shown with a colon between the numbers where the first is the numerator and the second the denominator 7:9 11:3

Proportion When one ratio is equal to another, it is called a proportion Notice that in a proportion, the “diagonal products” are

equal.

1 x 16 = 2 x 8 16 = 16 Finding the products of numerators and denominators of

alternate ratios and then setting them equal to each other is sometimes called “cross multiplication”

Proportional Reasoning Proportional reasoning involves the ability to

understand and compare ratios, and to predict and produce equivalent ratios.

It requires comparisons between quantities and also between the relationships between quantities.

Examples - Ratios There are 11 boys and 9 girls in a soccer club. Find the

following ratios:

a. Ratio of boys to girls

b. Ratio of girls to boys

c. Ratio of boys to all

members of the club

The number of boys is 11, it is the numeratorThe number of girls is 9, it is the denominatorThe ratio is

The number of girls is 9, it is the numeratorThe number of boys is 11, it is the denominatorThe ratio is

The number of boys is 11, it is the numerator The number of all members of the club is 20, it is the denominatorThe ratio is

Examples – Proportion 1. Find the value of n if

Solution: the diagonal products are equal so we will use cross multiplication

5 x 9 = 3 x n 45 = 3n 15 = n

2. If 4 L of juice are needed for 6 people, how many litres are needed for 9 people?

Solution: We set up a proportion and solve. The ratio of 4 litres to 6 people is equal to the ratio

of n litres to 9 people

litres litres

people people

4 x 9 = 6 x n

36 = 6n

n = 6

There are 6 litres of juice needed for 9 people

Choosing appropriate units of measure There are 2 systems of measurement and

many units to choose from when we are measuring distances: Metric System: millimeter (mm), centimeter (cm),

meter (m), kilometer (km) Imperial System: inches (in. or “), feet (ft. or ‘),

yards (yd.), miles (mi.)

Units of measure: Distance Metric Units

1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm

Imperial Units 1 mi = 1760 yd 1 yd = 3 ft 1 ft = 12 in

When should we use each unit? We should use different units when

measuring things of different sizes Example: GOOD!

The length of my walk to school is about 2 km, or 2000 m

Example: BAD! 555555! The length of my walk to school is about 200,000

cm, or 2,000,000 mm

When should we use each unit? Example: GOOD!

My pen is about 5.5 inches in length

Example: BAD! 555555! My pen is about 8.68 x 10^ -5 miles in length

Referents We can use everyday objects to act as

referents for various units of measure A referent is something we can use to

estimate a given unit of measure Example: We can use the width of our

pinkie finger as a referent for 1 cm

Homework In your Vocabulary books, make entries for

the words listed at the start of your handout Provide a referent for each unit of measure

listed at the end of your handout Pg. 11, #2-6, pg. 12, Reflect Question