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1. Converting units of measurement

Warm-up

Three quick questions from earlier lessons.

Discover

A table is 2 meters long. About how many centimeters is that?

200
0.02

Key idea

Smaller units: more of them.

To convert, use the rate between the units (1 ft = 12 in, 1 yd = 3 ft, 1 m = 100 cm, 1 kg = 1000 g). Converting to a smaller unit, multiply; to a larger unit, divide.

Worked example

Convert 54 inches to feet.

54 \div 12 = 4.5

Feet are larger: divide.

Your turn, guided

Your turn: 3 kg in grams.

3000

× 1000.

Practice ★

How many inches are in 5 feet? (1 foot = 12 inches)
How many inches are in 2 feet? (1 foot = 12 inches)

How many meters are 600 centimeters? (1 m = 100 cm)

Watch out!

Practice ★★

A rope is 11 yards and 2 feet long. How long is it in feet? (1 yard = 3 feet)
A rope is 10 yards and 2 feet long. How long is it in feet? (1 yard = 3 feet)

Exit ticket

Three questions to check you have got it.

How many inches are in 5 feet? (1 foot = 12 inches)
How many inches are in 2 feet? (1 foot = 12 inches)
How many inches are in 7 feet? (1 foot = 12 inches)

Summary card

Converting units

To a smaller unit: multiply; to a larger unit: divide.

Watch out: check that the answer makes sense in size.

2. Unit rates: price per item, speed

Warm-up

Three quick questions from earlier lessons.

Discover

4 tickets cost 20 dollars. Which describes the price?

5 dollars per ticket
0.2 dollars per ticket

Key idea

A unit rate is the amount for one.

A unit rate tells how much for one unit: dollars per pound, miles per hour. Every ratio

a : b

gives two unit rates,

\tfrac{a}{b}

and

\tfrac{b}{a}

; choose the one the question asks for. Compare prices or speeds with unit rates.

Worked example

150 miles in 3 hours.

50 miles per hour

150 ÷ 3.

In 5 hours:

5 \cdot 50 = 250

miles

Use the rate.

Your turn, guided

Your turn: 6 pens cost 9 dollars. Dollars per pen.

1.5

9 ÷ 6.

Practice ★

4 pounds of apples cost 20 dollars. What is the price per pound (dollars)?
3 pounds of apples cost 27 dollars. What is the price per pound (dollars)?

A car travels 175 miles in 5 hours at a constant speed. How far does it go in 9 hours?

Watch out!

Practice ★★

Store A sells 2 notebooks for 8 dollars. Store B sells 11 for 66 dollars. What is the lower price per notebook (dollars)?
Store A sells 6 notebooks for 24 dollars. Store B sells 10 for 90 dollars. What is the lower price per notebook (dollars)?

Boss problem. Mia runs 5 km in 30 minutes at a steady pace.

Exit ticket

Three questions to check you have got it.

4 pounds of apples cost 20 dollars. What is the price per pound (dollars)?
3 pounds of apples cost 27 dollars. What is the price per pound (dollars)?
8 pounds of apples cost 72 dollars. What is the price per pound (dollars)?

Summary card

Unit rates

Amount for one: divide by the number of units.

Watch out: which quantity is per what?

3. Mass and volume as a rate

Warm-up

Three quick questions from earlier lessons.

Discover

1 cm³ of iron has a mass of about 8 g. About what is the mass of 5 cm³ of iron?

40 g
13 g

Key idea

Mass and volume form a rate.

For one material, mass and volume are proportional. The mass per unit of volume (grams per cm³) is a unit rate: mass = rate × volume, and rate = mass ÷ volume. A larger rate means a denser material.

Worked example

A 6 cm³ piece of a metal has a mass of 54 g.

54 ÷ 6 = 9 g per cm³

Unit rate.

9 × 10 = 90 g

Mass of 10 cm³.

Your turn, guided

Your turn: 1 cm³ of a wood has a mass of 3 g. Mass of 12 cm³?

36

3 × 12.

Practice ★

Each cubic centimeter of a metal has a mass of 5 grams. What is the mass, in grams, of a piece with a volume of 6 cm³?
Each cubic centimeter of a metal has a mass of 4 grams. What is the mass, in grams, of a piece with a volume of 14 cm³?

A stone has a mass of 8 grams and a volume of 4 cm³. What is the mass of 1 cm³ of the stone, in grams?

Watch out!

Practice ★★

Block A has a mass of 108 g and a volume of 27 cm³. Block B has a mass of 12 g and a volume of 2 cm³. Which block is made of the denser material (more grams per cm³)?
1. Block B
2. Block A, because it has the greater mass
3. Neither: the masses tell us nothing about it
Block A has a mass of 84 g and a volume of 28 cm³. Block B has a mass of 14 g and a volume of 2 cm³. Which block is made of the denser material (more grams per cm³)?
1. Block B
2. Block A, because it has the greater mass
3. Neither: the masses tell us nothing about it

A stone has a mass of 8 grams and a volume of 4 cm³. What is the mass of 1 cm³ of the stone, in grams?

Exit ticket

Three questions to check you have got it.

Each cubic centimeter of a metal has a mass of 5 grams. What is the mass, in grams, of a piece with a volume of 6 cm³?
Each cubic centimeter of a metal has a mass of 4 grams. What is the mass, in grams, of a piece with a volume of 14 cm³?
Each cubic centimeter of a metal has a mass of 3 grams. What is the mass, in grams, of a piece with a volume of 4 cm³?

Summary card

Mass and volume

Grams per cm³ = mass ÷ volume; mass = grams per cm³ × volume.

Watch out: divide mass by volume, not volume by mass.

4. Percent of a quantity; finding the percent and the whole

Warm-up

Three quick questions from earlier lessons.

Discover

A class of 40 students: 25% walk to school. How many walk?

Key idea

Percent means per hundred.

p\%

of a whole is

\tfrac{p}{100} \times \text{whole}

. What percent? part ÷ whole × 100. Find the whole: part ÷

\tfrac{p}{100}

Worked example

30% of a number is 12.

\text{whole} = 12 \div 0.3

0.3 × whole = 12.

40

Divide.

Your turn, guided

Your turn: what percent of 80 is 20?

25

20 ÷ 80 × 100.

Practice ★

What is 50% of 340?
What is 5% of 300?

What percent of 140 is 7?

Watch out!

Practice ★★

60% of a number is 228. What is the number?
45% of a number is 162. What is the number?

Exit ticket

Three questions to check you have got it.

What is 50% of 340?
What is 5% of 300?
What is 40% of 400?

Summary card

Percent of a quantity

$p\% = p/100$ ; percent = part ÷ whole × 100.

Watch out: divide the part by the whole, not the reverse.

Answers: Unit rates and percentages

1. Converting units of measurement

$60$
$24$

$35$
$32$

$60$
$24$
$84$

2. Unit rates: price per item, speed

$315$

$48$

3. Mass and volume as a rate

$30$
$56$

Block B
Block B

$30$
$56$
$12$

4. Percent of a quantity; finding the percent and the whole

$170$
$15$

$380$
$360$

$170$
$15$
$160$