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1. Proportional relationships: y = kx

Warm-up

Three quick questions from earlier lessons.

Discover

3 pounds of apples cost 6 dollars. At the same price, what do 5 pounds cost?

10 dollars
8 dollars

Key idea

Proportional means $y = kx$ .

In a proportional relationship

y = kx

, where

k = \tfrac{y}{x}

is the constant of proportionality (the unit rate). Its graph is a line through the origin, and

k

is the slope.

Worked example

A printer prints 45 pages in 3 minutes.

k = 15

pages per minute

45 ÷ 3.

y = 15x

Equation.

120 pages in 8 minutes

x = 8.

Your turn, guided

Your turn: $y$ is proportional to $x$ , and $y = 28$ when $x = 4$ . Find $k$ .

7

28 ÷ 4.

Practice ★

$y$ is proportional to $x$ . When $x = 2$ , $y = 10$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 3$ , $y = 15$ . Find the constant of proportionality $k$ in $y = kx$ .

Which equation represents a proportional relationship?
1. $y = 2x$
2. $y = 2x + 9$
3. $y = x + 2$

Watch out!

Practice ★★

A train travels 21 miles in 3 minutes at a constant speed. How far does it travel in 12 minutes?
A train travels 30 miles in 6 minutes at a constant speed. How far does it travel in 10 minutes?

Exit ticket

Three questions to check you have got it.

$y$ is proportional to $x$ . When $x = 2$ , $y = 10$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 3$ , $y = 15$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 4$ , $y = 28$ . Find the constant of proportionality $k$ in $y = kx$ .

Summary card

Proportional relationships

$y = kx$ , $k = y \div x$ ; the graph passes through $(0, 0)$ .

Watch out: scale by multiplying, not by adding.

2. Proportional relationships in tables, graphs and equations; unit rates with fractions

Warm-up

Three quick questions from earlier lessons.

Discover

A graph is a straight line through $(0, 0)$ and $(4, 12)$ . Which point is also on it?

$(1, 3)$
$(1, 9)$

Key idea

Four ways to see a proportional relationship.

In a table,

y \div x

is the same in every row. On a graph, the points lie on a line through the origin, and the point

(1, k)

shows the constant. The equation is

y = kx

(or

x = \tfrac{1}{k} y

). The unit rate can be a fraction: divide the quantities even when they are fractions.

Worked example

A worker paints $\tfrac{2}{3}$ of a wall in $\tfrac{1}{2}$ hour.

\tfrac{2}{3} \div \tfrac{1}{2} = \tfrac{4}{3}

walls per hour

Walls ÷ hours.

y = \tfrac{4}{3} x

Equation.

Your turn, guided

Your turn: $y = 6x$ . Write $x$ in terms of $y$ .

x = \tfrac{y}{6}

Divide by 6.

Practice ★

The graph of a proportional relationship passes through $(5, 40)$ . What is $y$ when $x = 1$ ?
The graph of a proportional relationship passes through $(3, 27)$ . What is $y$ when $x = 1$ ?

For $x = 1, 2, 3$ , which list of $y$ -values makes a proportional relationship?
1. $6, 12, 18$
2. $7, 8, 9$
3. $6, 12, 24$

Watch out!

Practice ★★

A snail crawls $\tfrac{1}{ 7 }$ of a meter in $\tfrac{1}{ 4 }$ of an hour. What is its speed in meters per hour?
A snail crawls $\tfrac{1}{ 3 }$ of a meter in $\tfrac{1}{ 8 }$ of an hour. What is its speed in meters per hour?

Boss problem. A hose fills a tank at a constant rate: 33 liters in 3 minutes. Let $y$ be the liters after $x$ minutes.

Exit ticket

Three questions to check you have got it.

The graph of a proportional relationship passes through $(5, 40)$ . What is $y$ when $x = 1$ ?
The graph of a proportional relationship passes through $(3, 27)$ . What is $y$ when $x = 1$ ?
The graph of a proportional relationship passes through $(5, 10)$ . What is $y$ when $x = 1$ ?

Summary card

Tables, graphs, equations

Constant ratio; line through the origin; $(1, k)$ ; $y = kx$ .

Watch out: a constant difference is not a constant ratio.

Answers: Proportional relationships

1. Proportional relationships: y = kx

$y = 2x$

$84$
$50$

2. Proportional relationships in tables, graphs and equations; unit rates with fractions

$6, 12, 18$

$\frac{4}{7}$
$\frac{8}{3}$

$17$