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

Three quick questions from earlier lessons.

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

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.

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

x = \tfrac{y}{6}

Divide by 6.

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. $6, 12, 24$

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.

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 $x = 1$ ?

Tables, graphs, equations

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

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

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