Finding the constant of proportionality for directly proportional relationships | Maths

Starter quiz

What letter is used to represent the constant of proportionality?

'k' ✓
Which of the following equations represent $y$ being directly proportional to some form of $x$ ?

$y=5x+1$

$y={x\over5}$ ✓

$y=\sqrt{5}x$ ✓

$y=5x$ ✓

$y=5\sqrt{x}$
Match each equation to its correct value of $k$ .

$y={\sqrt4{x}}$

⇔

${\sqrt4}$ ✓

$y={3x\over4}$

⇔

$3\over4$ ✓

$y={\sqrt3{x}}$

⇔

${\sqrt3}$ ✓

$y={x\over4}$

⇔

$1\over4$ ✓

$y={5x\over6}$

⇔

$5\over6$ ✓
$y$ is directly proportional to $x$ . When $x$ = 70, $y$ = 245. Write an equation connecting $x$ and $y$ .

$y=3.5x+1$

$y=3.5x-1$

$y=3.5x$ ✓

$y=x+3.5$
$y$ is directly proportional to $x$ . When $x$ = 8, $y$ = 13.2. Write an equation connecting $x$ and $y$ and use this to find the value of $y$ when $x$ = 35.

'57.75' ✓
$y$ is directly proportional to $x$ . When $x$ = 8, $y$ = 13.2. Write an equation connecting $x$ and $y$ and use this to find the value of $x$ when $y$ = 66.

'40' ✓

Exit quiz

Which of the following equations represent $y$ being directly proportional to $x$ or a power of $x$ in some way?

$y=3x^2$ ✓

$y=3\sqrt{x}$ ✓

$y=3^x$

$y=2x^3$ ✓

$y={x^2\over3}$ ✓
Match each equation to its correct value of $k$ .

$a=5{b}^2$

⇔

5 ✓

$a=2{b}^3$

⇔

2 ✓

$a=3{\sqrt[3]{b}}$

⇔

3 ✓

$a=\sqrt{3}{\sqrt{b}}$

⇔

$\sqrt{3}$ ✓
Match the following:

$a\propto{b}$

⇔

$a$ is directly proportional to $b$ ✓

$a\propto{b}^3$

⇔

$a$ is directly proportional to ${b}^3$ ✓

$a\propto{\sqrt[3]{b}}$

⇔

$a$ is directly proportional to ${\sqrt[3]{b}}$ ✓

$a\propto{\sqrt{b}}$

⇔

$a$ is directly proportional to ${\sqrt{b}}$ ✓

$a\propto{b}^2$

⇔

$a$ is directly proportional to ${b}^2$ ✓
$y$ is directly proportional to $x^2$ . When $x$ = 6, $y$ = 108. Write an equation connecting $x$ and $y$ and use this to find the value of $y$ when $x$ = 11.

'363' ✓
$y$ is directly proportional to ${\sqrt[3]{x}}$ . When $x$ = 64, $y$ = 20. Write an equation connecting $x$ and $y$ and use this to find the value of $y$ when $x$ = 343.

'35' ✓
$y$ is directly proportional to ${\sqrt[3]{x}}$ . When $x$ = 64, $y$ = 20. Write an equation connecting $x$ and $y$ and use this to find the value of $x$ when $y$ = 25.

'125' ✓

Worksheet

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Lesson Details

Key learning points

y can be directly proportional to x^n
This is because x^n could be written as a different variable.

Common misconception

Setting up the initial proportion statement and hence the general form of the equation the wrong way around.

Pay particularly attention to the language used. Give pupils the opportunity to verbalise what different proportion statements mean. E.g. y ∝ x^2, 'read this aloud using the correct mathematical vocabulary'.

Keywords

Direct proportion - Two variables are in direct proportion if they have a constant multiplicative relationship