Tables and rules | Year 7 | Ascent Library

01 / LEARN

Understand the idea.

Match each input with its output

A table pairs each input with its output. For $y = 3 n + 2$ , replace $n$ with the input, multiply by $3$ , then add $2$ . Apply that same order at every row.

The table is a record of the supplied or calculated pairs. Several rows matching a rule show that the rule fits those rows; they do not prove it describes every other input unless the pattern assumption is supplied.

Rule y=3n+2
n	y
0	2
1	5
2	8
3	11

Find, test and reverse

For a simple linear pattern, compare outputs when the input increases by $1$ . A repeated increase of $5$ suggests a term $5 n$ . Then use one pair to find the fixed adjustment and check it against every given row.

To recover an input from a known output, reverse the rule. For $3 n + 2 = 17$ , subtract $2$ before dividing by $3$ . Reverse operations in reverse order so that the final value is the original input.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Complete a table

Find outputs for $y = 3 n + 2$ when n=1,2,3,4.

Completed pairs
n	y
1	5
2	8
3	11
4	14

Worked solution

1
Substitute the first two inputs.
$\begin{aligned} n = 1 : & y = 3 \times 1 + 2 = 5 \\ n = 2 : & y = 3 \times 2 + 2 = 8 \end{aligned}$
Multiplication is completed before adding 2 for each input.
2
Substitute the remaining inputs.
$\begin{aligned} n = 3 : & y = 3 \times 3 + 2 = 11 \\ n = 4 : & y = 3 \times 4 + 2 = 14 \end{aligned}$
The same rule gives the other two rows of the table.
3
Check the changes down the table.
$8 - 5 = 11 - 8 = 14 - 11 = 3$
Increasing the input by 1 adds one more group of 3 to the output.

Answer

5,8,11,14.

EXAMPLE 02

Find and reverse a rule

Inputs 1,2,3,4 give outputs 4,9,14,19. Find a simple rule and the input giving output 34.

Worked solution

1
Find the repeated output increase.
$\begin{aligned} 9 - 4 & = 5 \\ 14 - 9 & = 5 \\ 19 - 14 & = 5 \end{aligned}$
For these unit input increases, a simple linear rule has coefficient 5.
2
Determine and test the fixed adjustment.
$\begin{aligned} 5 \times 1 - 1 & = 4 \\ 5 \times 2 - 1 & = 9 \\ 5 \times 3 - 1 & = 14 \\ 5 \times 4 - 1 & = 19 \end{aligned}$
Subtracting 1 from 5 times the input fits every supplied pair, so use $y = 5 n - 1$ .
3
Set the output equal to 34.
$\begin{aligned} 5 n - 1 & = 34 \\ 5 n & = 35 \\ n & = 7 \end{aligned}$
Add 1 to both sides, then divide both sides by 5 to recover the input.
4
Check the recovered input.
$5 \times 7 - 1 = 34$
Input 7 produces the requested output under the chosen rule.

Answer

$y = 5 n - 1$ ; input 7.

03 / YOUR TURN

Try it yourself.

Check your understanding

For $y = 4 n - 3$ , find the output when n=5.

Show answer & reasoning

17.

Substituting $n = 5$ gives $y = 4 \times 5 - 3$ . Multiply first to obtain $20 - 3 = 17$ , so the output is $17$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

For $y = 2 n + 5$ , find the outputs for $n = 0, 1, 2$ .
2 marks
For $y = 6 n$ , find the output at $n = 4$ .
2 marks
Complete the missing output using the constant-addition pattern.
2 marks
Input–output pairs
Input Output
0 2
1 6
2 10
3 ?
Give a simple rule for the supplied pairs and check it at $n = 2$ .
3 marks
Pairs
$n$ $y$
0 2
1 6
2 10
3 14
For $y = 3 n + 1$ , find the input giving output 22.
3 marks
Does $y = 2 n + 4$ match every pair $(1, 6)$ , $(2, 8)$ , $(3, 11)$ ? Explain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

For $y = 2 n + 5$ , find the outputs for $n = 0, 1, 2$ .
2 marks
$2 \times 0 + 5 = 5$ ; $2 \times 1 + 5 = 7$ ; $2 \times 2 + 5 = 9$ .
5,7,9.
For $y = 6 n$ , find the output at $n = 4$ .
2 marks
$6 \times 4 = 24$ .
24.
Complete the missing output using the constant-addition pattern.
2 marks
Input–output pairs
Input Output
0 2
1 6
2 10
3 ?
Add 4 each time: 2,6,10,14.
14.
Give a simple rule for the supplied pairs and check it at $n = 2$ .
3 marks
Pairs
$n$ $y$
0 2
1 6
2 10
3 14
Each input increase of 1 adds 4 to output.
At $n = 0$ the output is 2.
$y = 4 n + 2$ ; at $n = 2$ it gives 10.
For $y = 3 n + 1$ , find the input giving output 22.
3 marks
$3 n + 1 = 22$ .
Subtract 1: $3 n = 21$ .
Divide by 3: $n = 7$ .
7.
Does $y = 2 n + 4$ match every pair $(1, 6)$ , $(2, 8)$ , $(3, 11)$ ? Explain.
3 marks
$2 \times 3 + 4 = 10$ .
Testing only the first two rows would miss the mismatch.
No. At input 3 it gives 10, not 11.

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Curriculum reference & lesson scope

Australian Curriculum v9.0 / South Australian Curriculum

Year 7 coordinate, pattern and authentic-graph interpretation scope. Includes prerequisite four-quadrant plotting; formal gradient/intercept techniques are not required here.

Official curriculum source ↗

Input–output pairs
Input	Output
0	2
1	6
2	10
3	?

Pairs
$n$	$y$
0	2
1	6
2	10
3	14

Input–output pairs
Input	Output
0	2
1	6
2	10
3	?

Pairs
$n$	$y$
0	2
1	6
2	10
3	14

In this lesson

Understand the idea.

Match each input with its output

Find, test and reverse

See the method in action.

Complete a table

Substitute the first two inputs.

Substitute the remaining inputs.

Check the changes down the table.

Find and reverse a rule

Find the repeated output increase.

Determine and test the fixed adjustment.

Set the output equal to 34.

Check the recovered input.

Try it yourself.

Check your understanding

Keep practising