Year 6 Maths Patterns & Rules

Determine a rule from a table of values

Q: What is a two-step rule?

A rule with two operations in order, such as multiply by 4, then add 3.

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Theory

A table of values pairs each input with an output made by the same rule. When the inputs go up by $1$, the gap between outputs tells you the multiplier; then you find what to add or subtract. This Year 6 maths page shows how to find one-step and two-step rules, test them on every row, and use them to find any output or to work back to an input.

An input goes into the rule and an output comes out. The rule must work for every row.

A one-step rule adds, subtracts, multiplies or divides once: input $\times 7$, or input $+ 9$.

A two-step rule does two operations in order, such as multiply by $4$, then add $3$.

When the inputs go up by $1$, the outputs go up by the multiplier. Outputs $7, 11, 15, 19$ go up by $4$, so the rule starts "multiply by $4$"; $1 \times 4 = 4$ and $7 - 4 = 3$, so it is multiply by $4$, then add $3$.

To work backwards, undo the steps in reverse order: subtract $3$, then divide by $4$.

The outputs go up by $4$: multiply by $4$. Then $1 \times 4 + 3 = 7$, so add $3$.

A two-step rule as a machine. Going back, undo the steps: $-3$ then $\div 4$.

Finding a rule when the inputs go up by 1:

\[ \text{multiplier} = \text{gap between outputs} \]

multiplier = gap between outputs

\[ \text{extra} = \text{output} - \text{input} \times \text{multiplier} \]

extra = output - input \times multiplier

\[ \text{output} = \text{input} \times 4 + 3 \]

output = input \times 4 + 3

Test every row. A rule that only fits the first row is not the rule.

          Inputs that skip. If the inputs go up by \(2\), the output gap is twice the multiplier.
        

Finding the rule

Check that the inputs go up by $1$ (if not, divide the output gap by the input gap).
Gap between outputs = multiplier.
Multiply the first input; add or subtract to reach the first output.
Test the rule on every row.

Working backwards from an output

Undo the last step first (add becomes subtract).
Then undo the multiply by dividing.

Example 1 — One-step multiply

Inputs $1, 2, 3$ give outputs $7, 14, 21$. Find the rule and the output for input $9$.

Solution

$9 \times 7$

$=$

$63$

9 \times 7 = 63

Multiply by $7$; output $63$.

Example 2 — One-step add

Inputs $1, 2, 3$ give outputs $17, 18, 19$. Find the rule and the output for input $25$.

Solution

$25 + 16$

$=$

$41$

25 + 16 = 41

Add $16$; output $41$.

Example 3 — Two-step rule

Use the first diagram. Find the rule and the output for input $12$.

Solution

$12 \times 4 + 3$

$=$

$51$

12 \times 4 + 3 = 51

Multiply by $4$, then add $3$; output $51$.

Example 4 — Test every row

Inputs $1, 2, 3$ give outputs $4, 7, 10$. Max says the rule is "add $3$". Is he right?

Solution

$2 + 3$

$=$

$5$

2 + 3 = 5, not 7

No: add $3$ gives $5$ for input $2$, not $7$. The outputs go up by $3$, so the rule is multiply by $3$, then add $1$.

Example 5 — Inputs that skip

Inputs $2, 4, 6$ give outputs $10, 16, 22$. Find the rule and the output for input $10$.

Solution

Inputs go up by $2$ and outputs by $6$, so the multiplier is $6 \div 2 = 3$. $2 \times 3 = 6$ and $10 - 6 = 4$.

$10 \times 3 + 4$

$=$

$34$

10 \times 3 + 4 = 34

Multiply by $3$, then add $4$; output $34$.

Example 6 — Work backwards

The rule is multiply by $4$, then add $3$. An output is $63$. What is the input?

Solution

$63 - 3$	$=$	$60$
$60 \div 4$	$=$	$15$

(63 - 3) \div 4 = 15

$15$. Check: $15 \times 4 + 3 = 63$.

Example 7 — Hiring bikes

Bike hire: $1$ hour costs $\$13$, $2$ hours $\$18$, $3$ hours $\$23$.
(i) Find the rule.
(ii) What do $8$ hours cost?

Solution

$8 \times 5 + 8$

$=$

$48$

8 \times 5 + 8 = 48

(i) Multiply the hours by $5$, then add $8$ (a $\$5$ hourly rate plus an $\$8$ fee).
(ii) $\$48$

Example 8 — Find the mistake

Inputs $1, 2, 3$ give outputs $6, 10, 14$. Leo sees the outputs go up by $4$ and says the rule is "multiply by $4$". Explain the mistake and find the output for input $15$.

Solution

$15 \times 4 + 2$

$=$

$62$

15 \times 4 + 2 = 62

Multiply by $4$ gives $4$ for input $1$, not $6$; she must also add $2$. Output $62$.

Common pitfalls

Testing only the first row. Check the rule on every row of the table.

Using the gap as the whole rule. Outputs going up by $4$ means multiply by $4$, then add or subtract something.

Ignoring skipped inputs. Divide the output gap by the input gap to find the multiplier.

Undoing in the wrong order. Work backwards: undo the add first, then the multiply.

Frequently asked questions

How do I find a rule from a table?

Find the gap between outputs (the multiplier), then add or subtract to match the first row. Test every row.

What is a two-step rule?

A rule with two operations in order, such as multiply by $4$, then add $3$.

How do I find the input from an output?

Undo the steps in reverse order.

What if the inputs do not go up by 1?

Divide the output gap by the input gap.

Why must the rule work for every row?

Many rules fit one row; only the true rule fits them all.

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\(63 - 3\)	\(=\)	\(60\)
\(60 \div 4\)	\(=\)	\(15\)

Determine a rule from a table of values

📖 Theory

Finding the rule

Working backwards from an output

Common pitfalls

Frequently asked questions

Theory