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Module 1 · L5 of 13 ~45 min ⚡ +90 XP available

Building Formulas from Patterns and Tables

Use starting values and repeated changes to construct formulas from tables, patterns and practical cost situations. Every linear formula hides a simple structure: one starting value and one repeated change.

Today's hook, A delivery company charges $\$10$ before travel begins, then adds $\$3$ for each kilometre. How could you write a formula for any number of kilometres? What if the per-kilometre charge changed to $4?
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Recall, your gut answer first

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

A delivery company charges $10 before travel begins, then adds $3 for each kilometre.

Without calculating how could you write a formula for any number of kilometres? Type your first formula and define the variables.

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02
The pattern you need to own
+5 XP to read

Every linear formula from a table shares the same hidden structure: a starting value (the output when input = 0) and a repeated change (how much the output shifts for each unit increase in the input).

Starting value is the amount present before the repeated change begins. Repeated change (rate) becomes the coefficient of the variable. The formula is output = starting value + rate × input.

The general form of a linear cost formula with an example.
$\text{output} = \text{starting value} + \text{rate} \times \text{input}$
Starting value
The output when input = 0. This is the constant term in your formula. Not always the first listed value in the table.
Repeated change
The constant difference between consecutive output values. This becomes the coefficient (rate) in your formula.
Always test
Substitute a known input-output pair into your formula to verify it is correct before using it.
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What you'll master
Know

Key facts

  • A starting value is the amount present before the repeated change begins.
  • A repeated increase or decrease becomes the variable term.
  • A formula should be tested against known table values.
Understand

Concepts

  • Not every increasing table has the same repeated change.
  • The first output is not always the starting value.
  • Context determines what each variable means.
Can do

Skills

  • Identify the repeated change in a table.
  • Write a formula from a pattern or practical context.
  • Test a formula using known values.
Beyond the syllabus. Building Formulas from Patterns and Tables is not a named requirement of Formulas and equations (MST-11-01). Constructing a linear rule from a table is genuinely useful, and it is the bridge into Linear relationships (MST-11-02) where it is assessed, but this focus area will not examine it for its own sake. What Formulas and equations does examine is substituting into and rearranging the formulas the syllabus actually names: blood alcohol content and its limitations, the Fried's, Young's and Clark's medication rules, speed, time and stopping distance, and rearranging a non-linear formula. Use this lesson to get fluent with the pattern, then make sure you can do those.
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Key terms
Starting valueThe output of a formula when the input equals zero; the constant term.
Repeated change (rate)The constant amount by which the output increases or decreases for each unit increase in the input.
Linear formulaA formula of the form output = a + r × input, where a and r are constants.
InputThe independent variable, the value you choose or measure (e.g. kilometres, weeks, boxes).
OutputThe dependent variable, the value calculated from the formula (e.g. cost, savings).
Testing a formulaSubstituting a known input value into the formula and checking the output matches the table.
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Look for equal input steps and equal output changes

A table supports a simple linear formula when equal increases in the input produce equal increases or decreases in the output.

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Look for equal input steps and equal output changes
core concept

A table supports a simple linear formula when equal increases in the input produce equal increases or decreases in the output.

A constant change per step gives a linear formula.

The starting value is $\$12$ when $b = 0$. The repeated change is $\$5$ per box, so $C = 12 + 5b$.

Common error: The first listed output is only the starting value if the input is 0. If the table starts at $b = 1$, you must back-calculate to find the output when $b = 0$.

A table of values is linear if the output changes by a constant amount for equal input steps. Calculate first differences (output changes): if all are equal, the relationship is linear. If not, the formula is not of the form y = mx + c.

Pause, copy the first-difference test: calculate the change in output for each equal-size input step; if all first differences are equal, the table is linear and the gradient m equals that constant difference into your book.

Did you get this? True or false: a table with outputs 5, 10, 20, 40 for inputs 1, 2, 3, 4 has a constant repeated change.

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Check the differences before assuming a linear formula

We just saw that a table is linear when the output changes by a constant amount for equal input steps, calculate first...

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Check the differences before assuming a linear formula
core concept

We just saw that a table is linear when the output changes by a constant amount for equal input steps, calculate first differences to check. That raises a question: a table can look like it's increasing steadily without actually being linear, what tells you it isn't, and what does it mean for the formula? This card answers it → if first differences are not all equal, the relationship is not linear; the simple y = mx + c model is wrong and you must look for a different pattern.

If the output changes by different amounts, the simple "starting value plus rate times input" model may not fit. Always calculate first differences before writing any formula.

Non-linear table example: outputs 2, 4, 8, 16 for inputs 1, 2, 3, 4 have differences of +2, +4, +8, these are not equal. Do not force a linear formula onto this data.

The table below shows a non-linear pattern:

Output
2
4
8
16
Change (not equal!)
-
+2
+4
+8

Always calculate the differences in the table before assuming a linear model. If first differences are not constant, the simple y = mx + c form is wrong. Look at second differences or other patterns to identify the correct model type.

Pause, copy the rule: always check first differences before writing a formula, if they are not constant, y = mx + c is wrong; non-constant first differences signal a non-linear relationship into your book.

Quick check: A table has outputs 7, 11, 15, 19 for inputs 1, 2, 3, 4. What is the repeated change?

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Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · BUILD FROM A DELIVERY TABLE

A delivery service charges a fixed fee of $\$10$ plus $\$3$ per kilometre. Build a formula for total cost $C$ after $k$ kilometres, and test it against a known value.

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Starting value $= \$10$ (cost when $k = 0$)
The cost is $10 before any kilometres are travelled. This is the starting value.
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PROBLEM 2 · BUILD FROM A SAVINGS PATTERN

Mia starts with $\$40$ and saves $\$15$ each week. Write a formula for her savings $S$ after $w$ weeks.

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Let $S$ be savings in dollars, $w$ be weeks.
Define both variables clearly. Context: $S$ is the savings amount and $w$ is the number of weeks elapsed.
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PROBLEM 3 · CHECK WHETHER A FORMULA FITS

A student claims the formula for a table with outputs 8, 14, 20, 26 for $n = 1, 2, 3, 4$ is $P = 6n + 2$. Is the student correct?

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Test $n = 1$: $P = 6(1) + 2 = 8$ ✓
The formula gives 8 when $n = 1$, which matches the first table value.
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Fill the gap: A hire company charges $\$30$ plus $\$8$ per hour. The starting value is and the rate is , so the formula is $C = 30 +$ .

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Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Trap 01
First listed output is not always the starting value
The starting value is the output when the input is zero. If a table starts at input = 1, you must back-calculate. Subtract the rate from the first listed output to find the starting value (where input = 0).
Trap 02
Forgetting to define variables
Always state what each variable represents and include units. Writing $C = 10 + 3k$ without saying "$C$ is cost in dollars and $k$ is kilometres" loses communication marks in exams.
Trap 03
Assuming linear when differences are unequal
Check that the output changes by the same amount each time before writing a linear formula. Outputs 2, 4, 8, 16 look increasing, but the differences (2, 4, 8) are not constant, so no simple linear formula applies.
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Revisit your thinking

Quick-fire practice · 4 calculations

Quick-fire practice · 4 calculations

1

A hire company charges $\$30$ plus $\$8$ per hour. Write a formula for total cost $C$ after $h$ hours.

2

A pattern has values 7, 11, 15, 19 for term numbers 1, 2, 3, 4. Write and test a formula.

3

A table has outputs 5, 10, 20, 40 for inputs 1, 2, 3, 4. Explain why a simple linear formula is not suitable.

4

A pool starts with 200 litres and loses 15 litres per hour. Write a formula for the volume $V$ after $h$ hours.

Match it: Which formula matches "starts at 3, increases by 6 each step" for input $n$?

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Revisit your thinking

The delivery company formula is $C = 10 + 3k$. The $\$10$ is the starting cost (before any kilometres) and the $\$3$ is the repeated cost for each kilometre.

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 44 marks

Q1. A printing company charges $\$25$ plus $\$2$ per page. Write a formula for total cost $C$ for $p$ pages, then find the cost for 18 pages. (4 marks)

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ApplyBand 44 marks

Q2. The values 9, 14, 19, 24 match term numbers 1, 2, 3, 4. Write a formula and test it using term 4. (4 marks)

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AnalyseBand 52 marks

Q3. Explain why checking differences is important before writing a formula from a table. (2 marks)

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📖 Comprehensive answers (click to reveal)

Drill 1: $C = 30 + 8h$  ·  2: Repeated change = 4; starting value = 3 (back-calculate: 7 − 4 = 3). Formula: $T = 3 + 4n$. Test $n=4$: $T = 3 + 16 = 19$ ✓  ·  3: Differences are 5, 10, 20, not equal, so no linear formula fits.  ·  4: $V = 200 - 15h$

Q1 (4 marks): Starting value = $\$25$ [1]. Rate = $\$2$ per page [1]. Formula: $C = 25 + 2p$ [1]. When $p = 18$: $C = 25 + 2(18) = 25 + 36 = \$61$ [1].

Q2 (4 marks): Repeated change = 5 [1]. Starting value: $9 - 5 = 4$ (when $n = 0$) [1]. Formula: $T = 4 + 5n$ [1]. Test $n = 4$: $T = 4 + 20 = 24$ ✓ [1].

Q3 (2 marks): Checking differences reveals whether the relationship is linear [1]. If differences are unequal, a linear formula will not accurately model the data and will give incorrect outputs for untested values [1].