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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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Worksheets

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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

Recall — your gut answer first

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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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The pattern you need to own

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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.

$\text{output} = \text{starting value} + \text{rate} \times \text{input}$

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.

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.

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.

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.

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 $b = 0$.

What to write in your book

Step 1: Check differences — are equal input steps producing equal output changes?
Step 2: Identify the starting value (output when input = 0).
Step 3: Identify the rate (the constant output change per unit input).
Formula structure: output = starting value + rate × input
Example: $C = 12 + 5b$ — starting cost $12, then $5 per box.

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.

Check the differences before assuming a linear formula

core concept

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

Change (not equal!)

What to write in your book

Before writing a formula from a table, always calculate first differences (consecutive output changes).
Equal first differences → linear formula is appropriate.
Unequal first differences → linear formula is NOT appropriate; the relationship is non-linear.
Example non-linear: outputs 2, 4, 8, 16 — differences are 2, 4, 8 (doubling each time).

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

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.

Starting value $= \$10$ (cost when $k = 0$)

The cost is $10 before any kilometres are travelled. This is the starting value.

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.

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.

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?

Test $n = 1$: $P = 6(1) + 2 = 8$ ✓

The formula gives 8 when $n = 1$, which matches the first table value.

What to write in your book

Worked Example 1: $C = 10 + 3k$ — starting $10, then $3 per km. Test: $k=3 \Rightarrow C = 19$.
Worked Example 2: $S = 40 + 15w$ — always define both variables with units.
Worked Example 3: $P = 6n + 2$ — always test at least two or three known values to confirm a formula.
Key routine: Starting value → Rate → Write formula → Test.

Fill the gap: A hire company charges $30 plus $8 per hour. The starting value is and the rate is , so the formula is $C = 30 +$ .

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.

Quick-fire practice · 4 calculations

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

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

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

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 $n$?

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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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.

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].

Boss battle · Pattern Builder

earn bronze · silver · gold

For each table, identify the starting value, the repeated change and one test value before choosing the correct formula. Beat the boss to bank a tier.

⚔ Enter the arena

Science Jump · platform challenge

Climb platforms by answering formula-from-table questions. Pool: lesson 5.

Mark lesson as complete

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← Lesson 4 · Rearranging Formulas Lesson 6 · Spreadsheet Formulas →

Module overview · Year 11 Maths Standard