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Module 1 · L9 of 13 ~45 min ⚡ +90 XP available

Coordinates, Tables and Linear Patterns

Represent practical relationships using tables and ordered pairs, then recognise linear patterns by checking for constant differences. Every graph starts with a table, master the table and the graph takes care of itself.

Today's hook, A taxi fare starts at $\$6$ and goes up $\$3$ per kilometre. Can you spot the pattern in the table before you even touch a graph?

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The key idea: linear pattern check

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

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

Think First, your gut answer first

+5 XP warm-up

A taxi fare starts at $6 and increases by $3 for each kilometre. How could a table show the fare for 0, 1, 2 and 3 kilometres?

Without calculating write the first few rows of the table. Make a prediction before the lesson reveals the answer.

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The key idea: linear pattern check

+5 XP to read

A table shows a linear pattern when equal input steps produce equal output changes. Check the differences if they're constant, the relationship is linear.

Ordered pairs connect a table to a graph. Each row of a table becomes one point $(x, y)$ on the plane. Constant differences mean a steady rate of change, the defining feature of a linear relationship.

Equal input steps + equal output changes = linear pattern

Ordered pair, input first

Write $(x, y)$ where $x$ is the input and $y$ is the output. Order matters: $(2, 12) \neq (12, 2)$.

Check the differences

Calculate the output change at each step. If all differences are equal, the table is linear.

Predict forward

If the difference is constant, add it to extend the table and predict any value in the sequence.

What you'll master

Know

Key facts

An ordered pair is written as $(x, y)$.
The $x$-value is the input and the $y$-value is the output.
A table is linear when equal input steps produce equal output changes.

Understand

Concepts

Tables, coordinates and graphs can represent the same relationship.
Constant differences show a steady rate of change.
An increasing table is not automatically linear.

Can do

Skills

Write ordered pairs from a table.
Identify whether a table shows a linear pattern.
Predict values using a constant difference.

Key terms

Ordered pair $(x, y)$A pair of numbers recording an input ($x$) and its matching output ($y$). Order matters.

Input ($x$)The independent variable, the value you choose or control.

Output ($y$)The dependent variable, the value that results from the input.

Linear patternA pattern where equal changes in input produce equal changes in output (constant first difference).

Constant differenceThe same change in output for every equal step in input, confirms a linear relationship.

QuadrantOne of four regions of the Cartesian plane divided by the $x$- and $y$-axes.

Ordered pairs connect tables and graphs

An ordered pair records an input and its matching output.

Ordered pairs connect tables and graphs

core concept

An ordered pair $(x, y)$ records an input and its matching output. In a distance-cost table, $x$ might represent kilometres and $y$ might represent cost.

The order matters: $(2, 12)$ is not the same as $(12, 2)$. The input comes first, then the output. Each row in a table gives exactly one ordered pair that can be plotted on the Cartesian plane.

Common error: Do not reverse the coordinates. The input comes first, then the output.

An ordered pair (x, y) records one input-output relationship. In a table, each row becomes one point. Plotting ordered pairs and joining them makes the pattern visible. A straight line indicates a linear relationship.

Pause, copy the ordered pair definition ((x, y) records one input-output pair), the graphing step (plot each pair and join, a straight line confirms linearity), and the visual check (a perfectly straight line means a linear relationship) into your book.

Quick check: A table has inputs 0, 1, 2, 3 and outputs 6, 9, 12, 15. What is the ordered pair for the row where the input is 2?

Increasing does not always mean linear

We just saw that ordered pairs connect table rows to graph points, and that plotting them makes the pattern visible.

Increasing does not always mean linear

core concept

We just saw that ordered pairs connect table rows to graph points, and that plotting them makes the pattern visible. That raises a question: just because outputs increase as inputs increase doesn't mean the relationship is linear, how do you distinguish a linear pattern from a non-linear one using a table? This card answers it → calculate first differences (the change in output for each equal input step); if all first differences are equal, the relationship is linear; if not, it is non-linear.

A table can increase without having a constant difference. To confirm linearity, calculate the change in output at each equal step, every difference must be the same.

$$\text{linear if: } \Delta y = \text{constant for equal steps in } x$$

Example of a non-linear table:

Input	1	2	3	4
Output	2	5	11	20
Change		+3	+6	+9

Reasoning check: The outputs increase, but the changes are not equal (+3, +6, +9), so this is not a linear pattern.

A table can increase without being linear. Test: calculate first differences (change in output for each equal input step). If all differences are equal, the table is linear. If not equal, the relationship is non-linear even if outputs all increase.

Pause, copy the first-difference linearity test: for equal input steps, calculate the change in output at each step; if all changes are equal (constant first difference), the table is linear; if the changes vary, the relationship is non-linear into your book.

True or false: A table where the outputs are 3, 7, 11, 15 (for inputs 0, 1, 2, 3) shows a linear pattern.

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

PROBLEM 1 · ORDERED PAIRS FROM A TABLE

Write ordered pairs from a taxi fare table.

Distance, km	0	1	2	3
Fare, $\$$	6	9	12	15

Identify input = distance (km), output = fare ($\$$).

In each ordered pair, input comes first: $(x, y) = (\text{distance}, \text{fare})$.

PROBLEM 2 · CHECK WHETHER A TABLE IS LINEAR

Week	0	1	2	3
Savings, $\$$	20	35	50	65

The input increases by 1 week each time.

Equal input steps: confirm the input column increases by the same amount each row.

PROBLEM 3 · PREDICT USING A LINEAR TABLE

A car travels at a constant speed. Predict the distance at 4 hours.

Time, h	0	1	2	3
Distance, km	0	80	160	240

Check differences: $80-0=80$, $160-80=80$, $240-160=80$.

Constant difference of +80 km per hour confirms a linear pattern.

Fill the gap: A savings table shows outputs 50, 65, 80, 95 for weeks 0, 1, 2, 3. The constant difference is dollars per week, so the prediction for week 4 is dollars.

Common errors · the 3 traps that cost marks

Trap 01

Reversing the coordinates

In $(x, y)$, the input always comes first. Writing $(12, 2)$ instead of $(2, 12)$ places the point in the wrong location on the graph and answers the wrong question.

Trap 02

Assuming increasing = linear

A table where outputs go 2, 5, 11, 20 is increasing but the differences (+3, +6, +9) are not equal, so it is not linear. Always calculate the differences to confirm.

Trap 03

Forgetting equal input steps

The linearity check only works when input steps are equal. If your inputs are 0, 1, 2, 4 (skipping 3), equal output changes would be misleading. Confirm equal input spacing first.

Revisit your thinking

Quick-fire practice · 4 questions

Write the ordered pairs for a table with inputs 0, 1, 2, 3 and outputs 4, 10, 16, 22.

Decide whether the table in question 1 is linear and explain your reasoning using differences.

A savings table is 50, 65, 80, 95 for weeks 0, 1, 2, 3. Predict the savings at week 4.

Explain why outputs 1, 4, 9, 16 for inputs 1, 2, 3, 4 are not linear.

Match the description: Which table shows a linear pattern?

Revisit your thinking

Earlier you wrote the first few rows of the taxi fare table. Let's check: starting at $\$6$ and adding $\$3$ each kilometre gives the table 6, 9, 12, 15 for distances 0, 1, 2, 3.

The ordered pairs are $(0, 6)$, $(1, 9)$, $(2, 12)$ and $(3, 15)$. The differences are all +3, confirming this is a linear pattern. The taxi fare table is linear because each extra kilometre adds $3, a constant rate of change.

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Final check: True or false: a table with outputs 6, 9, 12, 15 for inputs 0, 1, 2, 3 has a constant difference of +3 and shows a linear pattern.

Show what you have learned

Multiple choice, then short answer under exam conditions.

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 32 marks

Q1. Write the ordered pairs for inputs 0, 1, 2, 3 and outputs 8, 13, 18, 23. (2 marks)

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ApplyBand 32 marks

Q2. Decide whether the table in Question 1 is linear. Explain using differences. (2 marks)

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AnalyseBand 43 marks

Q3. A distance table shows 0, 90, 180, 270 km for times 0, 1, 2, 3 h. Predict the distance at 5 h and explain. (3 marks)

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📖 Answers (click to reveal)

Q1 (2 marks): $(0, 8)$, $(1, 13)$, $(2, 18)$, $(3, 23)$ [1 mark per two correct pairs].

Q2 (2 marks): Differences: $13-8=5$, $18-13=5$, $23-18=5$. Constant difference of +5 [1]. The table is linear [1].

Q3 (3 marks): Differences: all +90 km/h [1]. At 4 h: $270+90=360$ km [1]. At 5 h: $360+90=450$ km [1].

Drill 1: $(0,4)$, $(1,10)$, $(2,16)$, $(3,22)$ · Drill 2: Differences all +6, linear · Drill 3: Difference +15, savings at week 4 = $110 · Drill 4: Differences +3, +5, +7, not constant, so not linear.

Consolidate and move on

Sit the module quiz, then close the lesson off.

Boss battle · Linear or Not?

earn bronze · silver · gold

Check equal input steps, then check whether the output change stays constant. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

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Mark lesson as complete

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Module overview · Year 11 Maths Standard