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.
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
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 calculatingwrite the first few rows of the table. Make a prediction before the lesson reveals the answer.
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.
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.
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.
Skills
- Write ordered pairs from a table.
- Identify whether a table shows a linear pattern.
- Predict values using a constant difference.
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.
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?
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.
Example of a non-linear table:
| Input | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Output | 2 | 5 | 11 | 20 |
| Change | +3 | +6 | +9 |
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
Write ordered pairs from a taxi fare table.
| Distance, km | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Fare, $ | 6 | 9 | 12 | 15 |
| Week | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Savings, $ | 20 | 35 | 50 | 65 |
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 |
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
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?
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.
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.
Pick your answer, then rate your confidencethat tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Write the ordered pairs for inputs 0, 1, 2, 3 and outputs 8, 13, 18, 23. (2 marks)
Q2. Decide whether the table in Question 1 is linear. Explain using differences. (2 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)
📖 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.
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.
⚔ Enter the arenaClimb platforms by answering coordinates and linear pattern questions. Pool: lesson 9.
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