Coordinates, Tables and Linear Patterns

Represent practical relationships using tables and ordered pairs, then recognise linear patterns by checking for constant differences.

45 min Algebra Linear relationships Lesson 9 of 13
(x,y)

Choose how you work: type answers on screen, or work in your book.

Printable worksheet

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Use the printable version for ordered pairs, plotting tables and checking constant differences.

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Think First

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?

Type the first few rows of the table.

Write the first few rows of the table in your book.

Write your table in your book
Saved

Know

  • 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

  • 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

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

Linear Pattern Check

equal input steps
Check that the input increases by the same amount.
equal output changes
If the output change is constant, the table is linear.

Core Content

1. Ordered Pairs Connect Tables and Graphs

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 $(12, 2)$.

Common error: Do not reverse the coordinates. The input comes first, then the output.
Worked Example 1

Write ordered pairs from a taxi fare table

Distance, km0123
Fare, $691215

The ordered pairs are $(0, 6)$, $(1, 9)$, $(2, 12)$ and $(3, 15)$.

Each pair means: for this many kilometres, the fare is this many dollars.

Worked Example 2

Check whether a table is linear

Week0123
Savings, $20355065

The input increases by 1 week each time.

The output changes by +15 each time: 20 to 35, 35 to 50, 50 to 65.

The table is linear because the output change is constant.

Worked Example 3

Predict using a linear table

A car travels at a constant speed. The distance table is shown.

Time, h0123
Distance, km080160240

The distance increases by 80 km each hour.

At 4 hours, the distance is $240 + 80 = 320$ km.

2. Increasing Does Not Always Mean Linear

A table can increase without having a constant difference.

Input1234
Output251120
Change-+3+6+9
Reasoning check: The outputs increase, but the changes are not equal, so this is not a linear pattern.
Activity

Tables and Coordinates Practice

  1. Write the ordered pairs for a table with inputs 0, 1, 2, 3 and outputs 4, 10, 16, 22.
  2. Decide whether the table is linear and explain your reasoning.
  3. A savings table is 50, 65, 80, 95 for weeks 0, 1, 2, 3. Predict the savings at week 4.
  4. Explain why outputs 1, 4, 9, 16 for inputs 1, 2, 3, 4 are not linear.
Complete the coordinate and table practice in your book.

Revisit the Taxi Fare Table

The taxi fare table is linear because each extra kilometre adds $3. The ordered pairs can be plotted to show the relationship visually.

Explain the constant pattern in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Use ordered pairs and constant differences to justify your answers.

1. Write the ordered pairs for inputs 0, 1, 2, 3 and outputs 8, 13, 18, 23. 2 MARKS

Answer in your book.

2. Decide whether the table in Question 1 is linear. Explain using differences. 2 MARKS

Answer in your book.

3. 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

Answer in your book.

Linear or Not?

Check equal input steps, then check whether the output change stays constant.

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