Mathematics Standard • Year 11 • Module 1 • Lesson 9

Coordinates, Tables and Linear Patterns

Build fluency writing ordered pairs from tables, testing for a constant difference and predicting the next linear value.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 In the ordered pair (x, y), which letter is the input and which is the output?

x = ____________ y = ____________

Q1.2 A taxi table has inputs 0, 1, 2, 3 and outputs 6, 9, 12, 15. Write the four ordered pairs.

Q1.3 Circle TRUE or FALSE.

"Every increasing table is linear." TRUE / FALSE

Stuck? Revisit lesson § Ordered Pairs Connect Tables and Graphs — input first, output second.

2. Worked example — taxi fare table

Follow each line of working.

Problem. A taxi charges $6 flagfall plus $3 per km. Distances 0, 1, 2, 3 km give fares $6, $9, $12, $15. Write the ordered pairs and decide if the table is linear.

Step 1 — Read the table.

distance (input): 0, 1, 2, 3 fare (output): 6, 9, 12, 15

Step 2 — Pair each input with its output.

(0, 6), (1, 9), (2, 12), (3, 15)

Reason: input always comes first inside the brackets.

Step 3 — Test for constant differences.

Input change: 1, 1, 1 (constant). Output change: +3, +3, +3 (constant).

Reason: equal input steps producing equal output changes is the test for linear.

Conclusion. The table is linear because the output rises by a constant +$3 per +1 km.

3. Faded example — savings table

Weeks 0, 1, 2, 3 give savings $20, $35, $50, $65. Fill in each blank. 4 marks

Step 1 — Write the ordered pairs:

(0, ____), (1, ____), (2, ____), (3, ____)

Step 2 — Input change:

Each step: +____ week.

Step 3 — Output change:

20 → 35 = +____, 35 → 50 = +____, 50 → 65 = +____.

Step 4 — Linear?

YES / NO (circle one). The constant difference is +$____ per week.

Predict. At week 4, savings will be $____________.

Stuck? Revisit lesson § Worked Example 2 — Check whether a table is linear.

4. Graduated practice — pairs, patterns and predictions

Show working below each part.

Foundation — write the ordered pairs (4 questions)

Q	Problem	Answer
4.1 1	Inputs 0,1,2,3 → outputs 4,10,16,22. Write the pairs.
4.2 1	Inputs 1,2,3,4 → outputs 5,8,11,14. Write the pairs.
4.3 1	Inputs 0,1,2,3 → outputs 50,65,80,95. Write the pairs.
4.4 1	State which axis (x or y) the input goes on.

Standard — test for linearity and predict (6 questions)

For each table, give the input change, the output changes, decide linear or not, and (if linear) predict the next value.

4.5 0,1,2,3 → 4,10,16,22. Linear? Next at input 4? 2 marks

4.6 0,1,2,3 → 50,65,80,95. Linear? Next at input 4? 2 marks

4.7 Weeks 0,1,2,3 → savings 80,130,180,230. Linear? Predict week 5. 2 marks

4.8 Hours 0,1,2,3 → distance 0,80,160,240. Linear? Predict at h = 4. 2 marks

4.9 Inputs 1,2,3,4 → outputs 2,5,11,20. Linear? Why or why not? 2 marks

4.10 Inputs 1,2,3,4 → outputs 1,4,9,16. Linear? Why or why not? 2 marks

Extension — reverse the table (2 questions)

4.11 A linear table has output 12 at input 0, and output 47 at input 5. (a) Find the constant change per input step. (b) Write the four outputs at inputs 0, 1, 2, 3. 3 marks

4.12 A bank statement shows balances $1000, $980, $960 at weeks 0, 1, 2. Write the pairs, decide linear, and predict the balance at week 6 (assume the pattern continues). 3 marks

Stuck on 4.11? Output change over 5 input steps is 47 − 12 = 35. Divide by 5 to find the per-step change.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I wrote the input first inside each bracket: (0, 4), (1, 10), (2, 16), (3, 22).

For 4.5 I checked the differences: +6, +6, +6 → linear; next at input 4 is 28.

For 4.10 I noticed differences +3, +5, +7 are not equal → NOT linear (it's a squared pattern).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Input vs output

x = input. y = output.

Q1.2 — Taxi pairs

(0, 6), (1, 9), (2, 12), (3, 15).

Q1.3 — TRUE or FALSE

FALSE. A table can increase without the differences being equal (e.g. outputs 1, 4, 9, 16).

Q3 — Faded savings table

Step 1: (0, 20), (1, 35), (2, 50), (3, 65).
Step 2: input change +1 week each step.
Step 3: output changes +15, +15, +15.
Step 4: YES, linear; constant difference = +$15/week.
Predict: week 4 savings = $80.

Q4.1 — Pairs

(0, 4), (1, 10), (2, 16), (3, 22).

Q4.2 — Pairs

(1, 5), (2, 8), (3, 11), (4, 14).

Q4.3 — Pairs

(0, 50), (1, 65), (2, 80), (3, 95).

Q4.4 — Input axis

Input goes on the x-axis (horizontal).

Q4.5 — Linear test

Differences +6, +6, +6 → linear. Next at input 4: 22 + 6 = 28.

Q4.6 — Linear test

Differences +15, +15, +15 → linear. Next at input 4: 95 + 15 = 110.

Q4.7 — Savings prediction

Differences +50, +50, +50 → linear. Week 5: 80 + 5(50) = $330.

Q4.8 — Distance prediction

Differences +80, +80, +80 → linear. At h = 4: 240 + 80 = 320 km.

Q4.9 — Non-linear check

Differences +3, +6, +9 are NOT equal → not linear. The differences themselves are growing.

Q4.10 — Square numbers

Differences +3, +5, +7 are NOT equal → not linear. The outputs are 1², 2², 3², 4² (a quadratic pattern).

Q4.11 — Reverse a linear table

(a) Per-step change = (47 − 12) / 5 = +7 per input step.
(b) Outputs at inputs 0, 1, 2, 3 = 12, 19, 26, 33.

Q4.12 — Decreasing linear table

Pairs: (0, 1000), (1, 980), (2, 960). Differences −20, −20 → linear (decreasing). At week 6: 1000 − 6(20) = $880.