Year 7 Mathematics · AC9M7A05

Growing Patterns, Tables and Cartesian Graphs

We are learning to generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane

Ready to project and teach

Learning goalsSay it simply

We are learning to generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane.

A growing pattern connects a stage number x with a quantity y. Constant additive growth produces a linear rule whose coefficient is the growth per stage.

Build a value table, calculate first differences and use one ordered pair to determine the constant term; write and plot each pair in x-then-y order.

A pattern of separate stages is discrete even when its plotted points align. Verify the rule against every known stage before using it to predict a later stage.

Success criteria

  • I can derive a function rule from a growing pattern.
  • I can generate a table and plot ordered pairs correctly.
  • I can interpret the growth rate and constant term and verify a prediction.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Derive a growing-pattern rule

  1. Count stages as 4, 7 and 10 tiles.
  2. Identify the constant first difference of 3 tiles.
  3. Write y=3x+b and use stage 1.
  4. Solve 4=3+b to find the constant term b=1.

Final answer: y=3x+1

Check: At discrete stage 3 the rule gives 10 tiles.

Example 2

Generate and plot a table

  1. Use y=2x−1 for x=0,1,2,3.
  2. Calculate y=−1,1,3,5.
  3. Write the four ordered pairs.
  4. Plot discrete points with x first.

Final answer: (0,−1), (1,1), (2,3), (3,5)

Check: Each first difference is 2 when x rises by 1.

Example 3

Application problem 1

Problem: A growing pattern follows y=3x+1. Generate values for x=0,1,2,3, plot the ordered pairs mentally or on paper, and explain what 3 and 1 mean.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: The points are (0,1), (1,4), (2,7), (3,10). The 3 is the increase in y for each increase of 1 in x; the 1 is the starting value when x=0.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: The points are (0,1), (1,4), (2,7), (3,10). The 3 is the increase in y for each increase of 1 in x; the 1 is the starting value when x=0.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: Find the rule through (1,4),(2,7),(3,10), then predict stage 20.

  1. Plan: Use the first difference to find the coefficient, then the constant term.
  2. Work: The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61.
  3. Interpret: The complete model for “Find the rule through (1,4),(2,7),(3,10), then predict stage 20.” shows The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61.

Final answer: The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61.

Check: The complete model for “Find the rule through (1,4),(2,7),(3,10), then predict stage 20.” shows The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7A05 - Growing Patterns, Tables and Cartesian Graphs
Example 1

Example 1 Derive a growing-pattern rule Count stages as 4, 7 and 10 tiles. Identify the constant first difference of 3 tiles. Write y=3x+b and use stage 1. Solve 4=3+b to find the constant term b=1. Final answer: y=3x+1 Check: At discrete stage 3 the rule gives 10 tiles.

Example 2

Example 2 Generate and plot a table Use y=2x−1 for x=0,1,2,3. Calculate y=−1,1,3,5. Write the four ordered pairs. Plot discrete points with x first. Final answer: (0,−1), (1,1), (2,3), (3,5) Check: Each first difference is 2 when x rises by 1.

Example 3

Example 3 Application problem 1 Problem: A growing pattern follows y=3x+1. Generate values for x=0,1,2,3, plot the ordered pairs mentally or on paper, and explain what 3 and 1 mean. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: The points are (0,1), (1,4), (2,7), (3,10). The 3 is the increase in y for each increase of 1 in x; the 1 is the starting value when x=0. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: The points are (0,1), (1,4), (2,7), (3,10). The 3 is the increase in y for each increase of 1 in x; the 1 is the starting value when x=0. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: Find the rule through (1,4),(2,7),(3,10), then predict stage 20. Plan: Use the first difference to find the coefficient, then the constant term. Work: The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61. Interpret: The complete model for “Find the rule through (1,4),(2,7),(3,10), then predict stage 20.” shows The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61. Final answer: The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61. Check: The complete model for “Find the rule through (1,4),(2,7),(3,10), then predict stage 20.” shows The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61.

Curriculum examplesCopied content

Content description: generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Continue the pattern 5, 8, 11, __ by using its first difference.

    Check answer

    Answer: 14.

    Hint: Add the constant first difference.

    Why: Using the method “Add the constant first difference.” gives 14.

  2. 2. For y=4x+2, find y when x=3.

    Check answer

    Answer: 14.

    Hint: Substitute x=3.

    Why: Using the method “Substitute x=3.” gives 14.

  3. 3. Write the ordered pair for x=2 and y=7.

    Check answer

    Answer: (2,7).

    Hint: Write x before y when you record the ordered pair.

    Why: Using the method “Ordered pairs place x first.” gives (2,7).

  4. 4. For y=3x−2, list y when x=0,1,2.

    Check answer

    Answer: −2, 1, 4.

    Hint: Apply the rule to each input.

    Why: Using the method “Apply the rule to each input.” gives −2, 1, 4.

  5. 5. A pattern has 6 tiles at stage 1 and grows by 4. Find its rule.

    Check answer

    Answer: y=4x+2.

    Hint: Use 6=4(1)+b.

    Why: Using the method “Use 6=4(1)+b.” gives y=4x+2.

  6. 6. Do (1,5),(2,8),(3,11) fit y=3x+2?

    Check answer

    Answer: Yes. Substitution gives 3(1)+2=5, 3(2)+2=8 and 3(3)+2=11, so all three ordered pairs satisfy the rule.

    Hint: Substitute each x and compare y.

    Why: The reasoning for “Do (1,5),(2,8),(3,11) fit y=3x+2?” is complete because Yes. Substitution gives 3(1)+2=5, 3(2)+2=8 and 3(3)+2=11, so all three ordered pairs satisfy the rule.

  7. 7. Explain the constant term 5 in y=2x+5.

    Check answer

    Answer: The constant 5 is the output when x=0, because y=2(0)+5=5; graphically it is the vertical-axis intercept.

    Hint: Set x equal to zero.

    Why: The reasoning for “Explain the constant term 5 in y=2x+5.” is complete because The constant 5 is the output when x=0, because y=2(0)+5=5; graphically it is the vertical-axis intercept.

  8. 8. For the discrete rule y=5x−1, which whole-number stage first exceeds 30?

    Check answer

    Answer: A short table gives stage 6: y=5(6)−1=29 and stage 7: y=5(7)−1=34; because the discrete stages are whole numbers, stage 7 is the first output above 30.

    Hint: Test stages 6 and 7 in a short value table.

    Why: The complete model for “For the discrete rule y=5x−1, which whole-number stage first exceeds 30?” shows A short table gives stage 6: y=5(6)−1=29 and stage 7: y=5(7)−1=34; because the discrete stages are whole numbers, stage 7 is the first output above 30.

  9. 9. Find the rule through (1,4),(2,7),(3,10), then predict stage 20.

    Check answer

    Answer: The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61.

    Hint: Use the first difference to find the coefficient, then the constant term.

    Why: The complete model for “Find the rule through (1,4),(2,7),(3,10), then predict stage 20.” shows The first difference is 3, so y=3x+b; using (1,4) gives the constant term b=1, hence y=3x+1 and discrete stage 20 has y=3(20)+1=61.

  10. 10. A growing pattern follows y=3x+1. Generate values for x=0,1,2,3, plot the ordered pairs mentally or on paper, and explain what 3 and 1 mean.

    Check answer

    Answer: The points are (0,1), (1,4), (2,7), (3,10). The 3 is the increase in y for each increase of 1 in x; the 1 is the starting value when x=0.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Growth number used as complete rule.

Correction: Include the starting/fixed component.

Common mistake: x and y coordinates reversed.

Correction: Plot input first, output second.

Common mistake: Discrete points joined automatically.

Correction: Only join if intermediate inputs are meaningful.

Curriculum alignmentStart here

We are learning to generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane.

A growing pattern connects a stage number x with a quantity y. Constant additive growth produces a linear rule whose coefficient is the growth per stage.

Build a value table, calculate first differences and use one ordered pair to determine the constant term; write and plot each pair in x-then-y order.

A pattern of separate stages is discrete even when its plotted points align. Verify the rule against every known stage before using it to predict a later stage.

Success criteria

  • I can derive a function rule from a growing pattern.
  • I can generate a table and plot ordered pairs correctly.
  • I can interpret the growth rate and constant term and verify a prediction.
Related resourcesKeep teaching
Teach & ExplainTeaching slides and samples

Teach this topic step by step

Explore optional teaching slide packs for classroom lessons and explanations at home.

Browse Teach & Explain · Browse Print & Go

Teachers: follow SkillrHub on TPT, then email us to request a free sample before buying. Include the year, subject and topic or curriculum code.

Request a free sample

After trying the sample, honest feedback is welcome. A TPT review is optional, where available, and does not need to be positive.