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C1.2 · Represent number patterns with tables, expressions, and linear equations

Learn to represent number patterns with tables, expressions, and linear equations through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Algebra

Tiles, tables, and coordinates tell the same story

A pattern can be shown with objects, a table of values, and a graph. Each representation must describe the same rule. Consider a design with two fixed tiles and three more tiles for every new stage. The stage number is the input; the number of tiles is the output. We will list exact counts, plot ordered pairs, and write a short rule. The straight graph belongs to this particular constant-growth pattern, not to every pattern.

What you will learn

  • Build a table from a growing pattern.
  • Plot stage and tile-count pairs on labelled axes.
  • Explain the starting value and constant change.

Count from a clear starting value

At stage zero the design has two fixed tiles. Every step to the next stage adds three tiles. Stage one has five tiles; stage two has eight. The fixed two is the starting value, even though the first displayed stage may be one.

Translate the table to a graph

Put stage number along the horizontal axis and tile count along the vertical axis. A point such as (2, 8) means stage two contains eight tiles. Plot values at equal intervals so the graph does not distort the pattern.

Describe the rule in words and symbols

For stage n, start with two and add three n times. Write tiles = 3n + 2. This equation predicts a whole-number tile count for each whole-number stage; intermediate graph positions are a visual guide, not half-built stages.

Worked example

A design growing by three tiles

A design starts with two fixed tiles and gains three tiles per stage. How many tiles are at stage four?
Stage number and tile count
Stage number and tile countThe exact stage/count pairs are (0,2), (1,5), (2,8), (3,11), and (4,14); only whole-number stages represent designs.stage ntiles(0,2)(1,5)(2,8)(3,11)(4,14)

The exact stage/count pairs are (0,2), (1,5), (2,8), (3,11), and (4,14); only whole-number stages represent designs.

  1. Identify the parts
    The starting value is 2 tiles and each stage adds 3 tiles.
  2. Write the rule
    At stage n, tile count is 3n + 2.
  3. Substitute stage four
    3 × 4 + 2 = 14 tiles. The plotted point (4,14) agrees.
Answer: There are 14 tiles at stage four.
Check: The counts 2, 5, 8, 11, 14 rise by three each time.

Common mistakes and how to avoid them

Using 3n without the two fixed tiles.
Correction: Check stage zero: the rule must return two, so add the starting value.

Lesson summary

  • A table, coordinate plot, and rule must use matching values.
  • Constant growth is the same difference between consecutive stages.
  • The starting value is the count at stage zero.

Check your understanding

Question 1

The same pattern reaches which tile count at stage five?
  1. 17
  2. 15
  3. 16
  4. 20
Show answer and explanation
17
Use 3 × 5 + 2 = 17.

Key terms

Starting value
The output when the stage number is zero.
Ordered pair
An input and its matching output written in order, such as (2,8).

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation C1.2. It is a study resource, not an official curriculum publication.

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