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C1.2 · Represent patterns with tables, graphs, and expressions

Learn to represent patterns with tables, graphs, and expressions through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Algebra

Connect the figure number to the tile count

A growing pattern changes by a stated amount at each step. A table lists the step number and the quantity at that step. A coordinate graph plots each pair as a point. For this lesson, a hypothetical tile design starts with three tiles in figure 1 and adds two tiles for each new figure. The table, expression, and graph must all tell the same story; a slanted line alone is not enough to show why the pattern grows.

What you will learn

  • Make a table of values for a linear growing pattern.
  • Plot ordered pairs with labelled axes.
  • Use the rule to predict a later term.

Build the table

Figures 1, 2, 3, and 4 have 3, 5, 7, and 9 tiles. Each time the figure number increases by one, the tile count increases by two. This constant increase makes the growing pattern linear. Do not add two to the figure number; add two to the tile count.

State the rule

Two tiles are added per figure, and one extra tile is always present. If n is the figure number, the tile count is 2n + 1. Test n = 1: 2 × 1 + 1 = 3. Test n = 4: 2 × 4 + 1 = 9. Both match the table.

Plot and interpret

Put figure number on the horizontal axis and tile count on the vertical axis. Plot (1,3), (2,5), (3,7), and (4,9). Each point describes one actual figure in the pattern. The connecting stroke shows the trend, but fractional figure numbers do not represent physical tile designs.

Worked example

A tile pattern that adds two

A hypothetical pattern has 3, 5, and 7 tiles in figures 1, 2, and 3. How many tiles are in figure 4?
Figure number and tile count
Figure number and tile countOrdered points (1,3), (2,5), (3,7), and (4,9) match the rule 2n + 1. Only whole-number figure positions describe designs.Figure numberTiles(1,3)(2,5)(3,7)(4,9)Hypothetical pattern

Ordered points (1,3), (2,5), (3,7), and (4,9) match the rule 2n + 1. Only whole-number figure positions describe designs.

  1. Find the change
    The count rises by two each time: 3, 5, 7, ...
  2. Extend once
    7 + 2 = 9 tiles in figure 4.
  3. Check the rule
    2 × 4 + 1 = 9.
Answer: Figure 4 has 9 tiles.
Check: The ordered pair (4,9) appears at the expected position.

Common mistakes and how to avoid them

Putting tile count on the horizontal axis.
Correction: Label axes first: input figure number on x, output tile count on y.

Lesson summary

  • A table records input-output pairs.
  • A constant increase produces a linear growing pattern.
  • The graph and rule must match every listed pair.

Check your understanding

Question 1

How many tiles does figure 6 have under the rule 2n + 1?
  1. 13
  2. 12
  3. 11
  4. 8
Show answer and explanation
13
Replace n with 6: 2 × 6 + 1 = 13.

Key terms

Term
One value at a specified step of a pattern.
Ordered pair
An input and output written as (x,y).

Continue through Grade 6 Math

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation C1.2. It is a study resource, not an official curriculum publication.

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