DoAssignment.ca

DoAssignment study guide

C1.1 · Identify repeating, growing, shrinking, and linear patterns

Learn to identify repeating, growing, shrinking, and linear patterns through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Algebra

Ontario Grade 6 Mathematics — C1.1

A pattern is a set of objects or numbers arranged according to a rule. Before describing one, look at what happens from one part to the next. A colour, shape, or number might repeat. A number might get larger or smaller. Its change might stay the same or vary. In this lesson, you will identify repeating, growing, shrinking, and linear patterns.

What you will learn

  • Identify repeating patterns by finding what repeats.
  • Identify growing and shrinking patterns by comparing terms.
  • Recognize a linear pattern when its terms change by the same amount each step.
  • Use words, lists, and tables to describe patterns clearly.

Start by looking for what changes

A term is one item in a pattern. In a number pattern, each number is a term. In a pattern of shapes, a shape or group of shapes can be a term. The position tells where a term appears. The first term is in position 1, the next is in position 2, and so on.
A repeating pattern has a part that comes back in the same order. Imagine placing a blue tile, then a yellow tile, then a green tile, and repeating those three colours. The repeating part is blue, yellow, green. Look for a group that returns without changing its order.
A growing pattern gets larger as it continues. A shrinking pattern gets smaller. To identify either one, compare terms that are next to each other. Ask what changed. If you can, compare several steps rather than guessing from just one.
  • A term is one item or group in a pattern.
  • A repeating pattern returns to the same part in the same order.
  • Growing means getting larger; shrinking means getting smaller.

Compare the steps

A step is the move from one term to the next. In a number pattern, you can describe a step by saying how much the number changes. In the pattern 4, 7, 10, each step adds 3. In 18, 14, 10, each step subtracts 4. In 2, 5, 9, the changes are 3 and then 4.
A linear pattern is a number pattern with the same change at every step. It may grow by the same amount each time or shrink by the same amount each time. The key clue is that the change stays the same. A pattern can grow without being linear if its changes are not equal.
A table can help you compare terms. Put the position in one column and the term value in another. Read down the values and compare each step. The table helps you check the pattern; it does not change the pattern.
  • Compare each term with the term right before it.
  • Equal changes from step to step identify a linear pattern.
  • A growing or shrinking pattern does not have to be linear.

Describe the rule in clear words

A rule explains how a pattern is made or continued. For a repeating pattern, name the group that repeats. For a growing or shrinking number pattern, say how the terms change. For a linear pattern, name the equal change at each step.
Check your description against every step shown. If you say “add 2 each time,” the first move and the next move should both add 2. If the changes are not equal, do not call the pattern linear. You can still describe it as growing or shrinking.
When a pattern uses shapes or objects, describe only the rule shown. If a new tile is added at each step, say that one tile is added each time. Do not add details that are not given.
  • State the repeating group or the change between terms.
  • Check that your rule fits every step shown.
  • Use only details supplied by the pattern.

Try the method, then practise

First, notice what the pattern contains. Next, compare the groups or terms. Then identify it as repeating, growing, shrinking, or linear. If it grows or shrinks, check whether the amount of change stays equal.
Practise by making a short list of terms and writing its rule in words. Check the rule against each step. You can also explain which clue helped you identify a pattern someone else made.
  • Notice, compare, identify, and check.
  • A rule should fit all the terms shown.

Position and term value in the example

PositionTerm value
16
210
314
418

Worked example

Find the repeating group

A sequence of coloured tiles is blue, red, red, blue, red, red, blue, red, red. Identify the pattern and state what comes next.
  1. Look for a group that returns
    Read the tiles from left to right. The group blue, red, red appears again and again in the same order.
  2. Continue the group
    After the last red tile, the group starts again with blue. The pattern repeats, so the next tile is blue.
Answer: The pattern repeats blue, red, red. The next tile is blue.
Check: The same three-colour group fits the sequence from beginning to end.

Worked example

Compare growing and shrinking patterns

Two number patterns are shown. Pattern A is 2, 5, 9, 14. Pattern B is 21, 17, 13, 9. Identify each as growing or shrinking, and decide whether each is linear.
  1. Check Pattern A
    The numbers get larger, so Pattern A is growing. The changes are 3, then 4, then 5. Since the changes are not equal, it is not linear.
    +3, +4, +5+3,\ +4,\ +5
  2. Check Pattern B
    The numbers get smaller, so Pattern B is shrinking. Each step subtracts 4. Since the change stays the same, Pattern B is linear.
    −4, −4, −4-4,\ -4,\ -4
Answer: Pattern A is growing but not linear. Pattern B is shrinking and linear.
Check: Pattern A has unequal changes. Pattern B has the same change at every step.

Worked example

Identify a linear pattern in a table

A pattern has these position and term pairs: position 1 has value 6, position 2 has value 10, position 3 has value 14, and position 4 has value 18. Is it growing, shrinking, or repeating? Is it linear?
Position and term value
Position and term valueThese hypothetical values rise by 4 for each next position.PositionTerm value1, 62, 103, 144, 18Hypothetical values from the problem

These hypothetical values rise by 4 for each next position.

  1. Compare the term values
    The values increase from 6 to 10 to 14 to 18, so the pattern is growing. Each value is 4 more than the value before it.
    +4, +4, +4+4,\ +4,\ +4
  2. Decide whether it is linear
    The amount added is the same at every step. Therefore, this is a linear pattern. The position-and-value pairs show the order of the terms.
Answer: The pattern is growing and linear. Its rule is to add 4 at each step.
Check: Each successive value is 4 greater than the one before it.

Common mistakes and how to avoid them

Calling every growing pattern linear.
Correction: Compare the changes. A growing pattern is linear only when the change is the same at every step.
Looking at only one pair of terms before naming a pattern.
Correction: Check all the steps shown. The changes may vary.
Choosing a repeating group that changes order.
Correction: A repeating group must return in the same order each time.

Lesson summary

  • A repeating pattern returns to the same group in the same order.
  • A growing pattern gets larger; a shrinking pattern gets smaller.
  • A linear pattern has the same change at every step. It can grow or shrink.
  • Compare several terms and check that your description fits each step.

Check your understanding

Question 1

The number pattern is 7, 11, 15, 19. Which description fits best?
  1. Repeating only
  2. Growing and linear
  3. Shrinking and linear
  4. Growing but not linear
Show answer and explanation
Growing and linear
The terms grow by 4 each time. The equal change makes the pattern linear.

Question 2

The number pattern is 16, 13, 9, 4. What is the best description?
  1. Shrinking, with changes that are not equal
  2. Shrinking and linear
  3. Growing and linear
  4. Repeating
Show answer and explanation
Shrinking, with changes that are not equal
The values get smaller, but the changes are subtract 3, subtract 4, and subtract 5. The changes are not equal.

Question 3

Which phrase best describes red, green, red, green, red, green?
  1. A shrinking pattern
  2. A growing pattern
  3. A repeating pattern
  4. A linear number pattern
Show answer and explanation
A repeating pattern
The two-colour group red, green returns in the same order.

Key terms

Pattern
A set of objects or numbers arranged according to a rule.
Term
One item or group in a pattern.
Repeating pattern
A pattern in which a group returns in the same order.
Growing pattern
A pattern whose terms get larger as it continues.
Shrinking pattern
A pattern whose terms get smaller as it continues.
Linear pattern
A number pattern with the same change from one term to the next.

Continue through Grade 6 Math

View the complete Grade 6 Math Ontario Grade 6 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation C1.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question