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C1.3 · Extend patterns and use their rules to find missing values

Learn to extend patterns and use their rules to find missing values through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Algebra

Notice what changes, describe the rule, and use it to continue a pattern

A pattern is a list or arrangement that follows a rule. The rule tells how one part connects to the next, or how an input connects to an output. For example, if you start with 4 counters and add 3 each time, the number of counters grows in a predictable way. To find a missing value, first look carefully at the values you know. Then state a rule that fits them and test it. A pattern rule should explain all the given values, not just one pair.

What you will learn

  • Identify how terms in a pattern change.
  • Describe a pattern rule in clear words.
  • Extend a pattern by finding its next terms.
  • Use a rule to find a missing value and check that it fits.

1. Look for what changes

A term is one item in a pattern. In the list CAD 5, 8, 11, 14, each number is a term. Compare terms beside each other. From 5 to 8, the value increases by 3. From 8 to 11, it also increases by 3. This repeated change suggests the rule: add 3 each time.
A pattern can grow, shrink, or repeat. In CAD 20, 16, 12, 8, subtracting 4 each time explains the change. In a repeating pattern such as red, blue, red, blue, the same order starts again. Say the rule in words before trying to extend the pattern.
A growing pattern can also be shown with a table. The position tells where a term appears. The value tells the number at that position. Looking down the value column helps you notice the change. When a rule is not obvious, compare more than one pair of terms.
  • A term is one item in a pattern.
  • Compare nearby terms to find a repeated change.
  • A rule must fit every given term.

2. Use a rule to find a missing value

Sometimes a pattern has a blank. Do not fill it by guessing. Find the rule first, then use it to work from a known value. For a growing number pattern, moving forward might mean adding the same amount each time. Moving backward means undoing that change.
Some tables use a rule that connects an input to an output. For example, the rule might be “double the input, then add 1.” Apply both actions in that order to each input. The output values can then be checked against the rule.
A useful check is to use the rule again on another known value. If it does not fit, reconsider the rule. A rule that explains only one value is not enough. Use the simplest rule that matches all the information shown.
  • Find and state a rule before filling a blank.
  • Follow the actions in the order stated.
  • Check the result against another known value when possible.

3. Work through examples

In each example, the diagram or table shows stated values only. The points in a graph pair each position with its pattern value. The graph is another way to see how the values change; the rule still comes from the pattern.
  • Read the values carefully before deciding on a rule.
  • Use the rule to extend or complete the pattern.

4. Try it and check your thinking

For practice, extend 7,12,17,□7, 12, 17, \square by finding the change between terms. Then complete the input-output table with the rule “add 4 to the input, then double the result”: input 1 gives output 10, input 2 gives output 12, and input 3 is missing. State the rule you used and check your answer.
When you review your work, ask: Did I compare the right values? Does my rule explain every given term? Did I follow the rule in the correct order? These questions help catch errors before you finish.
  • Write the rule in words.
  • Apply it carefully to the missing position or input.
  • Check that the completed pattern makes sense.

A growing tile pattern

Design positionNumber of tiles
14
27
310
413

Worked example

Extend a growing pattern

A row of tiles has 4 tiles in the first design, 7 in the second, and 10 in the third. How many tiles are in the fourth design?
Design position and tile count
Design position and tile countEach point shows the stated design position and its number of tiles. The fourth point follows the add-3 rule.Design positionNumber of tiles1, 42, 73, 104, 13

Each point shows the stated design position and its number of tiles. The fourth point follows the add-3 rule.

  1. Compare the terms
    The number of tiles goes from 4 to 7, then from 7 to 10. Each change is an increase of 3.
    7−4=3,10−7=37-4=3,\quad 10-7=3
  2. Use the rule
    The rule is add 3 each time. Add 3 to the third design to find the fourth.
    10+3=1310+3=13
Answer: The fourth design has 13 tiles.
Check: The sequence 4, 7, 10, 13 increases by 3 each time.

Worked example

Find a missing term

Complete the pattern 30,25,□,15,1030, 25, \square, 15, 10.
Position and pattern value
Position and pattern valueThe missing value at position 3 is 20. The points show the completed pattern.PositionPattern value1, 302, 253, 204, 155, 10

The missing value at position 3 is 20. The points show the completed pattern.

  1. Find the change
    The first two terms decrease by 5. The last two terms also decrease by 5, so subtract 5 each time fits the pattern.
    30−25=5,15−10=530-25=5,\quad 15-10=5
  2. Fill the blank
    Subtract 5 from 25 to find the missing term. Check that subtracting 5 from the missing term gives 15.
    25−5=20,20−5=1525-5=20,\quad 20-5=15
Answer: The missing term is 20.
Check: The completed pattern is 30, 25, 20, 15, 10. Each term is 5 less than the one before it.

Worked example

Use a rule with two actions

A machine follows the rule “double the input, then add 1.” The input-output pairs are 1 to 3, 2 to 5, and 3 to a missing output. Find the missing output.
Input and output pairs
Input and output pairsThe first two pairs are given. The third pair uses the same rule: double the input, then add 1.InputOutput1, 32, 53, 7

The first two pairs are given. The third pair uses the same rule: double the input, then add 1.

  1. Follow the first action
    The input is 3. Double it before doing anything else.
    3×2=63\times 2=6
  2. Follow the next action
    Now add 1 to the result. The rule gives an output of 7.
    6+1=76+1=7
  3. Check the rule
    For input 2, doubling gives 4 and adding 1 gives 5, which matches the given output. This supports the rule.
    2×2+1=52\times 2+1=5
Answer: The missing output is 7.
Check: Applying the same two actions to input 3 gives 7.

Common mistakes and how to avoid them

Guessing a missing value without checking the change.
Correction: Compare the known terms and state a rule that fits them before filling the blank.
Using a rule that fits only one pair of terms.
Correction: Check the rule against all the given values, including another pair when possible.
Doing the actions of a rule in the wrong order.
Correction: Read the rule carefully and complete each action in the stated order.

Lesson summary

  • A pattern follows a rule.
  • Compare terms or input-output pairs to notice what changes.
  • Use the rule to extend a pattern or find a missing value.
  • Check that the rule fits the known values.

Check your understanding

Question 1

What number completes 6,10,14,□6, 10, 14, \square if the same change continues?
  1. 16
  2. 18
  3. 20
  4. correctIndex": 1, "explanation": "Each term increases by 4, so the next term is 14 + 4 = 18."}]}]}
Show answer and explanation
18
The pattern increases by 4 each time. Add 4 to 14 to get 18.

Question 2

A rule says “multiply the input by 3, then subtract 2.” What is the output for input 4?
  1. 10
  2. 12
  3. 14
  4. correctIndex": 0, "explanation": "Multiply 4 by 3 to get 12, then subtract 2 to get 10."}]}]}
Show answer and explanation
10
Multiply 4 by 3 to get 12, then subtract 2 to get 10.

Question 3

Which rule fits the pattern 21, 18, 15, 12?
  1. Add 3 each time
  2. Subtract 3 each time
  3. Subtract 2 each time
  4. correctIndex": 1, "explanation": "Each term is 3 less than the term before it."}]}]}
Show answer and explanation
Subtract 3 each time
Each term is 3 less than the term before it.

Key terms

Pattern
A list or arrangement that follows a rule.
Term
One item or value in a pattern.
Input
A starting value used with a rule.
Output
The value produced when a rule is used on an input.

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