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C1.3 · Use pattern rules to predict and find missing terms

Learn to use pattern rules to predict and find missing terms through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Algebra

A Grade 6 lesson on noticing how a pattern changes

A pattern is a list or arrangement that follows a rule. A term is one item in that pattern. For example, in the list 4, 7, 10, 13, each number is a term. The pattern rule tells how the terms are connected. You may already know how to count forward or add equal groups. We will use those skills to explain patterns, fill in gaps, and predict what comes next. A good rule must fit every term shown, not just the first pair.

What you will learn

  • Describe the rule that changes one term into the next.
  • Use a pattern rule to find a missing term.
  • Use a pattern rule to predict a term farther along in a pattern.
  • Check that a rule works for all the terms shown.

1. Notice what changes

Start with a simple pattern made from counters. Imagine one counter in the first group, three in the second group, and five in the third group. Each new group has two more counters than the group before it. The pattern rule is “add 2 each time.”
The numbers in a pattern are its terms. The first term is the first number, the second term is the next number, and so on. The place of a term in the list is its position. We can keep track of positions by counting: first, second, third, and fourth.
To find a rule, compare terms that are beside each other. Ask: Does the number increase or decrease? By how much? Is the same change repeated? A change from one term to the next is not enough evidence by itself if later terms do not follow it. Check the rule against all the given terms.
A pattern can also repeat rather than grow. For example, red, blue, blue, red, blue, blue repeats the group red, blue, blue. Look for what repeats or changes before you decide on a rule.
  • A term is one item in a pattern.
  • A pattern rule describes how the pattern is made.
  • Check a suggested rule against every term shown.

2. Keep track of positions

A table can help when a pattern has many terms. One row shows the position, and another row shows the term at that position. This makes it easier to see whether the rule still works as the pattern grows.
For the pattern that starts at 5 and adds 3 each time, the terms are 5, 8, 11, 14, and 17. Each move to the right adds 3. The fifth term is 17 because there are four moves of 3 after the first term.
You can also describe a pattern using a position-and-term table or a coordinate graph. On a graph, each point pairs a position with the term at that position. The graph is another way to show the same information; it does not replace checking the pattern rule.
next term = current term + change
  • The position tells where a term sits in the list.
  • A table connects each position to its term.
  • The first term is already at position 1, so count the moves after it carefully.

3. Find gaps and predict terms

A missing term is a place in the pattern where a number has been left out. Use the terms around the gap to work out what belongs there. Then check the rule from the start of the pattern through the end.
For example, in 6, 10, , 18, the change is 4 each time. The missing term is 14: it is 4 more than 10 and 4 less than 18. Both sides of the gap support the same rule.
To predict a later term, continue the rule in order, or use a table to keep track of the position. A prediction should follow the stated rule. If more than one rule could fit a short list, say which rule you are using and check whether the problem gives more information.
  • Use terms on both sides of a gap when they are available.
  • Continue the same rule to make a prediction.
  • A rule should explain the known terms as well as the missing one.

4. A careful pattern-solving routine

First, write the terms in order and mark any missing spot. Next, compare neighbouring terms. State the change in plain words, such as “add 4 each time” or “subtract 2 each time.” If the pattern repeats, name the repeating group.
Then use the rule to fill the gap or continue to the requested position. Finally, read the pattern from the beginning and check every step. This final check catches mistakes such as adding the right amount once and then changing the rule by accident.
If a pattern does not have one clear rule from the information given, do not pretend that it does. Explain what rule you tested. In a classroom pattern question, a stated rule or enough terms may help identify the intended pattern.
  • State the rule in words before using it.
  • Keep the rule the same at every step.
  • Check the completed pattern from beginning to end.

A pattern and its positions

Position1234
Term581114
Change from previous term—+3+3+3

Worked example

Example 1: Continue a growing number pattern

A pattern begins 7, 12, 17, 22. State the rule and find the next two terms.
Positions and terms
Positions and termsEach point pairs a position with the term at that position.PositionTerm1, 72, 123, 174, 225, 276, 32Pattern stated in the problem

Each point pairs a position with the term at that position.

  1. Compare neighbouring terms
    From 7 to 12, the pattern increases by 5. From 12 to 17, it also increases by 5. The next change is likely the same, and we will check it against the full list.
    12−7=512-7=5
  2. State the rule
    The repeated change is 5, so the rule is “add 5 each time.” Apply it to 22 to find the next term, then apply it once more.
    22+5=27,27+5=3222+5=27,\quad 27+5=32
Answer: The rule is add 5 each time. The next two terms are 27 and 32.
Check: The differences are 5, 5, 5, 5, and 5, so the same rule fits the whole pattern.

Worked example

Example 2: Find a missing term

Find the missing term: 31, 27, , 19, 15.
  1. Find the change
    The first move is from 31 down to 27, a decrease of 4. The last move is from 19 down to 15, also a decrease of 4. These terms support the rule “subtract 4 each time.”
    31−27=4,19−15=431-27=4,\quad 19-15=4
  2. Fill the gap
    Subtract 4 from 27 to get the missing term. Then check that subtracting 4 from the missing term gives 19.
    27−4=23,23−4=1927-4=23,\quad 23-4=19
Answer: The missing term is 23. The rule is subtract 4 each time.
Check: The completed list is 31, 27, 23, 19, 15. Every step decreases by 4.

Worked example

Example 3: Predict a farther term

A pattern starts at 4 and adds 3 each time. What is the sixth term?
Six terms in the pattern
Six terms in the patternThe hypothetical pattern starts at 4 and increases by 3 at each step.PositionTerm1, 42, 73, 104, 135, 166, 19Hypothetical pattern given in the problem

The hypothetical pattern starts at 4 and increases by 3 at each step.

  1. List the positions
    The first term is 4. To reach the sixth term, make five moves, because the first term is already at position 1. Add 3 at each move.
    4, 7, 10, 13, 16, 194,\ 7,\ 10,\ 13,\ 16,\ 19
  2. Read the requested position
    Count the terms in order. The sixth term is 19. The list also shows that each move follows the given rule. sixth term=19
Answer: The sixth term is 19.
Check: There are five increases of 3 after the first term: 4, 7, 10, 13, 16, 19.

Common mistakes and how to avoid them

Using the change only once, then switching to a different change.
Correction: Keep the same rule at every step, unless the problem states a rule that changes.
Counting the first term as a move when finding a later position.
Correction: The first term is already at position 1. To reach position 6, make five moves after it.
Checking only the terms beside a missing spot.
Correction: Read through the whole completed pattern and confirm that the rule works at every step.

Lesson summary

  • A pattern rule tells how terms are connected or how a group repeats.
  • Compare neighbouring terms to find a repeated change.
  • Use the rule to fill a gap or continue to a requested position.
  • Check every step before accepting an answer.

Check your understanding

Question 1

The pattern is 9, 13, 17, 21. What is the rule and the next term?
  1. Add 4 each time; 25
  2. Add 3 each time; 24
  3. Subtract 4 each time; 17
  4. Add 5 each time; 26
Show answer and explanation
Add 4 each time; 25
Each term is 4 more than the term before it. Adding 4 to 21 gives 25.

Question 2

Find the missing term: 20, 16, , 8.
  1. 10
  2. 12
  3. 14
  4. 15
Show answer and explanation
12
The rule is subtract 4 each time: 20, 16, 12, 8.

Question 3

A pattern starts at 6 and adds 2 each time. What is the fourth term?
  1. 10
  2. 11
  3. 12
  4. 14
Show answer and explanation
12
The terms are 6, 8, 10, 12. The fourth term is 12.

Key terms

Pattern
A list or arrangement that follows a rule.
Term
One item in a pattern.
Pattern rule
A description of how a pattern is made or how its terms are connected.
Position
The place of a term in an ordered pattern.
Missing term
A term left out of a pattern that needs to be found.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation C1.3. It is a study resource, not an official curriculum publication.

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