DoAssignment.ca

DoAssignment study guide

C1.3 · Extend patterns and predict unknown terms using linear rules

Learn to extend patterns and predict unknown terms using linear rules through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Algebra

Find what changes, describe the rule, and predict a term

A pattern is an ordered list of numbers or objects. A term is one number or object in that list. You may already know how to continue a pattern by looking at what changes from one term to the next. In this lesson, you will use that change to make a rule. A linear rule describes a pattern that changes by the same amount each time. This rule can help you find terms, including terms far along in the pattern.

What you will learn

  • Recognize a pattern that changes by the same amount each time.
  • Describe a linear pattern rule in words and symbols.
  • Use a rule to extend a pattern and find a term at a given position.
  • Check a prediction against the pattern.

1. Look for a steady change

Imagine building rows of tiles. The first row has 4 tiles. Each new row has 3 more tiles than the row before it. The tile counts begin 4, 7, 10, 13. The change between each pair of terms is 3.
The amount added each time is called the common difference. It is common because it stays the same throughout the pattern. A pattern with a constant increase or decrease can be described by a linear rule.
To find the common difference, compare neighbouring terms. Subtract the earlier term from the next term. For example, the change from 7 to 10 is 3. Check more than one pair. If the same change continues, it supports the rule. A pattern that decreases steadily has a negative change; for example, 18, 14, 10 changes by −4 each time.
The position tells where a term appears. The first term is at position 1, the second at position 2, and so on. Keep positions in order, even when the term values are not in order.
d = next term - previous term
  • A term is one item in a pattern.
  • A constant change between terms is the common difference.
  • The position counts how far along the pattern a term appears.

2. Write and use a linear rule

A rule can be described in words first. For the tile pattern, start with 4 tiles and add 3 tiles for every step after the first term. The phrase “after the first term” matters: the starting 4 is already there at position 1.
A compact rule uses the position number, often written as nn, and the term value, often written as TT. The position is multiplied by the common difference, then the starting adjustment is made. For a pattern beginning at 4 and increasing by 3, the rule is T=3n+1T = 3n + 1. At position 1, it gives 3(1)+1=43(1)+1=4. At position 2, it gives 3(2)+1=73(2)+1=7.
The adjustment in this rule is 1 rather than 4 because multiplying the position by 3 already gives 3 at position 1. Adding 1 makes the first term 4. This is a useful way to write the rule, not a different pattern.
You can also reason without first writing the compact rule. Find how many steps separate the requested position from position 1. Multiply that number of steps by the common difference, then add the first term. For position 6 in the tile pattern, there are 5 steps after the first term, so the term is 4+5(3)=194+5(3)=19.
A coordinate graph can show the same information. Each plotted point pairs a position with its term value. The horizontal axis shows position, and the vertical axis shows term value. The points for this pattern are (1,4)(1,4), (2,7)(2,7), (3,10)(3,10), and (4,13)(4,13). The graph helps you see that equal steps in position match equal increases in term value.
T=T1+(n−1)dT = T_1 + (n - 1)d
  • Use the position as the input to a rule and the term value as the output.
  • For a rule starting at the first term, count the steps after position 1.
  • A coordinate point lists position first and term value second.

3. Make a prediction and check it

When a position is missing from a list, first look at the terms on both sides of the gap. Test whether the pattern has one steady change. If it does, continue that change to fill the gap.
For a distant position, use the rule or count the number of steps from the first position. Do not multiply the common difference by the position number and then add the first term. That would count one extra change. The first term is already at position 1.
Check a prediction in two ways when possible. Substitute the position into the rule, and compare the answer with the nearby terms. If the pattern increases by 5, for example, each next term must be 5 greater than the last. A prediction that breaks that condition does not fit the rule.
Not every list of numbers must follow a linear rule. A few terms alone may allow more than one possible continuation. Use the stated pattern rule or the pattern information provided. Do not claim a steady change unless the terms or context support it.
  • Use nearby terms to test a missing value.
  • For a distant term, count changes from the first term.
  • Check that your answer still follows the common difference.

4. Try the method

For each pattern, write the common difference and extend the list by two terms. Then use the stated position to predict a term.
Pattern A begins 6,11,16,…6, 11, 16, \ldots. Its common difference is 5. The next two terms are 21 and 26. For a term at position 8, count 7 changes from the first term.
Pattern B begins 30,26,22,…30, 26, 22, \ldots. Its common difference is −4. The next two terms are 18 and 14. A negative difference means subtract 4 each time; it does not mean the term values are positions.
  • A linear pattern may increase or decrease.
  • Use the same change each time when extending the pattern.

Positions and tile counts

PositionTile count
14
27
310
413

Worked example

Find a distant term

A tile pattern has 5 tiles in its first design. Each new design adds 4 tiles. How many tiles are in design 9?
Tile count by design
Tile count by designHypothetical tile pattern. Each point pairs a design position with its tile count.Design positionNumber of tiles(1, 5)(2, 9)(3, 13)(4, 17)Hypothetical pattern stated in the problem

Hypothetical tile pattern. Each point pairs a design position with its tile count.

  1. Identify the pattern
    The first design has 5 tiles, and each step adds 4. The common difference is 4.
    d=4d=4
  2. Count the changes
    From design 1 to design 9 there are 8 changes. Multiply the number of changes by 4, then add the 5 tiles already in the first design.
    5+8(4)=375+8(4)=37
Answer: Design 9 has 37 tiles.
Check: The counts go 5, 9, 13, and continue to rise by 4. The rule also gives 5 at position 1.

Worked example

Fill a missing term

A sequence has the terms 31, 25, , 13, 7. Find the missing term and describe the rule.
  1. Find the change
    The change from 31 to 25 is −6. The change from 13 to 7 is also −6, so the missing term should keep the same decrease.
    25−6=1925-6=19
  2. Check both sides
    The missing term is 19. The changes are −6 from 31 to 25, from 25 to 19, from 19 to 13, and from 13 to 7.
    31, 25, 19, 13, 731,\ 25,\ 19,\ 13,\ 7
Answer: The missing term is 19. The rule is to subtract 6 each time.
Check: Every pair of neighbouring terms differs by −6.

Worked example

Write a rule and predict

A pattern begins 8, 11, 14, 17. Write a rule for the term at position n, then find the term at position 12.
Pattern values by position
Pattern values by positionThe plotted points show the first four terms of the stated pattern.PositionTerm value(1, 8)(2, 11)(3, 14)(4, 17)Pattern stated in the problem

The plotted points show the first four terms of the stated pattern.

  1. Describe the pattern
    The common difference is 3. Begin with 8 at position 1. At position n, there have been n−1 changes of 3.
    T=8+3(n−1)T=8+3(n-1)
  2. Use the position
    Position 12 is 11 changes after position 1. Add 11 groups of 3 to the first term.
    T=8+3(12−1)=41T=8+3(12-1)=41
  3. Check the rule
    At position 1, the rule gives 8. Each increase of 1 in position adds 3 to the term, matching the pattern.
    8, 11, 14, 178,\ 11,\ 14,\ 17
Answer: The rule is to start at 8 and add 3 for each step after position 1. The term at position 12 is 41.
Check: The rule gives 8 at position 1 and increases by 3 each time.

Common mistakes and how to avoid them

Adding the first term after multiplying the common difference by the position.
Correction: Count the changes after position 1. Use n−1 changes, because the first term is already at position 1.
Checking only one pair of terms before deciding on a rule.
Correction: Compare several neighbouring pairs when they are available. A linear rule needs the same change each time.
Treating a decrease as a positive change.
Correction: Keep track of direction. For 20, 16, 12, each change is −4, so subtract 4 each time.
Mixing up a position with a term value.
Correction: The position tells where a term is in the list. The term value is the number at that position.

Lesson summary

  • Find the change between neighbouring terms.
  • A steady change is the common difference in a linear pattern.
  • To predict a term, count the changes after the first term and apply the rule.
  • Check that the rule gives the first term and keeps the same change.

Check your understanding

Question 1

A pattern begins 9, 13, 17, 21. What is the term at position 7?
  1. 29
  2. 33
  3. 37
  4. 28
Show answer and explanation
33
The common difference is 4. Position 7 is 6 changes after the first term, so the term is 9+6(4)=33.

Question 2

Which rule describes 22, 19, 16, 13, ...?
  1. Start at 22 and add 3 each time.
  2. Start at 22 and subtract 3 each time.
  3. Start at 19 and subtract 3 each time.
  4. Start at 22 and subtract 4 each time.
Show answer and explanation
Start at 22 and subtract 3 each time.
Each term is 3 less than the term before it, so the rule is to start at 22 and subtract 3 each time.

Question 3

The first term is 6 and the common difference is 5. What is the term at position 4?
  1. 21
  2. 26
  3. 20
  4. 11
Show answer and explanation
21
Position 4 is 3 changes after position 1. Add 3 groups of 5 to 6 to get 21.

Key terms

Pattern
A list of numbers or objects arranged by a rule.
Term
One number or object in a pattern.
Position
The place of a term in a pattern, counted from 1.
Common difference
The amount added or subtracted between each pair of neighbouring terms in a steady pattern.
Linear rule
A rule for a pattern that changes by the same amount each time.

Continue through Grade 8 Math

View the complete Grade 8 Math Ontario Grade 8 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation C1.3. 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