Create patterns showing relationships among rational numbers

Ontario Grade 8 Mathematics

Algebra

Create and describe patterns using integers, fractions, and decimals

A pattern is a set of numbers arranged so that the numbers follow a rule. A rational number is a number that can be written as a fraction. Integers, fractions, and terminating decimals are examples. For instance,

-2

,

\frac{1}{2}

, and

0.75

are rational numbers. In this lesson, you will create patterns that make a relationship among rational numbers easy to see. You can use a number list, a table, or a number line. A clear pattern has a rule that explains how its numbers are connected.

What you will learn

Recognize rational numbers in different forms.
Create a number pattern that shows a clear relationship among rational numbers.
Describe a pattern in words and use it to find a missing or next value.
Check that a pattern rule works for every pair of entries.

Start with a concrete pattern

Imagine moving along a number line. Start at

-1

and move

\frac{1}{2}

unit to the right each time. The first few positions are

-1

,

-\frac{1}{2}

,

0

, and

\frac{1}{2}

. Each position is one half greater than the position before it.

The number line helps show both the order and the size of the steps. Moving right means the numbers increase. Moving left means they decrease. A pattern can also be shown as a list or table, but the rule should stay the same in every representation.

A rational number may be written in more than one way. For example,

\frac{1}{2}

and

0.5

name the same amount. Choose forms that make the relationship easy to see. If you mix forms, check that you are comparing equal amounts.

A pattern rule tells how entries are related.
On a number line, numbers farther right are greater.
Equivalent forms name the same rational number.

Build and describe a rule

A simple way to create a pattern is to choose a starting number and a repeated change. For example, begin at

-\frac{3}{4}

and add

\frac{1}{4}

each time. The pattern is

-\frac{3}{4}, -\frac{1}{2}, -\frac{1}{4}, 0

. The repeated change is the rule.

To check the rule, compare each number with the one immediately before it. From

-\frac{3}{4}

to

-\frac{1}{2}

is an increase of

\frac{1}{4}

. The next two steps also increase by

\frac{1}{4}

. The same rule works throughout the pattern.

You can also create a pattern from two related lists. For example, each value in one list might be double the value beside it. A table makes the paired relationship visible. State the relationship in words so another person can continue the pattern without guessing.

A useful description names the starting point and the change, or explains how the paired values are related. Avoid saying only that the numbers “go up.” Explain by how much, or describe the exact connection.

Choose rational numbers and a rule that connects them.
Apply the same rule each time.
Check every step or pair before presenting the pattern.

Use different forms to show relationships

A pattern can connect fractions, decimals, and integers. Before using different forms together, make sure they represent comparable amounts. For example,

0.25

is the same as

\frac{1}{4}

, so adding

0.25

each time is the same as adding

\frac{1}{4}

.

A number line is helpful when the pattern depends on order or direction. A table is helpful when you want to compare two lists of values. A written rule explains the connection in plain language. These forms can work together: the diagram shows where values lie, the table organizes them, and the words state the rule.

When a pattern has a missing entry, use the rule rather than guessing from the appearance of the list. Check the entry before and after the gap, if available. Both should fit the same relationship.

Fractions and decimals can represent the same rational number.
Choose a representation that makes the relationship clear.
A missing value must fit the pattern rule.

Create, test, and communicate

When you create your own pattern, first decide what relationship you want to show. You might use a repeated increase or decrease, or pair numbers so one is a stated multiple of the other. Pick values that make the rule easy to check.

Write enough entries for the relationship to be clear. Then test the rule on every step or pair. Finally, describe the rule in a full sentence. A classmate should be able to continue your pattern from that description.

Try this practice task: create four rational numbers that decrease by

0.4

each time, starting at

1.2

. Write the numbers and describe the rule. Then make a second pattern of paired rational numbers in which each second value is half its first value. Check every pair.

Plan the relationship before listing values.
Include enough entries to make the rule visible.
Explain and test the rule, not just the answers.

A paired pattern with a repeated relationship

First number	Second number	Relationship
$-\frac{1}{2}$	$-1$	Second is twice the first
$\frac{1}{4}$	$\frac{1}{2}$	Second is twice the first
$\frac{3}{4}$	$\frac{3}{2}$	Second is twice the first

Worked example

Repeated addition with fractions

Create the first five terms of a pattern that starts at

-\frac{5}{6}

and increases by

\frac{1}{3}

each time. State the rule.

Five terms on a number line

The values increase from left to right in steps of one third.

Use a common-sized step
The starting value and the change are both fractions. Rewrite the change as sixths so it is easy to add to the starting value. One third is two sixths.
$\frac{1}{3}=\frac{2}{6}$
Continue the pattern
Add two sixths to each term. The first increase gives negative three sixths, which is negative one half. The next increase gives negative one sixth, followed by one sixth and three sixths.
$-\frac{5}{6},\ -\frac{3}{6},\ -\frac{1}{6},\ \frac{1}{6},\ \frac{3}{6}$
State the rule
The entries rise by the same amount each time. The last two fractions can also be simplified, but keeping sixths makes the repeated change clear.
$-\frac{5}{6},\ -\frac{1}{2},\ -\frac{1}{6},\ \frac{1}{6},\ \frac{1}{2}$

Answer: The five terms are

-\frac{5}{6}, -\frac{1}{2}, -\frac{1}{6}, \frac{1}{6}, \frac{1}{2}

. The rule is to add

\frac{1}{3}

each time.

Check: Each consecutive pair differs by

\frac{1}{3}

, so the same rule works for all four steps.

Worked example

Connect decimals and fractions

Create four terms starting at

0.2

and increasing by

\frac{3}{10}

each time. Explain why the decimal and fraction forms can be used together.

Decimal pattern

The terms move right by three tenths each time.

Match the forms
Write the starting decimal in tenths. This shows that it is the same kind of amount as the increase.
$0.2=\frac{2}{10}$
Add the repeated change
Add three tenths at each step. The numerators increase by three while the denominator stays ten.
$\frac{2}{10},\ \frac{5}{10},\ \frac{8}{10},\ \frac{11}{10}$
Write the terms as decimals
Each fraction in tenths can be written as a decimal. The rule remains an increase of three tenths, or $0.3$ , each time.
$CAD 0.2,\ 0.5,\ 0.8,\ 1.1$

Answer: The four terms are CAD 0.2, 0.5, 0.8, 1.1. The pattern increases by

\frac{3}{10}

, which is

0.3

, each time.

Check: The difference between each neighbouring pair is

0.3

.

Worked example

Show a relationship in paired numbers

Create three pairs of rational numbers so that the second number in each pair is twice the first. Use first numbers

-\frac{1}{2}

,

\frac{1}{4}

, and

\frac{3}{4}

.

Find the related value
For each pair, double the first number. Doubling a fraction means adding it to itself. This keeps the relationship the same even when the first number is negative.
$-\frac{1}{2}+\left(-\frac{1}{2}\right)=-1$
Complete all three pairs
The second value for one quarter is one half. The second value for three quarters is six quarters, which is one and one half.
$\left(-\frac{1}{2},-1\right),\ \left(\frac{1}{4},\frac{1}{2}\right),\ \left(\frac{3}{4},\frac{3}{2}\right)$
Describe the relationship
Each second value is twice its paired first value. Check the positive pairs as well as the negative pair so the rule is verified throughout.
$2\left(\frac{1}{4}\right)=\frac{1}{2},\quad 2\left(\frac{3}{4}\right)=\frac{3}{2}$

Answer: The pairs are

\left(-\frac{1}{2},-1\right)

,

\left(\frac{1}{4},\frac{1}{2}\right)

, and

\left(\frac{3}{4},\frac{3}{2}\right)

. The second number in each pair is twice the first.

Check: Doubling each first value gives its partner, including doubling

-\frac{1}{2}

to get

-1

.

Common mistakes and how to avoid them

Changing the amount added partway through a pattern.

Correction: Use the same stated change at every step, then check each neighbouring pair.

Treating a negative number as positive when continuing a pattern.

Correction: Keep the sign and use the number line or fraction values to check the direction of change.

Assuming two different-looking numbers are different amounts.

Correction: Check whether they are equivalent forms, such as

0.5

and

\frac{1}{2}

.

Giving a list without explaining its rule.

Correction: State the repeated change or the relationship between paired entries.

Lesson summary

A rational-number pattern follows a rule that connects its entries.
Patterns can use integers, fractions, and decimals, including equivalent forms.
Describe the rule clearly and check that it works at every step or in every pair.
Use a number line, table, or list when it helps make the relationship visible.

Check your understanding

Question 1

A pattern starts at

-0.6

and increases by

0.2

each time. What are the next two terms?

$-0.4, -0.2$
$-0.8, -1.0$
$-0.4, 0$
$-0.2, 0$

Show answer and explanation

-0.4, -0.2

Adding

0.2

to

-0.6

gives

-0.4

. Adding

0.2

again gives

-0.2

.

Question 2

Which pair shows that the second number is twice the first?

$\left(\frac{1}{3},\frac{2}{3}\right)$
$\left(\frac{1}{3},\frac{1}{6}\right)$
$\left(-\frac{1}{2},\frac{1}{2}\right)$
$\left(\frac{3}{4},\frac{3}{4}\right)$

Show answer and explanation

\left(\frac{1}{3},\frac{2}{3}\right)

Twice

\frac{1}{3}

is

\frac{2}{3}

. The other pairs do not have a second value that is twice the first.

Question 3

Which description fits the pattern

\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1

?

Add $\frac{1}{4}$ each time.
Add $\frac{1}{2}$ each time.
Subtract $\frac{1}{4}$ each time.
Double each term to get the next term.

Show answer and explanation

Add

\frac{1}{4}

each time.

Each neighbouring pair differs by

\frac{1}{4}

, so adding one quarter gives every next term.

Key terms

Pattern: A set or list of numbers that follows a rule.
Rational number: A number that can be written as a fraction, including integers and terminating decimals.
Equivalent forms: Different ways to write the same value, such as $0.5$ and $\frac{1}{2}$ .
Term: One number in a pattern.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation C1.4. 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

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C1.4 · Create patterns showing relationships among rational numbers

Ontario Grade 8 Mathematics

Create and describe patterns using integers, fractions, and decimals

What you will learn

Start with a concrete pattern

Build and describe a rule

Use different forms to show relationships

Create, test, and communicate

A paired pattern with a repeated relationship

Repeated addition with fractions

Connect decimals and fractions

Show a relationship in paired numbers

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review