Represent rational-number patterns with tables, graphs, and equations

Ontario Grade 8 Mathematics

Algebra

Use tables, graphs, and equations to describe how values change

A pattern is a set of values that follows a rule. A rational number is a number that can be written as a fraction, such as

\frac{1}{2}

,

-\frac{3}{4}

, or

2

. Rational-number patterns can be shown with a table, a graph, and an equation. Each representation gives a different view of the same rule. In this lesson, you will start with values and their changes, organize them, and then show the pattern with points and symbols.

What you will learn

Recognize how a rational-number pattern changes from one term to the next.
Organize input and output values in a table.
Plot ordered pairs from a table on a coordinate graph.
Write and check an equation that represents a pattern.

1. Start with the values and their change

Suppose a pattern starts at

-1

and increases by

\frac{1}{2}

each time. The first values are

-1

,

-\frac{1}{2}

,

0

,

\frac{1}{2}

, and

1

. Each new value is the previous value plus

\frac{1}{2}

.

A table places related values in rows or columns. For a pattern, one column can show the term number, or input, and another can show the pattern value, or output. An input is the value you choose or start with. An output is the value the pattern rule gives for that input.

A graph shows each input and output as a point. An ordered pair is a pair of numbers written in a set order: input first, output second. For example, the table pair with input

2

and output

0

is written

(2,0)

. On a coordinate graph, the input is read across the horizontal axis and the output is read up or down the vertical axis.

An equation is a mathematical statement that says two quantities are equal. A letter can stand for a changing value. In this lesson,

n

stands for the input and

y

stands for the output. The rule for the example is

y=-1+\frac{1}{2}n

. It says to begin at

-1

and add half of the input. When the input increases by

1

, the output increases by

\frac{1}{2}

.

y=-1+\frac{1}{2}n

Track the input and output separately.
A graph point is written as (input, output).
The table, graph, and equation must describe the same pattern.

2. Build and read a table, graph, and equation

To build a table, choose input values, apply the rule, and record each output beside its input. Use a consistent set of inputs so the table is easy to read. If the pattern is about whole-number steps, the inputs might be

0

,

1

,

2

, and

3

. The outputs can still be fractions or negative numbers.

To graph the table, use each input and output as a point. First find the input on the horizontal axis. Then find the output on the vertical axis. Mark the point where those positions meet. Label both axes so a reader knows what each number means. For a pattern listed at separate steps, plot the points as separate points. Do not join them unless the situation says values between the listed steps belong in the pattern.

To find an equation from a pattern, look for what happens to the output when the input increases by

1

. This change is sometimes called the step change. Also check the output when the input is

0

, if that value is available. These observations help describe the rule. Then test the rule with more than one row of the table.

A pattern can have negative and positive values, or fractional values. Pay attention to signs and fractions as you calculate. For example, adding

\frac{1}{2}

to

-\frac{1}{2}

gives

0

. The values may cross zero without changing the pattern rule.

(n,y)

Use the same inputs in the table, graph, and equation.
Label the horizontal and vertical axes.
Check an equation by substituting table inputs and comparing the outputs.

3. Use representations to check your thinking

A table makes exact values easy to compare. A graph makes the direction and spacing of the pattern easier to see. An equation gives a short description that can be checked for any chosen input in the pattern.

Representations work together. If a table says that input

3

gives output

5

, the graph should include

(3,5)

and the equation should also give

5

when the input is

3

. If one representation disagrees, look for a calculation or plotting error.

Some patterns have a constant change in output for each increase of

1

in the input. In those patterns, the graph points line up. The equation can show the starting output and the repeated change. Do not rely only on the graph’s appearance: use the exact table values to check the pattern.

When you create a pattern, state the rule in words before writing an equation. For example, “start at three and subtract two each time” is easier to check than a string of numbers alone. Then write a table, plot its ordered pairs, and see whether the equation matches.

Compare all three representations to catch errors.
A graph helps show change, while the table preserves exact values.
Explain the rule in words before recording it as an equation.

Worked example

A pattern that increases by one-half

A number pattern begins at

-1

and increases by

\frac{1}{2}

for each step. Let

n

be the step number, starting at

0

, and let

y

be the pattern value. Represent the pattern with a table, graph, and equation for steps

0

through

4

.

Pattern values by step

Each point represents one step from the table. The vertical axis includes fractional and negative values.

Record the rule
The pattern starts at $-1$ . Each increase of $1$ in the step number adds $\frac{1}{2}$ to the value. This gives the equation.
$y=-1+\frac{1}{2}n$
Calculate the outputs
Use the rule for each step. At step $0$ , the value is $-1$ . At step $1$ , it is $-\frac{1}{2}$ . Continuing gives $0$ , $\frac{1}{2}$ , and $1$ . The input row is CAD 0, 1, 2, 3, 4, and the output row is $-1, -\frac{1}{2}, 0, \frac{1}{2}, 1$ .
Plot the ordered pairs
Plot each input with its output. The graph uses the points shown in the table. The points are separate because the pattern is recorded at step numbers.
$(0,-1),(1,-\frac{1}{2}),(2,0),(3,\frac{1}{2}),(4,1)$
Check
For example, the equation gives $1$ at step $4$ , which matches the table and the plotted point.
$-1+\frac{1}{2}(4)=1$

Answer: The input row is CAD 0, 1, 2, 3, 4, and the output row is

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

. The graph points are

(0,-1)

,

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

,

(2,0)

,

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

, and

(4,1)

. The equation is

y=-1+\frac{1}{2}n

.

Check: The output rises by

\frac{1}{2}

for each step, and the equation gives the listed output at every input.

Worked example

Use rational inputs in a rule

A pattern rule is “double the input, then add one.” Let

x

be the input and

y

the output. Represent the rule for inputs

-1

,

0

,

\frac{1}{2}

, and

2

.

Outputs for rule double and add one

The plotted coordinates use the four stated inputs, including the rational input one-half.

Write the rule
Doubling the input means multiplying it by $2$ . Then add $1$ to that result.
$y=2x+1$
Find the outputs
Substitute each input. For $x=\frac{1}{2}$ , doubling gives $1$ , and adding $1$ gives $2$ . The other outputs follow the same rule. The input row is $-1, 0, \frac{1}{2}, 2$ , and the output row is $-1, 1, 2, 5$ .
Make the graph points
Use the input first and output second in each ordered pair. Plot these exact points on a coordinate graph.
$(-1,-1),(0,1),(\frac{1}{2},2),(2,5)$
Verify a value
For input $2$ , doubling gives $4$ , and adding $1$ gives $5$ . This agrees with the table and point.
$2(2)+1=5$

Answer: The equation is

y=2x+1

. The input row is

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

, and the output row is

-1, 1, 2, 5

. The graph points are

(-1,-1)

,

(0,1)

,

(\frac{1}{2},2)

, and

(2,5)

.

Check: The equation matches each row when the input is used for

x

.

Worked example

Find a rule from a table

A table lists inputs

0

,

1

,

2

, and

3

, with outputs

3

,

1

,

-1

, and

-3

. Describe the pattern, write an equation, and give the graph points.

A pattern that decreases by two

The plotted points match the input-output pairs in the table.

Compare consecutive outputs
Each time the input increases by $1$ , the output decreases by $2$ . The output at input $0$ is $3$ . These two facts describe the pattern: start at $3$ and subtract $2$ for each increase of $1$ in the input.
$3,1,-1,-3$
Write an equation
The output begins at $3$ when the input is $0$ . The input contributes a decrease of $2$ for each step, so the equation is:
$y=3-2x$
List the points
Each row becomes an ordered pair, with the input first. These are the points to plot.
$(0,3),(1,1),(2,-1),(3,-3)$
Check the last row
Using input $3$ , subtract twice $3$ from $3$ . The result matches the final table output.
$3-2(3)=-3$

Answer: The pattern starts at

3

and decreases by

2

per increase of

1

in the input. Its equation is

y=3-2x

. Plot

(0,3)

,

(1,1)

,

(2,-1)

, and

(3,-3)

.

Check: The rule gives all four outputs in the table, including

-3

at input

3

.

Common mistakes and how to avoid them

Putting the output first when making a graph point.

Correction: Write and plot the input first, followed by the output:

(x,y)

.

Adding the step change even when the pattern decreases.

Correction: Read the sign of the change. A decrease of

2

means subtract

2

each step.

Using a graph point that does not match the table.

Correction: Check each point against its table row. The horizontal coordinate must be the input and the vertical coordinate must be the output.

Writing an equation that matches only the first row.

Correction: Test the equation with several rows, including an input other than

0

.

Lesson summary

A table records inputs and outputs side by side.
A graph shows each input-output pair as an ordered pair, with input first.
An equation describes how to get the output from the input.
Check that the table, graph, and equation all show the same pattern.

Check your understanding

Question 1

A pattern starts at

2

when

n=0

and decreases by

\frac{1}{2}

for each increase of

1

in

n

. Which equation and output at

n=2

are correct?

$y=2-\frac{1}{2}n$ ; the output is $1$ .
$y=2+\frac{1}{2}n$ ; the output is $3$ .
$y=\frac{1}{2}-2n$ ; the output is $-\frac{7}{2}$ .
$y=2-n$ ; the output is $0$ .

Show answer and explanation

y=2-\frac{1}{2}n

; the output is

1

.

The pattern begins at

2

and subtracts

\frac{1}{2}

for each step. At

n=2

, it subtracts

1

, giving output

1

.

Question 2

A table has input

-1

with output

4

. Which ordered pair should appear on its graph?

$(4,-1)$
$(-1,4)$
$(-1,-4)$
$(4,1)$

Show answer and explanation

(-1,4)

An ordered pair lists input first and output second, so the point is

(-1,4)

.

Question 3

The equation is

y=3-2x

. What output belongs with input

2

?

$7$
$1$
$-1$
$-4$

Show answer and explanation

-1

Substitute

2

for

x

:

3-2(2)=3-4=-1

.

Key terms

Rational number: A number that can be written as a fraction, including integers and fractions.
Input: The value used in a rule to find an output.
Output: The value produced by applying a rule to an input.
Ordered pair: Two values written in order as $(x,y)$ ; the first is the input and the second is the output.
Equation: A mathematical statement showing that two expressions have equal value.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation C1.2. 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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C1.2 · Represent rational-number patterns with tables, graphs, and equations

Ontario Grade 8 Mathematics

Use tables, graphs, and equations to describe how values change

What you will learn

1. Start with the values and their change

2. Build and read a table, graph, and equation

3. Use representations to check your thinking

A pattern that increases by one-half

Use rational inputs in a rule

Find a rule from a table

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