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C1.1 · Compare growing and shrinking patterns by rate and initial value

Learn to compare growing and shrinking patterns by rate and initial value through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Algebra

Use rate and initial value to describe how patterns change

A pattern is a set of values that follows a regular change. To compare patterns, look at two features. The rate tells how much the output changes for each step in the input. The initial value is the output when the input is 0. A pattern can start higher but change more slowly than another pattern. We will use tables, words, and coordinate graphs to compare these features.

What you will learn

Identify whether a pattern grows or shrinks.
Find a pattern’s rate and initial value from a table or description.
Compare two patterns by their rates and initial values.
Compare their outputs at the same input.

Notice the change in a pattern

Imagine a tile design that has 5 tiles at step 0. Each new step adds 3 tiles. The counts are 5, 8, 11, and 14. The pattern grows because the counts increase. Its rate is an increase of 3 tiles per step, and its initial value is 5 tiles.

Now imagine a collection that has 18 counters at step 0. Two counters are removed at each step. The counts are 18, 16, 14, and 12. This pattern shrinks. Its rate is a decrease of 2 counters per step, and its initial value is 18 counters.

The input is the step number or position. The output is the value at that input. The initial value is found at input 0, even if the pattern is described using other steps. The rate describes the change in output for an equal change in input.

Growing patterns have outputs that increase as the input increases.
Shrinking patterns have outputs that decrease as the input increases.
The initial value is the output at input 0.
The rate includes the amount and direction of the change.

Read rates and initial values from a table

Begin by checking the input values. When the input increases by 1 each time, compare neighbouring outputs. For example, outputs 6, 10, 14, and 18 increase by 4 each time. The rate is an increase of 4 per input step.

Then find the output paired with input 0. If that output is 6, the initial value is 6. A table makes both features visible: the first output at input 0 shows where the pattern starts, and the repeated output changes show its rate.

A rate can describe a decrease as well as an increase. If the outputs go from 15 to 12 to 9, they decrease by 3 for each one-step increase in input. Say that the rate is a decrease of 3 per step. This wording makes the direction clear without needing a general rule with letters.

Use equal input steps when comparing output changes.
For inputs that increase by 1, each neighbouring output change gives the rate.
State clearly whether the output increases or decreases.

Compare two patterns fairly

Compare the initial values to see which pattern starts higher. Compare the rates to see which pattern changes more per step and whether it grows or shrinks. These are different comparisons: starting higher does not mean changing faster.

A coordinate graph can help. The horizontal axis shows the input, and the vertical axis shows the output. Each plotted point pairs one input with its output. The point at input 0 shows the initial value. Points that rise as they move right show a growing pattern; points that fall as they move right show a shrinking pattern.

To decide which pattern has the greater output at a particular input, compare their outputs at that same input. One pattern may start higher, while the other later becomes greater. State both the starting comparison and the rate comparison before making a claim about a particular step.

Compare initial values to compare where patterns start.
Compare rates to compare how patterns change.
Compare outputs at the same input to decide which is greater at that step.

A clear comparison routine

First identify the inputs and outputs. Next find each output at input 0, which is the initial value. Then compare the output changes over equal input steps to find each rate. Finally describe which pattern starts higher, which grows or shrinks faster, and—if asked—which has the greater output at a chosen input.

If inputs do not increase by 1, do not treat every output difference as a one-step rate. Compare changes across matching input intervals. In the examples here, the tables use one-step increases, so the output differences can be read directly as changes per step.

Keep input and output meanings clear.
Use matching input intervals when comparing rates.
A complete comparison includes both initial values and rates.

Comparing two patterns

Pattern	Output at input 0	Change each step	Description
A	15	decreases by 3	Starts higher and shrinks
B	4	increases by 3	Starts lower and grows

Worked example

Find a rate and an initial value

A pattern has input-output pairs (0, 5), (1, 8), (2, 11), and (3, 14). Find its initial value and rate. Is it growing or shrinking?

Input and output values

Hypothetical pattern values. Each point shows an input and its output; the output at input 0 is 5.

Find the initial value
The initial value is the output paired with input 0. The table shows that output as 5.
$5$
Compare neighbouring outputs
The input increases by 1 each time. The output increases by 3 from 5 to 8, from 8 to 11, and from 11 to 14. The repeated change means the rate is an increase of 3 per step.
$8-5=3, 11-8=3, 14-11=3$
Describe the pattern
The outputs rise by the same amount each step, so the pattern is growing. It starts at 5 and increases by 3 per step.

Answer: The initial value is 5. The pattern grows at a rate of 3 per input step.

Check: Starting at 5 and increasing by 3 three times gives 14, which matches the output at input 3.

Worked example

Compare a shrinking pattern with a growing pattern

At inputs 0, 1, 2, and 3, Pattern A has outputs 15, 12, 9, and 6. Pattern B has outputs 4, 7, 10, and 13. Compare their initial values and rates. Which has the greater output at input 2?

Two patterns

Hypothetical values. A decreases from 15; B increases from 4. Compare their outputs at the same input.

Compare starting values
At input 0, Pattern A is 15 and Pattern B is 4. Pattern A starts higher.
$15>4$
Find the rates
Pattern A decreases by 3 at each step. Pattern B increases by 3 at each step. One shrinks while the other grows.
$12-15=-3, 7-4=3$
Compare at input 2
At input 2, A has output 9 and B has output 10. Although A started higher, B has the greater output at this input.
$10>9$

Answer: Pattern A starts higher and decreases by 3 per step. Pattern B starts at 4 and increases by 3 per step. At input 2, Pattern B is greater: 10 compared with 9.

Check: The listed outputs confirm the changes: A drops by 3 each step, while B rises by 3 each step.

Worked example

Compare two growing patterns

Pattern P starts at 7 and increases by 2 per step. Pattern Q starts at 2 and increases by 4 per step. Find their outputs at input 3. Then compare their initial values and rates.

Starting values and growth

Hypothetical values. P starts higher; Q increases more at each step.

Record what each description says
Pattern P starts at 7 and adds 2 at each step. Pattern Q starts at 2 and adds 4 at each step. These statements give the initial values and rates directly.
Find the outputs after three steps
For P, add 2 three times to 7. For Q, add 4 three times to 2. This follows each pattern’s stated change at every step.
$7+2+2+2=13, 2+4+4+4=14$
Compare the results
P starts higher, but Q increases more per step. At input 3, Q has the greater output.
$14>13$

Answer: P starts higher, at 7, while Q grows faster, increasing by 4 rather than 2 per step. At input 3, P is 13 and Q is 14, so Q is greater.

Check: The table values show three increases of 2 for P and three increases of 4 for Q.

Common mistakes and how to avoid them

Calling the first listed output the initial value even when its input is not 0.

Correction: Find the output paired with input 0. That is the initial value.

Comparing only the initial values.

Correction: Compare the rates too. A pattern that starts lower can grow faster.

Calling a pattern growing just because its outputs are positive.

Correction: Growing or shrinking describes how outputs change as the input increases. Positive outputs can still decrease.

Comparing output changes over unequal input intervals as if they were equal-step rates.

Correction: Compare changes over matching input intervals. In these examples, each input step increases by 1.

Lesson summary

The rate describes how the output changes for each equal input step.
A growing pattern’s outputs increase; a shrinking pattern’s outputs decrease.
The initial value is the output at input 0.
Compare both initial values and rates to describe two patterns.
To decide which pattern is greater at a particular step, compare outputs at the same input.

Check your understanding

Question 1

A pattern has outputs 12, 9, 6, and 3 at inputs 0, 1, 2, and 3. What are its initial value and rate?

Initial value 12; decreases by 3 per step
Initial value 9; decreases by 3 per step
Initial value 12; increases by 3 per step
Initial value 3; decreases by 3 per step

Show answer and explanation

Initial value 12; decreases by 3 per step

The output at input 0 is 12. Each step the output decreases by 3.

Question 2

Pattern A starts at 10 and grows by 2 per step. Pattern B starts at 6 and grows by 5 per step. Which statement is correct?

A starts higher, and B grows faster.
B starts higher, and A grows faster.
A starts higher, and A grows faster.
B starts higher, and B grows faster.

Show answer and explanation

A starts higher, and B grows faster.

A starts at 10 compared with B’s 6. B grows by 5 per step, more than A’s 2.

Question 3

A pattern starts at 18 and decreases by 4 per step. What is its output at input 2?

Show answer and explanation

After two steps, subtract 4 twice from 18. The output is 10.

Key terms

Input: The step or position used to find a pattern’s output.
Output: The value paired with an input in a pattern.
Rate: The amount and direction of output change for each equal increase in input.
Initial value: The output when the input is 0.
Growing pattern: A pattern whose outputs increase as the input increases.
Shrinking pattern: A pattern whose outputs decrease as the input increases.

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