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C1.1 · Compare repeating, growing, and shrinking patterns by rate and starting value

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

Ontario Grade 7 Mathematics

Algebra

Look at how patterns change and where they begin

A pattern is a list or arrangement that follows a rule. Some patterns repeat the same part. Some grow. Others shrink. To compare patterns, look for two things: how the values change from step to step, and where the pattern starts. The change from one step to the next is its rate of change. In this lesson, a rate is stated as a number of items per step. A repeating pattern returns to the same part, so it does not keep growing or shrinking. We can describe how often its cycle repeats.

What you will learn

Recognize repeating, growing, and shrinking patterns.
Describe how a pattern changes from one step to the next.
Compare patterns by their rate of change and starting value.

1. Notice the pattern and its starting value

A step is one position in a pattern. For a list, the first number is at step 1. Sometimes it is useful to also describe a starting value at step 0: the value just before the listed steps begin. We will say clearly which meaning we use.

Try a concrete model. Imagine counters arranged in rows. A pattern might have 4 counters at step 1, then 7, then 10. Each new step adds 3 counters. This is a growing pattern. The rate of change is 3 counters per step.

A shrinking pattern changes in the other direction. If a pile has 18 counters and loses 2 each step, its values are 18, 16, 14, and 12. Its rate of change is a decrease of 2 counters per step. Saying “decrease by 2” avoids confusion about the direction.

A repeating pattern cycles through a fixed order. For example, red, blue, blue, red, blue, blue repeats a three-colour cycle. It does not add or remove colours as the steps continue. Its cycle length is 3 steps: the number of steps before the same order begins again.

rate of change = change in value per step

Growing: the values increase by the same amount each step.
Shrinking: the values decrease by the same amount each step.
Repeating: a fixed part occurs again in the same order.
The starting value tells where a pattern begins; name the step used.

2. Find and compare rates

To find the rate of change, compare neighbouring values. Subtract the earlier value from the next value. If the difference stays the same, it tells how much the pattern changes each step. For example, in 5, 9, 13, 17, the differences are 4, 4, and 4. The pattern grows by 4 each step.

For 20, 17, 14, 11, the differences are decreases of 3 each time. The pattern shrinks by 3 per step. A pattern can have a larger starting value but a smaller rate. These are separate features, so compare them separately.

For a repeating pattern, look for the shortest group that repeats. State its cycle length and starting part. For example, in 2, 5, 2, 5, 2, 5, the cycle has two steps and begins with 2. Unlike a growing or shrinking pattern, the values do not keep changing in one direction.

\text{difference} = \text{next value} - \text{earlier value}

Compare consecutive values to find a steady change.
Use units, such as counters per step, when they are known.
Compare starting values separately from rates.
For a repeating pattern, describe the repeating cycle rather than claiming it grows or shrinks.

3. Represent patterns with a table and a graph

A table keeps the step number beside its value. This makes the starting point and changes easy to check. A graph shows the same information as points. The horizontal position shows the step, and the vertical position shows the value. Each plotted point must match a table entry.

For example, suppose a pattern starts at 6 counters at step 0 and adds 2 counters each step. Its values at steps 0, 1, 2, and 3 are 6, 8, 10, and 12. The starting value is 6 counters. The rate is an increase of 2 counters per step.

A second pattern could start at 10 counters and add 1 counter per step. It starts higher, but it grows more slowly. At step 3, the first pattern has 12 counters and the second has 13. A comparison at one step does not erase the difference between starting values and rates. State both when comparing.

value at step n =

\text{starting value}

+ (change per step)× n

Read across a table to compare steps and values.
Graph each table pair as one point.
A higher starting value does not always mean a faster-growing pattern.

4. A careful comparison

Use the same steps when comparing two patterns. First name each pattern type. Next state its starting value. Then state its rate or cycle length. Finally, compare what happens at a chosen step if the question asks for it.

A steady rate is useful only when the same change continues from step to step. Check more than one pair of values before naming the rate. If the differences vary, do not claim a constant rate. For a repeating sequence, check that the same group returns in the same order.

Keep the units consistent. If one pattern counts tiles and another counts counters, describe each rate with its own unit. A comparison is clearest when it names the pattern, the starting value, and the rate or cycle.

Check the pattern rule before comparing.
Use consistent step numbers and units.
Report starting value and rate as different pieces of information.

A quick pattern guide

Pattern type	What happens	What to compare
Repeating	A fixed group returns in order	Starting part and cycle length
Growing	Values increase by a steady amount	Starting value and increase per step
Shrinking	Values decrease by a steady amount	Starting value and decrease per step

Worked example

Compare two growing patterns

Pattern A has 5 tiles at step 0 and adds 3 tiles each step. Pattern B has 8 tiles at step 0 and adds 2 tiles each step. Compare their starting values and rates. Find their values at step 3.

Tile patterns by step

Each point shows step number and tile count. Pattern A starts at 5 and gains 3 each step; Pattern B starts at 8 and gains 2.

List the starting values
At step 0, Pattern A has 5 tiles and Pattern B has 8 tiles. Pattern B starts with 3 more tiles.
$5<8$
Compare the rates
Pattern A adds 3 tiles per step. Pattern B adds 2 tiles per step. Pattern A grows faster, even though it starts with fewer tiles.
$3>2$
Find the values at step 3
Three steps add three equal changes to each starting value. Pattern A reaches 14 tiles. Pattern B reaches 14 tiles as well.
$5+3\times3=14,\quad 8+2\times3=14$

Answer: Pattern B starts higher, but Pattern A grows faster. At step 3, both have 14 tiles.

Check: Pattern A gains 3 tiles in each of three steps, for a total gain of 9. Pattern B gains 6. The resulting values are both 14.

Worked example

Compare shrinking patterns

A water container has 24 cups at step 0 and loses 4 cups per step. A second container has 18 cups at step 0 and loses 2 cups per step. Which starts with more? Which shrinks faster? What is each value at step 2?

Container amounts by step

The plotted values show each container losing a steady number of cups per step.

Compare the starts
At step 0, the first container has 24 cups and the second has 18 cups. The first starts with more.
$24>18$
Compare the decreases
The first loses 4 cups each step. The second loses 2 cups each step. Losing more each step means the first shrinks faster.
$4>2$
Continue for two steps
After two steps, the first has lost 8 cups and has 16 left. The second has lost 4 cups and has 14 left.
$24-4\times2=16,\quad 18-2\times2=14$

Answer: The first container starts with more and shrinks faster. At step 2, it has 16 cups; the second has 14 cups.

Check: The values follow 24, 20, 16 and 18, 16, 14. Each list decreases by the stated amount each step.

Worked example

Recognize and compare a repeating pattern

Pattern A is 3, 6, 3, 6, 3, 6. Pattern B is 2, 5, 8, 11, 14, 17. Describe each pattern, its start, and its rate or cycle length.

Values at each step

Pattern A repeats its two values. Pattern B rises by 3 each step. The horizontal axis is the step number.

Find the repeating cycle
Pattern A alternates between 3 and 6. The group 3, 6 repeats, so its cycle length is 2 steps. Its first value is 3.
$3,6 | 3,6 | 3,6$
Check the other changes
Pattern B increases by 3 from one value to the next. It is growing, starts at 2, and has a rate of 3 per step.
$5-2=3,\quad 8-5=3$
Compare what is different
Pattern A repeats and has a two-step cycle. Pattern B grows steadily. Its starting value is 2, while Pattern A begins with 3. Their descriptions use different measures because only Pattern B keeps increasing.
$3\ne2$

Answer: Pattern A repeats a two-step cycle and begins with 3. Pattern B grows by 3 per step and begins with 2.

Check: The values in Pattern A return to 3, 6 after every two steps. Consecutive values in Pattern B differ by 3.

Common mistakes and how to avoid them

Calling the pattern with the largest first value the fastest-growing pattern.

Correction: Compare starting values and changes per step separately. A pattern can start lower and grow faster.

Describing a repeating sequence as growing because one value is larger than the previous one.

Correction: Check the whole sequence for a returning group. A repeating pattern may go up and down without continuing to grow.

Giving a rate without saying whether the pattern increases or decreases.

Correction: Say “increases by” or “decreases by,” and include the unit per step when possible.

Lesson summary

A pattern follows a rule across steps.
A growing or shrinking pattern has a steady change when consecutive differences stay the same.
A repeating pattern returns to the same group in the same order.
Compare starting values separately from rates or cycle lengths.

Check your understanding

Question 1

Pattern P is 7, 10, 13, 16. Pattern Q is 9, 11, 13, 15. Which statement is correct?

P starts lower and grows faster.
P starts higher and grows faster.
Q starts lower and grows faster.
Both start the same and grow at the same rate.

Show answer and explanation

P starts lower and grows faster.

P starts at 7 and increases by 3 each step. Q starts at 9 and increases by 2 each step.

Question 2

What is the cycle length of 4, 1, 4, 1, 4, 1?

1 step
2 steps
3 steps
6 steps

Show answer and explanation

2 steps

The group 4, 1 repeats after two steps.

Question 3

A pattern has 21 beads at step 0 and loses 5 beads each step. How many beads are there at step 2?

Show answer and explanation

It loses 5 beads twice, so 21 decreases by 10 and becomes 11.

Key terms

Pattern: A list or arrangement that follows a rule.
Step: One position in a pattern.
Starting value: The value at the named beginning step of a pattern.
Rate of change: How much a value increases or decreases for each step.
Cycle length: The number of steps in a repeating group.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation C1.1. It is a study resource, not an official curriculum publication.

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