Grade by Grade
C. Algebra · Patterns, Equations, Inequalities, Coding and Modelling

Pattern Rules: Predict, Justify and Find Missing Terms

C1.3 A1About 40 minutes
Learning goals
  • I can find the pattern rule for a repeating or growing pattern.
  • I can use a pattern rule to extend a pattern.
  • I can predict a term far away and explain (justify) my prediction.
  • I can find missing terms in a pattern.

Words to know

pattern rule
A rule that tells how a pattern starts and how it continues.
extend
Continue a pattern by adding more terms.
predict
Use a rule to tell what a term will be without building every term.
justify
Explain why your answer makes sense, with math reasons.
missing term
A term that has been left out of a pattern.

Remember from Grade 3

In Grade 3 you used pattern rules to extend patterns and find missing pieces. In Grade 4 you will predict terms that are far away, like the 20th or the 50th, and justify your predictions: explain why they must be right.

Let's start with a story

Mr. Singh's class plants a row of trees along the school fence: maple, spruce, birch, maple, spruce, birch, ... They plant 20 trees. What kind of tree is the 20th tree?

You could draw all 20 trees. But a pattern rule can tell you much faster!

Pattern rules for repeating patterns

The rule for a repeating pattern tells the core: maple, spruce, birch repeats. The core has 3 trees.

Every 3rd tree finishes a core. So trees number 3, 6, 9, 12, 15, 18 are all birch (the last tree in the core). Those are the multiples of 3.

Tree number123456...181920
Treemaplesprucebirchmaplesprucebirch...birchmaplespruce
Tree 18 is birch because 18 is a multiple of 3. Then the core starts again.
1M2S3B4M5S6B7M8S9B10M11S12B13M14S15B16M17S18B19M20Score 1core 2core 3core 4core 5core 6tree 20 is the 2nd tree of core 7: spruceM = mapleS = spruceB = birch
18 trees make 6 full cores, so tree 18 is birch. Tree 19 starts core 7 (maple), and tree 20 is spruce.
Worked example

Example 1: Predict far ahead in a repeating pattern

What is the 20th tree in the pattern maple, spruce, birch?

  1. The core has 3 trees. Find the biggest multiple of 3 that is not more than 20: 3 × 6 = 18.
  2. Tree 18 ends a core, so it is birch.
  3. Tree 19 starts a new core: maple.
  4. Tree 20 is the 2nd tree in the core: spruce.

The 20th tree is a spruce. Justify: 18 trees make 6 full cores, and 20 is 2 more, so it matches the 2nd tree in the core.

Pattern rules for growing patterns

For a growing (or shrinking) pattern, the rule has two parts:

  1. The start: the first term.
  2. The change: what you do to get the next term.

Start at 5 and add 4 each time gives 5, 9, 13, 17, 21, ...

Term number12345
Term value59131721
Start at 5, add 4 each time.

Predicting far ahead in a growing pattern

What if you want the 10th term? You can keep adding 4. Or use this trick:

  • From term 1 to term 10 there are 9 jumps (10 − 1 = 9).
  • Each jump is 4, so 9 jumps add 9 × 4 = 36.
  • Start at 5: 5 + 36 = 41.

So the 10th term is 41. You can check by counting on: 21, 25, 29, 33, 37, 41. ✔

A shortcut for some patterns

Look at 3, 6, 9, 12, ... The start is 3 and the jump is also 3. Each term is just the term number × 3: term 1 is 1 × 3 = 3, term 4 is 4 × 3 = 12. So term 20 is 20 × 3 = 60.

This shortcut works when the pattern starts with the same number it adds, like 5, 10, 15, ... (term number × 5) or 8, 16, 24, ... (term number × 8).

Worked example

Example 2: Predict and justify

Toothpick squares in a row use 4, 7, 10, 13, ... sticks. How many sticks does Figure 10 need?

  1. The rule is start at 4, add 3 each time.
  2. From Figure 1 to Figure 10 there are 10 − 1 = 9 jumps.
  3. 9 jumps of 3: 9 × 3 = 27.
  4. 4 + 27 = 31.

Figure 10 needs 31 sticks. Justify: Figure 1 has 4 sticks and each of the 9 new squares adds 3 sticks: 4 + 9 × 3 = 31.

Finding missing terms

To find a missing term, first find the rule using the terms you can see. Then use it to fill the gap.

024681012141618202224262830+6+6+6
6, 12, ☐, 24: the jumps must all be +6, so the missing term is 18.
Worked example

Example 3: A missing term in a growing pattern

Find the missing term: 6, 12, ☐, 24, 30

  1. Use terms that are side by side: 30 − 24 = 6, and 12 − 6 = 6. The rule is add 6.
  2. 12 + 6 = 18.
  3. Check: 18 + 6 = 24. ✔

The missing term is 18.

Worked example

Example 4: A missing term that is the first term

Find the missing term: ☐, 35, 31, 27, 23

  1. The jumps are 31 − 35 = −4, 27 − 31 = −4. The rule is subtract 4.
  2. To go backward, do the opposite: add 4.
  3. 35 + 4 = 39.

The missing first term is 39. The pattern is start at 39, subtract 4 each time.

Worked example

Example 5: Missing terms in a repeating pattern

Fill the gaps: 🔺🔵🔵🔺☐🔵🔺🔵☐

  1. The core is 🔺🔵🔵 (3 elements).
  2. The 5th spot is the 2nd element of the core: 🔵.
  3. The 9th spot is the 3rd element of the core: 🔵.

Both missing elements are 🔵: 🔺🔵🔵🔺🔵🔵🔺🔵🔵.

Be a pattern detective

  • Always check your rule with every term you can see, not just the first two.
  • To go forward, use the rule. To go backward, do the opposite (add ↔ subtract).
  • When you predict, say how you know. A good justification uses the rule: "9 jumps of 3 is 27, plus the 4 we started with."
Just for fun

A pattern detective's favourite question: "What comes next?"

A pattern detective's least favourite answer: "Lunch."

Hands-on activity

Predict, then build to check

You need: about 40 coins, buttons or blocks, and paper.

  1. Build a growing pattern: Figure 1 has 3 coins, Figure 2 has 5, Figure 3 has 7. Write the rule.
  2. Predict how many coins Figure 8 needs. Write your prediction and how you got it.
  3. Build Figures 4 to 8 (or keep adding to one pile) to check. Were you right?
  4. Now your grown-up covers one term in a written pattern, like 10, 15, ☐, 25. You find the missing term and explain.
  5. Switch roles. Make a repeating pattern of coins (heads, heads, tails ...) and ask your grown-up to predict the 15th coin.

What to look for: the prediction for Figure 8 is 17 (3 + 7 × 2 = 17), and your child can explain it using the rule.

Grown-up note: keep small items like these away from babies and toddlers.

Your turn

1

What is the rule for 8, 15, 22, 29? What is the next term?

2

Find the missing term: 11, 16, ☐, 26, 31

3

Find the missing term: ☐, 40, 36, 32

4

The core is A B C D. What is the 12th letter?

5

The core is 🐟🐟🦭. What is the 10th picture?

6

A pattern starts at 2 and adds 5. Predict the 10th term.

7

Explain how you would find the 100th term of 3, 6, 9, 12, ...

More practice: warm-up

1

Extend: 4, 8, 12, 16, ?, ?

2

Extend: 50, 45, 40, ?, ?

3

Find the missing term: 2, 4, ☐, 8, 10

4

What is the rule for 10, 13, 16, 19?

5

The core is up, down. What is the 6th move?

More practice: on your own

1

Find both missing terms: 7, ☐, 19, 25, ☐

2

Chairs around tables in a row: 4, 6, 8, 10, ... Predict the chairs for 15 tables.

3

The repeating pattern is 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, ... What is the 23rd term? Justify.

4

Find the missing term: 48, 42, ☐, 30, 24

5

Zoe says the 8th term of 5, 10, 15, 20, ... is 35. Is she right? Explain.

Challenge

1

A growing pattern has a 4th term of 20 and a 5th term of 26. It adds the same amount each time. What is the 1st term? Explain.

2

In the pattern ❄️⛄⛄❄️⛄⛄ ..., how many ⛄ are in the first 30 pictures? Justify.

3

A pattern starts at 9 and adds 4. Will 50 be a term in this pattern? Explain.

Check your understanding

Quiz

1.

What is the rule for 6, 13, 20, 27?

2.

Find the missing term: 15, 25, ☐, 45

3.

The core is circle, square, triangle. What is the 9th shape?

4.

A pattern starts at 1 and adds 3. What is the 10th term?

5.

Find the first term: ☐, 21, 18, 15

6.

Which is the best justification that the 10th term of 2, 4, 6, 8, ... is 20?

For parents and teachers

Big idea: The Ontario curriculum (C1.3) asks Grade 4 students to determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating and growing patterns.

Common difficulties:

  • Counting jumps wrong when predicting. From term 1 to term 10 there are 9 jumps, not 10. Drawing a short number line helps.
  • Going backward with the same operation. To find a term before, children must do the opposite (add instead of subtract).
  • Repeating patterns far ahead. Children may draw every element. Show them how multiples of the core length mark the end of each core.
  • Answers without reasons. Ask "How do you know?" and accept explanations that use the rule.

How to help: Use real patterns: "If you save $3 a week starting with $5, how much after 10 weeks?" Encourage your child to predict, then check by building or counting on. Celebrate a careful check even when the prediction was wrong (Strand A: reflecting and persevering).