Grade by Grade
C. Algebra · Patterns, Variables, Coding and Modelling

Pattern Rules: Extend, Predict and Find What Is Missing

C1.3About 40 minutes
Learning goals
  • I can find and say the rule of a repeating, growing or shrinking pattern.
  • I can use the rule to extend a pattern.
  • I can predict a term far along in a pattern and justify (explain) my prediction.
  • I can find missing shapes or numbers in a pattern.

Words to know

pattern rule
Words that tell exactly how a pattern works, like start at 3 and add 2 each time.
extend
To make a pattern longer by following its rule.
predict
To use what you know to say what will happen, before you check.
justify
To explain why your answer makes sense.
missing element
A part of a pattern that has been left out.

Rules are the secret of patterns

Every pattern has a rule. When you know the rule, you can make the pattern as long as you want, guess faraway parts, and fill in gaps. It is like having a super power!

Saying a pattern rule

A good rule tells two things: how the pattern starts, and what happens each time.

  • Repeating: The core is red, blue, blue. It repeats.
  • Growing: Start at 4. Add 3 each time. → 4, 7, 10, 13, ...
  • Shrinking: Start at 30. Take away 5 each time. → 30, 25, 20, 15, ...

Start AND change

Add 2 each time is not the whole rule. 1, 3, 5, 7 and 10, 12, 14, 16 both add 2, but they are different patterns! Always say where it starts too.

Worked example

Find the rule and extend

Find the rule. Then extend the pattern by 3 more terms.

6, 10, 14, 18, ___, ___, ___

  1. Is it getting bigger or smaller? Bigger, so something is added.
  2. Find the jumps: 6 to 10 is 4. 10 to 14 is 4. 14 to 18 is 4.
  3. The rule: start at 6 and add 4 each time.
  4. Extend: 18 + 4 = 22, 22 + 4 = 26, 26 + 4 = 30.

Rule: start at 6, add 4. The next terms are 22, 26, 30.

024681012141618202224262830+4+4+4+4+4+4
The jumps of 4 are all the same size. That is how you know the rule is add 4.

Making predictions

To predict means to figure out a term before you build it. Then you justify: you explain how you know.

A table is a great tool for predicting in growing patterns.

Term 1Term 2Term 3Term 4
A growing pattern of towers: 1, 3, 5, 7 squares.
Worked example

Predict and justify

How many squares will be in term 8 of the tower pattern above?

  1. Write the rule: start at 1, add 2 each time.
  2. Make a table and keep going until term 8.
  3. Term 1: 1, term 2: 3, term 3: 5, term 4: 7, term 5: 9, term 6: 11, term 7: 13, term 8: 15.
  4. Justify: I know because the rule is add 2, and I kept adding 2 in my table until term 8.

Term 8 will have 15 squares.

Term12345678
Squares13579111315
The table makes it easy to keep track of each term.
Worked example

Predict in a repeating pattern

In the pattern 🟥 🟦 🟥 🟦 🟥 🟦 ..., what colour is the 14th term?

  1. The core is 🟥 🟦. It has 2 parts.
  2. Blue is on terms 2, 4, 6, 8... That is counting by 2s!
  3. Keep counting by 2s: 10, 12, 14. So 14 is a blue term.
  4. Justify: Blue is always on the counting-by-2s numbers, and 14 is one of them.

The 14th term is blue 🟦.

Finding missing elements

Sometimes a part of a pattern is missing. Use the rule to find it. Look at the parts before and after the gap.

Worked example

What is missing?

Find the missing numbers: 45, 40, ___, 30, ___, 20

  1. 45 to 40 goes down 5. That might be the rule.
  2. Check with the parts we know: 30 to 20 has one gap in between. 30 − 5 = 25, and 25 − 5 = 20. It works!
  3. First gap: 40 − 5 = 35.
  4. Second gap: 30 − 5 = 25.

The missing numbers are 35 and 25. Rule: start at 45, take away 5.

Worked example

A missing shape

What is missing? 🔺 🟢 🟢 🟦 🔺 🟢 ___ 🟦 🔺 🟢 🟢 🟦

  1. Find the core. 🔺 comes again in the 5th spot, so the core is the first 4 shapes: 🔺 🟢 🟢 🟦.
  2. The second core is 🔺 🟢 ___ 🟦. Compare it with the core.
  3. The 3rd shape in the core is 🟢.

The missing shape is 🟢.

Just for fun

Riddle: What comes next? J, F, M, A, M, J, J, ___

A! They are the first letters of the months: January, February, March, April, May, June, July… August. 📅

Your turn

1

Find the rule: 5, 7, 9, 11

2

Extend: 12, 22, 32, ___, ___

3

Extend: 50, 48, 46, ___, ___

4

What is missing? 🍎 🍐 🍐 🍎 ___ 🍐

5

What is missing? 3, 6, ___, 12, 15

6

In 🟡 🟣 🟡 🟣 ..., what colour is the 10th term?

7

Towers grow 2, 4, 6, 8. Predict term 6.

Hands-on activity

Hide and predict

You need: cups or blocks, paper, a pencil.

  1. A grown-up builds a growing pattern with towers of blocks, like 2, 5, 8, 11.
  2. You say the rule, then predict how many blocks term 6 will need. Write it down.
  3. Build terms 5 and 6 to check. Were you right?
  4. Now the grown-up hides one tower under a cup. Figure out how many blocks are hiding, then lift the cup to check.
  5. Switch roles!

What to look for: your child says the start and the jump, and explains how they know.

Math talk

Use the sentence I know because... every time you predict. Example: I know term 6 has 17 because I added 3 each time: 2, 5, 8, 11, 14, 17.

More practice: warm-up

1

Extend: 20, 25, 30, ___

2

What is missing? 10, 20, ___, 40

3

What is missing? 🌙 ⭐ 🌙 ___ 🌙 ⭐

4

Extend: 9, 8, 7, ___

5

What does predict mean?

More practice: on your own

1

Find the rule and the missing number: 14, 18, ___, 26, 30

2

Find the missing numbers: 70, ___, 50, ___, 30

3

In ⬛ ⬛ ⬜ ⬛ ⬛ ⬜ ..., what colour is the 9th tile? Justify.

4

Kim says the rule for 2, 4, 6, 8 is add 2. Ben says the rule for 10, 12, 14, 16 is also add 2. Are they the same pattern? Explain.

5

A growing pattern of squares has 4, 7, 10 squares. Predict term 5.

Challenge

1

Ella saves money. Week 1 she has 5¢. Week 2 she has 15¢. Week 3 she has 25¢. She keeps the same pattern. How much will she have in week 6? Show your table.

2

In 🔴 🔵 🟢 🔴 🔵 🟢 ..., what colour is the 15th term? Justify.

3

Find the two missing numbers: 11, ___, 27, ___, 43

Check your understanding

Quiz

1.

What is the rule? 8, 11, 14, 17

2.

What is missing? 25, 30, ___, 40

3.

What is missing? 🟥 🟥 🔵 🟥 ___ 🔵

4.

Towers have 3, 5, 7 cubes. How many cubes are in term 5?

5.

In 🍁 ⛄ 🍁 ⛄ ..., what is the 8th term?

6.

What does justify mean?

7.

Extend: 60, 55, 50, ___

For parents and teachers

This lesson asks children to state a pattern rule fully (where it starts and how it changes), then use it to extend, predict far terms and fill in missing parts.

Common mistakes

  • Looking only at the last term to guess the next one. Have your child check the jump between every pair of terms.
  • Giving only the change (add 2) without the start.
  • Losing track when predicting far terms. A table, a number line or counting on fingers helps.
  • For missing numbers in the middle, guessing without checking against the numbers after the gap.

How to help

  • Always ask How do you know? and expect a reason that uses the rule.
  • Practise with money (nickels and dimes), skip counting on walks, or stairs.
  • Let your child make up missing-number puzzles for you, and pretend to get one wrong so they can explain your mistake.