C. Algebra · Patterns, Equations, Coding and Modelling

Pattern Rules: Extend, Predict and Find Missing Elements

C1.3 A1About 40 minutes

Learning goals

I can write a pattern rule that tells the start and what repeats.
I can use a pattern rule to extend a pattern.
I can predict an element far along in a pattern and justify (explain) my prediction.
I can find missing elements in repeating, growing and shrinking patterns.

Words to know

pattern rule
Words that tell exactly how a pattern works, like Start at 4. Add 5 each time.

extend
Make a pattern longer by continuing it.

predict
Use what you know to say what will come later.

justify
Explain why your answer makes sense.

missing element
A part of a pattern that has been left out.

Remember from Grade 2

In Grade 2 you said pattern rules, extended patterns and filled in missing parts. In Grade 3 the patterns are bigger, the numbers are bigger, and you will make predictions far along a pattern and explain how you know.

Writing a pattern rule

A good pattern rule tells two things:

How it starts.
What repeats: the core, the operation or the movement.

❄️ ⛄ 🧣 ❄️ ⛄ 🧣 → The core ❄️ ⛄ 🧣 repeats. It has 3 elements.
4, 9, 14, 19 → Start at 4. Add 5 each time.
90, 80, 70 → Start at 90. Subtract 10 each time.

A rule must work every time

Check your rule on every step, not just the first one. For 2, 4, 8, 16, the rule add 2 works for the first step (2 + 2 = 4), but not the next (4 + 2 is 6, not 8). The real rule is start at 2, double each time.

Predicting in a repeating pattern

Look at this pattern: ❄️ ⛄ 🧣 ❄️ ⛄ 🧣 ❄️ ⛄ 🧣 …

The core has 3 elements. That means the core ends at positions 3, 6, 9, 12, 15, … These are the numbers we say when we skip count by 3s. So 🧣 is in every one of those positions.

Position	1	2	3	4	5	6	7	8	9
Element	❄️	⛄	🧣	❄️	⛄	🧣	❄️	⛄	🧣

🧣 is at positions 3, 6 and 9, the skip-counting-by-3 numbers.

The core ❄️ ⛄ 🧣 has 3 elements, so the 🧣 is at every skip-counting-by-3 position. That makes it easy to predict the 14th element.

Worked example

Example 1: Predict far along

In the pattern ❄️ ⛄ 🧣 ❄️ ⛄ 🧣 …, what is the 14th element?

The core has 3 elements, so 🧣 is at 3, 6, 9, 12, 15, …
12 is the closest of those numbers that is less than 14. Position 12 is 🧣.
Then the core starts again: position 13 is ❄️, position 14 is ⛄.

The 14th element is ⛄. I know because 🧣 is at position 12, so the core starts over at 13.

Predicting with movements

An arrow turns a quarter turn to the right each time: up, right, down, left, up, right, down, left, … The core has 4 movements. The arrow points left at positions 4, 8, 12, 16 (skip counting by 4s).

Worked example

Example 2: Where will the arrow point?

The arrow pattern starts with up. Which way does arrow number 10 point?

The core is up, right, down, left, so left is at 4, 8, 12, …
Position 8 is left.
Position 9 starts the core again: up. Position 10: right.

Arrow 10 points right.

Predicting in a growing pattern

Pattern rule: Start at 4. Add 5 each time. To predict the 8th number, we can extend the pattern, or think about how many jumps we need.

Start at 4 and make 7 jumps of 5: 4, 9, 14, 19, 24, 29, 34, 39. The 8th number is 39.

Worked example

Example 3: Predict and justify

Rule: Start at 4. Add 5 each time. What is the 8th number?

The 1st number is 4. To get to the 8th number, we make 7 jumps (8 − 1 = 7).
7 jumps of 5 is 5, 10, 15, 20, 25, 30, 35.
4 + 35 = 39.
Check by extending: 4, 9, 14, 19, 24, 29, 34, 39. ✔

The 8th number is 39. I know because there are 7 jumps of 5 after the first number, and 4 + 35 = 39.

Worked example

Example 4: Will this number be in the pattern?

Pattern: 3, 6, 9, 12, … (start at 3, add 3). Will 25 be in the pattern? Will 30?

The pattern is the skip-counting-by-3 numbers.
Keep counting: 15, 18, 21, 24, 27. It jumps from 24 to 27, right over 25.
Keep going: 27, 30. 30 is there!

25 will not be in the pattern (it goes 24, 27). 30 will be in the pattern.

Just for fun

Why was the snowman so calm about pattern predictions?

He already knew he'd be standing at position 14!

Finding missing elements

To find a missing element, first find the rule using the parts you can see. Then use the rule to fill the gap. Always check that the rule works all the way through.

Worked example

Example 5: Missing numbers

Find the missing numbers: 81, □, 63, 54, □

Look at two numbers that are next to each other: 63 to 54 is −9 (63 − 9 = 54).
Test the rule: 81 − 9 = 72, and 72 − 9 = 63. ✔
Last box: 54 − 9 = 45.

81, 72, 63, 54, 45. The rule is start at 81, subtract 9 each time.

Worked example

Example 6: A gap of two steps

Table of values: Figure 1 → 4, Figure 2 → □, Figure 3 → 10, Figure 4 → 13. Find the missing value.

Figure 3 to Figure 4 is 10 to 13: +3.
Figure 1 to Figure 3 is 4 to 10. That is 2 steps and a change of 6. 3 + 3 = 6, so each step is +3. ✔
Figure 2: 4 + 3 = 7.

Figure 2 has 7. The rule is start at 4, add 3.

Your turn

Write a pattern rule for 6, 11, 16, 21.

Extend the pattern by 3 numbers: 200, 190, 180, …

In 🔺 🔵 🔺 🔵 …, what is the 10th element? Explain.

Find the missing number: 12, 18, □, 30

Find the missing element: A B C A □ C A B C

Rule: Start at 2. Add 10. Will 52 be in the pattern?

The arrow pattern goes up, right, down, left, … Which way does arrow 8 point?

Hands-on activity

Pattern prediction game

You need: paper, a pencil, and 2 kinds of small things (like dry pasta and buttons, or 2 colours of LEGO).

A grown-up secretly makes a pattern rule, like pasta, button, button or start at 5, add 4.
The grown-up shows only the first 4 elements.
You write the pattern rule.
Now predict the 12th element or number. Write your prediction and how you know.
Build or write the pattern out to the 12th spot to check.
Switch jobs: you make a rule and the grown-up predicts.

Math talk: "I predict ___ because ___." If a prediction is wrong, that's okay! Find where the thinking went off track. Mistakes help our brains grow.

Grown-up note: keep small items away from younger children who might put them in their mouths.

More practice: warm-up

Write a pattern rule for 10, 20, 30, 40.

Extend: 35, 30, 25, … (3 more numbers)

Find the missing number: 2, 4, □, 8, 10

Find the missing element: 🍁 🍂 🍁 □ 🍁 🍂

What is the 6th element of A B A B A B …?

More practice: on your own

Find the missing numbers: □, 14, 21, 28, □

Rule: Start at 3. Add 4 each time. What is the 6th number? Justify.

In ❄️ ⛄ 🧣 ❄️ ⛄ 🧣 …, what is the 20th element? Explain.

Pattern: 5, 10, 15, 20, … Will 48 be in the pattern? Explain.

Find the missing number: 100, □, 60, 40 Explain.

Challenge

Table: Figure 1 → 5, Figure 2 → □, Figure 3 → □, Figure 4 → 20. Each step adds the same amount. Find the missing values.

A pattern of letters goes A A B C A A B C … What is the 17th letter? Explain.

Rule: Start at 1. Double each time. Leo predicts the 7th number is 14. Is he right? Justify.

Check your understanding

Quiz

What is the pattern rule for 7, 10, 13, 16?

What is the missing number? 25, 50, □, 100

In 🍎 🍌 🍇 🍎 🍌 🍇 …, what is the 9th element?

Rule: Start at 2. Add 5. What is the 5th number?

Find the missing number: 48, 40, □, 24

Pattern: 4, 8, 12, 16, … (add 4). Which number will be in the pattern?

For parents and teachers

Big idea: A pattern rule lets us predict far ahead without drawing everything (C1.3). For repeating patterns, the size of the core helps: if the core has 3 elements, the last element is at positions 3, 6, 9, 12, … For growing patterns, children can extend a table, count jumps, or use skip counting.

Common difficulties:

Counting jumps. To get from the 1st to the 8th number takes 7 jumps, not 8. A number line helps.
Rules that only work once. Remind your child to test the rule on every pair.
Missing numbers with a two-step gap. Show that a change of 6 over 2 steps means +3 each step.
Predicting in repeating patterns. Writing the position numbers under the elements makes it much easier to see.

How to help: Play "What comes 12th?" with patterns made from cutlery, socks or toys. Always ask, "How do you know?" Celebrate a clear explanation even when the answer needs fixing.