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

Pattern Rules: Predicting Far Ahead and Finding Missing Elements

C1.3 A1About 40 minutes

Learning goals

I can find a pattern rule and use it to extend the pattern.
I can write a term-number rule that connects each term number to its term value.
I can make a prediction about a term far away and explain why it is right.
I can find missing elements in repeating, growing and shrinking patterns.

Words to know

pattern rule
A rule that tells how a pattern works.

start-and-change rule
A rule that tells the first term and how to get from one term to the next, like start at 5 and add 4 each time.

term-number rule
A rule that tells how to get the term value straight from the term number, like term number × 4 + 1.

predict
Use a rule to say what a term will be before you build or count it.

justify
Explain why your answer must be right.

missing element
A term in a pattern that has been left out.

Remember from Grade 4

In Grade 4 you used rules like start at 5 and add 4 each time. To find a far-away term, you counted the jumps: the 10th term has 9 jumps after the first term. You also noticed that a pattern like 3, 6, 9, 12 is just term number × 3.

In Grade 5 you will write a term-number rule for patterns that don't start with the jump number, like 5, 9, 13, 17. It lets you jump straight to the 50th or 100th term.

Let's start with a story

The Grade 5 class is setting up tables for a Family Math Night. Each table is a square. Tables are pushed together in a long row, and chairs go around the outside.

1 table: 4 chairs
2 tables: 6 chairs
3 tables: 8 chairs

They expect enough families for 25 tables. How many chairs do they need? Adding 2 twenty-four times would take a while. Let's find a faster rule.

Each table adds a chair on its top and bottom side, and the row always has one chair at each end. That is why the rule is term number × 2 + 2.

Two kinds of pattern rules

Start-and-change rule: start at 4 and add 2 each time. Good for finding the next term.
Term-number rule: connects the term number to the term value. Good for jumping far ahead.

How to find a term-number rule

For a pattern that adds the same amount each time:

Find the change. Here it is +2.
Multiply each term number by the change: 1 × 2, 2 × 2, 3 × 2.
Compare with the real term values. How much do you add (or subtract) to fix it?

Term number (tables)	Term number × 2	Term value (chairs)
1	2	4
2	4	6
3	6	8
4	8	10

Every term value is 2 more than term number × 2.

Each term value is 2 more than term number × 2. So the term-number rule is:

chairs = term number × 2 + 2

For 25 tables: 25 × 2 + 2 = 50 + 2 = 52 chairs.

Justify: each table has a chair on its top side and its bottom side (25 × 2), and there is one more chair at each end of the row (+ 2). The rule matches the real tables!

Worked example

Example 1: Write a term-number rule

Find a term-number rule for 5, 9, 13, 17, ... Use it to find term 50.

The change is +4, so multiply each term number by 4: 4, 8, 12, 16.
Compare: 5 is 4 + 1, 9 is 8 + 1, 13 is 12 + 1, 17 is 16 + 1. Each value is 1 more.
Rule: term number × 4 + 1. Check term 3: 3 × 4 + 1 = 13 ✔
Term 50: 50 × 4 + 1 = 200 + 1 = 201.

Term 50 is 201. The rule term number × 4 + 1 works for every term we can see, so we can trust it.

Worked example

Example 2: A shrinking pattern far ahead

The pattern 50, 46, 42, 38, ... subtracts 4 each time. What is term 10?

Way 1 (count jumps): term 1 to term 10 is 9 jumps. 9 × 4 = 36. 50 − 36 = 14.
Way 2 (term-number rule): term number × 4 is 4, 8, 12, 16. The values 50, 46, 42, 38 are what you get when you take those away from 54: 54 − 4 = 50, 54 − 8 = 46. Rule: 54 − term number × 4.
Term 10: 54 − 10 × 4 = 54 − 40 = 14 ✔

Term 10 is 14. Both ways agree, which is a great way to justify a prediction.

Worked example

Example 3: Predict in a repeating pattern

A string of lights repeats 🔴🟡🟢🔵. What colour is the 50th light?

The core has 4 elements.
Every 4th light (4, 8, 12, ..., 48) ends a core. 50 ÷ 4 = 12 R2.
That means 12 full cores (48 lights), then 2 more lights.
The 2nd element of the core is 🟡.

The 50th light is 🟡 yellow. The remainder tells you the position inside the core.

The core red, yellow, green, blue has 4 lights. Twelve full cores use 48 lights, so light 49 starts a new core and light 50 is the 2nd light: yellow.

Worked example

Example 4: Missing elements in a growing pattern

Find the missing terms: 8, ☐, ☐, 26, 32

Look at terms you know side by side: 26 → 32 is +6.
From 8 to 26 there are 3 jumps. 26 − 8 = 18, and 18 ÷ 3 = 6 ✔ The jump is always 6.
Fill in: 8 + 6 = 14, 14 + 6 = 20. Check: 20 + 6 = 26 ✔

8, 14, 20, 26, 32

Worked example

Example 5: A missing first term

Find the missing term: ☐, 21, 29, 37, ...

The change is +8 (21 → 29 → 37).
To go backward, do the opposite: subtract 8. 21 − 8 = 13.

The first term is 13. Check: 13 + 8 = 21 ✔

Justify like a mathematician

A good justification says why:

"I found the change was +4, and 9 jumps of 4 is 36, so 50 − 36 = 14."
"My rule works for every term I can see, so I can trust it for term 50."
"I checked my answer a second way and got the same number."

If your two ways disagree, don't worry. That's useful! It tells you to look again.

Just for fun

If the class pushed 100 tables into one row, the rule says 100 × 2 + 2 = 202 chairs. That row would be so long that the family at one end would need a walkie-talkie to ask the family at the other end to pass the snacks!

Hands-on activity

Predict, then build

You need: about 35 coins, buttons or blocks, some forks and spoons, paper and a pencil.

Part 1. Build a growing pattern: Figure 1 has 4 coins, Figure 2 has 7, Figure 3 has 10.

Find the change and write a term-number rule. (Hint: term number × 3, then fix it.)
Predict how many coins Figure 10 needs. Write your prediction and how you know.
Build Figures 4, 5 and 6 to test the rule. Does Figure 6 have the number your rule says?

Part 2. Lay out a repeating pattern: fork, spoon, spoon, fork, spoon, spoon, ... Predict what the 20th item will be. Then lay out 20 items to check.

What to look for: the rule is term number × 3 + 1. Figure 6 has 19 coins and Figure 10 has 31 coins. The 20th item is a spoon (20 ÷ 3 = 6 R2, the 2nd item in the core).

Grown-up note: use forks and spoons only (no knives), and keep coins away from babies and toddlers.

Your turn

The rule is start at 6 and add 5 each time. What is term 10?

Write a term-number rule for 3, 5, 7, 9, ... Find term 20.

Write a term-number rule for 4, 8, 12, 16, ... Find term 25.

A pattern repeats 🍎🍌🍇. What is the 20th fruit?

Find the missing terms: 12, 19, ☐, 33, ☐

Find the missing term: 80, 74, 68, ☐, 56

Find the missing terms: 10, ☐, ☐, 31

More practice: warm-up

What is the next term: 2, 4, 6, 8, ☐?

Write a start-and-change rule for 30, 27, 24, ...

Find the missing term: 5, 10, ☐, 20

A pattern goes ⬆️⬇️⬆️⬇️... What is the 9th element?

The term-number rule is term number × 10. What is term 7?

More practice: on your own

Write a term-number rule for 7, 11, 15, 19, ... Find term 30.

100, 93, 86, 79, ... What is term 8?

A pattern repeats ❄️⛄🧣🧤. What is the 37th element?

Find the missing first term: ☐, 21, 29, 37

Lee says term 10 of 5, 8, 11, ... is 35, because 10 × 3 + 5 = 35. Is he right?

Challenge

A pattern has the rule term number × 6 − 2. Write the first 4 terms. Is 100 in the pattern?

A snow fort pattern uses 5, 8, 11, ... blocks. Is there a term with exactly 50 blocks? With exactly 75?

The pattern 95, 88, 81, ... subtracts 7 each time. Which term is the first one less than 30?

Check your understanding

Quiz

What is a term-number rule for 5, 8, 11, 14, ...?

What is term 20 of term number × 4 + 1?

A pattern repeats ★●▲. What is the 11th element?

Find the missing term: 9, 16, ☐, 30

60, 55, 50, ... What is term 9?

Which is the best justification that term 10 of 2, 5, 8, ... is 29?

For parents and teachers

Big idea: Grade 5 students determine pattern rules and use them to extend patterns, make and justify predictions, and find missing elements in repeating, growing and shrinking patterns (C1.3). The big step from Grade 4 is the term-number rule (like term number × 4 + 1), which jumps straight to a far-away term. It also prepares for algebra expressions later in this unit.

Common difficulties:

Counting one jump too many. From term 1 to term 10 there are 9 jumps, not 10. Draw a short number line to show it.
Writing "× 4" and forgetting the fix-up. Always test the rule on two or three terms you can see.
Repeating patterns: ignoring the remainder. 50 ÷ 4 = 12 R2 means the 2nd element of the core; R0 means the last element.

How to help: Ask "How do you know?" after every prediction, and ask for a second way to check. Social-emotional learning (A1): when two methods disagree, treat it as a puzzle, not a failure.