Rules for Patterns Using Words and Tables

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1 Getting Started
2 Watch and Notice
3 Try It Together
4 Build the Table in Your Copy
5 Class Challenge
6 What Did We Notice?
7 What's Next
Pupil practice

1 - Getting Started ~4 mins

Illustration for Getting Started Here is a growing tile pattern: pattern 1, then pattern 2, then pattern 3, each one a little bigger than the last. Pattern 1 uses 3 tiles, pattern 2 uses 5 tiles, pattern 3 uses 7 tiles.

Big question

Now the big question: how many tiles would pattern 10 need? Could you work it out without drawing all ten patterns?

Teacher Notes

Display the three growing patterns as pupils settle. Give five seconds of quiet think-time before any hands go up, then take three hands-up answers. Do not confirm a correct answer yet — the point is to leave the question open so the table feels useful in the next step.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~8 mins

To help us, we use a machine: a machine that takes a term-number in and gives a value out using a hidden rule. We feed in 1, 2, 3 and read off the values it hands back, then we work out the hidden rule.

Our first machine: terms 1, 2 and 3 give us 3, 5 and 7

Watch as we feed the term-numbers 1, 2 and 3 through the machine and read off the values 3, 5, 7. Look at the jump between each value: 3 to 5 is up 2, and 5 to 7 is up 2 again, so it grows by 2 for every extra term. That means term 1 has 1 lot of 2, term 2 has 2 lots of 2, term 3 has 3 lots of 2 — so the value is 2 times the term. Check term 1: 2 × 1 = 2, but we have 3, so there is 1 extra. The rule is tiles = 2 × term + 1.

The times-three rule

Before I send these through, predict the jump: how much do you think the values go up by each time? Here the machine takes a term-number and gives back three times as much: term 1 gives 3, term 2 gives 6, term 3 gives 9 — up 3 each time, so 3 lots for every extra term. Check term 1: 3 × 1 = 3, and we have 3, so there is nothing extra to add. The rule is tiles = 3 × term.

The times-four minus one rule

Predict the jump again before we feed these in. The values are 3, 7, 11, 15 — going up in fours, so 4 lots for every extra term, and the value is 4 times the term. Check term 1: 4 × 1 = 4, but we have 3, so we are 1 short — we take one off. The rule is tiles = 4 × term − 1. Notice how the step between values matches the number we multiply by.

Teacher Notes

Walk all three worked examples aloud, one at a time, building the step-to-multiplier link on the board as the description does.

Example 1 — point at the +2 between each value, say it grows by 2 for every extra term, so the value is 2 lots of the term; then show the 1 extra to reach 3 at term 1.
Example 2 — ask the class to predict the jump first, then show it grows by 3, so the value is just three lots of the term with nothing added.
Example 3 — ask 'what number do the values jump by?' before you feed them in, link that 4 to the multiplier, then check term 1 to find the minus one.

The key insight to revoice from a pupil: the step between values tells you the number you multiply by.

👇 Mark your steps as done as you complete them.

+10 XP

3 - Try It Together ~10 mins

Today we work through this growing pattern together: term 1 gives 4, term 2 gives 7, term 3 gives 10. I will send the next term-numbers through the machine and build a two-row table on the board as the values land, while you watch and call out each value. Then we will agree on the rule in words together.

Find the rule: 4, 7, 10

Teacher Notes

This round is for talking it through together — pupils take turns at the board and the class agrees or corrects out loud.

Send term 4 and term 5 through the machine; write each term-number and value into a two-row table on the board as it lands, so pupils see the table take shape before they copy it next step. Ask 'what is the step between values?' (3) before anyone names the full rule, then ask how many lots of the term that gives. Revoice a strong answer as 'so it is three times the term number, then add one'. Watch for pupils who say the rule is just add three — that continues the pattern but cannot leap to a far term; steer them to the three times the term form.

+15 XP

4 - Build the Table in Your Copy ~3 mins

COPYBOOK MOMENT

Illustration for Build the Table in Your Copy In your maths copy, draw a two-row table with the top row labelled term-number and the bottom row labelled value. Fill in terms 1 to 5 for our pattern, then write the rule as a sentence underneath the table.

Teacher Notes

Walk the room glancing for two things: the table is drawn with two clear rows, and the rule sentence underneath uses times the term number, not just add three. This is whole-class copybook practice, not marking.

+10 XP

5 - Class Challenge ~8 mins

Today we crack the rule for a few of these patterns by sending term-numbers through the machine and watching the values. Once we have the rule, we will use it to leap straight to a far term, without drawing every step in between.

Crack the hidden rule

Teacher Notes

This round is the practice bank — pupils take turns at the board, check each answer, and the class confirms before moving on. Keep the board work brisk rather than over-explaining.

For each of the three challenges, have a pupil probe one more input to test their guess, then build and check the rule. After the rule is confirmed, ask the class 'so what would term 10 be?' so they practise leaping to a far term. The final challenge has a minus part — watch for pupils stopping at four times the term and forgetting to take 2 off.

+15 XP

6 - What Did We Notice? ~3 mins

MATHS TALK

How does the table help us see the rule that the picture hides? Which was easier for you: reading the rule from the row of patterns, or reading it from the table of numbers?

Teacher Notes

Listen for pupils naming the step between values as the clue to the multiplier. Revoice a strong answer: so once we know the step, we know the times-number, and the rest is just the adjustment. Head off the idea that add the step each time is a full rule — push them to see why times the term is what lets you leap to term 50 without listing every term in between.

+10 XP

7 - What's Next ~3 mins

What we learned

A two-row table of term-number against value lays a pattern out so its rule is easy to spot.
The step between values tells you the number to multiply the term by; the rest of the rule is the adjustment that sets where the pattern starts.
Once you have the rule in words, you can leap straight to any term without drawing every step.

Coming up

Next we look at the properties of operations and the distributive law — how multiplication shares out over addition, so a tricky product becomes two easy ones.

Teacher Notes

Keep this brisk. Recap the three bullets, then point forward to the distributive law lesson.

+5 XP

Pupil practice

Module 10 · Algebra: Patterns, Expressions and Equations Mixed

Lesson 105 · Rules for Patterns Using Words and Tables

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Teacher Class Feed

1 - Getting Started ~4 mins

2 - Watch and Notice ~8 mins

Our first machine: terms 1, 2 and 3 give us 3, 5 and 7

The times-three rule

The times-four minus one rule

3 - Try It Together ~10 mins

Find the rule: 4, 7, 10

4 - Build the Table in Your Copy ~3 mins

5 - Class Challenge ~8 mins

Crack the hidden rule

6 - What Did We Notice? ~3 mins

7 - What's Next ~3 mins

What we learned

Coming up

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