Year 6AlgebraEquations and relationshipsYear 6 Maths

Patterns, rules and the coordinate plane

What this lesson covers

Inequalities can also include 'or equal to' (≤, ≥) to show a relationship that allows for the possibility of equality.
Tables provide a way of organising the positions and elements of a pattern to reveal relationships or rules.
A coordinate plane is formed when two perpendicular number lines intersect at (0, 0); usually the coordinate plane consists of a horizontal x-axis and a vertical y-axis.
Coordinates are represented as (x,y), where the x-value represents horizontal movement and the y-value represents vertical movement. Plotting points on a coordinate plane can help to visualise numeric patterns.

Key idea

Square tables pushed into a row

One table on its own seats four people.
Push a second one against it and the row seats six.
Not eight: two seats get lost where they join.
So how many does a row of twenty-five tables seat?

Key idea

Keep going and watch what happens

Three tables seat eight. Four seat ten.
Every extra table adds exactly two more seats.
One along each side, and none on the joins.

Key idea

Put it in a table to see it

One column for the position, one for the answer.
Now the pattern is lined up instead of remembered.
A table is how you make a rule findable.

Tables	Seats
1	4
2	6
3	8
4	10

Key idea

The gap is always two

4, 6, 8, 10, 12: up two every time.
A constant difference means a simple rule exists.
And the two is the two side seats per table.

Tables	Seats	Gap
1	4
2	6	+2
3	8	+2
4	10	+2

Key idea

The rule, in words

Two seats for every table: that is the "up two each time".
Plus two more, one at each end of the row.
So: double the number of tables, then add two.
Those two end seats are the "plus two". You can point at them.

Tables	Double it	Add 2	Seats
3	6	+2	8
5	10	+2	12

Worked example

Now answer the original question

Twenty-five tables. Double it: fifty.
Add two: fifty-two seats.
We never drew twenty-five tables. The rule did it.
That is a conjecture: a prediction the rule lets you make.

Tables	Working	Seats
10	20 + 2	22
25	50 + 2	52

Key idea

Always test a rule on a case you know

Try the rule on 2 tables: double is 4, plus 2 is 6.
And we counted 6 at the start. It agrees.
A rule that fails a case you already know is wrong.

Tables	Rule says	We counted
1	4	4
2	6	6
3	8	8

Key idea

Less than, and less than or equal to

You know < and >. They mean strictly smaller and strictly bigger.
≤ and ≥ add three words: "or equal to".
The little line underneath is the "or equal to".

Symbol	Means
<	smaller than
≤	smaller than, or the same
>	bigger than
≥	bigger than, or the same

Key idea

Why those three words matter

Is 5 < 5? No. Five is not smaller than five.
Is 5 ≤ 5? Yes, because "or equal to" allows it.
Same two numbers, opposite answers.
So the symbol has to be read carefully, not skimmed.

Sentence	True?
5 < 5	false
5 ≤ 5	true

Worked example

Is 8 × 7 ≤ 8 × 5 + 4² true?

Work out each side, respecting the order of operations.
Left: 8 × 7 = 56.
Right: the square first, 4² = 16. Then 8 × 5 = 40. So 40 + 16 = 56.
56 ≤ 56 is true, and only because of the "or equal to".

Side	Working	Value
Left	8 × 7	56
Right	40 + 16	56

Worked example

One symbol, opposite answers

Is 3 × 6 > 2 × 9? Both sides are 18.
18 > 18 is false. Eighteen is not bigger than itself.
But 3 × 6 ≥ 2 × 9 is true.
Same six numbers. One extra line. Different answer.

Sentence	Becomes	True?
8 × 7 ≤ 8 × 5 + 4²	56 ≤ 56	true
3 × 6 > 2 × 9	18 > 18	false
3 × 6 ≥ 2 × 9	18 ≥ 18	true

Worked example

Completing an open sentence

Fill the gap: 5 × ? = 45 − 5.
Do the side you can first: 45 − 5 = 40.
So 5 × ? = 40, which means ? is 8.
Simplify one side, then the gap is obvious.

Step	Sentence
Given	5 × ? = 45 − 5
Simplify the right	5 × ? = 40
Answer	? = 8

Key idea

Two number lines, crossed

Take a horizontal number line and a vertical one.
Cross them at zero, at right angles.
That is a coordinate plane, and the crossing point is (0, 0).

Key idea

The x-axis and the y-axis

The horizontal one is the x-axis.
The vertical one is the y-axis.
x goes across. y goes up. That order never changes.

Key idea

Along the corridor, then up the stairs

A point is written (x, y), always in that order.
The first number is how far across.
The second is how far up.
Along the corridor, then up the stairs.

Worked example

Plotting (3, 5)

Start at (0, 0). Count 3 across.
Then count 5 up from there.
Mark the point. That is (3, 5).
Get the order wrong and you land on (5, 3), somewhere else entirely.

Worked example

Points that sit on an axis

(4, 0) means 4 across and none up: it sits on the x-axis.
(0, 3) means none across and 3 up (on the y-axis).
A zero does not mean the point is missing.
It means no movement in that direction.

Worked example

Now plot the tables

Tables across, seats up. Each row of the table is one point.
1 table and 4 seats is the point (1, 4).
Then (2, 6), (3, 8), (4, 10), (5, 12).

Key idea

They make a straight line

Every point lies on one straight line.
That is what a constant difference looks like when you plot it.
And it means you could read off 6 tables without calculating.
This is why plotting a pattern is worth doing.

Your turn

Four to try

1. A pattern goes 5, 8, 11, 14. What is the rule in words?
2. Is 6 × 4 ≥ 5 × 5 true or false?
3. Fill the gap: 7 × ? = 30 + 5.
4. Which point is on the y-axis: (0, 6) or (6, 0)?

Show answer

Answers: 1. Three times the position, plus two: 3, 6, 9, 12 plus 2 gives 5, 8, 11, 14. Accept "add 3 each time" but push for the position rule. 2. False: 24 ≥ 25 is not true, and the ≥ does not rescue it because they are not equal. 3. 5, since 30 + 5 is 35. 4. (0, 6): no movement across, so it stays on the y-axis. If they pick (6, 0) they have the axes swapped, and beat 15 is the one to revisit.

Summary

Pattern, table, rule, plot

A table reveals the rule. The rule makes predictions.
Plotting turns the pattern into a picture.
And the little line under < means "or equal to".

Step	What it gives you
Draw the pattern	the first few answers
Put it in a table	the constant difference
Write the rule	any answer you like
Plot the points	a straight line

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