Year 7AlgebraEquations and relationshipsYear 7 Maths

Patterns, rules and the coordinate plane

What this lesson covers

  • Algebra has its own specialised notation to express relationships and operations concisely, including 3b in place of b + b + b.
  • A coordinate plane extends to 4 quadrants that meet at the origin (0, 0).
  • Linear patterns have a constant increase or decrease, can be described by the rule t = a × n + d, and can be graphed as a straight line on a coordinate plane.
Key idea

What is the 50th number in this pattern?

  • 3, 5, 7, 9, 11...
  • You could keep going. It would take a while.
  • Or you could find the rule and jump straight there.
Position nValue t
13
25
37
49
511
Key idea

Put it in a table first

  • Position on the left, value on the right.
  • Position is called n. Value is called t.
  • Now the pattern is easier to interrogate.
Position nValue t
13
25
37
49
511
Key idea

Find the step

  • How much does it go up each time?
  • Every single gap is 2.
  • That constant step is what makes it linear.
FromStep
3 to 5+ 2
5 to 7+ 2
7 to 9+ 2
9 to 11+ 2
Key idea

The shape of the rule

  • Every linear pattern fits t = a × n + d.
  • a is the step.
  • d is the bit that shifts it up or down.
PartWhat it is
nthe position
athe step
dthe starting adjustment
tthe value you get
Key idea

a is the easy one

  • The step is 2, so a is 2.
  • So far the rule is t = 2n + something.
  • Now for the something.
FromStep
3 to 5+ 2
5 to 7+ 2
7 to 9+ 2
9 to 11+ 2
Key idea

Go back one more step

  • Position 1 holds 3. What would position 0 hold?
  • One step back: 3 − 2 = 1.
  • That is d. The rule is t = 2n + 1.
StepWorkingResult
The stepis 2so a = 2
Go back to n = 03 − 2so d = 1
The rulet = 2n + 1
Worked example

Test it before you trust it

  • n = 1 gives 2 × 1 + 1 = 3. Correct.
  • n = 5 gives 2 × 5 + 1 = 11. Correct.
  • Two checks is enough to be confident.
PositionWorkingValue
n = 12 × 1 + 13
n = 32 × 3 + 17
n = 52 × 5 + 111
Worked example

So the 50th number is...

  • n = 50, so 2 × 50 + 1.
  • 101.
  • No counting. That is what a rule is for.
PositionWorkingValue
n = 12 × 1 + 13
n = 32 × 3 + 17
n = 52 × 5 + 111
n = 502 × 50 + 1101
Key idea

Algebra writes things short

  • b + b + b is written 3b.
  • 3 × b is also 3b; the times sign is dropped.
  • And b × b is b².
LongShort
b + b + bwritten as3b
3 × bwritten as3b
a × bwritten asab
b × bwritten asb²
Key idea

Why drop the times sign

  • Because × looks far too much like the letter x.
  • 3 × x + 2 is a nightmare to read.
  • 3x + 2 is not.
With the signWithout
3 × x + 23x + 2
a × b × cabc
Key idea

Like terms

  • Terms with the same letter can be added together.
  • 3a + a + a is 5a: five lots of a.
  • Different letters cannot join.
ExpressionWhySimplifies to
3a + a + aall a terms5a
3b − 2ball b termsb
Worked example

4x + 3y + 2x

  • The x terms go together: 4x + 2x = 6x.
  • 3y has nothing to join.
  • 6x + 3y. That is as simple as it gets.
ExpressionWhySimplifies to
3a + a + aall a terms5a
3b − 2ball b termsb
4x + 3y + 2xx with x only6x + 3y
2a + 3a²different powerscannot join
Key idea

Rearranging a formula

  • A = lw gives the area from length and width.
  • What if you know the area and want the width?
  • Same balance rule as last lesson.
Rearranging a formula
Worked example

Making w the subject

  • w is being multiplied by l.
  • So divide both sides by l.
  • w = A ÷ l.
StepWorkingResult
A = lwl is multiplying w
Divide both by lA ÷ l = w
Sow = A ÷ l
Key idea

The plane goes four ways

  • You have used the top-right corner for years.
  • The axes carry on the other way too.
  • Four regions, meeting at the origin, (0, 0).
The plane goes four ways
Key idea

Which quadrant is which

  • The signs of the two numbers tell you.
  • Both positive is the first, top right.
  • Then anticlockwise from there.
Which quadrant is which
Key idea

The four, in order

  • First: both positive.
  • Second: x negative, y positive.
  • Third: both negative. Fourth: x positive, y negative.
QuadrantSignsExample
Firstx positive, y positive(3, 2)
Secondx negative, y positive(−3, 2)
Thirdx negative, y negative(−3, −2)
Fourthx positive, y negative(3, −2)
Worked example

Now plot the pattern

  • Position across, value up. (1, 3), (2, 5), (3, 7)...
  • Join them.
  • A straight line, which is why it is called linear.
Now plot the pattern
Key idea

Four views of one thing

  • The pattern, the table, the rule, the line.
  • The step of 2 is in all four of them.
  • It is the gap, the a, and how steep the line is.
ViewWhere the 2 shows up
The numberseach gap is 2
The tableeach row goes up 2
The rulet = 2n + 1
The graphup 2 for every 1 across
Your turn

Your turn: four questions

  • Pause here and try all four.
  • On question 2, find the step first, then go back to zero.
  • On question 4, look at the signs.
Question
1Simplify 5a + 3a − 2a.
2A pattern goes 4, 7, 10, 13. What is the rule?
3Make l the subject of A = lw.
4Which quadrant is (−2, 3) in?
Show answer

1. 6a. All three are a terms: 5 + 3 − 2 = 6. 2. t = 3n + 1. The step is 3, and going back one step from 4 gives 1 at position zero. 3. l = A ÷ w. Divide both sides by w. 4. The second quadrant: x is negative and y is positive, so it is top left.

Worked example

The answers

  • 5a + 3a − 2a is 6a. And the rule is t = 3n + 1.
  • l = A ÷ w. Divide both sides by w.
  • (−2, 3) is the second quadrant, top left.
Answer
16a
2t = 3n + 1
3l = A ÷ w
4second quadrant
Summary

What to take away

  • The step is the number in front of n.
  • d is what would sit at position zero.
  • Only terms with the same letter can be collected.
Remember
A linear patterngoes up by the same amount each time
That stepis the number in front of n
dis what would sit at position zero
Like termssame letter, same power; only those join
Four quadrantsthe signs of x and y tell you which

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