Year 5AlgebraEquations and relationshipsYear 5 Maths

The equals sign, inequalities and growing patterns

What this lesson covers

  • The relative size of expressions involving numbers can be communicated using “greater than” (>), “less than” (<), and equals (=).
  • Growing patterns can increase or decrease by the addition or subtraction of a constant (arithmetically) or multiplication or division by a constant (geometrically).
Key idea

You can answer this without adding

  • Is 2,456 + 203,938 bigger or smaller than 3,456 + 231,930?
  • Both parts on the right are bigger than both on the left.
  • So the left is smaller, and you added nothing. Read, do not calculate.
both parts bigger, so the whole is bigger
leftrightbigger?
2,4563,456right
203,938231,930right
Key idea

What the equals sign actually means

  • Equals does not mean "here comes the answer".
  • It means the two sides are the same. It balances.
  • Which is why it reads in both directions.
one of these readings is a dead end
reading "=" aswhat that costs you
"here comes the answer"you reject 7 + 5 = 6 + 6
"the same as"you accept it: both sides are 12
Key idea

All three of these are true

  • 7 + 5 = 12. Nobody argues with that one.
  • 12 = 7 + 5 is equally true. The sign is not an arrow.
  • 7 + 5 = 6 + 6 is true too. No single answer in sight.
if it balances, it is true
sentenceis it true?
7 + 5 = 12true
12 = 7 + 5true: it reads both ways
7 + 5 = 6 + 6true: both sides are 12
Key idea

When it does not balance

  • Then one side is greater than the other, and you say so.
  • > is greater than. < is less than.
  • The pointed end always aims at the smaller number.
the point aims at the small end
symbolmeansexample
=the same as6 + 6 = 12
>greater than13 > 12
<less than11 < 12
Worked example

Now check the opening question

  • 2,456 + 203,938 is 206,394.
  • 3,456 + 231,930 is 235,386.
  • So the symbol is <, exactly as the comparison said.
2,456 + 203,938 < 3,456 + 231,930
sidetotal
2,456 + 203,938206,394
3,456 + 231,930235,386
Worked example

A harder one, still no calculating

  • 2,456 × 2 against 1,228 × 4.
  • 2,456 is double 1,228. And 2 is half of 4.
  • Double one, halve the other, and the product does not move.
so the two sides are equal
first numbersecond number
left2,4562
right1,2284
changehalveddoubled
Worked example

Check it, once

  • 2,456 × 2 = 4,912.
  • 1,228 × 4 = 4,912.
  • Equal. The reasoning was right, and it was quicker.
2,456 × 2 = 1,228 × 4
sidevalue
2,456 × 24,912
1,228 × 44,912
Worked example

Is this sentence true?

  • Work out each side separately, then compare them.
  • 60 − 15 = 40 is false, because the left side is 45.
  • 9 × 4 = 40 − 4 is true. Both sides come to 36.
a false sentence is not a wrong sum. It is a lie.
sentencetrue or false
45 + 30 = 75true
60 − 15 = 40false: it is 45
9 × 4 = 40 − 4true: both are 36
Worked example

An open sentence has something hidden

  • 999,999 − ⬚ = 899,999. What is in the box?
  • The balance has to stay level, so ask what was taken away.
  • 999,999 down to 899,999 is a drop of 100,000.
An open sentence has something hidden
Key idea

A pattern that adds a constant

  • 3, 4.5, 6, 7.5 and so on.
  • Each term is 1.5 more than the one before it.
  • Plot it and the points sit on a straight line.
A pattern that adds a constant
Key idea

A pattern that multiplies

  • 3, 6, 12, 24 and so on.
  • Each term is double the one before, not 3 more.
  • Plot that and it curves, getting steeper as it goes.
A pattern that multiplies
Key idea

Both kinds run downhill too

  • A pattern can shrink by a constant: subtract 3 each time.
  • Or shrink by a constant multiple: halve it each time.
  • To describe any pattern, give the first term and the rule.
start at 20, subtract 3. Start at 80, halve.
patternthe rule
20, 17, 14, 11 ...subtract 3
80, 40, 20, 10 ...divide by 2
Your turn

Your turn

  • 1. Put in <, > or =: 4,000 + 500 and 4,500 + 1.
  • 2. Put in <, > or =: 6 × 8 and 12 × 4.
  • 3. Fill the box: 450 + ⬚ = 1,000.
  • 4. Continue and describe: 2, 5, 8, 11 ... and 1, 3, 9, 27 ...
Show answer

1. < (the right is one more). 2. = (halve 12 and double 4 gives 6 and 8, so both are 48). 3. 550. 4. 14, 17: start at 2, add 3. Then 81, 243: start at 1, multiply by 3.

Summary

What to take away

  • Equals means balance, and it reads in both directions.
  • Often you can decide a sentence by comparing, not calculating.
  • Describe a pattern with its first term and its rule.
the closing chart
jobthe move
=both sides the same
> or <the point aims at the smaller side
a hidden numberask what keeps it balanced
a growing patternfind what is done each time

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