Year 6

Autumn Transition Therapy

Can add and subtract increasingly large numbers mentally

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be freely used within the member school.All opinions and contributions are those of the authors. The contents of this resource are not connected with nor

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Commissioned by The PiXL Club Ltd.July 2020

© Copyright The PiXL Club Limited, 2020

Choose a word, discussing its word class and meaning. Pass a beanbag around the classroom from pupil to pupil. Pupils have 5 seconds from when they catch the beanbag to suggest a synonym for the previous word. At any point the teacher can shout ‘All change’ and change the word that pupils are considering. Make it more or less challenging by selecting ambitious language or shortening/lengthening the time pupils have to respond. This can also be played in small groups.

LIN

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additionsubtractioncompensate

roundadjust

Hot potato

The starting word is…

Hot potatoLI

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IT

additionsubtractioncompensate

roundadjust

Mental calculation

Can you think of situations where you

need to be able to add and subtract

mentally?

Mental calculation

Being able to add and subtract mentally is an important life skill. You might need it to:

Add up the cost of a few items

of shopping Check change when buying something

Plan a party

Rank it!

With a partner, order these calculations from easiest to hardest. The answers are given so you can focus on discussing how you would solve

them. Would you do them mentally or would you need a written method?Give reasons for your ranking.

1. 345 – 89 = 2562. 112,410 – 400 = 112,0103. 45.679 – 20 = 25.6794. 1,765 – 802 = 9635. 399 – 244 = 155

The bigger the number, the more

difficult it is to solve mentally. Do you

agree?

Selecting appropriate methods

Part of being a successful mathematician is being able to decide on the best method to calculate. This session will help you to

understand that it is the properties of the numbers, rather than their size, which make numbers easier or harder to work with

when calculating mentally.

It is a bit like being a detective – looking for clues in numbers to guide you as to how to

solve mathematical problems!

Using place value

One useful strategy is to use place value to calculate mentally. This is useful for when numbers don’t bridge

another place value column. For example:

126 + 3 = 129 (the ones total less than 9, so it doesn’t bridge the tens)

126 + 6 = 132 (the ones total more than 9, so they do bridge the tens)

Let’s look at how we can use place value to solve:

12,405 + 400

Using place value

In this case, as we are adding a multiple of

one hundred to 12,405, I can just

focus on the hundreds column.

405 + 400 = 805

So, 12,405 + 400 = 12,805

Let’s try a more challenging example.

12,405 + 1,400

Using place value

In this case, we are adding thousands as well as hundreds.

12,405 + 1,400

So, we need to add 1 to the thousands and 4 to the hundreds.

So:12,405 + 1,400 = 13,805

Use your knowledge of place value to complete these additions mentally.

Your turn

345 + 20012,456 + 3,00045,876 + 1,100

Use your knowledge of place value to solve this problem.

Your turn

At a rugby match there are 12,456 home supporters. If 3,000 away supporters join them, how many people are watching the match

altogether?

Let’s move onto another strategy – using near-multiples of 10, 100 or 1,000.

Near-multiples

If I look closely at the numbers, I notice that 399 is near to a multiple of a hundred – in this case, 400. So, we can use rounding and adjusting as a

mental calculation strategy. This is sometimes called compensating.

53,593 + 399

Near-multiples

As 399 is a near-multiple of one hundred, I can use the strategy of rounding and adjusting to calculate mentally.

This is when I might make a few jottings as I go.

53,593 + 399

Round 399 to 400. This gives me an easier calculation. I can add 5 hundreds to 4 hundreds to

make 9 hundreds. I will jot down 53,993.53,593 + 400 = 53,993

To round 399 to 400, I added 1. I need to adjust for that and subtract 1 from the answer.

53,993 – 1 = 53,992

So, 53,593 + 399 = 53,992

Near-multiples

We can do the same for near-multiples of 1,000. 16,246 + 2,998

Talk to your partner:

Look carefully at the numbers. Can you detect a near-multiple? How could

you use rounding and adjusting to solve this?

Near-multiples

2,998 is a near-multiple of 1,000, so we can use rounding and adjusting.

Add 2 to 2,998 to round it to 3,000. Add 6thousands to 3 thousands to make 9 thousands. Jot

down 19,246.16,246 + 3,000 = 19,246

Because we added 2 to round 2,998, we need to adjust the answer by subtracting 2. 19,246 – 2 = 19,244

So, 16,246 + 2,998 = 19,244

16,246 + 2,998

Near-multiples

So, one strategy to add numbers mentally is to look for numbers that are near-multiples of 10, 100 or 1,000. Then, use rounding to make the numbers easier to

work with and adjust the numbers to find the answer.

Steps: identify if one of the numbers to be added is a near-

multiple of 10, 100, 1,000 etc; round the number to make it easier to work with; add the numbers; reverse the adjustment you made; check the final calculation.

6,239 + 69399 + 3,704

£72,246 + £2,999

Your turn

Can you identify any near-multiples in these numbers? Use rounding and adjusting to find the total.

Your turn

Select a suitable strategy to calculate this mentally.

A recipe for a crumble topping requires 3,380g of flour and 410g

of sugar. What is the combined weight of these two ingredients?

Subtracting mentally

Now, let’s look at some subtractions where we can use place value to calculate mentally.

How might you go

about this?

35,385 – 80

Using place value

35,385 – 80 80 – 80 = 0 tensComparing the place value

columns, I can see that I simply need to subtract 8 tens.

So, 35,385 – 80 = 35,305All of the other digits would remain the same as there is no exchanging

involved.

Using place value

24,464 – 1,200

4,400 – 1,200 = 3,200

So, 24,464 – 1,200 = 23,264

Comparing the place value columns of the thousands and hundreds, I can see that there would be no

exchanging involved, so this could be done mentally.

Having completed this simple subtraction, we can then adjust the equivalent digits in the

answer.

All of the other digits would remain the same as there was no exchanging

involved.

672 – 403,385 – 200532 – 410

4,358 – 200

Your turn

Use your knowledge of place value to find the difference between these numbers.

Think about whether the subtraction involves

bridging a place value column.

Near-multiples

Using near-multiples can also work for subtraction. 1,998 is a near-multiple of 1,000, so

we can use rounding and adjusting.

Add 2 to 1,998 to round it to 2,000. 7 thousands subtract 2 thousands is 5 thousands. Jot down

65,347.67,347 – 2,000 = 65,347

Because we added 2 to round 1,998, we subtracted 2 too many, so we need to adjust the

answer by adding 2 on.

65,347 + 2 = 65,349

So, 67,347 – 1,998 = 65,349

67,347 – 1,998

Have you noticed that lots of shop prices tend to be near-multiples of

10, 100 and 1,000?

For example: £0.99, £12.98 or £223,900

Why do you think prices are often set like this?

Near-multiples

In which country would you find these labels?

Near-multiples

£3.99

£12.98

Jamal buys a baseball cap and a table tennis bat.

What is the total cost of the two items?

What strategy would you use to solve this mentally?

£12.98 + £3.99 = ?

When numbers are close to a multiple of 10, 100 or 1,000, we can round and adjust them to make the numbers easier.

For this method you may find it useful to jot a few numbers

down as you go.

Near-multiples

If we add 1p to £3.99, it makes £4.00. This is a much easier

number to work with.

Let’s start with the baseball cap.

£3.99

£12.98 Let’s round the price of the bat in

the same way.

If we add 2p to £12.98, it makes £13.00. This is also a much

easier number to work with.

Near-multiples

Now we have rounded the numbers, the calculation is:£4.00 + £13.00 = £17.00

Because we rounded by adding 3p altogether, we need to adjust the numbers (or compensate) and use

the inverse to subtract 3p from the total.

£17.00 – 3p = £16.97 So:£12.98 + £3.99 = £16.97

Near-multiples

Your turn

Choose 2 items and find the total cost mentally using

the rounding an adjusting strategy.

£3.99

£12.98

£325.96

£18.99

£22.97

Your turn

Choose 2 items and find out the difference in price between them using the rounding and adjusting strategy.

£3.99

£12.98

£325.96

£18.99

£22.97

Reflection

Making good decisions about how to calculate is all about being a number detective and looking carefully at the properties of the

numbers.

Compare place value columns – if there is no bridging involved, it can be easy to calculate mentally.

Look for near-multiples of 10, 100 or 1,000 and use rounding and adjusting to help to calculate mentally.

Just because a number is large, doesn’t mean it is harder to calculate mentally! It’s all about the properties of the numbers.