Get Calculating! It’s all a matter of scale

A lesson for Year 6 pupils to familiarise themselves with the key features of calculators and encourage appropriate and effective use whilst also exploring and reasoning about key aspects of mathematics. The activities can be used in isolation or as a whole lesson. A PowerPoint to support the activities is also available to download from the MEI website. The lesson is focussed on calculation and particularly on developing understanding of multiplicative relationships and how multiplication can be understood as scaling. It also provides opportunities to discuss the interpretation of answers on a calculator and to use rounding skills. There is also a talking point around what we mean by ‘as big as’. Do we mean length, area, mass or something else? Does it make a difference what we compare?

Rationale Since calculators are no longer needed for KS2 SATs, their use has not been such a key focus. These ‘Get Calculating’ activities aim to develop familiarity with calculators whilst not detracting from the need for pupils to develop efficient mental and written calculation strategies. The activities can be done using non-scientific or scientific calculators, which pupils will be encouraged to use as they start Year 7.

Resources Fact file slide from the PowerPoint or individual information strips from Appendix 1 Matching triples cards for cutting up Appendix 2A or 2B Talk prompts sheet [Slide 12] Calculators – scientific or non-scientific – enough for 1 between 2 pupils NB Check to see if the number of decimal places displayed can be changed. Some 4 operation (non-scientific calculators) can be put into modes where numbers are rounded automatically which gives misleading answers

Learning Intentions/Outcomes The only reference to calculators in the Primary National Curriculum concerns making effective use of them, so this unit aims to use a calculator to explore some key aspects of the curriculum, which will support transition from Year 6 to Year 7.

The aspects covered are:

Estimating and rounding

Multiplication as scaling

Inverse operations

Interpreting answers in context

Problem solving strategies o being systematic o trial and improvement o applying properties of number

Talk Prompts [Slide 12] All activities provide opportunities for pupils to reason about the numbers involved and to discuss approaches. Sentence prompts to encourage this discussion are included on the PowerPoint and this slide could be printed and displayed for use by all during the lesson: It could be this because … It can’t be this because … This answer represents … If we do this then … I wonder what will happen if … I know this so … ‘… is … times bigger than …’ ‘… is … a [fraction] the size of …’ The scale factor is …

Teacher Note – Multiplication as Scaling What do we mean by ‘4 times as big’? In most cases, multiplication is introduced as repeated addition. The examples in this lesson lead pupils to consider multiplication as scaling. ‘The BFG was four times the size of a normal man’. This leads to a discussion about what we mean by ‘four times the size’. Was he 4 times as tall? If he was 4 times as tall, how wide and deep would he be? Was his mass 4 times that of a normal man? When we consider scaling as multiplication, we need to establish what it is that is being scaled. Using precise language, we can describe what is happening to the numbers involved. If we think about a multiplication 6 x 4 = 24, 6 is the multiplicand, 4 is the multiplier and 24 is the product. When multiplication is considered as repeated addition, the multiplier (4) indicating the number of groups of the multiplicand (6). When multiplication is considered as scaling, the multiplier (4) is a scale factor, indicating how many times bigger the product will be than the multiplicand. Identifying what the product relates to will be key. In the case of the BFG, what is being multiplied is a measurement of length (height of a normal man). The result is then the height of the BFG. Identifying what property of the object in question is being compared and scaled will be important in being able to interpret the result of the calculation. The image on the title slide comes from the NCETM Mastery PD Materials. Segments 2.17 and 2.27 provide further ideas to support the teaching of multiplication as scaling

Introduction We are comparing dimensions using multiplication and the key language of: ‘… is … times the size of …’ Or ‘… is … [a fraction] the size of …’ It will however be really important to interpret what is being scaled. The words ‘the size of’ will be replaced by ‘the length’, ‘the area’, ‘the cost’ etc. in line with the context of the problem. The chapter in Roald Dahl’s The BFG where they have breakfast with the Queen is a lovely context for the idea of scaling up as they have to find other things to create his table, chair and place setting. The BFG was 4 times the size of a normal man! He wasn’t 4 men standing on each other’s shoulders, he was one man, four times as big as a normal man. This leads us to think about multiplication in a different way; multiplication as scaling. It also requires us to think about what aspect is being scaled. Multiplying his height by 4 will make him 4 times as tall but at the same time he would cover 16 times the area and be 64 times the volume of a normal man. [See teacher note above]. Some of the contexts will require the pupils to research or you could provide the fact file included in the PowerPoint. Adding images to the PowerPoint will also support thinking. The calculator can support the handling of large numbers. There will need to be time to discuss what the answer represents. If you divide the larger value by the smaller one, the answer will be the scale factor. If you select the correct values, when you multiply or divide by the scale factor, the answer will be the other value.

True, false or it could be Aims: To solve problems involving the relative sizes of two quantities To solve problems [involving similar shapes] where the scale factor is known or can be found Introduce the statement [Slide 3] and discuss what they know about it to start with. Encourage the pupils to draw the dog and cat and discuss the relationship between their heights. What might be three times the size of a dog? What might a cat be three times the size of? Ask the pupils what sort of calculation they think they will have to do in order to decide. Introduce the vocabulary of scale factor and discuss that this does not have a unit of measure like the other values in the statements. The ‘three times’ in this statement is a scale factor. Ask the pupils what the statement would be if we started with A cat is … one third as tall as a dog [Slide 3] This slide is animated to highlight the scale factor language and to reverse the statement using a fractional scale factor. [Slide 4] gives a further four statements to investigate and [Slide 9] gives relevant facts which are also available in a table in Appendix 1. You could give the information all at once or cut the information into strips and only supply facts when they are asked for by the pupils. [Slides 5, 6, 7 and 8] break down the four statements and provide prompts to support thinking. You may want to use these slides, adding images and maps or you may decide to provide all the information at once on [Slide 4]. A. A dog is three times as tall as a cat B. Land’s End to John o’ Groats is four times the distance from London to Manchester C. Ben Nevis is one fifth the height of Everest D. Wales is half as big as Scotland

E. A large fries is 11

2 times the size of a regular fries

Questions How will you decide if it is/isn’t true? What situation could it be true/false if you are saying ‘it could be’? What calculation can you do? What is the answer on the calculator showing you? Do you need to round your answer? Does this answer show that the statement is true or false? What is the scale factor in this situation?

Solutions: Depending on the data you use, the answers may vary but based on the data here: A. The figures we have used say no, it is less than three times at just over twice the height but

as dog heights vary widely (think Chihuahua and Great Dane!) you could find cases where the dog is three times as tall as the cat and possibly the other way round too!

B. Yes it is roughly 4 times the distance C. Ben Nevis is less than a fifth the height of Everest – what fraction of the height is it? 6.5

times roughly – between 𝟏

𝟔 and

𝟏

𝟕 would be a good range

D. Wales is a quarter the area but over half the length – interesting discussion point! E. ‘Large’ are under 1.5 times ‘Regular’ in both mass and cost – challenge to work out how

much bigger without adding. More like 1.25 times in both cases.

Matching Triples Aims: To solve problems involving the relative sizes of two quantities To solve problems [involving similar shapes] where the scale factor is known or can be found There are two versions of this activity. Version A [Appendix 2A and Slide 6] contains more straightforward values than Version B [Appendix 2B and Slide 7].

The aim of this activity is the same but this time the context has been removed and the pupils need to apply their understanding in an abstract context. There are values and scale factors to arrange in groups of 3. There is more than one way to complete some of the statements but only one way to use all the cards at the same time.

Encourage some pupils to create individual statements to gain confidence. Others may be able to tackle the whole challenge straight away, aiming to use all the cards at the same time. Encourage them to estimate first, applying understanding of place value, before using the calculator to check that their estimate is correct. Ensure that they are confident in interpreting the calculator display and know how to handle an answer which is not a whole number. Going Deeper Ask the pupils to change the positions of the values in their statements. How do they need to alter the scale factor to make the statement true again? e.g. 1 000 is one tenth of 10 000 so 10 000 is ten times 1 000

Reflection Encourage pupils to reflect on what the lesson has highlighted, both in the use of calculators and also in the links between operations. The benefit of using rounding to estimate answers before calculating The need to repeat calculations in order to check that numbers have been entered correctly The language of scale factors The fact that scale factors link to multiplication and division

Optional follow-up activity A Sample from the SATs (no calculators allowed!) Aims: To solve problems involving the relative sizes of two quantities To solve problems [involving similar shapes] where the scale factor is known or can be found To use efficient mental strategies to solve problems The aim here is to apply the learning from the calculator activity to solve these problems using mental and written methods. There are 4 questions from recent KS2 SATs papers which require pupils to apply their understanding of scaling. The questions are in Appendix 3 at the end of this document

For further information about MEI’s Primary and Transition activities, please

visit our website:

mei.org.uk/Primary

Appendix 1

Fact File

Average cat height 24cm

Average dog height 56cm

Land’s End to John O’ Groats 874 miles

London to Manchester 211 miles

Height of Everest 8 848 metres

Height of Ben Nevis 1 345 metres

Area of Wales 20 735 km2

Area of Scotland 80 077 km2

Length of Wales 170 miles

Length of Scotland 274 miles

Large fries 150 grams

Regular fries 114 grams

Cost of large fries £1.39

Cost of regular fries £1.09

Appendix 2A: Matching Triples Cards Version A (mixed up for cutting)

9 21.5 101

10000 30 1

10

42 21 3

2121 3 1

2

75 1000 2.5

40 750 20

150 140 1

4

160 7 5

is times

x =

÷ =

Matching Triples Solution A

21 2121 101

10000 1000 1

10

30 9 3

21.5 43 1

2

75 30 2.5

140 7 20

160 40 1

4

150 750 5

Appendix 2B: Matching Triples Cards B (mixed up for cutting)

160 7 101

120 240 3

10

27 6000 30

16.8 400 2

5

750 51 2.5

175 5151 25

300 72 3

4

42 0.9 15

is times

x =

÷ =

Matching Triples Solution B

51 5151 101

72 240 3

10

27 0.9 30

16.8 42 2

5

750 300 2.5

175 7 25

160 120 3

4

400 6000 15

Appendix 3

KS2 SATs 2018 Paper 3

KS2 SATs 2019 Paper 3