Pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.
(National Curriculum page 10 )
Connections within Mathematics
Making connections to other topics within this year group
Fractions (including decimals and percentages)
- multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams
- solve problems involving number up to 3 decimal places
When working on multiplication and division and/or fractions (including decimals and percentages), there are opportunities to make connections between them, for example:
You could give the children strips of paper and ask them to fold them to show you different proper and mixed fractions, for example, 5⁄8, 1 3⁄4. Next ask them to multiply these fractions by single digit numbers. They could use the strips to help them: 1 5⁄8 x 6
1x 6 = 6
5⁄8 x 6 = 30⁄8 or 3 6⁄8
1 5⁄8 x 6 = 6 + 3 6⁄8 = 9 6⁄8 or 9 3⁄4
Numbers with decimals are frequently seen in real life, for example when using money, so give the children opportunities to multiply these in context. For example, you could give them take-away menus and ask them to find out how much it would cost to buy four of a meal deal or a particular course. You could give them the total cost of six of the same dish and ask to work out which dish you chose.
You could ask the children problems that involve multiplying numbers up to 3 decimal places and link to measures, such as:
- Jessie had eight lengths of rope. Each was1m 36cm. If he put them side by side what would the total length be?
- Paddy had 12 cartons of orange juice. Each carton contained 0.750l. How much juice did he have altogether?
- Suzie, the baker, was making 14 loaves of bread for the local supermarket. For each loaf she needed 1.275kg of flour. What is the total amount of flour that she needed?
- India took part in a sponsored bike ride at her school. She cycled 25 times around the perimeter of the school playground. The perimeter is 105.34m. How far did she travel?
When working on multiplication and division and/or measurement there are opportunities to make connections between them, for example:
You could give the children opportunities to rehearse multiplying by 10, 100 and1000 by converting, for example, millimetres to centimetres, centimetres to metres, metres to kilometres. They could then multiply lengths, masses and capacities of different sizes, for example, 14.75kg by 8. You could then put these into problem format, for example:
- Benji, a party organiser, was going to make a fruit punch. For each guest he needed 0.250ml of orange juice and 0.250l of mango juice. If there are 25 guests coming to the party, what is the total amount of juice Benji needs?
You could give the children an approximate equivalence between miles and kilometres, for example1.6km is approximately 1 mile. Then they multiply this amount to find approximate equivalences for other miles, for example 5 miles, 8 miles, 10 miles, 14 miles. The children could make a spider diagram for this and other equivalences.
You could give the children lengths of one side of different regular polygons, for example, pentagon, octagon, decagon, dodecagon and ask them to find their perimeters by multiplying each length by the number of sides the polygon has.
You could also give the children the lengths of different sized rectangles and ask them to find their areas, for example, a rectangle 28cm by 12cm.
Set problems involving time and money for the children to use, for example:
- Samir spent 45 minutes completing his homework. It took Pete three times as long. How long did it take Pete to complete his homework?
- It took Carol 1 ½ hours to drive from Oxford to London. It took Lorna a third of that time. How long did it take Lorna to travel to London?
- Harry is given £3.75 a week as pocket money. He is saving it to buy a computer game. How much will he have saved over 8 weeks? What about 12 weeks?
- Georgie saved £2.25 of her pocket money each week. How much will she have saved over 9 weeks?
- Penny had saved £75 over a period of 12 weeks. She saved an equal amount every week. How much did she save each week?
Making connections to this topic in adjacent year groups
- recall multiplication and division facts for multiplication tables up to 12 × 12
- use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together 3 numbers
- recognise and use factor pairs and commutativity in mental calculations
- multiply two-digit and three-digit numbers by a one-digit number using formal written layout
- solve problems involving multiplying and adding, including using the distributive law to multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects
Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.
Pupils practise mental methods and extend this to 3-digit numbers to derive facts, (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6).
Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers (see Mathematics Appendix 1).
Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60.
Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or 3 cakes shared equally between 10 children.
- multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
- divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context
- divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context
- perform mental calculations, including with mixed operations and large numbers
- identify common factors, common multiples and prime numbers
- use their knowledge of the order of operations to carry out calculations involving the 4 operations
- solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why
- solve problems involving addition, subtraction, multiplication and division
- use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy
Pupils use the whole number system, including saying, reading and writing numbers accurately Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division (see Mathematics Appendix 1).
They undertake mental calculations with increasingly large numbers and more complex calculations.
Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.
Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50, etc, but not to a specified number of significant figures.
Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.
Common factors can be related to finding equivalent fractions.
Cross-curricular and real life connections
Learners will encounter number and place value in:
Within the geography curriculum there are opportunities to connect with multiplication and division, for example in the introduction of the Key Stage 2 Programme of Study it states that pupils should extend their knowledge and understanding beyond the local area to include the United Kingdom and Europe, North and South America. This will include the location and characteristics of a range of the world’s most significant human and physical features. Children could, for example, find out about the currencies used in a selection of countries. They could then make up a currency converter using mental calculation strategies and then check using multiplication, for example:
£1= 1.20 Euros
£2 = 2.40 Euros
£3 = 3.60 Euros
£4 = 4.80 Euros
£5 = 6 Euros