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National Curriculum Resource Tool

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Year 6 - Multiplication and Division

New Curriculum

  • 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 four 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.

Non-Statutory Guidance

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 in this document).

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.

Links and Resources

Teachers should use every relevant subject to develop pupils’ mathematical fluency. Confidence in numeracy and other mathematical skills is a precondition of success across the national curriculum.
Teachers should develop pupils’ numeracy and mathematical reasoning in all subjects so that they understand and appreciate the importance of mathematics.

– National Curriculum in England Framework Document, September 2013, p9

Connections within Mathematics

Making connections to other topics within this year group

Fractions

  • identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places
  • multiply one-digit numbers with up to two decimal places by whole numbers
  • use written division methods in cases where the answer has up to two decimal places
  • divide proper fractions by whole numbers [e.g. ⅓ ÷ 2 = ⅙]
  • multiply simple pairs of proper fractions, writing the answer in its simplest form

When working on multiplication and division and/or fractions there are opportunities to make connections between them, for example:

Multiply numbers such as:

  • 245.25 by 10, 100 and 1000
  • 1.35 by 8
  • ¼ x ½

Divide numbers such as:

  • 12 578 by 10, 100 and 1000
  • 237 by 5
  • ⅓ ÷ 2

Ratio and proportion

  • solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts

When working on multiplication and division and/or ratio and proportion there are opportunities to make connections between them, for example:

Convert the ingredients in this lasagne recipe for 4 people so that it will serve 12:

  • 350g minced beef
  • 1 onion
  • 1 clove garlic
  • 600g tin of tomatoes
  • 2 tablespoons tomato puree
  • 175g lasagne sheets

Algebra

  • express missing number problems algebraically
  • use simple formulae

When working on multiplication and division and/or algebra there are opportunities to make connections between them, for example:

Solve missing number problems, e.g. 6(a + 12)  =  144
multiply out the equation: 6a + 72  =  144
balance by -72: 6a + 72 – 72  =  144 – 72
  6a  =  72
Use known division facts: a  =  72 ÷ 6
  a  =  12

Find perimeters and areas of rectangles using the appropriate formulae, e.g. a square field has sides of 24.75m. What is its perimeter? What is its area?

Measurement

  • solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal places where appropriate
  • convert between miles and kilometres

When working on multiplication and division and/or measurement there are opportunities to make connections between them by solving problems such as;

  • 1 pint = 0.57 litres, how many litres in 8 pints? How many pints in 12 litres?
  • Dan was driving between two cities in France. The sign said the distance was 185km. He wanted to know what that was in miles. How can he find out? How many miles is it?

Statistics

  • calculate and interpret the mean as an average

When working on multiplication and division and/or statistics there are opportunities to make connections between them, for example:

Solve problems, e.g. find the mean monthly temperature for Reykjavik, Iceland

Monthly temperatures for Reykjavik
Jan Feb March April May June July Aug Sept Oct Nov Dec
-2°C -1°C 3°C 6°C 10°C 13°C 14°C 14°C 11°C 7°C 5°C -2°C

Making connections to this topic in adjacent year groups

Year 5

  • identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
  • solve problems involving multiplication and division including using their knowledge of factors and multiples, squares and cubes
  • know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
  • establish whether a number up to 100 is prime and recall prime numbers up to 19
  • multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers
  • divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context
  • multiply and divide numbers mentally drawing upon known facts
  • multiply and divide whole numbers and those involving decimals by 10, 100 and 1000
  • recognise and use square numbers and cube numbers, and the notation for squared (2) and cubed (3)
  • solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign
  • solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates.

 

Key Stage 3

  • use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
  • use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
  • use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
  • recognise and use relationships between operations including inverse operations
  • use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations
  • interpret and compare numbers in standard form A x 10n 1≤A<10, where n is a positive or negative integer or zero
  • define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%
  • interpret fractions and percentages as operators
  • use a calculator and other technologies to calculate results accurately and then interpret them appropriately

Cross-curricular and real life connections

Learners will encounter multiplication and division in:

Art & Design

Within the art and design 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 be taught to develop their techniques, including their control and their use of materials, with creativity, experimentation and an increasing awareness of different kinds of art, craft and design. This could include designing and creating life size models of, for example a Barbara Hepworth sculpture or a Van Gogh painting where the children need to find realistic measurements and then scale them down using division.

Geography

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. Work on multiplication and division could include converting between miles and kilometres and vice versa when looking at distances between countries or famous locations, making currency converters for pounds stirling and the currency in the country they are investigating.

History

Within the history 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 ‘in planning to ensure the progression described above through teaching the British, local and world history outlined below, teachers should combine overview and depth studies to help pupils understand both the long arc of development and the complexity of specific aspects of the content’. The history curriculum requires that pupils should ‘compare aspects of life in different periods’, suggesting comparisons between Tudor and Victorian periods, for example. Scale models could be one way of learning about life in different periods.

This research outlines the strategies used by Year 5 and 6 children to solve division calculations. They found that children with a more flexible approach are often more successful than those with only one or two strategies.

This research highlights the problems students may have if solving multiplication and division calculations using informal methods alone. In the study these methods dominated in Year 6 and there was evidence of a lack of conceptual understanding of multiplication and division.

Programme of Study statements Activities
A B C D E F G
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          
perform mental calculations, including with mixed operations and large numbers            
identify common factors, common multiples and prime numbers            
solve problems involving multiplication and division use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy            

Activity A

Teachers could ask the children to create their own numbers to multiply using digit cards and dice. Make the link between the grid method and the compact method, for example, asking the children what is the same and what is different about the two methods:

Digit cards example 

Activity B

Set problems such as:

Naomi had a stamp album; it had 135 pages. On each page there were 45 stamps. How many stamps did she have?

This Nrich activity entitled ‘Long Multiplication’ requires children to solve a multiplication puzzle where all except one of the digits are missing. A wonderful challenge!

Activity C

Set problems such as:

Yukesh was given a box of 755 books to put on the shelves in the library. He put 24 books on each shelf. How many shelves did he fill? How many shelves did he need for all the books?

Courtney had a collection of 1256 coins. She put them into piles of 16. How many piles did she have? How many were left?

Activity D

Give the children a set of 0-9 digit cards. They choose four of the cards, make a four-digit number and write it down. They then use two of those cards to make the divisor. They calculate the answer and then make up a linked problem using their numbers.

Activity E

Provide the children with problems that involve using mental calculation strategies, such as:

  • x4 by doubling and doubling again,
  • x5 by x10 and halving
  • x20 by x10 and doubling
  • x9 by multiplying by 10 and adjusting
  • x6 by multiplying by 3 and doubling

Activity F

‘Abundant Numbers’, an activity from Nrich requires children to explore factors of numbers. ‘Factors and multiples’ is another Nrich game, perfect for practising skills.

Activity G

Provide problems such as:

  • Milly is saving £2.75 a week to buy a pair of jeans. The jeans cost £37. For how many weeks does she need to save?
  • In Sports 4 U, there are 18 larges boxes each containing 136 footballs. How many footballs are there altogether?

For each problem, ask the children to check their answer by rounding. They could also use this method to estimate what the answer might be before solving the problem.

Examples of what children should be able to do, in relation to each (boxed) Programme of Study statement

multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
  • Look at long-multiplication calculations containing errors, identify the errors and determine how they should be corrected.
  • Solve problems such as:Printing charges for a book are 3p per page and 75p for the cover. I paid £4.35 to get this book printed. How many pages are there in the book? Write down the calculations that you did. Seeds are £1.45 for a packet. I have £10 to spend on seeds. What is the greatest number of packets I can buy?
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
  • Every day a machine makes 100 000 paper clips, which go into boxes. A full box has 120 paper clips. How many full boxes can be made from 100 000 paper clips?
    • Each paper clip is made from 9.2 centimetres of wire. What is the greatest number of paper clips that can be made from 10 metres of wire?
  • A DJ has two different sized storage boxes for her CDs. Small boxes hold 15 CDs. Large boxes hold 28 CDs. The DJ has 411 CDs. How could the DJ pack her CDs?
solve problems involving multiplication and division
use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy

Children should be able to:

  • Give the best approximation to work out 4.4 × 18.6 and explain why. Answer questions such as: roughly, what answer do you expect to get? How did you arrive at that estimate? Do you expect your answer to be greater or less than your estimate? Why?
perform mental calculations, including with mixed operations and large numbers
  • Use mental strategies to calculate in their heads, using jottings and/or diagrams where appropriate. For example, to calculate 24 × 15, they multiply 24 × 10 and then halve this to get 24 × 5, adding these two results together. They record their method as (24 × 10) + (24 × 5). Alternatively, they work out 24 × 5 = 120 (half of 24 × 10), then multiply 120 by 3 to get 360.
identify common factors, common multiples and prime numbers
  • Children should be able to answer questions such as:
    • How can you use factors to multiply 17 by 12?
    • Start from a two-digit number with at least six factors, e.g. 72. How many different multiplication and division facts can you make using what you know about 72? What facts involving decimals can you derive?
    • What if you started with 7.2? What about 0.72?
    • Which three prime numbers multiply to make 231?
    • use their knowledge of the order of operations to carry out calculations involving the four operations
  • Children should be able to find answers to calculations such as 5.6 ⬜ = 0.7 or 3 × 0.6, drawing on their knowledge of number facts and understanding of place value. They should be able to approximate, use inverses and apply tests of divisibility to check their results.
  • Children should know the square numbers up to 12 × 12 and derive the corresponding squares of multiples of 10, for example 80 × 80 = 6400.

Moving from grid to a column method

This video shows how children can easily move from the grid method to the compact method, whilst maintaining conceptual understanding.