A progression of understanding in mathematics

1. A progression of understanding in mathematics To progress in Mathematics, we have to understand the basics and then build upon these basics a layer at a time. As each topic in Maths is interlinked, if we have a gap in our understanding then it can effect many areas, not just one aspect. For example, if we do not have a method or an understanding of multiplication, we would find it difficult to interpret questions on some aspects of real life problem solving, division, ratio and proportion, ordering fractions… the links go on!

2. The Village Academy have developed a progression of understanding for the four operations over a number of years – our 5 a day. Our progression concentrates on the introduction of standard symbols, the use of the empty number line as a jotting to aid mental calculation and on the introduction of pencil and paper procedures. It is important that children do not abandon jottings and mental methods once pencil and paper procedures are introduced because mental methods and jottings may be more appropriate at times.

3. Our long term aim is for children to be able to select an efficient method of their choice (whether this be mental, written or upper Key Stage 2 using a calculator) that is appropriate for a given task. They will do this by asking themselves: • Can I do this in my head? • Can I do this in my head using drawings or jottings? • Do I need to use a pencil and paper procedure? • Do I need a calculator? This meeting focuses on how we develop this understanding for multiplication and division.

4. In the Foundation Stage, Robins engage in many different practical activities to become familiar with numbers. Some of them may begin looking at 5 a day in the final term. In Key Stage One, Swallows are currently focussing on setting up addition and subtraction 5 a day, but will be moving onto multiplication and division soon. In Key Stage Two, Owls and Woodpeckers have been looking at multiplication this term, and will focus on division in the following term.

5. Multiplication Step 1 • To do this, a child needs to: • Be able to count in 1s, 2s, 5s and 10s • Read and understand multiplication number sentences: 4 x 2 = 4 lots of 2 • Arrange objects to do what the number sentence is asking.

6. Multiplication Step 2 • To do this, a child needs to: • Practise counting in 2s, 5s and 10s as a minimum. • Understand multiplication as repeated addition e.g. 3 x 4= 4 + 4 + 4. • Arrange up to 20 objects into arrays. • Be able to draw arrays to represent a number sentence.

7. Multiplication Step 3 • To do this, a child needs to: • Practise times tables and know 2s, 5s and 10s as a minimum. • Recap that multiplication is repeated addition. • Repeatedly add along a number line to find unknown multiplications (u x u) • Use facts they already know to shorten this process

8. Multiplication Step 4 • To do this, a child needs to: • Practise times tables and know 2, 3, 4, 5 and 10s. • Know how to multiply any number by 10 • Be able to partition numbers into tens and units • Use this partitioning to multiply numbers to 19 initially • Be able to mentally multiply a number by a multiple of 10

9. Multiplication Step 5 • To do this, a child needs to: • Be able to multiply a single digit by a multiple of 100 • Be able to partition numbers into hundreds, tens and units • Be able to multiply a multiple of 10 by another • Be able to set out multiplication number sentences in a grid

10. Multiplication Step 6 • To do this, a child needs to: • Be able to divide any number by 10 and 100 • Be able to multiply single digits by tenths • Be able to multiply single digits by hundredths • Be able to set out number sentences in a grid

11. Multiplication Standard method

12. Division Step 1 • To do this, a child needs to: • Be able to count in 1s and 2s • Be able to share objects between 2 people • Be able to group objects into 2s • Read and understand division number sentences: 4 ÷ 2 = 4 grouped into 2s • Arrange objects to do what the number sentence is asking.

13. Division Step 2 • To do this, a child needs to: • Use equipment to group objects into 2s, 3s, 4s or 5s up to 20 objects. • Use equipment to share objects by 2, 3, 4 or 5 up to 20 objects. • Arrange objects into arrays • Be able to draw arrays to represent a division number sentence.

14. Division Step 3 • To do this, a child needs to: • Use a number line to repeatedly add 2 to divide by 2. • Same with 5s • Same with 10s • Experience what to do with remainders

15. Division Step 4 • To do this, a child needs to: • Use a number line to divide by 3 or 4 up to 10 groups • Identify numbers bigger than 10 times the number you are dividing by. • Use a number line to add a group of 10 first, then use repeated addition for left over groups • Use a number line to add more than one group of 10 if need be

16. Division Step 5 • To do this, a child needs to: • Find more that one group of 10 and then use repeated addition or table facts to divide by 6. • Same with 7s • Same with 8s • Same with 9s

17. Division Step 6 • To do this, a child needs to: • Use step 5 method to divide HTU by numbers up to 19 • HTU divided by any TU number • Divide decimal numbers by units

18. Division Standard Steps

19. Supporting your child at home • Practise times tables thoroughly – in order, out of order, division facts, ten times bigger, hundred times bigger, ten times smaller. • Show children that Maths IS linked to real life (half price sale, money etc) • If you want, you can use some of the games in the hand-out to help your child feel secure in their number and calculation work.