Welcome to multiplication and division Fawkham C.of E. Primary School Explaining how we teach mathematics, and what you can do to help your child become a confident mathematician. Welcome to our next half, which we will focus on multiplication and division. Welcome to multiplication and division Fawkham C.of E. Primary School

Multiplication and Division Calculation methods Using and applying: using the calculation methods in situations that have meaning. Vocabulary: vital and constantly referred to through all our teaching. We use a consistent, progressive approach KS1 – Focus on mental calculation strategies KS2 - Mental calculation strategies and a standard written method Again we focus on calculation methods, using and applying in other words using the calculations in real or pretend situations. Vocabulary is vital and constantly referred to through all our teaching.

Starting to understand multiplication: Begin to relate addition to combining 2 groups of objects, counting all the objects. Activities: Count out 3 cakes. Now count out 3 more cakes. How many cakes are there altogether? There are 4 cars in the garage 4 more arrive. How many cars are in the garage now. Even in pre-school and the reception class children should start grouping objects and making up stories like these. Some of the vocabulary can be introduced e.g. Lots of, double, groups or, share, halve ... Know how to use the relevant vocabulary:

Understanding multiplication Double and halve dart board Understanding multiplication Know by heart addition doubles of all numbers to at least 10 then to 20. Activities: There are 6 counters in the red cup and 6 counters in the blue cup. How many counters are there altogether? Can you record this to show me what you did? What is double 4? Which two numbers would make a total of 8, 12, 14? What is double 10? What is double 4? How could we work out double 14? Once in KS1 the children will start learning to record what they are doing when they add groups together or take groups away.

Understanding multiplication Understand the operation of multiplication as repeated addition or as describing an array. Understanding multiplication is taught using repeated addition and arrays. An array is an arrangement of rows and columns that shows groups put together. For example a row of 2, underneath another row of 2 etc, showing 10 put into 5 rows of 2, the children should then see that it is the same as two rows of 5.

How could I arrange them into equal rows? Arrays are a useful visual tool for multiplication and division I have 12 counters. This slide shows how arrays can be used for multiplication and division, which can be taught together as they are inverses. How could I arrange them into equal rows?

What number sentences could you write to go with this array? 6 + 6 = 12 2 + 2 + 2 + 2 + 2 + 2 = 12 2 x 6 = 12 6 x 2 = 12 We can also say that 12 ÷ 6 = 2 and 12 ÷ 2 = 6

Can you think of any other ways to arrange the 12 counters? And another way of arranging 12 counters.

What number sentences could you write to go with this array? 4 + 4 + 4 = 12 3 + 3 + 3 + 3 = 12 3 x 4 = 12 4 x 3 = 12 We can also say that 12 ÷ 4 = 3 and 12 ÷ 3 = 4

Multiplication using jumps along a number line (repeated addition). 2 x 5 means jumps of 2 made 5 times 0 2 4 6 8 10 +2 or 2 jumps of 5 (5 x 2) Number lines can be used to show multiplication and division. These show a jump of 2 made 5 times (2 x 5) And a jump of 5 made twice (5 x 2) - notice how confusing the vocabulary can be, instead of saying ‘a jump of 5 made two times’ our language says twice – children need to know the vocabulary in a mathematical context to be able to understand what is going on – and this is an example of why some children, particularly those with limited vocabulary find maths confusing. 0 5 10 +5

Multiplication facts are absolutely vital for progress with maths. The children need to know them thoroughly, so that they can use the knowledge not just in multiplication problems, but also in division, fractions, percentages, ratio etc. It is important that they learn division alongside multiplication. Multiplication can be seen as repeated addition 4 + 4 + 4 + 4 + 4 = 4 x 5 (make 20) Division can be seen as repeated subtraction. 20 – 4 – 4 – 4 – 4 – 4 = 0 (How many 4s have we taken away?) so 20 ÷ 4 = 5 0 x 6 = 0 1 x 6 = 6 2 x 6 = 12 3 x 6 = 18 4 x 6 = 24 5 x 6 = 30 6 x 6 = 36 7 x 6 = 42 8 x 6 = 48 9 x 6 = 54 10 x 6 = 60 0 x 2 = 0 1 x 2 = 2 2 x 2 = 4 3 x 2 = 6 4 x 2 = 8 5 x 2 = 10 6 x 2 = 12 7 x 2 = 14 8 x 2 = 16 9 x 2 = 18 10 x 2 = 20 Multiplication facts are absolutely vital for progress with maths. The children need to know them thoroughly, so that they can use the knowledge not just in multiplication problems, but also in division, fractions, percentages, ratio etc. It is important that they learn division alongside multiplication. Multiplication can be seen as repeated addition 4 + 4 + 4 + 4 + 4 = 4 x 5 (make 20) Division can be seen as repeated subtraction. 20 – 4 – 4 – 4 – 4 – 4 = 0 (How many 4s have we taken away?) so 20 ÷ 4 = 5

Times table awards Times table awards 0 x 2 = 0 1 x 2 = 2 2 x 2 = 4 3 x 2 = 6 4 x 2 = 8 5 x 2 = 10 6 x 2 = 12 7 x 2 = 14 8 x 2 = 16 9 x 2 = 18 10 x 2 = 20 0 x 2 = 0 1 x 2 = 2 2 x 2 = 4 3 x 2 = 6 4 x 2 = 8 5 x 2 = 10 6 x 2 = 12 7 x 2 = 14 8 x 2 = 16 9 x 2 = 18 10 x 2 = 20 Times table awards By the time the children come into year 3 they should already be confident with the 2, 5 and 10 times tables. To help motivate the children to learn their times tables in class 2 we have times table awards. For being able to recite the times table they earn a bronze sticker. For being able to answer random questions from the times table they earn a silver sticker. For being able to answer division questions, word problems, fraction questions using that times table they earn a certificate with a gold sticker that is presented in front of the school. 0 x 6 = 0 1 x 6 = 6 2 x 6 = 12 3 x 6 = 18 4 x 6 = 24 5 x 6 = 30 6 x 6 = 36 7 x 6 = 42 8 x 6 = 48 9 x 6 = 54 10 x 6 = 60 Times table awards By the time the children come into year 3 they should already be confident with the 2, 5 and 10 times tables. To help motivate the children to learn their times tables in class 2 we have times table awards. For being able to recite the times table they earn a bronze sticker. For being able to answer random questions from the times table they earn a silver sticker. For being able to answer division questions, word problems, fraction questions using that times table they earn a certificate with a gold sticker that is presented in front of the school.

Speed races By the time the children leave year 2 they should know their number bonds (as listed last week), doubles etc. In Class 2 we revise these, then move onto ensuring that they can recall mixed up times tables with the added pressure of time. They answer 50 questions in 3 minutes, if they get them all right they move onto the next list next time. If they get less than 48 correct they get the same list again next time. They are rewarded with merit marks when they improve their best score. Speed races are another tool that we use to regularly practise and motivate the children to learn their number bonds and times tables. By the time the children leave year 2 they should know their number bonds (as listed last week), doubles etc. In Class 2 we revise these, then move onto ensuring that they can recall mixed up times tables with the added pressure of time. They answer a set number of questions in 3 minutes, if they get them all right they move onto the next list next time. If not they get the same list again next time. They are rewarded with merit marks when they improve their best score.

Know by heart the multiplication tables then use these facts to derive new facts Two numbers multiplied together make 20. What could the two numbers be? A baker puts 5 buns in each of 10 rows. How many buns are there? How would you write 5 + 5 + 5 + 5 as a multiplication calculation? Using the numbers 2, 4, 15, 30 and 60. Work out six different multiplication calculations. Do you know 4 x 3? Can you work out 4 x 30? 40 x 30? Children need to know by heart their multiplication tables They then use these facts to derive new facts Two numbers multiplied together make 20. What could the two numbers be? A baker puts 5 buns in each of 10 rows. How many buns are there? How would you write 5 + 5 + 5 + 5 as a multiplication calculation? Using the numbers 2, 4, 15, 30 and 50. Work out six different multiplication calculations. How do you work out 4 x 30? 40 x 30? Hit the button Tables wheel

Multiplying and dividing by We learn how to move the digits. The bully 10 or 100 pushes the digits over. Multiply by 10 move the digits one place to the left Multiply by 100 move the digits two places to the left Divide by 10 move the digits one place to the right Divide by 100 move the digits two places to the right. 10 Multiplying and dividing by 10 and 100 We learn how to move the digits. The bully 10 or 100 pushes the digits over. Multiply by 10 move the digits one place to the left Multiply by 100 move the digits two places to the left Divide by 10 move the digits one place to the right Divide by 100 move the digits two places to the right. We use sliders that the children can physically slide the numbers to the left or right to multiply (make larger) or divide (make smaller). Moving the digits over demonstration

Written multiplication method This shows how we teach the column multiplication method. If we multiply a two digit number by a one digit number we need to understand that the two digit number must be partitioned and each part multiplied by the one digit number. The ‘denes blocks’ demonstrate this. Understanding how we can multiply a tens number by a one digit number is essential for full understanding of this method. E.g. 4 x 2 is 8 so 40 x 2 is 80. Multiply the 3 by the 2 to make 6, then the 40 by the 2 to get 80, and then add the numbers together. This shows how we can set this out in columns down the page (the children don’t need to write the bits written in brackets, but it is a reminder of what they are doing at each row to find these numbers).

This shows how we can then extend to multiplying 3 digit by one digit, resulting in 3 rows of answers that then need adding. The compact method can be used if the children are confident with putting the tens number under the next column and remembering to add it. We cross it out once we’ve remembered to add it. Then 2 digit by 2 digit number.s

Division as grouping or chunking (repeated subtraction) Division goes alongside multiplication Division as sharing To start with the children learn to share objects into equal groups Then write the number sentence; 6 shared between 2 is 3 each, 6 ÷ 2 = 3 We use visual images, record jottings and use number sentences for division. Division as grouping or chunking (repeated subtraction) Using a number line 20 ÷ 5 = 4 20 divided into groups of 5 equals 4 Division on a number line 5 10 15 20 Or this one Division goes alongside multiplication In KS 1 children learn division as sharing To start with the children learn to share objects into equal groups Then write the number sentence; 6 shared between 2 is 3 each, 6 ÷ 2 = 3 We use visual images, record jottings and use number sentences for division. Children will find division as grouping, the inverse of multiplication, (repeated subtraction) easier to understand for working with bigger numbers. Using a number line 20 ÷ 5 = 4 20 divided into groups of 5 equals 4 The family of sums 4 x 5 = 20 5 x 4 = 20 20 ÷ 4 = 5 20 ÷ 5 = 4 The array shows this family of calculations. 20 spots in groups of 5 (columns). 20 spots in groups of 4 (rows). The family of sums 4 x 5 = 20 5 x 4 = 20 20 ÷ 4 = 5 20 ÷ 5 = 4 Using arrays 20 spots in groups of 5 (columns). 20 spots in groups of 4 (rows).

Understanding how the division facts can be used for other numbers. E Understanding how the division facts can be used for other numbers. E.g. 8 ÷ 2 is 4, so 80 ÷ 2 is 40. Yet again – we will have to continually come back to the importance of vocabulary. Regular use and exposure to these words is essential. And how to do simple division in your head, with the aid of partitioning.

Division methods with remainders. If the children know their multiplication facts and therefore the inverses, most of the division calculations that they will need to work out are actually really easy. Once we start working with dividing larger numbers then we teach them the short (bus stop) method.

The children need to know when to round the remainder up or down, e.g. How many boxes do we need? How many boxes will we fill? We also teach them how to express the remainder as a fraction, then in year 6 they will learn how to work out the remainder as a decimal. The big issue for division with remainders is when we use and apply the division in problems that mean we have to either round the answer up or down to get a whole number. Children do seem to find this difficult so the more practise and experience the better. If the division results in the answer 26 remainder 4 e.g. How many boxes do we need? The answer would be 27 How many boxes will we fill? The answer would be 26

We also teach the long division method using the process of repeatedly dividing, multiplying and subtracting. The children also need lots experience of working with decimals and of course money. Money is of course similar to decimals, but the children must realise that when working with money we always have two decimal places – tenths and hundredths.

I ate and I ate till I was sick on the floor 8 times 8 is 64. Useful tricks: Knowing that the two times table is also the doubles number bonds that they should already know (all answers are even numbers). 4 times table is double the two times table (all answers are even numbers). 8 times table is double the four times table (if they can double two digit numbers quickly in their head this is a useful skill) (all answers are even numbers). 6 times table is double the 3 times table (all answers are even numbers). 2 x 6 = 12, 4 x 6 = 24, 6 x 6 = 36, 8 x 6 = 48 (this only works for the even multiples) Multiples of the 5 times table always end in 0 or 5. Little rhymes that help with some of the trickiest multiplication facts: 5, 6, 7, 8 56 is 7 x 8 I ate and I ate till I was sick on the floor 8 times 8 is 64. Useful tricks that we teach to help the children remember their times tables (and therefore their division facts): Knowing that the two times table is also the doubles number bonds that they should already know (all answers are even numbers). 4 times table is double the two times table (all answers are even numbers). 8 times table is double the four times table (if they can double two digit numbers quickly in their head this is a useful skill) (all answers are even numbers). 6 times table is double the 3 times table (all answers are even numbers). 2 x 6 = 12, 4 x 6 = 24, 6 x 6 = 36, 8 x 6 = 48 (this only works for the even multiples) Multiplies of the 10 times table always end in 0 Multiples of the 5 times table always end in 0 or 5. Little rhymes that help with some of the trickiest multiplication facts: 5, 6, 7, 8 56 is 7 x 8 I ate and I ate till I was sick on the floor 8 times 8 is 64

These are some of the activities that we carry out, trying to make learning tables fun and keep the children motivated. Number grids Function machines Matching pairs

8 x table +10 -2 +8 When the children can add 10 quite quickly in their head and can take away 2 they can work out their 8 times table. Add 10 and take away 2 to work out the answers to the 8 times tables. Learning the 8 times table can be made easier if the children can add 10 and take away 2 quickly in their head.

9 x table Digit sums 0 x 9 = 0 9 0 + 9 = 9 1 x 9 = 9 18 1 + 8 = 9 9 0 + 9 = 9 18 1 + 8 = 9 27 2 + 7 = 9 36 3 + 6 = 9 45 4 + 5 = 9 54 5 + 4 = 9 63 6 + 3 = 9 72 7 + 2 = 9 81 8 + 1 = 9 90 9 + 0 = 9 0 x 9 = 0 1 x 9 = 9 2 x 9 = 18 3 x 9 = 27 4 x 9 = 36 5 x 9 = 45 6 x 9 = 54 7 x 9 = 64 8 x 9 = 72 9 x 9 = 81 10 x 9 = 90 The nine times table has lots of tricks. The answers are an easy pattern – the units go down one, the tens go up ten. The digits in the answers always add up to make 9 (in the multiples up to ten) The finger method can be seen in this u-tube clip. Finger trick

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