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# Teaching Multiplication KS2: A Guide For Primary School Teachers From Year 3 To Year 6

Multiplication in KS2 maths focuses on having pupils build on their understanding of multiplication from Key Stage 1; using the times tables they have already learnt to explore more complex multiplication facts.

This post will show you what that progression looks like from Year 3 to Year 6, what your students/child should be able to do before moving on, and finally offer some practical suggestions as to how some objectives could be taught in the classroom.

#### What is multiplication?

Multiplication is one of the ‘Four Operations’ – the four core maths concepts children need to tackle the rest of the subject, along with addition and subtraction, and division. If addition is the act of combining two numbers, multiplication is combining multiple groups of a number.

For example, the five times table describes the results of combining numerous groups of the number five e.g. 5×1 = 5, 5×2 = 10 (5+5). It is sometimes referred to as being ‘repeated addition’ for this reason.

### Multiplication KS1

By the end of KS1, pupils should already be familiar with the 2, 5 and 10 time tables, know the abstract symbol for multiplication and demonstrate the commutative property of multiplication (5×4 = 4×5).

### Teaching multiplication KS2 – Before you begin…

A consistent theme that I hope has come through this series is the need to check the prerequisite knowledge of the younger year groups. Since writing some of these posts, the National Centre for Excellence in the Teaching of Mathematics (NCETM) and the government have released non-statutory guidance on getting the most out of the National Curriculum, and understanding the prerequisites is a continuous theme within that guidance.

It is crucial that each teacher knows what the objectives are for the previous year group and checks that pupils have mastered those objectives before continuing with the objectives of the year group.

Looking to help pupils catch up on previous years’ work? Take a look at our collections of place value worksheets and addition and subtraction worksheets!

Students in Year 4 will also have to take the multiplication check from the 2020/2021 academic year, so Autumn 2020 is an excellent time for maths leads to review the teaching of multiplication across the whole school. Students achieving automaticity in their multiplication tables will undoubtedly understand many more mathematical ideas and concepts as they progress, not just through the rest of primary school but throughout their secondary mathematical education too.

### Teaching multiplication KS2: The theory

Multiplication is a key area of mathematics and is vital to pupils’ educational success within the subject, as the conceptual understanding of multiplication runs through many other areas of the subject – much like place value, addition etc.

There are three core structures of multiplication that students should be made aware of: repeated addition, scaling and Cartesian product. Students knowing these structures along with multiplication facts up to 12×12 will greatly aid not only in their understanding of multiplication but other areas of mathematics also. While there is no time to go into depth of the structures here, I will provide a brief outline.

Repeated addition is a structure that many teachers will be aware of. It is simply the understanding that some multiplication is the result of continuously adding the same integer continuously. To save time when communicating our mathematics, we use the multiplication symbol. Hence 2 + 2 + 2 = 2 x 3. It is commonly visualised as an array. This structure is evoked when we ask students to solve questions such as There are 3 pencils in a pack and I buy 5 packs. How many pencils are there altogether?

We can see above how teachers may bring this structure to life by getting students to see that within each pack are 3 pencils and after purchasing 5 packs I will have to count 3 five times or 3 x 5.

Scaling increases an amount by a given scale factor. This is where one amount  has increased in a multiplicative relation to another amount. For example, Jenny has 2 pens but Amy has 3 times as many pens. How many pens does Jenny have?

So as we can see here, the amount of Jenny’s pens have been scaled by a factor of 3. It is not the case that Amy has added two more groups of two one-by-one to Jenny’s original amount as it would be the case with multiplication as repeated addition. But this scaling of having three times the original amount happens at once.

The next step to this would be scaling both sets again. So if Jenny had 4 pens and the scale factor remained the same, how many would Amy have? Understanding this type of multiplication is important when it comes to converting measurements and currencies in UKS2 but students rarely recognise this structure.

The final structure, one that is rarely used in primary school (though hinted at in the non-statutory guidance) is Cartesian product. This is where we find the number of possible combination from two or more sets. While rare, it is still something worth looking into. This relates to questions such as ‘On a menu there are fries and sweet potato fries and both come in either a medium or large size. How many different ways are there to serve both types of fries?

As we can see from the table above, we have taken sets available to us in the question and found out that there are 4 possible ways that both types of fries could be served at this particular establishment. Writing this mathematically would simply be 2 x 2 = 4. Multiplication Lessons Resource Pack

Plug gaps and help conquer common upper KS2 misconceptions in multiplication Download Free Now!

### Multiplication Year 3

In the National Curriculum for maths in England, for each area of maths outlined, there is both a statutory requirement and a non-statutory requirement. The statutory requirements in KS2 maths are as follows:

• Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables
• Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods
• Solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m object

Non statutory notes and guidance:

• Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables
• Pupils develop efficient mental methods, for example, using commutativity and associativity (for example, 4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240) and multiplication and division facts (for example, using 3 × 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷ 3) to derive related facts (30 × 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3)
• Pupils develop reliable written methods for multiplication and division, starting with calculations of two-digit numbers by one-digit numbers and progressing to the formal written methods of short multiplication and division
• Pupils solve simple problems in contexts, deciding which of the 4 operations to use and why. These include measuring and scaling contexts, (for example 4 times as high, 8 times as long etc) and correspondence problems in which m objects are connected to n objects (for example, 3 hats and 4 coats, how many different outfits?; 12 sweets shared equally between 4 children; 4 cakes shared equally between 8 children).

#### Multiplication activities Year 3

This won’t cover a single lesson but will give an idea of the progression of teaching a new time table for the first time.

At the early stages of Year 3, students are still unlikely to have the necessary mental models in their long-term memory for many of their multiplication facts. Having these mental models deeply embedded and being able to retrieve quickly will greatly help all students in not on becoming more fluent in their mathematics but improve the far transfer of multiplication to solving problems.

This section will explore how we can provide those models to students using the Concrete Pictorial Abstract approach. While this will focus on learning the 4 timse table, this should be done after you have revisited the 2, 5 and 10 times tables.

Students should already be familiar with the idea that the act of multiplication is the equivalent of repeated addition from learning the 2, 5 and 10 times tables from previous year groups so teaching new time tables should continue with that structure of multiplication.

When beginning teaching a new time table students may need to keep a 1:1 correspondence with whatever concrete resources you use. For example, as can be seen below, it may be necessary for students to create groups of 4 from four separate items. When teaching new times tables, remember to keep a 1:1 correspondence between concrete resources and pictorial representations.

Depending on the students they may need to count each counter individually first so that they know the amount that has been counted. If this is the case, teachers should reinforce that this does not need to be done once the students are familiar with this idea and where possible reinforce the counting as part of the time table after each individual set of 4 has been counted. For example:

Child: (Counting the individual counters) one, two, three, four.

Teacher: Good. That is one group of four or one times four. It is equal to four. How many counters where there in one group of four?

Child: Four

Teacher: One times four is equal to four. Repeat.

Child: One times four is equal to four.

Teacher: If there were four in the last group, which number will we start to count from for the next group?

Child: Five. Five, six, seven, eight.

Teacher: Good. That is two group of four or two times four. It is equal to eight. One group of four was four and two groups of four was eight.

*This can then repeat for as long as needed*

What is important though is that we quickly move students away from this 1:1 correspondence where they can just see the 4 as one set. As soon as students are familiar with counting in 4s, I would then start incorporating chants and songs of the four-multiplication table as the counting will provide some conceptual understanding, though not enough.

We then need to apply this chanting to counting physical objects as one group of four. Using unifix cubes already in blocks of four is one way to do this. A good tip is when counting up in jumps of 4 is to start the count over at the beginning every time a new block is placed.

Here, more is more. We want to ensure that students see as many representations of the multiplication table as possible. Here are some manipulatives that can be used to show the 4 time tables in a concrete ways. Notice the progression from having manipulatives that can show 1:1 correspondence to having the abstract digit 4 to knowing that the manipulative represents the number 4.

Die:

Dienes Blocks:

Dominoes:

Directed counters:

Bar model:

Cuisenaire rods:

Counting stick:

All the above relate to the concrete aspect of learning mathematics. When looking at pictorial representations, students will draw on their understanding of the concrete, repeated addition and the commutative law to complete tasks like the one below:

4 + ___ + ____ =____               3 x ___ =____

Students would be expected to create the two additional groups of four, recognise the repeated addition calculation along with the multiplication calculation.

4 + 4 + 4 = 12              3 x 4 = 12

Again, this needs to be repeated using lots of different pictures to represent the required product.

Once students are familiar with this, I would then use arrays to reinforce the commutative properties of multiplication

4 + 4 = 8     2 x 4 = 8

2 + 2 + 2 + 2 = 8      4 x 2 = 8

Once students are familiar and confident with the above, I would then move to an abstract only approach with the students. This would take the form of jumping on a number line as shown below before moving on to just the calculations without any scaffolds.

It is worth noting that throughout the process of teaching a new multiplication table that contains conceptual understanding and not merely rote learning, formative assessment must be used to decide when a student is ready to move on to the next step within the Concrete-Pictorial-Abstract approach.

#### Multiplication word problems Year 3

A typical example of a word problem that students may be expected to solve by the of this period of teaching would look like this:

A book costs £4. How much do 6 books cost?

Depending on the students, they may decide to draw supports to help them with this problem or (ideally) they recognise the structure as one of repeated addition and apply their knowledge of the four times table to solve it.

4 x 6 = 24

6 books at £4 each would cost £24.

#### Multiplication: reasoning and problem solving Year 3

There is, of course, more to the learning of maths than just learning these objectives and reasoning and problem solving should not just be limited to word problems.

The blue ribbon is 4cm. The yellow ribbon is 5 times its size. What is the difference between the blue and yellow ribbon?

Here students are seeing another structure of multiplication – scaling. They need to be able to see that the yellow ribbon is 5 times the size meaning that you could fit 5 of the blue ribbon across the yellow. This allows them to see that one of the calculations needed to be successful is 4 x 5 and that is equal to 20.

Next, students need to see that they have to find the difference between the length of the yellow and blue ribbon. As there are two ribbons, this relies on the comparative structure of subtraction where the students need to compare the two lengths to find the difference between them. This would need to be solved through subtracting  4 from the 20 to get a difference of 16cm.

### Multiplication Year 4

• 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

Non statutory notes and guidance:

• 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.

#### Multiplication activities Year 4

For this lesson idea I will focus on the objective ‘recognise and use factor pairs and commutativity in mental calculations.’ It is worth remembering that throughout Y4, all children need to learn their multiplication fact up to 12×12.  This objective can be taught alongside the learning of those time tables facts though I believe works best once students have some degree of familiarity with all the multiplication facts.

Students need to know and use the mathematical terms for the numbers within multiplication calculations. I.e. in 4 x 5 = 20, the numbers multiplied together, in this case the 4 and the 5, are called factors and that together they are called a factor pair. The answer, in this case 20, is known as a product.

Fluency in this understanding at Year 4 will help students when they get to Year 5 and come across other terms such as prime, prime factors, highest common factor and lowest common factor. There is no reason why these terms could not be introduced during Year 3.

For the lesson, students should be challenged to make all the ways that they can to come up with a specific product. They should have resources like counters so that they can make arrays should they wish.

It should be demonstrated to the pupils how these arrays all have the same product (12) and how each array can be grouped to show the factors. I have always found it helpful to model to students working systematically for such tasks so that I know that I do not accidentally miss a factor pair out.

For example, while I may know that 3 and 4 are factors of 12, I will always start with a factor of one and find its corresponding pair and move up the number system. I would also explicitly teach students how to know to stop looking for pairs by looking at the number between the factor pairs.

For example, with the factor pairs of 2 and 6 the numbers 3, 4 and 5 are between them so checking those numbers for factors must happen. With the factor pair of 3 and 4, there are no whole numbers between them so there is no need to carry on trying to find anymore factors.

While a common approach to factors is the use of ‘factor bugs’, as ‘the bug’ does not really help to illuminate any of the mathematics I would caution against its use. Rather the use of Carroll or Venn diagrams should be used as these will have transferability to other areas of maths and other subjects. Note that at this stage it is not expected that students need to find common factors of 2 or more numbers.

#### Multiplication word problems Year 4

Due to the nature of the content, it would not be appropriate to have a ‘word problem’ more so than one merely stating that students should find all the factors for a given number.

#### Multiplication: reasoning and problem-solving Year 4

A good reasoning problem that students could explore is whether the larger the number the greater number of factors it has. This can lead students to finding patterns and setting the scene for prime numbers in Year 5.

In order to prove this, students would have to pick examples of numbers where a larger number has fewer factors than a smaller number. So they could choose 20 and 23 and recognise that 23 has only two factors where as 20 has six altogether. Conversations about the number of factors around odd and even numbers can also take place.

### Multiplication Year 5

• Identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers
• 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
• Multiply and divide numbers mentally, drawing upon known facts
• Multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000
• Recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)
• Solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes
• 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

Non statutory notes and guidance:

• Pupils practise and extend their use of the formal written methods of short multiplication and short division (see Mathematics appendix 1). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.
• They use and understand the terms factor, multiple and prime, square and cube numbers.
• Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (for example, 98 ÷ 4 =   = 24 r 2 = 24   = 24.5 ≈ 25).
• Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1,000 in converting between units such as kilometres and metres.
• Distributivity can be expressed as a(b + c) = ab + ac.
• They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9² x 10).
• Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example 13 + 24 = 12 + 25; 33 = 5 x ?).

#### Multiplication activities Year 5

My blog on the long multiplication method looks in depth at how to teach the following objective: 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.’

By now all students should be familiar with the multiplication facts up to 12×12. As can be seen by the sheer amount of objectives from the National Curriculum, now is the time that students really begin to apply this knowledge in novel ways or build on it e.g. identifying square and cubed numbers.

It is of great importance that on entering Y5 students who are not yet at the point of automaticity with their times tables are given additional support where needed.

#### Multiplication word problems Year 5

At this point, students should be experiencing multiple step word problems within a familiar context. A suitable problem for Year 5 at this stage would be the following:

Pencils come in boxes of 52. A school bought 540 boxes.
Pens come in packs of 30
A school bought 270 packs.
How many more pencils were ordered than pens?

To solve this, pupils need to appreciate that there are two underlying structures in the problem. The multiplication element and the final subtraction element.

Pupils should be confident at multiplying 52 and 540 together using a formal method and getting 28,080. Next they will need to multiply 30 by 270. While some may be able to perform this mentally, others will need to rely on the formal method. Multiplying those factors together will give the product 8,100.

The final step will be to find the difference between the pencils and pens. By this stage, it would be expected that students do not need to use a bar model to detect the underlying structure of the problem and should be able to use a method of their choosing to reach the answer 19,800.

#### Multiplication: reasoning and problem-solving Year 5

A common approach to check if students have mastered content is to get them to check for errors in work. By explaining what the error is, students not only demonstrate their understanding of the content but strengthen their own understanding of the content.

Teachers can use their experience to predict common errors within different content and create questions that use this misconception to prevent students from internalising the misconception themselves.

For example, in the above question, the student has not ‘dropped a zero’ when multiplying the 20 by the 163. This is a common misconception that often arises when students first learn this formal method.

### Multiplication Year 6

• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
• 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 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

Note: In the Year 6 curriculum, all four operations come under ‘number’. The ones above are only relevant to the multiplication aspect.

Non statutory notes and 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).
• 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.

#### Multiplication activities Year 6:

This lesson will focus on the objective: identify common factors, common multiples and prime numbers but will look only at identifying common factors.

Pupils should already be familiar with the vocabulary of factors and finding factors of products from Year 4. In Year 6, students take that learning one step further and look at to find common factors of 2 or more numbers.

By this stage, it would be expected that students have a strong conceptual grasp of all the multiplication facts up to 12×12 and so the use of arrays, as were used in Year 4, should not be promoted. Of course, your judgement is best used and if there are pupils who do need to use concrete resources still, they should not be denied them.

Venn diagrams are a good tool to use to help students to organise their thinking and if they have used them when looking at factors of one number, they should already be familiar with them.

Students should be encouraged to work systematically first to find all the factors of each product, in this case 24 and 36. They should do this in a systematic way starting with 1 and making their way up the numbers.

1 – 24

2 – 12

3 – 8

4 – 6

1 – 36

2 – 18

3 – 12

4 – 9

6 –

Next students can arrange all the factors into the appropriate part of a Venn diagram.

It is worth telling students that numbers that are not factors of either 36 and 24 can be written outside of the two circles.

Much of the multiplication objectives at Y6 are a continuation of those done in Year 5 with a few limited exceptions. It is therefore worthwhile for the Year 6 teacher to look back at the objectives in Year 5 and plan their work to ensure those skills are revisited and assessed during this time.

#### Multiplication: reasoning and problem-solving Year 6

Lloyd has 42 rugby cards. He shares them equally amongst his friends. How many friends could he possibly have?

This question is a variation on what pupils tend to come across as one of the unknowns is the number of friends that he has. Here students need to see that they are being invited to establish a range of answers and not simply finding one definitive answer as the question does not allow that to happen. Using their knowledge of factors, students could ascertain that Lloyd could have 1, 2, 3, 6, 7, 14, 21, 42 friends.

Multiplication is one of the first more complex topics pupils will encounter in maths, and it is one of the most important for their progression. Hopefully this post has given you some good ideas to help your pupils approach it with confidence, no matter which KS2 year they are in!

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x #### Multiplication Lessons Resource Pack

Plug gaps and help conquer common upper KS2 misconceptions in multiplication 