# Multiplication Tables: testing for fluency and teaching for understanding

Written by Louise Pennington, Professional Development Leader for maths at Oxford University Press.

# The Multiplication Tables Check

The Standards Testing Agency (STA) published the Multiplication Tables Check Assessment Framework in November last year. This document contains some key information about the Year 4 testing that will be statutory from June 2020:

• Voluntary pilot 10-28th June 2019
• 3 week window in June 2020 to administer 1:1, group or whole class
• Focused on fluent recall of facts to 12×12 in a×b= format (no problem solving/empty box questions)
• Not equal spread of all multiplication tables tested – no 1×, more from the middle table facts of 6×, 7×, 8×, 9× and 12× tables with no repeats or commutative questions asked
• The test asks 25 questions and will take up to 5 minutes with a 6 second response time per question
• Children can practise before taking the test

However, even with this multiplication tables check coming into force for Year 4 children in England, primary school teachers are not just focusing on fluent recall, but also on ensuring that children ‘understand’ their learning. Understanding multiplicative reasoning is key for life and not just the Year 4 multiplication test!

Many teachers report that fluent recall of facts is a common issue in their school and if children are expected to ‘know’ multiplication facts to 12×12 by the end of Year 4 (June), then there are a lot of ways for teachers to teach and support practice in multiplication across all primary age classes, to build knowledge in a structured way focusing on the CPA approach. Here are some examples, using a variety of manipulatives, to illustrate multiplication in a visual and concrete way to promote understanding.

# Building physical number patterns

Rather than simply colouring squares on a multiplication grid to highlight patterns, use counters or Numicon pegs on a number track to illustrate the relationship between times tables (e.g. double 3s for the 6s and double again for the 9s; the link between 2, 4 and 8 times tables) and find common multiples and patterns within. Building these patterns physically helps children ‘see’ the patterns and how they do not need to learn each multiplication table as a bank of new facts. To see this illustrated for the 3, 6 and 9 multiplication tables using Numicon pegs and the card track, follow this link.

# The ‘phone key pattern’

Another way to explore patterns in multiplication tables is to use the ‘phone key pattern’. This method explores patterns in the abstract but illustrates them pictorially. An example for 3, 6 and 7 multiplication tables can be found here.

If you are working with children in the early stages of developing understanding of multiplication, then you will probably be focusing on repeated addition. Doing this in a concrete way is important as children can actually ‘see’ what happens when we multiply. If we want to think about examples of activities that utilise manipulatives, then something readily available that can easily make a fixed set is interlocking cubes. Use these cubes fixed together in equal groups of (for example) 5, to make towers all the same size. Then roll a dice and pick up that many lots of the towers. The children can count in 5s to find the product and record their number sentences using ‘lots of’ or ‘groups of’.

Numicon is already arranged as a fixed image for a number and this is particularly useful when looking at repeated addition. Children could select a 5 shape and then generate a ‘lots of/groups of’ multiplier by throwing dice or spinning a spinner. The child then selects the resulting number of 5 shapes and lays them on the 10s Number line to find the product. Follow the link here for a quick video illustrating how Numicon can be used in this way for repeated addition.

# Arrays and commutativity

Generally, children will learn about multiplication through arrays after getting to grips with repeated addition. Arrays enable a child to lay out and see commutativity. Models and images in this structure of multiplication are closely linked to area and the grid method for long multiplication. Arrays are important also for learning about the relationship between multiplication and division. Children can use arrays to break down more complicated multiplication calculations later using the distributive property when multiplying with brackets.

Arrays are hard for young children to draw and keep in line, so a good way to explore arrays is with counters on grids or squared paper, or by using containers such as egg boxes, ice cube trays and muffin tins, and a selection of items. These can be laid out and studied for the multiplication and division facts that are represented. For example:

This muffin tin can be used to explore the facts 3×4, 4×3, 12÷3 and 12÷4.

You can also explore arrays in concrete ways with bingo dabber pens, finger paints on fingertips and corks dipped in paint for example.

The Cuisenaire rods can be used in a similar way to illustrate arrays and also commutativity. This video clip shows the relationship between 2 collections of rods by asking what is the same and what is different.

After exploring this, children could then go on to record multiplication and division facts relating to these models. Then, investigate other pairs of rod calculation such as 6×7 and 7×6.

Commutativity can also be explored with Numicon and a pan balance. Due to the fact that Numicon is weighted, children can explore and draw their own conclusions from ‘balancing’ equations such as 2×7 = 7×2 by putting two 7 shapes in one side of a pan balance and seven 2 shapes in the other. The product can also be checked by laying the Numicon shapes on the 10s number line so the child can be sure that these calculations are commutative and therefore result in the same product.

Jo Boaler,  Professor of Mathematics Education at the Stanford Graduate School of Education has a lot of activities to support understanding of multiplication on her website Youcubed. Here she suggests some interesting games and investigations focusing on multiplicative reasoning. One such investigation asks children to work with pennies and children are asked to explore grouping to find out how many pennies there could be.

Again, Jo Boaler’s Youcubed website is a favourite of mine when exploring arrays in multiplication called ‘How close to 100’ instructions. A template can be found here.

# The scaling structure for multiplication

Finally, to explore the scaling structure for multiplication – which children actually meet early in England as they are exposed to doubling and halving – in the foundation stage. Children could find ways to model and represent the scaling in questions such as:
Lyle has a torch which shines 5 times further than mine. If my torch beam lights up 3 metres of the path, how far does Lyles’ shine?

Cuisenaire rods and a rod track would be useful manipulatives here:As would Numicon shapes and the 10s number line:

Children need opportunities to use their knowledge in different situations, and so word problems and puzzles are useful to assist children in consolidating and remembering what they have learnt.

For example, to draw the structures of multiplication together children can be asked to make posters or concept maps such as this:

Image from http://theteachingthief.blogspot.com/2012/09/representing-multiplication-multiple.html?m=1

I hope this post has given you lots of ideas for how the CPA approach and manipulatives can help your children, preparing them not just for the multiplication test, but for truly understanding multiplication as a key life skill.