Number Lines

Starting out

• Count up together.
• Count down together.
• After a few rounds, remove the number line from the screen to see if they can still count together.
• How big is each step?
• How many steps are you taking to get to the end?
• What's the same/different about the two number lines?

Tip: Set one of the factors to 12 so that you can count all the way through the tables.
Also, try counting in front of them and making a deliberate mistake to see if they can spot it.

Ready to move on

• Count beyond 12× (i.e. beyond the reach of the number line).
• Make it explicit that the two number lines "end" in the same number. That's because both multiplications give the same answer. (See 'Commutativity' below.)
• Draw a number line for...[give them a calculation to draw in their books or on a mini whiteboard.]

Division

• When you set the tools to "Division", the number line will allow you to count down by default.
• Starting at the last number on the line, how many times do we count down to get to 0?
  Help pupils to make the connection between the amount we're counting down and the divisor and between the number of times we count down (subtract repeatedly) and the quotient.

Deriving other facts

• Explore how 2 lots of something can be doubled to give 4 lots and doubled again to give 8 lots. Similarly, double 3 lots to 6 lots and triple to give 9 lots.
• Explain the "one more/less lot" principle. For example, when you know ten lots of a number you can add one more lot or take one lot away to derive 11 or 9 lots respectively.
  The same principle applies to any fact obviously, which you can show them, and you can also extend it by looking at adding/taking away two lots or more. If you really way, you could talk about the distributive property of multiplication.
• Explain how the fact on display (e.g. 5 × 6) is related to 50 × 6 or 5 × 60 or 0.5 × 6 or 0.05 × 6.

Making links to other representations

• Explain how counting along the number line is related to the other representations on the screen. For example, that counting up is like the repeated addition and counting down is like repeated subtraction in the Abstract widget. It can also be related to Groups, Number Frames and Bar Models quite well.

Multiples

• Explain how, as you count up, each of the numbers are multiples of the number you're counting up in.
• Similarly, the other number line is multiples of the other number.
• Do the two number lines ever meet on the same number? Do they meet more than once?
  Wherever they meet is called a "common multiple". Use examples and non-examples of common multiples. For example, 30 is a common multiple of 5 and 6 but 25 and 24 are not. 25 is a multiple of 5 and 24 is a multiple of 6 but 25 isn't a multiple of 5 AND 6 and neither is 24.
• What is the lowest common multiple of the two numbers you're looking at?

Commutativity

• What's the same and what's different about the two calculations? (Same numbers, same operation but the order in which the factors are written has swapped.) Even though the numbers have swapped, we get the same answer. Two calculations give the same answer. Have we seen that before? (They might mention addition.) This could be interesting and useful. Let's see.
• I wonder what happens if we change one of the numbers...do we still get two calculations with the same answer? Yes. Show that it works for any two factors. This is called the commutative property of multiplication or commutativity for short. Commutativity tells us we can change the order of a multiplication and still get the same answer (product).
• Commutativity can very useful, especially when we're learning the tables. For example, if you already know the answer to 2 × 9 then you also know the answer to 9 × 2. So if you're ever stuck on a times tables question, ask yourself if you already know the commutative fact.
• Let's see if you can work out the commutative facts for each of these table questions. 7 × 12 is 84, what is 12 × 7? (Or similar examples.)

Number Frames

Starting out

• Count up together along the top row.
• Count up together along the bottom row.
• What's the same/different with the top row of number frames and the bottom row?

Tip: Try counting in front of them and making a deliberate mistake to see if they can spot it.

Ready to move on

• How big is each number frame in the top row?
• How many groups are along the top row?
• How big is each number frame in the bottom row?
• How many groups are along the bottom row?
• Can you see a link between the groupings?
• Make it explicit that when you join up the two sets of number frames, they are both the same size. That's because both multiplications give the same answer. (See 'Commutativity' below.)
• Show a set of number frames for...[give them a calculation to draw in their books or on a mini whiteboard, or using numicon.]

Division

• When you set the tools to "Division", the number frames will be joined by default.
• Start with the number frames joined as a group, and count the total number in each shape.
  Make it explicit that they are both the same, and that this is the dividend.
  Click split, and ask students to count the number of groups on the top row. What size is each group?
  Do the same for the bottom row.
  Help pupils to see that the number of groups and group size are also the divisor and the quotient.

Deriving other facts

• What do students notice about odd / even number frames?
• Explain how the fact on display (e.g. 5 × 6) is related to 50 × 6 or 5 × 60 or 0.5 × 6 or 0.05 × 6.

Making links to other representations

• Explain how counting up/down using the number frames is related to the other representations on the screen. For example, that it matches what we do when using the bar model - the size of each block is the number in each frame. It's also the same with the grouping tool.

Multiples

• Explain how, as you count up, each of the numbers are multiples of the number you're counting up in.
• Similarly, the other number frames show multiples of the other number.

Commutativity

• What's the same and what's different about the two calculations? (Same numbers, same operation but the order in which the factors are written has swapped.) Even though the numbers have swapped, we get the same answer. Two calculations give the same answer. Have we seen that before? (They might mention addition.) This could be interesting and useful. Let's see.
• I wonder what happens if we change one of the numbers...do we still get two calculations with the same answer? Yes. Show that it works for any two factors. This is called the commutative property of multiplication or commutativity for short. Commutativity tells us we can change the order of a multiplication and still get the same answer (product).
• Commutativity can very useful, especially when we're learning the tables. For example, if you already know the answer to 2 × 9 then you also know the answer to 9 × 2. So if you're ever stuck on a times tables question, ask yourself if you already know the commutative fact.
• Let's see if you can work out the commutative facts for each of these table questions. 7 × 12 is 84, what is 12 × 7? (Or similar examples.)

Rods

Starting out

Use the sort button to group the rods together.

Use the scatter button to break the groups apart again.

Use the stack button to place one group of rods on top of the other. You can then switch which group is on top.



• Count how many squares are in each type of rod.
• Count how many of each type of rod there are.

Tip: Use the stack and switch buttons to show students that each group of rods is the same.

Ready to move on

• Link to the area of a rectangle. When the rods are grouped, you can see that the total number of squares (the area) is
  the number of rods × the number in each rod.
• Show me you understand by drawing two matching rectangles on squared paper, and showing the two different ways it could be split into rods. For example, a 3 by 5 rectangle can be split into 3 rows of 5 or 5 columns of 3.

Deriving other facts

• Explain how the fact on display (e.g. 5 × 6) is related to 50 × 6 or 5 × 60 or 0.5 × 6 or 0.05 × 6.

• Relate Rods to Number Frames, Arrays, Repeated Addition and Worded.

Arrays

Starting out

What is an array?

• An arrangement of objects, pictures, or numbers in columns and rows is called an array.
  Each row of an array must be equal.
  Each column of an array must be equal.
• Use the sort buttons to show that the dots can be arranged in either x rows of y, or y rows of x.
• An array shows the full family of facts. For example, in a 3 × 4 array, you can see that:

  • 3 lots of 4 makes 12 in total
  • 4 lots of 3 makes 12 in total
  • 12 can be arranged into rows of 3, with 4 rows.
  • 12 can be arranged into rows of 4, with 3 rows.
• Note: It's not an array if it isn't a full rectangle, (so if any of the rows / columns are different to the rest).

Ready to move on

• Here is an array. What multiplication fact does it represent? What are all the calculations it represents? What calculations can you derive from it?
  Tip: Press 'Hide' on the bottom toolbar to hide the calculation and then press 'Random'. Can they now tell you what calculations this mystery array represents?
• How many arrays can you make from exactly 2 dots? 3 dots? x dots?
• What's special about 'prime arrays'? Square numbers?
• Link to the area of a rectangle. When the dots are in an array, you can see that the total number of dots (the area) is
  the number of rows × the number of columns.
• If you have 12 dots (i.e. an area of 12) arranged in rows of 3, how many columns are there?
• Show you understand by drawing an array for [give an example multiplication, like 2 × 3] on paper or on a mini whiteboard. How would you change that array to show 3 × 3 or 4 × 3?

Tip: Squared paper really helps here!

Division

• When you set the tools to "Division", the dots are arranged into an array rather than scattered.
• Ask students to look at the array and the multiplication represented by it, and tell you what part of the array (total, rows, or columns) represents the divisor, which represents the quotient. and which represents the dividend.

Deriving other facts

• What is the same/different about the arrays for, say, 6×5 and 5×6? (See 'Commutativity' below.)
• One more/one less. Show them how the array for 10×7 is one more row/column than 9×7. That means if you know the answer to 10×7 you can work out 9×7 by taking away one group of 7 from the answer.
• Doubling the number of columns or rows, doubles the area. This can be related to the numbers. If you double one of the factors, you double the answer. For example, 10×4 is double 5×4. That means if you know 2 times a number you can double the answer to get 4 times that number.
  Or if you know 10 times a number you can halve it to get 5 times that number.
  Note: if you double the number of rows and columns, you'll quadrulple the area. Maybe show that to pupils who are ready. Can they explain why?
• How about if you had ten lots of the whole array? Explain how the array for the fact on display (e.g. 5 × 6) would change for 50 × 6 or 5 × 60.
  Note: if you told them that the array for 50 × 6 is 10 lots of the array for 5 × 6, that can be written out as 10 × 5 × 6 or 5 × 10 × 6 or 5 × 6 × 10, you're making use of the associative property of multiplication.

  Put their understanding to the test by asking them to sketch an array for 3 × 6 and then 30 × 6. If they know the answer to 3 × 6, how do they use it to work out the answer to 30 × 6?

Making links to other representations

• Discuss how the array you're looking at relates to...
  • groups - an array is simply a tidier way of grouping the same number of objects
  • bar model - the long bar gives the area of the array. The number of small bar segments and the number inside each one are the rows/columns of the array.
  • repeated addition and number line - skip counting along the number line is like skip counting along the columns or down the rows.
  • abstract - the factors in the calculation give you the number of rows and the number of columns

Distributive property (adding two arrays-of-equal-height together)

With two arrays of the same height side by side (eg 5×3 and 5×1) show how they can be added together to give you an array for 5×4.

What two arrays would you draw for 16×5 to help you do it in your head? Perhaps 10×5 and 6×5. Well, that may be a good time to introduce the distributive property of multiplication.

Multiples

• Show an array and the calculation. Label factors and multiples.
• Explain how, as you count along the tops of the columns, each of the totals are multiples of the number you're counting up in. For example, if each column has 5 dots, then as you count along the columns (5, 10, 15, 20...), these are the multiples of 5.
• Take a snapshot of several arrays that have the same height, e.g. 6×4 and 7×4. Discuss how 4 is a common factor of 24 and 28. How is that clear from the two arrays?
  If you display two fresh arrays, can they work out the common factors?

Commutativity

• What's the same and what's different about the two possible arrays? (Same area but the number of rows and columns swap.) Even though the rows and columns have swapped, we get the same answer. Two calculations give the same answer. Have we seen that before? (They might mention addition.) This could be interesting and useful. Let's see.
• I wonder what happens if we change one of the factors (use the tool at the bottom)...do we still get two arrays with the same area? Yes. Show that it works for any two factors. This is called the commutative property of multiplication or commutativity for short. Commutativity tells us we can change the order of a multiplication and still get the same answer (product).
• Commutativity can very useful, especially when we're learning the tables. For example, if you already know the answer to 2 × 9 then you also know the answer to 9 × 2. So if you're ever stuck on a times tables question, ask yourself if you already know the commutative fact.
• Let's see if you can work out the commutative facts for each of these table questions. 7 × 12 is 84, what is 12 × 7? (Or similar examples.)

Grouping and Sharing

Starting out

• In 'Multiplication' mode, you can choose to group the squares in two different ways.
  Eg, if you are doing 3 × 4, you can group in 3's or in 4's.
• In 'Division' mode, you can choose whether you are sharing or grouping.
• When sharing, you can share between either of the factors. You'll get that many boxes to share the squares between.
• When grouping, you'll get enough boxes to put all the squares into groups of whichever factor you chose. For example, in 2 × 9, you'll get 9 boxes for 9 groups of 2.
• Using the language of 'groups of' can really help students to understand the concepts, particularly when you ask for one more / one less group of something.

Ready to move on

• Ask students to predict how many squares will be in each group.
• Show me you understand by taking 20 cubes / counters etc and sharing them into groups of 5.
  How many groups do you have?
  What if you shared them into groups of 4?
  Repeat for different calculations.

Divison

• What is the link between sharing and grouping?
  Do they do the same thing? What's the same? What's different?
• Can students think of real life questions where you might share things?
  Eg. When you have a particular number of things, like biscuits in a pack, and you want to share them equally.
• Can students think of any real life questions where you might group things?
  Eg. When you need a particular number of people in each team.

Deriving other facts

• Look at simple examples where one factor is even. Can students see why 5 groups of 6 is the same as 10 groups of 3?
  Why 3 groups of 10 is the same as 6 groups of 5?
• Explore how 2 groups of something can be doubled to give 4 groups and doubled again to give 8 groups. Similarly, double 3 groups to 6 groups and triple to give 9 groups.
• Explain how the fact on display (e.g. 5 × 6) is related to 50 × 6 or 5 × 60 or 0.5 × 6.

Making links to other representations

• Explain that each group works like the number frames tool. The frames are just a pre-grouped representation of the same thing.
• Make the link between the different groupings and the bar model - how are they the same?
• Ask students how these link to the rods and the arrays - it is easier to 'see' in the arrays, but the rods are simply a group of squares put together in a line.
• How do the groups link to the abstract representation? Which number is represented by what?

Commutativity

• How do you know you'll be able to share /group exactly with either factor?
• If it groups into 4 groups of 7, it will also group into 7 groups of 4 (or whichever factors you choose). Show that it works for any two factors. This is called the commutative property of multiplication or commutativity for short. Commutativity tells us we can change the order of a multiplication and still get the same answer (product).
• Commutativity can very useful, especially when we're learning the tables. For example, if you already know the answer to 2 × 9 then you also know the answer to 9 × 2. So if you're ever stuck on a times tables question, ask yourself if you already know the commutative fact.
• Let's see if you can work out the commutative facts for each of these table questions. 7 × 12 is 84, what is 12 × 7? (Or similar examples.)

Bar Models

Starting out

• Bar models are a great way to represent lots of different types of problems in maths.
• The bars are equal in length, but split differently, to show the different ways a number can be made.
• You can show or hide a label on each bar by clicking on it.
• Explain that the total can be made up in two ways, shown by the two models. Each bar is split into equal groups.
• Try hiding the total and counting up along the bar to get to it.

Ready to move on

• How do the bars relate to the abstract question?
• Show me you understand. Can you draw a bar model for 4 × 3 = 12 in your book or on a mini whiteboard?

Division

• How many blocks of 5 will make a bar of 30?
• How many blocks of 6 will make a bar of 30?
• Repeat for other calculations. What do students notice?

Deriving other facts

• If we added one more group on the end, what would we have to do to the total so the bars still match?
• If we take one group away from the bottom row, how many less are there now in the bottom row than in the top bar?
• Explain how if you double the number of groups / blocks, it also doubles the total. You could then double it again to get four times the original total.
• Show how we can use the existing fact to find other facts, eg. 4 × 5, 4 × 50, 40 × 5 and so on.

Making links to other representations

• How is this the same as the groups model?
  How about the number frames? Can you see how it's the same thing?
  Now the arrays and number rods - how are they the same thing?
• How does this show us the answers to the worded model?
• What about the abstract model? Which part is which on the bar model?

Multiples

• Any total that can be split into 2's with nothing left over is a multiple of 2, and so on.
• A bar model is a great way to show that one number is a multiple of another.

Commutativity

• Why are the totals always the same? That's because multiplication is commutative. Commutativity tells us we can change the order of a multiplication and still get the same answer (product).

Times Tables Grid

Starting out

• Click in the top right corner of the tool to expand it to full screen.
• Hover over a blank square to highlight it. Ask students what the number should be, then ask them how they know.
• Count up along a row with students.
• Count down along a row with students.
• What patterns can they see in the row?
• What patterns can they see in the column?
• There are two squares that are bright green. What do they have in common? Why?

Ready to move on

• Start by revealing the first square in a blank row. What happens to the total every time you move right one square? What are you adding on?
• Start by revealing the top square in a blank column. What happens to the total every time you move down one square? What are you adding on?
• Do the same, but starting from the final number in a row / column. What are you subtracting each time?
• If there are orange squares highlighted on the grid, what goes in them? Why are they the same as the green squares?

Division

• When you go to division mode, the totals are now labelled, but the labels on the edge of the grid, showing the factors are now hidden.
• Any equal totals in the grid will be in bold - for example, 12 can be made by doing 1 × 12 or 2 × 6 or 4 × 3.
• Discuss with students that a number can have more than one way to make it, depending on how many factors it has.
• Make it explicit that any number can be divided exactly by any of its factors.

Deriving other facts

• Help students to see that you can find new facts by using ones you know. For example, highlight 12 × 5, and ask students how they would now find 13 × 5.
• Highlight 5 × 6, and show that 10 × 6 is double.
• Highlight 10 × 4, and then show that 5 × 4 is half (as is 10 × 2)

Multiples

• The list of numbers in each column or row is the list of multiples of that number.

Commutativity

• This is a good way to emphasise that the row and column match. This is because 3 × 4 = 4 × 3 and so on.
• Give students the answer to 7 × 12, and ask them to give you the answer to 12 × 7. Make it really clear that if you know one, you also know the other! Students can use this to help out with tables they haven't learned yet, by filling in the matching fact from tables they have already learned.

Worded

Starting out

• Link the words to the multiplication question.
• Reveal one missing number and ask students to tell you the other.
• Switch which number you reveal!
• Ask students to tell you both numbers, once they are used to the style of question.
• When using division mode, ask students why we have the same missing number sentence twice.
  Why can we fill it in two different ways?
  Tip: Expand the tool to full screen to see all 4 sentences.

Ready to move on

• Show me you understand by writing a worded question for 8 × 6 (repeat for other calculations) on paper or on a mini whiteboard.
• Challenge yourself to write all 4 worded division sentences for a calculation.

Division

• Give concrete examples (perhaps using the groups tool above) of sharing objects to help students link the words to the concept.
• Use the language of sharing and groups of, and discuss the difference between them.
  Sharing between (the number of groups) gives (the size of each group).
  In groups of (the size of the group) gives (the number of groups).

Making links to other representations

• You can use the groups tool to represent these questions with concrete objects.

Abstract

Starting out

• You can click to show each question as a repeated addition.
• Use the swap button to show that it doesn't matter which way you do the repeated addition.
• Ask: Why might it be useful to use one repeated addition instead of the other?
  What about in 2 × 11? Is it easier to work out 11 + 11, or add on 2, 11 times?
• Reveal one answer. Ask students to tell you the other answer.

Ready to move on

• Without giving any clues, ask students to tell you the answer. Next, ask how they know.
• Ask students to represent the question with objects or pictures on paper or a mini whiteboard. You could choose a particular representation, or let them choose for themselves.
• Ask what other facts can we work out if we know this one? Eg. one more lot, one less lot, double, half.

Division

• Division mode allows you to represent the question as a repeated subtraction.
• Start by revealing all the missing numbers, including the repeated subtractions. Ask students to explain how the numbers in the repeated subtraction link to the abstract question.
• Show me you understand by writing 35 divided by 7 as a repeated subtraction on paper on on a mini whiteboard.

Making links to other representations

• Take some time to look at each of the tools in turn. How do they show the abstract question?