 # Number & Computation B Years 2 - 6(8) A Mathematics Centre Resource You are encouraged to contribute to these notes. Comments from teachers or students, photos and student work samples will enrich the professional experience for all of us.

Introduction

### Print Resources ### Materials

• Students supply their own timing device

The images in this menu are based on Poly Plug, but unlike Number & Computation A it is not expected that this equipment is present. Instead, this menu of Picture Puzzles is intended as a bridge towards meaningful symbolic representation of Times Tables and automatic recall of them.  Picture Puzzles have multiple levels of success. They do not have to be 'finished'. They can be revisited and continued. ### Introduction

Of course we want students to learn their times tables and be automatic in recalling them. A well developed skill tool box is part of the story of working like a mathematician. But this doesn't have to be achieved by rote learning; in fact it is low level learning to be able to repeat answers to the times tables without an associated brain picture (visualisation) to represent them and to support exploring their interconnections. Maths Over The Fence (below) is one story which supports this point of view.

All the times tables that students need to know exist within this array. Use drinking straws, or pencils, to 'fence them in'.

Knowing how to represent each times table as an array and exploring the links between the arrays, provides the flexibility that empowers the requirement to 'know your tables'.

Number & Computation B is based on Times Tables Torture from Calculating Changes (which requires separate membership). The 100 array model to represent all times tables is explored further in Exploring Times Tables from the same site.

Maths Over The Fence is published in both Calculating Changes Members' Stories, where it is published under its original title of Jesse & the Multiplication Garden, and Maths300 Lesson 97, Tackling Times Tables (which also requires separate membership).

### The Puzzles

There are six Picture Puzzles in this menu.   They are all structured in the same way.   Students challenge themselves to complete 20 times tables tortures against the clock. They have total control over the number of seconds each slide is displayed.
As one player times and changes slides the other only has to write the answer in a box on the Time Tables Torture Recording Sheet,
or mark the box with a slash ( / ) if they don't know the answer in the time.   • Player A is challenged with 20 slides and then the stop sign is reached.
• Here players swap roles and Player B faces a different 20 slides until the next stop sign is reached.
• All answers are checked by returning to Player A's first slide, then working through together to find at least two ways to know the answer.
• Alternatives are discussed and recorded. Each player only records for the slides they faced, but the emphasis is only helping each other know. All times tables arrays can be known in at least two ways. Even this one. I might know it because I see 2 rows of 1 or I might know it because I know that it gives the same result as 1 row of 2. Although 2 rows of 1 and 1 row of 2 give the same answer (2) they are not the same thing. This is an application of the Commutative Law of Multiplication which results in almost halving the number of tables facts to be learned. With this menu of Picture Puzzles students have enough time to discover this short cut for themselves. The array model of multiplication has commutativity built in and students can move the viewing position in their mind (or rotate their screen) to transform any array from a x b to b x a. The total doesn't change but the image does. This is the power of the array model and why the language of 'rows of...' is critical - row is a self-referenced word. The commutative possibility simply isn't present in the 'frogs on lily pads' model of 'groups of'. Counting in groups limits children to sequence counting on their fingers to work out their times tables. A necessary step in learning perhaps, but it is not multiplicative thinking.

The link between 'rows of...' and 'times' comes as simply as this:

Tell me what rows you see.
• Two rows of 1
So two rows of one equals...?

Now, how many times do you see 1?

• Two.
So, two times one equals...? In a picture like this one we certainly see 4 times 6 (that is, we see 6 four times), but placing pencils or straws on the screen, will also help us see: 2 x (2 x 6) ...or... 3 x (4 x 2) ...or... 6 x (2 x 2) which are applications of the Associative Law of Multiplication - the straws help us associate (collect together) numbers that help us complete the calculation. We also see: (2 x 5) + (2 x 1) which is an application of the Distributive Law. So now there are several ways of knowing because we have found mathematical principles underpinning what may have originally seemed like 100 isolated facts.

Exploring these ways of knowing with a peer and recording them on the sheet, develops student confidence. They often want to return to the same menu and shorten their chosen time to view each slide. Eventually, and for some soon, they recognise that the only clue they need is the grey times tables 'question' beside the image. Focussing on that as the slides change often results in a one per second completion rate. Automatic response has been developed, but is forever linked to a brain picture that can be called on if necessary. Why pair 5 & 7? For most students, the easiest tables to learn are 2s, 5s, and 10s. Among the most difficult are the 7s. However, any times table represented by rows of 7 can equally be represented by rows of 5 plus rows of 2. For example, in this image: 8 rows of 7 = 8 x 7 = 8 rows of 5 + 8 rows of 2 = (8 x 5) + (8 x 2) = 40 + 16 = 56 A very useful application of the Distributive Law which encourages a little mental arithmetic based on the students' confidence with the easier tables. Further, if the times tables question comes to you as 7 x 8, applying the Commutative Law tells you 8 x 7 gives the same answer; so calculate 7 x 8 by converting it to 8 x 7. (Yes, you could partition 7 x 8 into rows of 5 and rows of 3, but then you would have to be confident with your 7s to start with.) Why pair 9 & 10? Again it's to support students to use the Distributive Law, even if informally because they 'just see it', but this time with subtraction. 6 rows of 9 = 6 x 9 = (6 x 10) - (6 x 1) - now you see why the blue plugs are outlined = 60 - 6 = 54 Perhaps they will even begin to verbalise something like: To multiply by 9, multiply by 10 then take off the number you started with. What happens if the question is 9 times something? What happens if the number you started with is bigger than 10?

• Do I teach the students the words Commutative, Associative & Distributive?
Up to you.
Wow Denise, that thing you just did where you answered 2 x 8 by changing it to 8 x 2 was great!
How did you know to do that? ...

Would you like to learn the mathematician's name for what you did?

### More The more section of each Times Tables Torture comes after both students have tackled their 20 questions and all 40 questions have been discussed and recorded. The structure in each slide show is the same as shown here. This mini-challenge is about putting times tables to work to find factors. The Commutative Law means that there will always be at least two ways and the numbers have been chosen so there are others. In this case: 16 = 2 x 2 x 2 x 2 ... 16 = 4 x 4 ... 16 = 8 x 2 ... and variations 56 = 8 x 7 ... 56 = 2 x 4 x 7 ... 56 = 2 x 2 x 2 x 7 ... and variations 90 = 9 x 2 x 5 ... 90 = 3 x 3 x 2 x 5 ... 90 = 3 x 6 x 5 ... and variations

### Extensions

1. Backwards Questions
• I have 8 equal rows of yellow plugs, but I have covered them all up. I will tell you there are 72 plugs in total. How many are there in each row? Draw a picture to explain your answer.
• I have some made some equal rows of yellow plugs and covered them all up. There are 6 plugs in each row and 42 plugs in total. How many rows have I made? Draw a picture to explain your answer.

2. Building Numbers
Everyone knows that you can start with 1 and add 1 and make a new number called 2. Then add 1 to make a new number called 3 and continue this process forever building an infinity of numbers that are each represented by a series of adding 1.

But what happens if we try to build all the Natural Numbers from 1 using multiplication? Well, you don't get very far because:

 1 x 1 = • 1 ... which gets nowhere, so let's try starting with 2 as a building block. 1 x 2 = • • 2 ... Yay! we're off 2 x 2 = • • • • 4 ... Oops! We missed 3, so we have to include 3 as a building block. Start again.

 1 x 1 = • 1 1 x 2 = • • 2 1 x 3 = • • • 3 2 x 2 = • • • • 4 2 x 3 = • • • • • • 6 2 x 4 = 2 x 2 x 2 = • • • • • • • • 8 3 x 3 = • • • • • • • • • 9 ... but we missed 5 and 7 so we have to put those in as building blocks. Start again.

 1 x 1 = • 1 1 x 2 = • • 2 1 x 3 = • • • 3 2 x 2 = • • • • 4 1 x 5 = • • • • • 5 2 x 3 = • • • • • • 6 1 x 7 = • • • • • • • 7 2 x 4 = 2 x 2 x 2 = • • • • • • • • 8 3 x 3 = • • • • • • • • • 9 2 x 5 = • • • • • • • • • • 10
Continue the investigation.

• Can you predict which numbers will be the building block numbers for multiplication?
• The smallest building block number in multiplication is 2. What is the largest building block number you can find?
• Would you like to learn the name mathematicians give to these building block numbers for multiplication?
• What is the largest building block number for multiplication that mathematicians have found so far?
• How do mathematicians use these special numbers?

3. Consider introducing Eratosthenes Sieve and Prime Factorisation in this way... Mmm, perhaps we already have. 