ALG 05

Week 5: Here we find ways of counting the elements in a pattern by representing the structure of the pattern as an expression that can be fed into a calculator. In each case, we have chosen patterns that are multiplicative, ie which can be expressed most neatly as a series of multiplications.
[Note: In each of the tasks, we ask, "How could you use a calculator...". We do this as a way of getting pupils to make the operations that they would use explicit. We don't actually require them to have a calculator! We leave it to you to decide whether they should use one.]
-
Monday: This task makes use of the familiar dot patterns found on dice. However, pupils might not spot all the dice-patterns within dice-patterns formed by the dots in this task.

If we start by focussing on individual dots, we can see that the dots are grouped in 6s, and these are grouped in 4s, and these are grouped in 5s. So we have 6×4×5 dots altogether.
Or, starting from the 'outside', we can see a pattern of 5 elements, each of which is composed of 4 elements, each of which is composed of 6 dots, So we have 5×4×6 dots altogether.
At some stage, here or with a later task, you might want to tease out the fact that in a multiplicative expression like 5×4×6, each number represents something different. This is in contrast to an additive expression like 5+4+6, where the numbers all represent the same thing, be it dots or apples, or whatever.
-
Tuesday: This is fairly straightforward. The circles, each containing 4 dots, form an array.

There are 3 rows of 14 circles, each with 4 dots, making 3×14×4 dots altogether.
Or, there are 14 columns of 3 circles, each with 4 dots, making 14×3×4 dots altogether.
-
Wednesday: This kind of 'tree diagram' might be unfamiliar to pupils and thus quite challenging.

Starting from the bottom of the 'tree', we can get this expression for the total number of dots: 2×4×3.
Starting at a single dot, we can get this: 3×4×2.
Note that we only get such simple multiplicative expressions, because we have imposed a strict symmetry at each node. In real life things might be less neat, as well as more numerous!

-
Thursday: Here we have a 3-D array of 2p coins.

We can express the number of coins as 2×4×3, for example - an expression we used for the dots in the 'tree' in the previous task. There is a correspondence between the array of coins and the tree, although it may not be immediately apparent!
It is likely that some pupils will simply come up with 8×3 rather than 2×4×3, as the 8 coins in the top layer are so clearly visible. This is perfectly fine, but alert pupils to the more extensive expression, to make the point that we are primarily interesting in describing the structure of the arrangement, rather than in the total number of elements.
[Note: Our 3-D array is very close to a 'volume' model. Thus we could replace each coin by a unit cube, with the cubes forming a cuboid whose volume would then be given by length × breadth × height. This formula is fairly easy to remember and some pupils will know it for the volume of a cuboid. However, they may struggle to remember or realise that what the formula is essentially doing is counting unit cubes. It is likely that pupils will find it easier to make sense of the formula in the context of our array of coins.]
-
Friday: This is similar to Monday's pattern, except it is recursive - we can imagine it going on and on....

The multiplicative expression here is 4×4×4×4, or 4⁴, whether we work from the whole pattern inwards, or from a single dot outwards.
We could imagine producing the whole pattern 4 times, or cutting each dot into 4 smaller dots, to give
4×4×4×4×4, or 4⁵. And so on! "How many steps would it take to get over 1 million dots?"