Doplication

Age
7 to 11
Challenge level
filled star empty star empty star

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Thinking mathematically

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Explaining, convincing and proving

Mathematical mindsets

Being curious Being resourceful Being resilient Being collaborative

Problem

While I'm sitting in front of the computer I also have a dice and it is showing a $5$. Rather like this:-

 

4 black dots arranged in a square with one black dot in the centre.

 

This is quite a popular arrangement for this number.

 

Going around I see that some people wear ties or dresses that are very colourful and full of lovely patterns. The patterns may be made up of teddy bears, rockets, flowers and all sorts of other things. However their arrangement - if I represent each item just as a black dot then the pattern is usually :-

 

A 3-by-5 grid of black dots. In the middle of each square of 4 black dots is another black dot.

 

The arrangements usually sit quite well into a rectangular arrangement. So we might call this last one a $3$ by $5$ or a $5$ by $3$ arrangement. Squares are just special rectangles and so we could have a $4$ by $4$ :-

 

A 4-by-4 grid of black dots. In the middle of each square of 4 black dots is another black dot.

 

When we have the $3$ by $5$ we can count the dots and see that we have $23$ altogether. Now $3$ times $5$ IS NOT $23$. So we cannot write it as:- $3×5= 23$

So we (you) will have to invent another name instead of times and a sign to go with it. Perhaps you may choose "?" BUT you can choose anything that you like!

Now I can write:-

$3 ? 5 = 23$

and when I look at the $2$ by $2$ and the $4$ by $4$ I can write them as:-

$2 ? 2 = 5$ and $4 ? 4 = 25$

If you now look at a bigger arrangement that follows the same kind of pattern you'll find various ways of counting up the number of dots needed.

So let's look at this one $8 ? 4$

An 8-by-4 grid of black dots. In the middle of each square of 4 black dots is another black dot.

 

Count them up, calculate, or do whatever so that you find out how many there are altogether and we have:- $8 ? 4 = 53$. Invent some more questions of your own and work out the answers.

 

Now if this were an ordinary thing like multiplication, which you'd use in an ordinary rectangle like:-

 

A 3-by-5 grid of black dots.

 

You probably would know the answer straight away or you'd be able to work it out from things that you knew before. When people learn multiplication tables they often write them out in a big table. So you could explore what happens if you try a similar idea with doing "?" instead of multiplication and make a ?Table instead of a TimesTable.