Download shape + data handling

Shape Times Shape
http://nrich.maths.org/public/viewer.php?obj_id=5714
Stage:2
Challenge Level 1:
The coloured shapes stand for eleven of the numbers from 0 to 12. Each shape is a different number.
Can you work out what they are from the multiplications below?
Why do this problem?
This problem helps children become familiar with the idea of a symbol (in this case a shape) representing a number. It is also an
opportunity for pupils to practise using multiplication and division in a challenging context.
Possible approach
Ask partners to talk to each other about how they would work out what each shape stands for in the calculations above and
share their ideas amongst the whole group. In this case the last sum is actually the most helpful to start with.
You could then introduce the Shape Times Shape problem by displaying it on the whiteboard and explaining the task. (It might
also be useful for pairs of children to have a paper copy.) Ask the children to think on their own about where they might start.
Invite them to share their ideas with a partner, then discuss the options amongst the whole group. Look out for good reasoning,
based on their knowledge of number properties, and encourage everyone to strive for this level of explanation.
Set them off in pairs to tackle the problem indicating that the plenary will focus on how they went about solving it. You may
want to stop them after a few minutes to find out how they are recording their work and to share some efficient ways.
Key questions
Which sum is useful to start with? Why?
Now that we know that shape, which sum could we look at next? What does that tell us? How do we know?
How are you keeping track of what you have done?
Possible extension
Learners could make up their own problem using shapes as symbols and test it on a friend. The problem What's it Worth?
requires similar thinking processes to this problem and would be a good one for pupils to try next.
Possible support
Children might find it easiest to have numbered counters or cards available so that they can physically form the sums to check
their reasoning. You might want to support their recording by giving out a sheet with each of the shapes drawn on, like this .
What's it Worth?
Stage:2 Challenge Level:
Each symbol has a numerical value. The total for the symbols is written at the end of each row and column. Can you find the
missing total that should go where the question mark has been put?
Possible Support material
Where could you start?
What can you tell from the first multiplication with three purple squares and a yellow semi-circle?
Perhaps you can use what you now know to help you with another sum?
How will you remember what you have tried and what you have found out? You could use this sheet to keep a note of what you
think the shapes stand for
SOLUTION
The upside down isosceles triangle is 0 because every time it is in a question it is the answer. The only number that rule applies
to is 0.
The diamond is 1 because every time it is in a question the other number in the question is the answer. This can only happen if
the number is one.
The square is 2 because something cubed is equal to something. All the numbers must be under twelve so the squares must be 2
or 1 because when those numbers are cubed, the answer is under twelve. It can't be 1 because 1 cubed is one and the answer
was a different shape from the shape in the question. The semi-circle was 8 because 2 cubed is 8.
The oval was 4 because the second question on the left hand row was 2 times the oval is 8 and 2 times 4 is 8.
The circle is double the triangle because the rectangle times 4 is equal to the circle while the rectangle times 2 is equal to the
triangle. Now we need to find a number and its double. It can't be 7 and 14 because all the numbers must be twelve or below. It
can't be five and ten because the triangle and the circle are in sums which involve the 2 times tables and the 4 times tables and 5
isn't in any of them. It isn't 4 and 8 because the semi-circle and oval are those numbers and you can't have two different
symbols which are the same number. It can't be 3 and 6 because they aren't in the 2 or 4 times table. It can't be 2 and 4 or 1 and
2 because all those numbers are other symbols. It also can't be 0 and 0 because they are different symbols so they must be
different numbers. The only possible pair is 6 and 12 so the triangle is 6 and the circle is 12.
The circle is 12 and the oval is 4 so 12 divided by 4 is the value of the rectangle. 12 divided by 4 is 3 so the rectangle is 3.
The second sum on the top is rectangle times rectangle is equal to something. Since the rectangle is 3, it is 3 times 3 which is
equal to 9. So the star with more sides is 9.
That leaves the hexagon and the other star. It says 2 times star equals hexagon (see the second question on the right hand side)
so we have to search for a pair of doubles. The only remaining pair of doubles under 12 is 5 and 10.
Some of you said that you worked out the solution by trial and improvement - trying one number for a particular shape to see
whether it worked. That can be a useful strategy although it might take a little longer than some of the ways already described.
Here is a summary picture of the value of each shape: