Dynamic Number Line
Beginning
I started by showing the pupils the dynamic number line sketch I had made using
Geometers Sketchpad. You can drag point A or point B anywhere and the
resulting product, AxB, moves dynamically.
- I’ve got something to show you: a number line. It goes as low as you please
and as big as you please. At the moment, this is one number – pointing to 2.00,
A, and this is another number, B, 4.00 – what’s happening to get this number ...
8.00?
A conversation followed that established that this was about multiplication. I
asked a question to try and focus pupils on thinking about multiplying by zero.
- Is there any way I can move A so AxB doesn’t move?
Different pupils come out to the board to try and find a way of solving the
challenge. Many suggestions are offered by a number of pupils and there seems
genuine excitement and intrigue around the class. An answer was finally given:
~When you put the A on the zero you can move the B, the triangle won’t move
because it’s on the zero. You can move B anywhere and A doesn’t move.
A common language develops naturally between the pupils. The product AxB is
referred to as the triangle. This helps pupils to feel more comfortable taking about
what they see when they can use a language that they have developed. I adopt
their language to help make my questions clear.
- Is there any way of having B on the zero without having the triangle on the
zero?
~ No.
- Why not?
~ When you move that on the zero it’s a kind of a magnet and the triangle goes
to zero.
~ It’s like it’s attracted there.
- Tell me a rule.
~ It’s like a magnet!
The conversation develops and after some time the pupils conclude that
whatever you multiply by zero, the answer will always be zero. Example
questions are given, 10 x 0, 25 x 0 etc and pupils are comfortable giving the
answer.
Where are we going?
Similar questions and activities were carried out looking at different properties of
multiplication. The next thing we looked at was multiplication by one.
- Is there any way that I can move B so that whenever I move A the triangle is
always the same as A?
This again leads to lots of conversation about multiplication building on pupils
understanding, again letting them use their ideas about magnetism and
attraction.
The other properties to look at are multiplying by one negative number and
eventually multiplying two negative numbers together.
Where the activity could go
The next page of the sketch is called Mystery Machine. Pupils need to try and
find the position of zero and one by dragging the points to experiment. This tests
their understanding of multiplying by zero and one.
The sketch can be easily changed so that instead of showing multiplication it
shows any of the other operations. Looking at what happens when you perform
calculations with negative numbers is always interesting and often surprises
pupils.
Mathematical Aims
-
Multiplying by zero and one
Calculations involving one or more negative numbers
Communicating mathematical ideas
Problem Solving