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This problem follows on from
Arrow Arithmetic 1
We're trying to find a way of drawing numbers that will also help
us draw not just the number, but also make sense of number
operations such as addition and multplication.
In Arrow Arithmetic 1, we tried using a single arrow to represent a
number. This has some features we desire since we can draw long
arrows to represent big numbers and short arrows for small numbers.
The direction of the arrow should allow us to distinguish positive
from negative numbers.
But as we saw, there is a problem with using a single arrow, which
is captured in this question: 'How long does an arrow need to be to
represent the number 1, and which way should it point? '
Once we answer this question by choosing a unit arrow, it becomes
very easy to represent other numbers with arrows. 2 is an arrow
twice as long as the unit arrow, pointing the same way. -3 is an
arrow three times as long as the unit arrow, pointing in the
opposite direction.
These thoughts led me to the idea of using two arrows to picture a
number. I'm going to call this pair of arrows a t
wizzle. The thicker arrow in the
twizzle is the
unit arrow
- which we can choose to be any length we like and to point in any
direction. The other arrow in the twizzle is the
number arrow. It determines the value
of the twizzle by comparison to the unit arrow. Play with this
animation until you understand how it works, and how the twizzle
respresents a number. Check with the
hints to ensure you have discovered
all the ways to manipulate twizzles.
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In the next animation I am combining the blue and green twizzles
together to make a grey twizzle.
I now have two questions for you:
How is the value of the grey twizzle related to the value of the
blue and green twizzles?
How would you construct the grey twizzle from the blue and green
twizzles using rotations, translations, and enlargements?
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How would the value of a twizzle change if you rotated its number
arrow through 180 degrees?
Explain how you could use this fact to construct twizzle
subtraction.
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