Arrow Arithmetic 1

The first part of an investigation into how to represent numbers using geometric transformations that ultimately leads us to discover numbers not on the number line.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



This problem forms the first part of an investigation into how to represent numbers using geometric transformations that will ultimately lead us to discover numbers that are not on the number line.

I'm going to assume that you are happy with the idea of numbers that lie on the number line. Just to remind you, here is a picture of a small part of the number line.

Image
Arrow Arithmetic 1


The plan of action is to develop pictures or geometric representations of the numbers we know about already. We'll also develop ways to add, subtract, multiply and divide using just those pictures. Then we'll change the picture very slightly...

Here's my first attempt at a picture of the number 1. It's simply a blue arrow.

Image
Arrow Arithmetic 1


I don't think this is a good enough picture yet, but it's helpful to think about its shortcomings.

Let's try to do arithmetic with these arrows. Try three simple additions.

Image
Arrow Arithmetic 1


I tried these with three friends and here are the answers they gave me

Sam Hannah Tim
1) 3 3 3
2) -1 3 3
3) 1 3 2


They all had good explanations for their answers. Can you guess what they were?


My arrow picture obviously has some shortcomings! How would you improve it?