Vector Journeys

Why do this problem?

This problem offers a simple context for exploring vectors that leads to some interesting generalisations that can be proved with some vector algebra.
 
Here is an article that describes some of the background thinking that informed the creation of this problem. 

Possible approach

You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.

This problem requires students to draw tilted squares reliably. This interactivity might be helpful to demonstrate to students what a tilted square looks like. Students could play Square It until they can reliably spot tilted squares on a dotty grid.
 
A possible start which involves the minimum of teacher input is to draw the vector $\pmatrix{3\cr 1}$ and say:
"Imagine we are drawing squares using vectors with whole numbers.
This vector could be the side of a square, or the diagonal of a square.

Find the vectors that describe the journeys around the squares that include this vector as either a side or a diagonal."

This leads on to the challenge "In a while, I am going to ask you to find the vectors that describe the journeys around squares that could be drawn using a different vector as either the side or the diagonal. The challenge will be to answer without doing any drawing."
 
 
Alternatively, start by showing the picture of Charlie's walk.
"If the black vector is $\pmatrix{3\cr 1}$ what are the other three vectors?"
Once everyone is confident with vector notation, ask students to draw a square park of their own on dotty paper, making sure the vertices are on lattice points, and to work out the vectors that would describe Charlie's journey.
 
On the board, draw a table to collect together some of the vector journeys the students have devised. After the first few, can they start predicting what the second, third and fourth vectors will be once they know the first vector of a journey? Is there more than one possibility?
Give students some time to work on their own or in pairs to test any conjectures they make.
 
"Could we have worked out the vectors if we'd been given a diagonal of the square instead of a side?"
Show Alison's diagonal walk, and ask students to consider this question with regard to the squares they drew earlier on. After a short while, the diagonal vectors could be added to the information already collected on the board.
Then set students the three questions from the problem:

• Can they describe any relationships between the vectors that determine Alison's and Charlie's journey, for any square park?  
• Given the vector that describes Alison's journey, how can they work out the first stage of Charlie's journey?
• If all square parks have their vertices on points of a dotty grid, what can they say about the vectors that describe Alison's diagonal journey?  

Key questions

How can I use the first vector to work out the other three vectors which describe a journey around a square?

Possible support  

The interactivity in Square Coordinates helps students to visualise tilted squares.

Possible extension

Vector Walk challenges students to explore relationships between vector algebra and geometry, and to consider the points that can be reached on a grid using a set of vectors.