Challenge Level

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.

*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?

Finally bring the class together to share their ideas and justify their findings.

One technique for testing ideas at the end is to set a specific challenge, for example, to find the vectors describing Charlie's route if Alison's diagonal route is given by the vector $\pmatrix{35 \cr 15}$

Use of dynamic geometry software such as the free-to-download GeoGebra can help students to develop insights into the structure of this problem. The example below shows a construction which could be shared with students. Alternatively, an extension activity might be to encourage students to create their own constructions.

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

How can I use the diagonal vector to work out the four vectors which describe a journey around a square?

Is there a quick way to determine whether a given vector could be the diagonal of a square with corners on the lattice points of a square grid?

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

Opposite Vertices explores similar mathematical ideas but without vector notation.

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.