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
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.
Key questions
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?
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.
Possible support