Why do this problem?
This problem encourages students to think about vectors as representing a movement from one point to another. The need for coordinate representation of points will emerge automatically and the problem naturally requires an interplay between geometry and algebra.
The problem is available as a handout here
Set students the challenge to investigate possible end points when combining steps of vectors $b_1$ and $b_2$ in a vector walk. Some students will prefer to work algebraically while others will wish to represent the problem geometrically; by encouraging students to work in groups with others who have different preferred methods, rich mathematical thinking can emerge.
Students should aim to describe geometrically the set of points which can be made by combining the two vectors.
Once students have successfully described the set of points made from combinations of $b_1$ and $b_2$, set them the two challenges - to find other pairs of basic vectors which yield the same possibilities, and to find a pair of basic vectors which will never lead to the points found in the first part of the question.
What do the points you can reach with $b_1$ and $b_2$ have in common?
Can you describe the resulting set of points geometrically (i.e. describe them clearly without algebra)?
Work systematically combining $b_1$ steps with $b_2$ steps, recording the points visited.
Investigate the effect of changing the order in which the steps are taken.
Polygon Walk explores vector walks which form polygons around the origin.