Week 03: Squares (and other shapes) on a grid

Here we present children with parts of a square (or other simple shape). The shapes' vertices are grid-points and the aim is for children to analyse the grid-points and how they are related with sufficient care to be able to tell whether a desired shape 'looks right'? [Some children might take the analysis further and turn an observation such as this into a general rule: "If a line goes 4 to the right and 3 up, when I turn it through 90˚ it will go 4 up and 3 to the left". However, that is not the central aim of this week's tasks.]

Children can use a ruler if they wish, though you might want to encourage them to sketch the shapes (once the vertices have been carefully placed) as this is an important skill.

Task 03A: We can think of the line from A to B as going 4 units to the right and 2 up (or 2 to the right and 1 up, twice). Similarly the line from B to C goes 4 up and 2 to the left (or twice 2 up and 1 to the left).

We can locate D by going 4 up and 2 to the left from A (the same as going from B to C). Or we can start at C and go 4 to the left and 2 down (the same as going from B to A).

Task 03B: This is more demanding than Task 03A in that we are only given a line segment in one direction, not also in the perpendicular direction. There are two possible positions for vertex C (and D) but one of these leads to a square that is only partly on the given grid.

We can get from A to B by going 3 to the right and one up. This means C lies 3 up and 1 to the left from B (or 3 down and 1 to the right, though this would mean that D is not on the given portion of the grid).

Task 03C: One way to solve this task is by a form of trial and improvement, by, for example, trying to place B such that AB = BC and ∠ABC is a right angle. This has its merits as it encourages children to visualise.

An alternative, more analytic approach, is first to locate the other diagonal by using the properties of a square that its two diagonals are equal, bisect each other and are at right angles.

Task 03D: As children play around with this, they might discover that it is quite easy to generate more parallelograms by using some of the parallelogram properties. For example, the fact that parallelograms have rotational symmetry means that if, say, point B is moved 1 unit up the page, D has to move one unit down.

It is not easy to find the desired rectangles (at least not if their vertices are grid-points). Here are two (I don't think there are any more (apart from the square from Task 03C) where the vertices are grid-points).

Task 03E: This can be solved in the manner of Task 03C by first drawing the diagonal AC (which passes through X). Alternatively, as children play around with this, they might discover that whatever the position of point A, we can generate the vertices B, C and D by moving point A through successive rotations of 90˚ with X as the centre of rotation. Here, it might help to first draw the line segment XA.