These tasks involve visualising lines and points in the Cartesian plane. They give children the opportunity to estimate by eye but also to adopt a precise analytic approach.
Task 14A: Children will probably be able to estimate by eye that the straight line will cut the x-axis somewhere between the 10 and 15 marks. A straight edge might help children make a better estimate, or they could find the exact location by working analytically: a point traveling on the line from B goes 4 units to the right for every 1 unit down, so it cuts the x-axis 8 units to the right of point B, namely at (5+8, 0) or (13, 0).
Task 14B: Children might be able to make a good estimate of the location of the desired point by eye. However, the fact that this point lies outside the given grid might encourage more children to think analytically. The line through C and D is 1 unit above the previous straight line, so a point traveling along the line from D descends 3 rather than 2 units to reach the x-axis, so will travel 12 rather than 8 units to the right, thus cutting the axis at (5+12, 0) or (17, 0).
Or: the line through A and B cuts the x-axis at (13, 0), so the line through C and D will go through the point with coordinates (13, 1) and hence through (13+4, 0).
Task 14C: Children might at first find this to be more complex than the previous tasks as it involves two slanting lines. However, they might notice that for every 4 units that a point on each line moves to the right, the vertical distance between them reduces by 1 unit. As the points on the lines with x-coordinate 9, are 1 unit apart vertically, the points on the lines with x-coordinate 9+4 will be 0 units apart vertically, that is, they will coincide. The y-coordinate of this point will be 2 units more than the y-coordinate of (9, 5) [or 1 unit more than the y-coordinate of (9, 6)]. So the point has coordinates (13, 7).
Task 14D: These are the same green lines as in the previous task, and we can see that they meet at (13, 7). The point of intersection of the two red lines is way off the grid, which might make the task challenging for some children. However, if children adopt an analytic approach, they might notice that the red lines are 8 units to the right of the green lines. So their point of intersection, even though it is not visible on the grid, will be 8 units to the right of (13, 7), which is (21, 7).
Task 14E: If children give themselves time they might well be able to find the desired grid-points by eye, and then perhaps use a ruler to lend support to their choices - but how convincing a case can they make that the various points are correct?
Points Q, R, S and T in the diagram below are the other grid points on the given grid that are equidistant from A and B. Point Q is perhaps the easiest point to find, and perhaps the most crucial. As with the given point P, its distance from A and from B is equal to the diagonal of a 1 unit by 2 units rectangle.
Points A, P, B and Q form a square with diagonals AB and PQ. So PQ is at right angles to AB and bisects it. All other points equidistant from A and B will lie on the straight line through P and Q - and all the equidistant grid points will be spaced apart in steps of 3 units to the right and 1 unit down (or 3 to the left and 1 up).
