Estimating answers (age 13+)

Activities to try

1. 1

   Up-and-across reading with the scale stated first

   Why it works: A read-off value depends on the axis scale, so checking the scale comes before trusting any number; a value between grid lines is an estimate given to a sensible interval, not an exact figure.

   1. Before reading anything, state aloud what one grid square represents on each axis (e.g. 'each square is 1 unit' or 'each square is 2 units').
   2. For a plotted curve, draw a line up from a given x-value to the curve, then across to the y-axis, and record the estimated y.
   3. Read a value to the nearest sensible fraction of a grid square, writing it as a band (e.g. 'between 2 and 2.5, about 2.25') rather than a single over-precise figure.
   4. Re-read the same physical point after the grid interval changes (e.g. squares now worth 2) and note how the number changes while the point does not.
2. 2

   Finding both roots of a parabola

   Why it works: 'Find where y = 0' can have two answers for a parabola, located symmetrically; stopping at the first crossing misses a root, and the x-axis crossings are different from the y-intercept.

   1. On a parabola such as y = x² − 4, find every x-value where the curve meets the x-axis (where height = 0), not just the first one encountered.
   2. Use the symmetry of the curve: after finding one crossing, look on the opposite branch for its partner (a parabola that only touches the x-axis at its vertex has just one, and one that never reaches the x-axis has none).
   3. Given a target height (e.g. y = 5), find both symmetric x-values that reach it, and read each as an estimate.
   4. Distinguish 'y = 0' (where the curve meets the x-axis) from the y-intercept (where the curve meets the y-axis, here −4).
3. 3

   Intersections as shared solutions, and checking read-offs

   Why it works: An intersection of two graphs gives the value(s) common to both; a graphical read-off is approximate and can be wrong if the plot is inaccurate, so solutions should be checked and sense-checked against context.

   1. Read the intersection point of two plotted graphs (e.g. a curve and the line y = x) as the value(s) satisfying both.
   2. Distinguish where a single line crosses an axis from where two graphs cross each other.
   3. After reading a solution from a graph, substitute it back to check that it approximately satisfies the equation(s) (both sides come out close, not necessarily exactly equal), especially when the plot may be inaccurately drawn.
   4. Interpret a real-world graph (e.g. distance–time) to answer a contextual question, stating units and sense-checking against a realistic range.