Trial and Improve is a strategy used at key stage 34 to solve an equation, such as x2 + x = 100, to a specified degree of accuracy. We take an initial "guesstimate" and then increase or decrease our attempt at finding a solution, depending on whether our answer is "too small" or "too large".
For the equation, x = 9 is too small (81 + 9 = 90) and x = 10 is too large (100 + 10 = 110) so, step-by-step, we zoom in to find a solution, perhaps to one decimal place, by considering values between 9 and 10.
Although perceived as a relatively straightforward and mechanistic topic, without an imaginative introduction to associate it with everyday life, it can be difficult to present Trial and Improve in a way that captures imaginations. The 999 Challenge always makes a positive difference and generates enthusiasm for subsequent Trial and Improve work.
When I introduce the strategy, I ask if the emergency phone number, 999, can be made as one, two-digit whole number times another two-digit whole number. Students are invited to start with 33 x 33 and systematically increase one number and decrease another until the "magic 999" is found.
Obtaining answers close to 999 further enhances interest and increases motivation. Children want to be the first to find the solution and see the emergency phone number shining brightly on their calculator screen.
Parents who were invited to a "Help your son or daughter through their GCSEs" evening were also set the challenge. When the winner collected his cake, he said that he started off with 999 and obtained the solution using division. If a student finds the answer quickly, it is helpful to have a few other numbers waiting in the wings.
Trial and Improve could be used in other contexts so students can get a sense of moving towards a solution by taking advantage of information gained in previous attempts. For example, when is the trajectory of a tennis ball a certain displacement from its point of projection?