This prompt comes from one of Don Steward's booklets in the Median series. He suggests using the flow chart to begin a structured worksheet investigation. However, in my experience, the prompt can initiate open inquiry that encourages and draws upon the creativity of students.
The prompt is suitable for all classes of secondary school age (11 to 16). It invites students to place a number in the larger circle and calculate the results of taking the steps in the two paths. For less experienced inquirers, the teacher might label the larger circle 'input' and the two others 'output A' and 'output B'. Even if the teacher opts for the more open prompt, communication is easier when the class develops labels for the three circles.
When asked for comments or questions, students invariably reach the conclusion that, in the case of the prompt, one output is always three more than the other. This realisation acts as a sort of preliminary phase to the main part of the inquiry. The selection of a regulatory card at this point often reveals a great deal about students' mathematical thinking. My classes have tended to select one of two cards.
- Either students select find more examples, by which they mean to experiment with different pairs of operations. (The teacher of younger classes is advised to stipulate that only multiplication, addition and subtraction are permissible in the initial phases.) Students go onto induce a relationship between two outcomes.
- Or students choose prove the prompt is always true, which can lead to the development of algebra directly from one numerical case. If we start with four, for example, the top path gives 4 + 4 + 3 or 2 x 4 + 3 as an output. The bottom path gives 4 + 3 + 4 + 3 or 2 x (4 + 3). The algebraic expressions, then, are 2n + 3 and 2(n + 3), where n is the starting number. As 2(n + 3) expands to 2n + 6, it becomes clear that the difference between the outputs will always be three.
Students have shown great enthusiasm for finding algebraic expressions. They have then substituted into the expressions to deduce the difference between the outcomes for a particular starting number.
Guided poster Devised by Emma Morgan to guide students when presenting their inquiry (see opposite at top).