This prompt can spark an open, exciting and multi-faceted inquiry that combines Pythagoras' Theorem and trigonometry with algebraic expressions for sequences and area. Sometimes it does this too well, and the teacher has to guide the inquiry to ensure students have access to necessary mathematical concepts. The prompt can develop along different pathways based on the students' questions or observations:
- The sides of the triangles form ascending linear sequences. What is happening to the hypotenuse? Is its length also increasing in a linear sequence? Can the length be made to increase linearly?
- The angles in each triangle are (not) the same. Is the angle in the bottom right-hand (or top) corner getting bigger or smaller? If bigger (smaller), can you make it go smaller (bigger) with ascending sequences? What happens with descending sequences? Can you find two sequences that keep the angles the same? What happens when you use other types of sequences?
- The area increases in a quadratic sequence. How would you find an expression for the nth term of the sequence? Is there another set of triangles that has the same expression for the nth term of the area sequence?
The inquiry can be used to develop a conceptual understanding of the tangent ratio and, by calculating the length of the hypotenuse, the sine and cosine ratios as well. Students come to appreciate that the size of an angle is determined by the ratio of the lengths of two sides of the triangle. This explains how both sides can get longer, yet the angle gets smaller. Indeed, it is possible in the initial stages to use a unit ratio before introducing trigonometry in a formal way. The bottom right-hand angle in the prompt, for example, is increasing because the opposite side increases at a faster rate than the adjacent (as shown in the table below).
|Triangle in prompt|| 1||2 ||3 ||4 |
| Ratio |
| Unit ratio|| 1.33:1||1.4:1 ||1.43:1 || 1.44:1|
Additionally, finding an expression for the area can be accomplished by considering the general form of the triangles in the prompt (right), which leads to ½(2n+1)(3n+1). If students derive the expression 3n2 + 2.5n + 0.5 by taking differences between the terms in the area sequence (6, 17.5, 35, 58.5), then reconciling the two expressions has proved rewarding. The inquiry often ends with groups of students presenting conjectures, methods, and examples from their pathways to their peers.
Promethean flipchart download
Smartboard notebook download
The notebook and flipchart below were created by Colm Sweet (a UK maths teacher) for a structured inquiry he ran as part of a collaborative exploration of inquiry teaching with teachers of six other subjects. Colm restricted students' choice to nine regulatory cards.
Promethean flipchart (structured) download