Here I am on term holiday marking my year 9’s Pythagoras’ test. It wasn’t a particularly hard or long test and I am pleased to see that most students have done well. Because the test was short I decided to also test student’s problem solving ability with a short investigation into pythagorean triads:
Below are two rules to generate pythagorean triads. Use these rules to generate as many different triads as you can.
1. Start with an odd number greater than 1. Square the number then halve the answer. The two whole numbers either side of this result, together with the original odd number, form a Pythagorean triad.
2. Start with an even number greater than 2. Square this number then divide the answer by 4. The two whole numbers either side of this result, together with the original even number, form a Pythagorean triad.
This caused even my better students trouble. “I can’t do it sir”, “I don’t know what to do”, “How many do I have to do?”, were some comments. I checked that students knew what odd and even numbers were. They could also square numbers and do the simple arithmetic required. Some had trouble writing down two whole numbers either side of say 12.5. Eventually a list of triads started to appear. One student found four triads and then wrote on their paper: “Sorry, I don’t feel like doing more of these.”
I can see that next term I will have to give these students more practice at solving word problems where they have to read, read again, interpret, try a simpler problem, etc, etc.
This statue was erected in the harbour of Pythagorio, on the island of Samos, where Pythagoras and Aristarchus were born.