Always on the lookout for good challenges that will help students experience the disorientation of misfiring heuristics, I stumbled upon a new one, (courtesy of Mr Elliott, my daughter’s maths teacher.) Here it is:
Beautiful, isn’t it? Something is wrong, and the anomalies cannot be ignored.
Ideally, you would let small groups of students struggle with the problem while keeping some record of the debate so that you can elaborate on the processes that were involved, and consider what dynamics might help arrive at the correct solution. But this challenge lends itself to much more than considerations of counterdeception.
Let’s first consider the “solution”.
It’s a beautiful case of deception, isn’t it? Two seemingly identical triangles – ABC and DEF – are, in fact, neither identical nor triangles. They are different quadrilaterals. (Just triple the size of all the different components of ABC and DEF, and you will see the difference between the two.) The “slope” of the hypotenuse of the dark green and the red triangles are (slightly) different and neither is the same as the slope of the hypotenuse of hypothetical triangles ABC and DEF. (Apologies to all mathematicians who could no doubt explain this much more succinctly and elegantly.)
Now, clearly a lot of work has gone into making the deception as effective as possible. How did the drafters of this challenge maximise the deceptive nature of the challenge? For example, they made sure the red and green triangles would always meet on a point on one of grid squares to support the illusions. Reviewing the characteristics that would make this challenge more deceptive would be a good way to launch a discussion of how to cook up a good deception.
But of course, the greatest thing about challenges like this is that they are fun, and the greatest lessons that can be learnt from them is to appreciate a good deception, even if the joke is on you, and then learn from your experience.
