Two Wheels and a Belt (and Why I am Convinced That the Standards for Mathematical Practice Must Be How We Do Math)
One of the last tasks that we gave our geometry students this year was Two Wheels and a Belt from Illustrative Mathematics.
A certain machine is to contain two wheels, one of radius 3 centimeters and one of radius 5 centimeters, whose centers are attached to points 14 centimeters apart. The manufacturer of this machine needs to produce a belt that will fit snugly around the two wheels, as shown in the diagram below. How long should the belt be?
The correct answer is 53.42 cm, which several students got, in more than one way.
Some used correct mathematical reasoning to get the correct answer.
Others used incorrect mathematical reasoning to get the correct answer.
It is unfortunate that the incorrect reasoning produced the correct answer. What is more unfortunate is that this incorrect reasoning goes unobserved when our focus is only on answers. When our focus is on how we do the math and not just on what we get as an answer, students and teachers can learn more about mathematics.
As soon as the students began to construct viable arguments and critique the reasoning of others, the misconceptions in the incorrect solution became evident.
And so the journey to help students know and understand mathematics continues …