At a workshop last week, the following task caused a bit of confusion:
If a small gear has 8 teeth and the big gear has 12 teeth and the small gear turns 96 times, how many times will the big gear turn?
Several participants were convinced it was 144.
As we discussed the problem, it was clear that those who were getting 144 were setting up the proportion below and cross-multiplying.
Small Gear 8 = 96
Big Gear 12 x
They weren’t trying to reason about the situation. They weren’t thinking about whether it made sense that the big gear turned more times than the small gear. Instead, they identified this as a proportion problem, set up a proportion and solved for x because it was the method they were “supposed to use” to solve proportion problems.
It was a great example of why you should hold off on teaching students the cross-multiply algorithm until after they have had experience reasoning about ratios using their own strategies.
It also points to the fact that students need to develop meaning for the algorithms they use or they won’t know how or when to apply them.
Lobato and Ellis (2010) describe a student who set up a proportion and used cross-multiplication to correctly solve the following problem:
How much water would drip from a leaky faucet in 4 minutes, given that it dripped at a steady rate of 6 ounces in 8 minutes?
Although the student could provide a correct numerical answer to the problem, she could not explain why it was correct or why it made sense.
Her limited ability to reason proportionally was even more apparent when she could not make sense of the following problem:
One faucet was dripping 6 ounces in 20 minutes. A second was dripping 3 ounces in 10 minutes. Was the first one dripping slower, faster, or at the same pace as the second?
Her inability to solve the second task, as well as her inability to explain her answer in the first task, indicate that she was using a procedure that held no meaning for her. When presented with a new task that she couldn’t solve in the same way, she didn’t know how to reason about it.
The research also shows that student have a hard time making sense of the cross-multiplication algorithm. It doesn’t necessarily connect to their intuitive strategies. If the algorithm is introduced before they have developed meaning for ratios and proportions, it can interfere with and even hinder their ability to develop proportional reasoning.
So how do we help ratios make sense to students so that they can begin to reason proportionally? I’ll talk more about that next time, but if you want to know know, take a look at the resources below.
Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 629–67). Charlotte, NC: Information Age Publishing.
Lobato, J., & Ellis, A.B. (2010). Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics, Grades 6–8. National Council of Teachers of Mathematics.
Smith, J. (2002). The development of students’ knowledge of fractions and ratios. In B. Litwiller & G. Bright (Eds.), Making Sense of Fractions, Ratios, and Proportions (pp. 3–17). Reston, VA: National Council of Teachers of Mathematics.