Part 2:

The table shows some of the distances and times that Frog traveled. Is he going the same speed the whole time, or is he speeding up or slowing down? How can you tell?

Distance

3.75 cm 7.5 cm 12 cm 15 cm 40 cm

Time

1.5 sec 3 sec 4.8 sec 6 sec 16 sec

Larissa, Maria, Dani, and Julie created a shorthand 15:12 unit on their papers. They first multiplied the 15 cm:12 s unit by 2 to obtain 30 cm in 24 s, and they then began to multiply the 15:12 unit by other numbers to create other same-speed pairs. The students engaged in the generalizing action of extending by operating on the 15:12 unit, because they operated on the pair to create new instantiations of the ratio. The students then determined that they could multiply the 15:12 unit by any whole number to obtain a new same-speed pair. This resulted in the reflection generalization of the definition of a class of same-speed pairs.

The students then realized that all multiples of 5 cm in 4 s would work, as evidenced by Maria’s written comment: “Multiple of 5 & 4 has to be a form of 15 cm & 12 seconds.” Their recognition of the 5 cm:4 s pair likely occurred as a result of their prior experiences with gears. Every time the students encountered a nonuniform table of pairs of gear rotations, they focused on the smallest whole-number pair they could find in each table. Eventually, they generalized that this pair represented the gear ratio. The students’ fixation on the smallest whole-number pair, in combination with other studies showing that students in speed situations do not behave this way (Lobato & Siebert, 2002; Lobato & Thanheiser, 2000, 2002), suggests the influence of prior reasoning with gears. If so, we see here another reflection generalization, the influence of a prior idea on the speed situation. Because the microphone was not recording the girls’ conversation at this time, there is not enough evidence to determine if they consciously performed the generalizing action of relating by connecting back to the gears situation. However, Larissa’s language below suggests that at least for her, she connected back to the gears. This connection to the gears situation appeared to help the girls refine their definition of a class of same-speed pairs to a larger, more accurate class.

If used, the gear problem should be followed up by the frog problem since the ratios in the gear problem only make sense for integers, which could mislead students if they do not then follow up with an example in which the ratio can be between any two real numbers.