Middle school students at three stages of units coordination learn to make same speeds

A classroom study was conducted to understand how 18 seventh-grade students at three stages of units coordination constructed ways to make two cars travel the same speed as part of a unit on proportional reasoning co-taught by the first author and classroom teacher. Students at all three stages developed an understanding of how distance and time affected speed. However, when determining how to make two cars travel the same speed, students demonstrated differences related to their stages of units coordination. Three focus students show these differences. The student at stage 1 constructed doubled journeys as lengths with relative size. The student at stage 2 constructed same speeds by multiplying and by dividing but did not create a multiplicative structure with the results of division. The student at stage 3 constructed same speeds as multiples of the smallest distance-time pair. Comparisons with the other students in the class are made.