Grass Seedlings
Task
The students in Raulâs class were growing grass seedlings in different conditions for a science project. He noticed that Pabloâs seedlings were $1\frac12$ times a tall as his own seedlings. He also saw that Celinaâs seedlings were $\frac34$ as tall as his own. Which of the seedlings shown below must belong to which student? Explain your reasoning.
IM Commentary
For the sake of this commentary, let $R$ be the height of Rualâs seedlings (of course students do not need to use a letter to represent any number in this task). Student must compare the size of each of two products ($1\frac12 \times R$, and $\frac34 \times R$) to one factor ($R$) on the basis of the other factor without performing any computations. An extension to this problem would be to ask students how many times larger Pabloâs seedlings are than Celinaâs.
The wording â$1\frac12$ times as tallâ may be a clue to students that multiplication is the relevant operation (i.e., $1\frac12 \times R$ is the height of Pabloâs seedlings). The student should also realize that the phrase â$\frac34$ as tall asâ also signals multiplication â in fact, this phrase can be read as â$\frac34$ times as tall as.â In both cases, a scale factor is being specified and students should connect that to the interpretation of multiplication as scaling.
The solution shown below is in adult language. Responses by fifth grade students will vary widely in their level of precision
Solution
Since Pabloâs seedlings are $1\frac12$ times a tall as Raulâs, Pabloâs seedlings must be taller than Raulâs. Pabloâs seedlings looks like Raulâs seedlings scaled by a factor larger than 1.
Since Celinaâs seedlings are $\frac34$ as tall as Raulâs, Celinaâs seedlings must be shorter than Raulâs. Celinaâs seedlings looks like Raulâs seedlings scaled by a factor smaller than 1.
Celinaâs seedlings are shorter than Raulâs and Pabloâs are taller than Raulâs. Thus, the seedlings in pot A belong to Celina, the seedlings in pot B to Raul, and the seedlings in pot C belong to Pablo.
Grass Seedlings
The students in Raulâs class were growing grass seedlings in different conditions for a science project. He noticed that Pabloâs seedlings were $1\frac12$ times a tall as his own seedlings. He also saw that Celinaâs seedlings were $\frac34$ as tall as his own. Which of the seedlings shown below must belong to which student? Explain your reasoning.