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# Comparing Growth, Variation 1

Alignments to Content Standards: 4.OA.A

There are two snakes at the zoo, Jewel and Clyde. Jewel was six feet and Clyde was eight feet. A year later Jewel was eight feet and Clyde was 10 feet. Which one grew more?

## IM Commentary

The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount (an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example of "multiplicative thinking"). This would set the stage for a comparison of the two perspectives. In the case were the students don’t bring up both arguments, the teacher can introduce the missing perspective.

In later grades, students will learn that "which grows more" means "which has the greater absolute increase?" and "which has the greater growth rate?" means "which has the greater increase relative to the starting amount?" but students won't see this type of language for two or three years. Teachers need to be aware of this and work to ask questions as unambiguously as possible; for example, when asking for multiplicative comparisons, use language such as, "How many times greater is $x$ than $y$." They should also be prepared to address this potential for confusion along the way.

## Solution

Viewing this additively, both snakes grew 2 feet and therefore grew the same amount. Viewing it multiplicatively, Jewel grew $\frac{2}{6}$ its length, while Clyde grew $\frac{2}{8}$ its length. From this perspective, Jewel grew more. Given the purposeful phrasing of the problem, both interpretations are reasonable, but the goal is to understand the two perspectives, thus the difference between additive and multiplicative reasoning.

#### Janie says:

almost 2 years

I love this task as it is a great way to transition students to multiplicative reasoning. I work with lots of teachers and this task helps me help them understand the difference!

#### Cam says:

It seems to me that prompt would be unlikely to generate much interesting discussion as to the ambiguity of the question, which in the end is the point of this task. Of course, I completely agree that bringing up potentially confusing ideas should be done in the safe confines of a teacher-monitored discussion, and certainly not assigned as a homework assignment with students and parents left to fend for themselves.

#### Kathy says:

Asking "which one grew more" implies that there is one exact answer. This confuses children (and parents). Why not pose a performance task such as "Discuss the snakes' growth."

#### Martha says:

over 3 years

While I agree that students need to "see" and understand different ways of computing and reasoning out their answer, I still feel that math should have an exact answer. And, from what I am reading with the common core, it isn't that we are to be teaching this understanding differently. So, with that understanding in mind, I find this confusing for children because which one is the correct answer? Even though there might be to understand the perspectives between the two types of reasoning, what exactly is the correct answer? So, in my opinion because you have not shown which one is the exact answer and proven why it is then I do not think this is a "healthy" problem to use with children who are just discovering. As a teacher it is not my goal to confuse but to teach and to do so clearly.

#### Cam says:

over 3 years

I think there's ample room to accommodate the "math should have an exact answer" philosophy with this problem -- indeed, a real punchline of this problem is that the question "Which one grew more?" is ambiguous. There are two interpretations of how to convert that english sentence into a mathematical question, and those two mathematical questions each have unambiguously correct answers. I think it is very important to emphasize, even at an early age, the practice of asking mathematically precise questions -- differences between absolute growth and percent growth are tremendously relevant in the mathematical literacy we hope our kids will possess as they start making decisions and becoming informed citizens. Part of that skill is recognizing when questions aren't mathematically precise (e.g., Is cereal A is "better than" cereal B?). That said, I of course agree that teacher discretion is required to make sure that students aren't handed potentially confusing tasks before they're ready for it, giving new and fragile concepts time to soildify before being poked and prodded.

#### Sumner says:

over 3 years

At the fourth grade level it is appropriate to use the term "benchmark." Perhaps this might give rise to the opportunity of using this concept to solve the problem, as well.