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# Comparing Growth, Variation 2

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. When asked which one grew more, students gave varying answers.

Mia said, “Since the two snakes both grew two feet ($8-6=2$ and $10-8=2$) then I would say that they grew the same amount.”

Raul said, “They both grew 2 feet, but Jewel was only 6 feet to start with while Clyde was 8 feet to start with. That means Jewel grew more compared to her original length (2 is a larger part of six than it is of eight).“

Compare the two arguments. Describe the difference in the way the two students are thinking about the problem. Suppose a one-foot snake grew two feet and a 20-foot snake grew two feet. Could the two students still make the same type of argument?

## IM Commentary

The purpose of this task is to assess students’ understanding of multiplicative and additive reasoning. We would hope that students would be able to see identify that Student A is just looking at how many feet are being added on, while the Student B is comparing how much the snakes grew in comparison to how long they were to begin with. The follow up question is meant to highlight the difference in the two ways of thinking about growth. Some students will be more likely to agree with Student B when the difference in relative growth is accentuated as it is in the follow-up problem.

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 answers are correct, but the goal is to see if the student understand the two perspectives, and thus the difference between additive and multiplicative reasoning.

#### Kate Abell, NYC says:

We are going to try out the following adaptation in a study of student thinking around multiplicative comparison. after trying out version 2 in a video-taped argument between two teachers. The 4th gr. students who watched were unanimous in thinking that the teacher comparing additively was "correct."

In our problem , the snakes are pictured as bars with "foot" markings, with their original sizes shaded and their growth unshaded.

Snake Stories

Clyde

Jewell

Clyde was 1 foot long. After one year he was 5 feet long.

Jewell was 2 feet long. After one year she was 6 feet long.

Lorrie said both snakes grew the same.

Derek said Jewell grew more because she is the longest.

Serena said Clyde grew the most because he is now 5 times longer than he was, and that's not true of Jewell--she isn't 5 times longer than she was.

What do the three students mean? Explain their thinking.