# Task: Concrete to Abstract Multiplicative Comparison

While the fourth-grade students were still enjoying their lunch recess, the teacher, the student teacher, and I chatted about what the students could do during math that day.

Struggles with Models. The teacher mentioned that students still seemed to struggle when they drew a model (see examples images below: yellow, green, and red).

Suppose the context of a situation mentioned one person had 3 Pokémon cards (because those have proven to be immensely popular as example!) and a second person had four times as many cards. Then one student might draw the yellow model which represents the person with 3 cards in the first row, and the person with four times as many cards in the second row. Together, they have 15 cards.

Another student might draw a similar model to the green, but the box for the first person is as long as the four boxes of the second person combined. By itself, that’s fine. But it might lead to confusion about who has more, or confusion later about fractions, etc.

Finally, many students would draw something similar to the red model, with three groups of five cards each instead of five groups of three cards each, which represents one person with twice as many cards as the other. This model is especially problematic when the students are asked how many more cards the second person has than the first.

Struggles with Moving Between Additive and Multiplicative Comparison. The teacher also mentioned that students seemed to struggle to answer an additive comparison question after thinking about multiplicative comparison. (For example, in the question above, the second person has 9 more cards than the first person, but four times as many.)

We brainstormed about tasks the students could do that might help them draw the models correctly, but – most importantly – understand and use the models to make sense of a situation. We decided to return to a more concrete representation of the model, so we created two tasks: one for the Teacher Time center and the other for the Hands-On center. The student teacher and I agreed to split the students at Hands-On and each take half of a group.

In the introduction to the math lesson, and during the Teacher Time Center, the teacher had students represent a situation using foam cubes and small baskets.

For example, using the example above, students would put three foam cubes in one basket to represent the 3 Pokémon cards for the first person. Then they would put 3 foam cubes in each of four other baskets to represent the second person with 4 times as many Pokémon cards.

The teacher then drew or had them draw the diagram modeling the context of the situation. She asked them: “How many Pokémon cards do they have all together? How many more Pokémon cards does the second person have? etc.”

She then would ask a question starting with the whole amount: “If two people have 30 Pokémon cards together, and one person has 5 times as many as the second, then how many cards does the second person have? How many more cards does the first person have?”