# Teaching Multiplicative Inverse through a Comparison Model

**Scenario**

Imagine you are a fourth-grade teacher and your students have learned the equal-groups meaning of multiplication in third grade. Now, you’re planning to teach them the comparison model using the following group of word problems that are listed in the

CCSS table:

- • A blue hat costs $6. A red hat costs 3 times as much as a blue hat. How much does the red hat cost?
- • A red hat costs S18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost?
- • A red hat costs $ 18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat?

How will you teach this lesson? Use the questions below to guide your planning:

- 1
*What are the big ideas you expect students to understand?* - 2
*What representations will you use during teaching*? - 3
*Which deep questions will you ask during teaching?*

The above word problems ask students to find the bigger quantity, the smaller quantity, and the multiple, respectively (see more explanations in Section 3.2). Readers likely agree that the big idea inherent to this group of problems is inverse relations. In each problem, the concept of “3 times” is a key point to teach. For representations, one may guide students to either use manipulatives or draw discrete (separate objects) or continuous models (tape diagrams/bar models or number lines) to illustrate the problem situations. For deep questions, one may ask questions such as, “What does ‘3 times’ mean?”, which demands students’ prior knowledge of the equal groups meaning of multiplication. Given that multiplicative comparison is a hard concept for students to grasp, and that an understanding of which is a prerequisite for understanding multiplicative inverse, below I share insights from two Chinese classrooms and one U.S. classroom. Although inverse relations were not a direct focus in these lessons, these teaching examples illustrate how the key concept of “times” can be conceptually taught so discussions of multiplicative inverses can be built on the comparison model.

## Insights from Chinese Lessons

The two Chinese third-grade teachers (T1 and T2) presented one type of multiplicative comparison problem: finding the smaller quantity. Students in these classes had prior lessons about how to “find the multiple” and “find the bigger quantity.” Common features across both lessons were the teachers’ deep questions on representations (e.g., manipulatives or number lines), which likely linked students’ understanding of “times” to their prior knowledge of equal groups.

### Deep Questioning about “Times” When Reviewing Multiplication

The lesson in Tl’s class started with the following review problem projected onto the board:

The 1st row: 3 sticks (in a picture)

The 2nd row: Is 4 times as many as the 1st row (in words)

The 2nd row placed ( ) sticks

This task is similar to the first CCSS item in this section’s opening Scenario. Students knew the smaller quantity (number of sticks in the 1st row) and the multiple (the relationship between the 1st and 2nd rows). They were expected to find the bigger quantity (number of sticks in the 2nd row). The teacher began by asking, “Without placing the sticks, do you know how many sticks we should put in the second row?” Students quickly responded with 12 because 3x4= 12.^{6} The teacher then said, “Since you all know how to place them, I won’t ask you to do it hands- on.” Next, she proceeded to ask a set of deep questions tackling the meaning of “4 times.”

T: Ok, here, in which row should the number of sticks be considered as one group?

SI: I consider the number of sticks in the first row as one group.

T: So how many groups would you consider the second row as having? You, please.

S2: I consider the second row as having 4 groups.

T: Why do you think there are 4 groups in the second row? Yes, please. You have a seat.

S3: Because the second row has 4 times as many as the first row.

T: Agree?

Ss: Yes.

T: The second row has 4 times as many as the first row. So, you should consider the first row as one group; the second row then contains 4 groups of the same thing.

In the above conversation, the teacher first asked students which row was viewed as “one group.” In multiplicative comparison, this is a “referent.” The teacher then asked how many groups were in the second row (4 groups) and why there were 4 groups in the second row. Most importantly, the teacher explicitly summarized the observations at the end, emphasizing the importance of identifying the single referent group, and how the multiple (4 times) indicates repeating that referent group. This is a clear statement about the meaning of “4 times” that was linked to students’ prior knowledge of equal groups. Such an explicit summary is critical because it draws students’ attention to the key knowledge point.

60 *Inverse Relations between Multiplication and Division*