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Learning Math: Number and Operations

Meanings and Models for Operations Homework

Problem H1
Explain the diagrams that illustrate 124 using the following models:
a. The partitive (or equal groups) model
b. The quotative (or measurement or repeated subtraction) model

Problem H2
Which two of the following models represent the same multiplication problem?

Model (2) illustrates two groups of six, and model (4) illustrates six groups of two.

Problem H3
Show how to compute 19513 using two different models: the area model and long division. Show how the area model relates to long division.

Problem H4
Explain how adding zeros with the colored-chip model helps you understand subtraction problems like +2 – (-4).

Problem H5
Why doesn’t the colored-chip model work for all division problems with positive and negative integers?

Suggested Readings
Chapin, Suzanne and Johnson, Art (2000). Chapters 3 and 4 in Math Matters: Understanding the Math You Teach, Grades K-6 (pp. 40-72). Sausalito: CA: Math Solutions Publications.

Reproduced with permission from the publisher. © 2001 by Math Solutions Publications. All rights reserved.

Download PDF File:
Math Matters: Understanding the Math You Teach, Grades K-6


Problem H1

In the partitive model, we interpret 124 as dividing 12 into four equal groups. The result is the number in each group. Here, there are four columns, so a natural division is to group each column. The result is 3, the number in each column.

In the quotative model, we interpret 124 as dividing 12 into groups containing fours. The result is the number of groups. Here, there are four in each row, so a natural division is to group each row. The result is 3, the number of groups required to make 12.

Problem H2
The correct answer is (2) and (3), which each illustrate the multiplication problem 2 • 6 (i.e., two groups of six). The first model, (1), illustrates the problem 3 • 4 (or 4 • 3), and the last model, (4), illustrates the problem 6 • 2 (i.e., six groups of two). Note that all four models represent the same solution, 12, which does not imply that they represent the same problem.

Problem H3
Using the area model, we start with one flat, nine longs, and five units, and we wish to make a rectangle with 13 rows.

This cannot be done without first exchanging one of the longs for 10 units.

After doing this, we can arrange a 13-by-15 rectangle, so the result of the division is 15.

Using long division, 13 goes into 19 once, with a remainder of 6. Carry down the 5 for 65. Thirteen goes into 65 five times evenly. The quotient is 15.

The area model relates to long division in that it matches the long division algorithm. In long division, you first subtracted 130, or 10 • 13, and then subtracted 65, or 5 • 13, for a total result of 15. Thus, 15 • 13 = 195. Similarly, the area model gave you the rectangle 13 by 15, which consists of two rectangles: 10 • 13 and 5 • 13.

Problem H4
In problems like this one, there is no way to “subtract” chips that are not there. Here, adding four zero pairs does not change the value of +2, but it gives us the four red chips we need in order to “subtract.”

Problem H5
To use the colored-chip model for division, we need to use either partitive or quotative division to do the computations. When dividing a positive number by a negative number, we cannot use either of those models. We cannot “partition” a group into a negative number of parts, nor can we “count” the number of negatives within a positive number.


Series Directory

Learning Math: Number and Operations


Produced by WGBH Educational Foundation. 2003.
  • ISBN: 1-57680-678-2