 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 4, Part C:
Colored-Chip Models

In This Part: Addition | Subtraction | Multiplication | Division

So far we have used this model for addition, subtraction, and multiplication. Let's see what happens when we use it for division.

To find the quotient of (+6) (+2), start with six black chips.

The quotient of (+6) (+2) could be found in one of two ways:

Method 1: Finding the number of sets of positive two that are contained in positive six (measurement, repeated subtraction, or quotative division)

Method 2: Finding the number of items in each set when positive six is split into two equal sets (equal groups or partitive division)

Using Method 1, we group the black chips into sets of two: There are three sets of two black chips contained within six black chips: (+6) (+2) = +3.

Using Method 2, we divide the chips into two equal sets: There are three black chips in each of the two equal sets: (+6) (+2) = +3.

To find the quotient of (-6) (+2), start with six red chips.

The quotient of (-6) (+2) cannot be determined by Method 1 since there are no sets of positive two contained within negative six (i.e., within your six red chips, there are no black chips).

But...

Using Method 2, we can find the number of items in each set when negative six is divided into two equal sets: There are three red chips in each of the sets: (-6) (+2) = -3.

To find the quotient of (-6) (-2), start with six red chips.

The quotient of (-6) (-2) cannot be determined by Method 2 since we cannot divide a set into negative two equal groups.

But...

Using Method 1, we can find the number of sets of negative two contained within negative six: There are three sets of two red chips within the six red chips: (-6) (-2) = +3.

To find the quotient of (+6) (-2), start with six black chips.

The quotient of (+6) (-2) cannot be determined by Method 1 since there are no sets of negative two in positive six. And the quotient of (+6) (-2) cannot be determined by Method 2 since we cannot divide a set into negative two equal groups.

So...

Our model falls apart at this point! Note 11 Therefore, division in this case must be computed by writing a multiplication problem with a missing number in the following way:

 (+6) (-2) = ? means (-2) • ? = +6   Video Segment Watch this video segment for a quick demonstration of why this model falls apart when dividing a positive number by a negative one. Note that the participants in this segment use yellow chips instead of black ones to represent positive values. If you are using a VCR, you can find this segment on the session video approximately 19 minutes and 25 seconds after the Annenberg Media logo.    One of the interesting things about using models in mathematics is that no one model works for everything. In this case, the colored-chip model worked for addition, subtraction, multiplication, and most division problems. However, when we tried to divide a positive number by a negative one, the model fell apart. No model is going to carry over to all computations. Problem C7 Draw or make a diagram to show the colored-chip model for each of the following:

 a. (+8) (+4) b. (-4) (+2) c. (-8) (-2) d. (+4) (-2) Next > Homework  Session 4: Index | Notes | Solutions | Video