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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.

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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.

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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:

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:

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