Meanings and Models for Operations Part C: Colored-Chip Models (35 minutes)

In This Part: Addition
In this section, we will use manipulatives to model addition, subtraction, multiplication, and division with positive and negative integers. By working with these models, we will see why the laws for signed numbers (positive and negative) work the way they do. See Note 9 below.

We’ll begin with models for addition in which we will represent positive numbers with black chips and negative numbers with red chips:

Using red and black chips, every number can be represented in a variety of ways. For example, here are some ways to represent positive two (+2):

•Two black chips:

•Three black chips and one red chip:

In this case, one black chip and one red chip together sum to zero: +1 + (-1) = 0. What remains are two black chips, or +2.

•Five black chips and three red chips:

And so on! Now let’s explore some addition problems using these chips.

•To find the sum of +3 + (+1), join three black chips (three positives) with one black chip (one positive), for a total, or sum, of four black chips (four positives, or +4):

•To find the sum of -5 + (-2), join five red chips (five negatives) with two red chips (two negatives), for a sum of seven red chips (seven negatives, or -7):

•To find the sum of -4 + (+6), join four red chips (four negatives) with six black chips (six positives):

Because one black chip and one red chip together sum to zero, four pairs of black and red chips zero out. What remains are two black chips (two positives, or +2):

•To find the sum of +2 + (-5), join two black chips (two positives) with five red chips (five negatives):

Because one black chip and one red chip together sum to zero, two pairs of black and red chips zero out. What remains are three red chips (three negatives). The sum is negative three (-3):

Draw or make a diagram to show the colored-chip model for each of the following:

Problem C1
+4 + (-6)

Problem C2
-4 + (-6)

In This Part: Subtraction
When doing subtraction using this model, you cannot take away things that are not present. Adding pairs of zeros gives us the actual pieces we need to take away. This can be shown symbolically as follows: +6 – (-4) = (6 + 4) – (4 + -4) = 10 – 0 = 10. Notice that +4 was added to both numbers, which is the same as adding and subtracting 4. See Note 10 below.

• To find the difference of +3 – (+1), begin with three black chips (three positives). Subtract one black chip (one positive). Record what is left (two positives, or +2):

• To find the difference of -5 – (-2), begin with five red chips (five negatives). Subtract two red chips (two negatives). Record what is left (three negatives, or -3):

• To find the difference of +4 – (+6), begin with four black chips (four positives):

Your next step would be to subtract six black chips (six positives), but there are not enough black chips to subtract six. So put in zeros (pairs of positives and negatives) until there are enough black chips to subtract six:

(Note that the collection is still worth four black chips.) Now subtract six black chips, and record what is left (two negatives, or -2):

• To find the difference of +2 – (-3), begin with two black chips (two positives):

Since there are no red chips to subtract, put in zeros (pairs of positives and negatives) until there are enough red chips to subtract three:

(Note that the collection is still worth two black chips.) Now subtract three red chips, and record what is left (five positives, or +5):

Draw or make a diagram to show the colored-chip model for each of the following:

Problem C3
+5 – (-6)

Problem C4
-5 – (-6)

In This Part: Multiplication
In this section, multiplication is defined as “putting in” to a circle, or “taking out” from a circle. Remember, in Part A we described 3 • 4 as three groups of four. Three could be thought of as the operator — it tells us what to do with the second number. In this section, we need to know the sign of the operator. If the sign is positive, you put things into the circle. If the sign is negative, you take things out of the circle. In this case, you must put in zeros until you have enough to follow instructions.

•To find the product of +2 • (+3), start with an empty circle:

Since the 2 is positive, +2 • (+3) means that you should “put in” “two” sets of “three black chips”:

Record what is left; i.e., six positives (+6):

+2 • (+3) = +6

•To find the product of +2 • (-3), start with an empty circle. Since the 2 is positive, +2 • (-3) means that you should “put in” “two” sets of “three red chips”:

Record what is left; i.e., six negatives (-6):

+2 • (-3) = -6

•To find the product of -2 • (+3), start with an empty circle. Since the 2 is negative, -2 • (+3) means that you should “take out” “two” sets of “three black chips”:

But there are no black chips. So put in zeros (pairs of positive and negative chips) until there are enough black chips for you to take out two sets of three. Then take them out:

Record what is left; i.e., six negatives (-6):

-2 • (+3) = -6

•To find the product of -2 • (-3), start with an empty circle. Since the 2 is negative, -2 • (-3) means that you should “take out” “two” sets of “three red chips.”

But there are no red chips. So put in zeros (pairs of positive and negative chips) until there are enough red chips for you to take out two sets of three. Then take them out:

Record what is left; i.e., six positives (+6):

-2 • (-3) = +6

Draw or make a diagram to show the colored-chip model for each of the following:

Problem C5
-3 • (-3)

Problem C6
+3 • (-3)

In This Part: 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! See Note 11 below. 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.

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)