7.1: The Meaning of Division

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You will need: Centimeter Strips (Material Cards 16A-16L) and Base Blocks (Material Cards 3 - 15)

In these first few exercises, you'll be using c-strips to explore division.

Exercise 1

In this first exercise, we will divide $H \div Y$. H, the part before the division sign, is called the DIVIDEND, and Y, the part after the division sign, is called the DIVISOR. One way to do this division is to find the maximum number of times the divisor (Y) can be subtracted from the dividend (H). Take out a hot pink c-strip (H). To figure out how many yellow c-strips could be subtracted, make a train consisting of as many yellow c-strips (Y) as possible such that the train you make is shorter or equal to the length of H, and if you were to add one more yellow c-strip, the train would be longer than the hot pink c-strip.

a. How many yellow c-strips did you use? ______

This number is called the QUOTIENT.

The quotient is the maximum number of times the divisor can be subtracted from the dividend.

b. If the train you made is shorter than H, what color c-strip could
you add to that train to make it equal in length to the hot pink c-strip? ______

This piece, which must be shorter than the divisor, is called the REMAINDER.

Put the hot pink strip next to the train you formed. It should look like this:

Screen Shot 2021-05-30 at 2.38.27 PM.png

To compute $H \div Y$, we need to figure out what goes in the blanks to make the equation in part c, below, true. The first blank must contain the largest c-strip possible that will make the equation true, and the second blank must be shorter than the yellow c-strip. We made a train of yellows, with at most one extra piece to form the length of H. Notice this is a train of yellows (which can be formed into a rectangle) plus an extra piece. Form the two yellow strips into a rectangle and find the c-strip that goes across the width. That c-strip goes in the first blank, and the extra piece (R) goes in the second blank. Fill in the blanks of the equation below. This equation should make sense by looking at the picture above showing H and the train are the same length.

c. H = ____ $\cdots Y$ + ____

The QUOTIENT is in bold, and the REMAINDER is underlined.

We can now conclude that: $H \div Y$ = $\mathbf{R}$ remainder $\underline{R}$, since $H = \mathbf{R} \cdots Y$ + $\underline{R}$

This can be written as a statement about numbers by translating each c-strip to a number.

Using numbers in the above equation, $12 \div 5$ = $\mathbf{2}$ remainder $\underline{2}$, since 12 = $\mathbf{2} \cdots 5$ + $\underline{2}$

Exercise 2

For a - e, do the division using the c-strips to make a train of as many of the divisor as possible such that the length of the train you make is less than or equal to the length of the dividend, and if you were to add one more of the divisor to the train, it would be longer than the dividend. If the train ends up being the same length as the dividend, there will be no remainder and the second blank will be empty. Draw a diagram showing the c-strip that is the dividend and under it, draw the train you formed. Do this the same way we did it for exercise 1. Follow the example below.

Example: $H \div Y$

$H \div Y$ = $\mathbf{R}$ remainder $\underline{R}$, since H = $\mathbf{R}$ $\cdots$ Y + $\underline{R}$

$12 \div 5$ = $\mathbf{2}$ remainder $\underline{2}$, since 12 = $\mathbf{2}$ $\cdots$ 5 + $\underline{2}$

Show diagram below: