21 The Meaning of Division

There are many ways to view division and it depends on the situation as to which visualization is the most useful. For example, if you have 20 students in your class and want to split them into 4 groups, it is different from wanting to split them into groups of size 4. But, both situations show that 20 divided by 4 equals 5. It’s important for students to be comfortable with both visualizations for success in their later mathematics. Let’s dive into these two methods.

At first, you’ll be using c-strips to explore one visualization of division.

Exploration Activity

In this activity, we will divide H by Y, which we write as [latex]H \div Y[/latex]. First, lay out one hot pink c-strip (H). Separately, line up yellow c-strips (Y) one-by-one until you have come as close as possible to matching the hot pink c-strip without being longer than it. You know you are done if adding one more yellow c-strip makes the train longer than the hot pink c-strip.

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

Solution

Two yellow c-strips.

b.

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:

What did we just do? We took our first number (H) and subtracted the second number (Y) over and over until we could not subtract anymore without getting a negative number. Then, we figured out how much of H was leftover.

We have just visually represented [latex]H \div Y[/latex].

Definition – Dividend, Divisor, Quotient, Remainder

Consider the division problem [latex]H \div Y[/latex].

• The part before the division sign, H, is called the dividend.
• The part after the division sign, Y, is called the divisor.

If H and Y are whole numbers with [latex]Y \leq H[/latex]

• The quotient is the maximum number of times the divisor can be subtracted from the dividend without resulting in a negative number.
•  The remainder is the difference between the dividend and the product of the quotient and divisor.

Teaching tip: Avoid using phrases like “the dividend is the larger number and the divisor is the smaller number” or “big divided by small.” This is not true as students advance in mathematics and need to compute things like ¼ ÷ 5.

 

To compute [latex]H \div Y[/latex], we need to figure out what goes in the blanks to make the equation in part c, below, true. The first blank will contain the number of yellow c-strips used, and the second blank is the length of the c-strip used for the remainder. 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.

We can now conclude that: [latex]H \div Y[/latex] = [latex]\mathbf{Q}[/latex] remainder [latex]\underline{R}[/latex], since [latex]H = \mathbf{Q} \cdot Y[/latex] + [latex]\underline{R}[/latex]. The QUOTIENT is in bold, and the REMAINDER is underlined. This can be written as a statement about numbers by translating each c-strip to a number given by the strip’s length.

Writing in the value of our strips, our picture transforms into the following.

Using numbers in the above equation, [latex]12 \div 5[/latex] = [latex]\mathbf{2}[/latex] remainder [latex]\underline{2}[/latex], since 12 = [latex]\mathbf{2} \cdot 5[/latex] + [latex]\underline{2}[/latex]

Classroom Connection: You ask students to represent [latex]29\div4[/latex] and one of them draws the following picture. What is their mistake?

Solution

The remainder, 5, is larger than the divisor, 4. The student stopped subtracting 4’s too soon. They should have used one more piece of length 4 before figuring out the remainder.

 

 

Exercise 1

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 zero. 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 the exploration activity. Follow the example below.

 

Example: [latex]\color{white} H \div Y[/latex]

[latex]H \div Y[/latex] = [latex]\mathbf{2}[/latex] remainder [latex]\underline{2}[/latex], since H = [latex]\mathbf{2}[/latex] [latex]\cdot[/latex] Y + [latex]\underline{2}[/latex]

[latex]12 \div 5[/latex] = [latex]\mathbf{2}[/latex] remainder [latex]\underline{2}[/latex], since 12 = [latex]\mathbf{2}[/latex] [latex]\cdot[/latex] 5 + [latex]\underline{2}[/latex]

Show diagram below:

 

Solution(Diagrams)

1. 1. 
   2. 
   3. 
   4. 
   5. 

A lot of elementary school curricula introduce division through base blocks. The blocks are used to visually represent the division problem. Get out your base blocks and let’s explore division, like your future students might do.

Exercise 6

Compare your picture from exercise 5 with those below. Was your answer similar to one of these pictures? Which pictures are correct?

Picture 1

     Picture 2

Picture 3

Solution

All three pictures are correct. The first two show three groups of size 16. The third shows both 3 groups of size 16 and 16 groups of size 3 depending on if you look at rows or columns.

Note that there are other possible pictures, such as 36 circles, each with 3 blocks inside.

 

Based on Exercise 6, we see that there are multiple ways of thinking about and visualizing division. The two word problems below both describe the division [latex]18\div 3[/latex], but the problems are really quite different.

Both require the division problem [latex]18 \div 3[/latex], and the answer to each problem is 6. But the picture used to solve each is very different.

In Word Problem A, you are told how many kids there are, and you are asked to figure out how many cookies each child should be given. Imagine 3 children in front of you. You need to disburse the cookies amongst the 3 children so that each receives the same number of cookies. To begin, you might draw a model to represent three kids. It might look something like this:

You know there are 3 kids and each will be given the same number of cookies so that the total number of cookies they all have is 18. After disbursing, you might show the picture like this:

This is a way to illustrate that each of the 3 children has 6 cookies. If you prefer, you can actually draw 6 dots or 6 cookies in each circle. It’s up to you.

Note that if you look at the model drawn, it is simply the multiplication problem 3 [latex]\cdot[/latex] 6, since there are 3 sets/groups of 6 added together. The model of division Word Problem A represents is called:

Partitioning into (Equal) Subsets Model of Division

In this model, the divisor is the number of subsets/groups in which to disburse something. The dividend, 18, must be divided amongst 3 equal subsets (3 is the divisor). The quotient (answer to the division problem) will be the number of elements (6) placed in each subset (the kids, in this case.) The task is to partition 18 cookies into 3 equal subsets. In this model, you are given how many subsets there are (from reading the problem in this case, each kid represents one subset), but you do not how many are in each subset (that is the question being asked in this case, how many cookies each child should receive). You do the division to figure out how many elements will end up in each subset. After doing the division, the number of elements IN each subset is the answer to the problem.

One way to distinguish that a problem is suggesting this model is to consider whether the information in the problem provides you with the number of subsets. If it does, as in Word Problem A, then it is the partitioning into equal subsets model.

Okay, let’s look at Word Problem B now.

In Word Problem B, you don’t know how many subsets there will be. In fact, you are being asked  how many subsets you will be able to make, so the partitioning into subsets interpretation wouldn’t apply.

What you do know is how many of something will be in each subset. You would place three cookies in a baggie, then 3 more in another baggie, and so on. You won’t know how many subsets you will have until you finish disbursing the cookies. You would take three cookies away at a time and put them in a baggie, until you ran out of cookies. After doing this, a picture you might draw to illustrate this procedure might look like this:

Again, if you prefer to show three dots in each circle, as opposed to just writing the numeral 3, in each circle, that is perfectly fine. Either way, it should be clear from the drawing what procedure was used.

After disbursing all the cookies, you would count up how many baggies you were able to make, and that would give you the answer. Note the answer is 6, in this case. The model shown represents the multiplication [latex]6 \cdots 3[/latex], since there are 6 baggies of 3 cookies each.

The Model of division Word Problem B represents is called:

Repeated Subtraction or Measurement Model of Division

In this model, the divisor is the number of elements that will go into each subset/group. Each time the divisor (3) is subtracted from the dividend (18), a subset is formed. After all elements have been disbursed, the quotient (answer) is found by counting how many subsets were formed. This model is used when you are given how many elements are in each subset, and you are trying to find out how many subsets can be made (that is the question being asked). You do the division (by using repeated subtraction) to figure out how many subsets can be made. After doing the division, the number of subsets you made is the answer to the problem.

One way to distinguish that a problem is suggesting this model is to consider whether the information in the problem provides you with how many elements go into a subset. If so, as in Word Problem B, then it is the repeated subtraction model.

Word Problems requiring division generally fall into the category of Repeated Subtraction or Partitioning into Subsets. After reading a word problem, you should be able to analyze and interpret it as a problem that requires Partitioning into Subsets or Repeated Subtraction to solve the problem, and then be able to draw a picture using that model. Also, for any division problem given, you should be able to write two distinct word problems, one for each model, and draw a picture for each model, and be able to explain and show all the steps necessary to solve the problem.

If you are not presented with a word problem, but are simply given a division problem, you can use either model to think about and solve the problem. Many times, it is easier to use one model over another, especially if you need to draw a picture to show how you solved the problem.

Consider the following division problem: [latex]26 \div 2[/latex]. Let’s say the 26 represented 2 pennies. Would it be easier to partition 26 pennies into 2 piles and count how many are in each pile; or would it be easier to make several equal subsets containing 2 pennies each from the 26 pennies, and then to count how many subsets there are? The picture you draw to represent each model is quite different. Look at the difference:

Partition:

Repeated Subtraction:

While these models are a great way to visualize division, they are not technical definitions. The actual definition relies on the fact that division is the operation that undoes multiplication. Students can rely on this fact to learn basic division facts. For instance, since 6 [latex]\cdot[/latex] 7 = 42, then [latex]42 \div 6 = 7[/latex] and [latex]42 \div 7 = 6[/latex]. Below is a formal definition of division

Instead of writing c, it is sometimes easier to write a blank or box in the equations to see if a unique solution exists. If you can put a unique solution in the blank to the multiplication problem, then you can go back and fill in the solution to the division problem.

Exercise 20f illustrates that any number divided by 1 is itself.

In other words,

Exercise 20g illustrates that zero divided by any nonzero whole number is 0.

In other words,

Why did we specify nonzero values of a? You have always heard “You cannot divide by zero,” but why is that? Let’s explore division by zero using our models and our definition.

In exercise 20, you showed that dividing a nonzero number by 0 was undefined. Now, let’s explore what happens if we divide zero by zero.

Problem: Prove that division by zero is undefined

Solution: The goal is to show that [latex]a \div 0[/latex] is undefined for all values of a. There are two cases to consider, either [latex]a \neq 0[/latex] or [latex]a = 0[/latex].

Case 1: Let [latex]a \neq 0[/latex]. There is no unique number that could be put in the blank to make the equation, [latex]a = 0 \cdot\underline{\qquad}[/latex], true because the left-hand side of the equation is not zero (since [latex]a \neq 0[/latex]), and any number put in the blank will make the right hand side of the equation zero. Therefore, when [latex]a \neq 0[/latex], [latex]a \div 0[/latex] is not defined.

Case 2: Let [latex]a = 0[/latex]. There is no unique number that can be put in the blank to make the equation [latex]0 = 0 \cdot\underline{\qquad}[/latex] true, since any number will make it true. Therefore, [latex]0 \div 0[/latex] is indeterminate.

It takes practice writing this proof. You’ll need to practice writing it several times to master the problem. As you rewrite it over and over, focus on the flow of the logic until you really get the idea. Being able to express oneself precisely in mathematical language shows a deep level of understanding necessary to be a high-quality teacher.

So far, when using the definition, we have limited ourselves to problems where the answers were whole numbers or the problem divided by zero and was undefined. If you were asked to use the definition to find only whole number solutions, then some non-zero divisions, such as [latex]15\div 2[/latex] would be undefined under the whole numbers. There may, however, be rational or irrational solutions depending on the problem. For example, [latex]15=2\cdot 7.5[/latex]. Even though there is a possible solution in the rational numbers, we say [latex]15\div 2[/latex] is undefined under the whole numbers, because the solution is not a whole number.