25 Module 4.1 Definition and Properties

You will need: Centimeter Strips

Multiplication is one of the most important operations in mathematics, and understanding how it works helps us solve problems more easily and efficiently. As we explore multiplication, we can discover several special properties that make it powerful and flexible: the commutative, associative, and distributive properties. These properties show us that the order and grouping of numbers in multiplication don’t change the result, and that multiplication connects closely with addition. By investigating these patterns through examples and exercises using C-strips, we can see how multiplication follows consistent rules that work every time.

Here is a reminder of the different C-Strips we'll be working with, along with the abbreviation used for each, highlighted in bold:

Now it's your turn to try an exercise.

Exercise 2

Exercise 3

Note: A train can consist of one or more C-strips. In other words, a single white (or any other color) C-strip is actually a train. There doesn't have to be a caboose and engine!

 

Exercise 4

Follow the steps to fill in the blanks to each of the following problems.

• Identify the length of the C-strip after the multiplication sign using the chart.

• Build a train equal to the C-strip on the right side of the equation.

• Form it into a rectangle.

• Find a C-strip that fits across the width of the rectangle (this will be the missing factor).

• Write the abbreviation/color for that C-strip.

Exercise 5

a. Take out the Hot Pink (H) C-strip. Use your C-strips (one color at a time) to see if you can form a train made up of C-strips of the same color equal in length to the hot pink C-strip. In other words, see if you can do it with all whites (always possible for any train), or all reds, or all light greens, etc. You should be able to make six different trains. Each train is made up of a single color. Draw a picture of each of these trains under the Hot Pink one shown. I've drawn two trains for you already, one is simply the hot pink strip (a train consisting of just one c-strip), and the second is made up of three purple C-strips.

Solution

[Click here.]

 

 

b. Your work from exercise 4 should help complete this exercise.

• Take each train, one at a time, and put each strip in a vertical position, but side by side to form a rectangle.
• Find a C-strip that fits across the width (or top) of the rectangle. This should tell you from which multiplication problem each train was formed.
• Write an equation using C-strips and then translate it to an equation with numbers. For instance, for train 2, first I would make a rectangle out of the three purple C-strips. Next, I would try to find a C-strip to fit across the top, which would be light green. Therefore, train 2 was formed from the multiplication [latex]L \times P[/latex]. Remember that since the train is formed with purple C-strips, P is the second letter in the multiplication. So, the equation in C-strips is [latex]L \times P = H[/latex], and the numerical equivalent is [latex]3 \times 4 = 12[/latex].
• Follow this same procedure for the other four trains you made in part a.

Train 1 illustrates the multiplication [latex]W \times H = H[/latex], or [latex]1 \times 12 = 12[/latex]. (Note that if the hot pink strip is vertical, the white c-strip fits across the top.)

Train 2 illustrates the multiplication [latex]L \times P[/latex] = H, or [latex]3 \times 4 = 12[/latex]

Exercise 6

For train 2 in exercise 5, I made a train of purple C-strips equal in length to the hot pink C-strip. Now, make a train of purple C-strips equal in length to the brown (N) C-strip. Take the train of the purple C-strips and make it into a rectangle.

 

Now, we'll explore more about the equations formed in Exercise 4. Consider part a. [latex]D \times R = H[/latex] is analogous to the statement [latex]6 \times 2 = 12[/latex]. But, if you actually look at the original train that was formed to find the answer, you would see that there were 6 red C-strips that made a train equal in length to the hot pink C-strips. This gives rise to one of the whole number definitions of multiplication. Multiplication is really just the same thing as Repeated Addition.

Definition: Multiplication of Whole Numbers using the repeated addition approach

 

If a and b are whole numbers, then a [latex]\times[/latex] b = b + b + b + ... + b, where there are a addends of b in this sum. 

 

Examples of Definition Application

3 [latex]\times[/latex] 4 means 3 fours added together or 3 [latex]\times[/latex] 4 = 4 + 4 + 4 = 12

4 [latex]\times[/latex] 3 means 4 threes added together, or 4 [latex]\times[/latex] 3 = 3 + 3 + 3 + 3 = 12

5 [latex]\times[/latex] 2 means 5 twos added together, or 5 [latex]\times[/latex] 2 = 2 + 2 + 2 + 2 + 2 = 10        

2 [latex]\times[/latex] 5 means 2 fives added together, or 2 [latex]\times[/latex] 5 = 5 + 5 = 10

Note: Although 3 [latex]\times[/latex] 4 gives the same answer as 4 [latex]\times[/latex] 3, it does not mean the same thing! One means 4 + 4 + 4 and the other means 3 + 3 + 3 + 3.

Let's look at a real world example of this.

Maria has 3 friends. She wants to give each friend 4 balloons for her party.

Maria could think of this as: Each friend gets 4 balloons → 4 + 4 + 4 = 12

Maria could also think of this as: There are 4 different balloon colors and each color goes to her 3 friends → 3 + 3 + 3 + 3 = 12

 

Before we delve any further into this discussion, I want you to use the repeated addition definition to compute the following:

Exercise 7

Use the repeated addition definition of multiplication to compute the following. First, write out the meaning of the multiplication, and then compute the answer.

Now, it may seem obvious to you that 3 [latex]\times[/latex] 4 is the same answer as 4 [latex]\times[/latex] 3. That's because you've no doubt been using the commutative property of addition for years. Whether or not you really thought about why it was true is another thing. But, although 3 [latex]\times[/latex] 4 gives the same answer as 4 [latex]\times[/latex] 3, it does not mean the same thing. One means 4 + 4 + 4 and the other means 3 + 3 + 3 + 3. To a child just learning about addition and multiplication, there is no reason for them to conclude that one sum (4 + 4 + 4) happens to give the same answer as the other sum (3 + 3 + 3 + 3). For instance, if I were asked to compute the following two sums, it wouldn't be evident to me at all that the answer to each would end up being the same:

17 + 17 + 17 + 17 and 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4

Actually, I find it quite amazing that they are equal. But, indeed they are, because the first sum comes from the multiplication problem 4 [latex]\times[/latex] 17 (since there are 4 seventeens added together), and the second sum comes from the multiplication problem 17 [latex]\times[/latex] 4 (since there are 17 fours added together). Later, when we discuss a more geometrical definition for multiplication, it becomes obvious why the commutative property of multiplication is true.

 

The very exciting thing about the commutative property of multiplication is that, given a problem like 100 [latex]\times[/latex] 7, where it would take a long time to write out and compute the answer using the strict definition of multiplication by adding 7 over and over again, we can use our knowledge of the commutative property, that is 100 [latex]\times[/latex] 7 = 7 [latex]\times[/latex] 100, and compute the second multiplication instead. Wouldn't you agree that 100 + 100 + 100 + 100 + 100 + 100 + 100 is quicker and easier to compute than adding together 100 sevens? Thank goodness for the commutative property of multiplication!!!

 

Let's recall that the commutative property requires two numbers (they don't have to be two different numbers), and there are two sides of an equation to verify. The difference between the expressions a [latex]\times[/latex] b and b [latex]\times[/latex] a is that the order of the numbers is different. Remember that the commutative property means there has been an order change. To verify the commutative property of multiplication, you need to use the repeated addition definition of multiplication and add from left to right, one number at a time. The next example shows how to verify the commutative property of multiplication, given a set of two numbers.

Examples

a.  Verify the commutative property of multiplication is true for the given set of numbers. You'll have to remember how to add in base six to do part b.

a. {5, 3}

Solution

Since [latex]5 \times 3[/latex] = 3 + 3 + 3 + 3 + 3 = 6 + 3 + 3 + 3 = 9 + 3 + 3 = 12 + 3 = 15 and since [latex]3 \times 5[/latex]= 5 + 5 + 5 = 10 + 5 = 15, then [latex]5 \times 3 = 3 \times 5[/latex], because both equal 15.

 

b. {\(3_{six} \), \(4_{six} \)}

Solution

Since \(3_{\text{six}} \times 4_{\text{six}} \)= \(4_{\text{six}} + 4_{\text{six}} + 4_{\text{six}}\) = \(12_{\text{six}} + 4_{\text{six}}\) = \(20_{\text{six}}\) and since \(4_{\text{six}} \times 3_{\text{six}}\) = \( 3_{\text{six}} + 3_{\text{six}} + 3_{\text{six}} + 3_{\text{six}}\) = \(10_{\text{six}} + 3_{\text{six}} + 3_{\text{six}}\) = \(13_{\text{six}} + 3_{\text{six}} = 20_{\text{six}}\), then \(3_{\text{six}} \times 4_{\text{six}} = 4_{\text{six}} \times 3_{\text{six}}\), because both equal \(20_{\text{six}}\).

 

 

 

 

Exercise 8

Verify the commutative property of multiplication by writing an equation that must be true and using the repeated addition definition of multiplication, is true using the given set of numbers. Simplify each expression separately; then show they are equal.

a. {4, 3} Solution Since [latex]4 \times 3[/latex] = 3 + 3 + 3 + 3 = 6 + 3 + 3 = 9 + 3 = 12 and since [latex]3 \times 4[/latex]= 4 + 4 + 4 = 8 + 4 = 12, then [latex]4 \times 3 = 3 \times 4[/latex], because both equal 12.

b. {\(2_{five} \), \(3_{five} \)} Solution Since \(2_{\text{five}} \times 3_{\text{five}} = 2_{\text{five}} + 2_{\text{five}} + 2_{\text{five}} = 4_{\text{five}} + 2_{\text{five}} = 11_{\text{five}}\) and since \(3_{\text{five}} \times 2_{\text{five}} = 3_{\text{five}} + 3_{\text{five}} = 11_{\text{five}}\), then \(2_{\text{five}} \times 3_{\text{five}} = 3_{\text{five}} \times 2_{\text{five}}\), because both equal \(11_{\text{five}}\).

c. Make up a set in a base other than 10 and verify the commutative property of multiplication. (Various answers)

We can also use the repeated addition definition of multiplication to multiply in other number systems. Study the following examples and then work through the problems in Exercise 10.

Example in Tally

Example in Egyptian

Example in Mayan

Notice that all of these examples are done completely in the numeration system given. All of the steps are shown in that numeration system. This is how you should do the problems in Exercise 9. If you want, you can check the answer after you complete the problem by converting to base ten. For instance, the Mayan example in base ten is 2 [latex]\times[/latex] 132. So the answer should be 264. If you convert the final answer to base ten, it is also 264.

 

Exercise 9

Use the repeated addition definition of multiplication to compute the following. Make sure you write out the meaning of the multiplication in the system given, showing all of the work and exchanges (if necessary), to compute the answer. Do not do the problem in base ten, although you can use base ten to check the answer when you are done.

a.

b.

c.

Now we are ready to explore another property that we will use to multiply three trains.

Note: If this were a question on a test, you would only have to verify one of the six equations shown; there are always six possibilities one might choose, depending on which numbers you put in for a, b, or c.

Time to work on another property...

Now, it's time for more practice verifying the distributive property of multiplication over addition. The distributive property requires three numbers (they don't have to be three different numbers), and there are two sides of an equation to verify. Note the difference between the left and right sides. To verify, you need to use the order of operations when simplifying each side. In the order of operations, you always simplify what is in parentheses first. If each expression simplifies to the same thing, the equation is true.

The following are some examples of how to verify the distributive property of multiplication, given a set of three numbers.To verify the distributive property of multiplication is true by example, first you must write an equation that must be true using any three numbers (or by using three particular numbers given to you). Then, by doing the order of operations on each side (each individual expression), you must show that each side of the equation gives the same result.

Remember, if this were a question on a test, you would only have to verify one of the six equations shown. There are always six possibilities one might choose, depending on which numbers you put in for a, b, or c.

Someone who needed to multiply [latex]57 \times 102[/latex] might find it easier to think of the problem as [latex]57 \times (100 + 2) = (57 \times 100) + (57 \times 2) = 5700 + 114 = 5814[/latex].

Someone who needed to compute [latex]47 \times 38 + 47 \times 62[/latex] might realize this is really just the same as [latex]47 \times (38 + 62) = 47 \times 100 = 4700[/latex].

This next example illustrates that multiplication also distributes over subtraction:

Someone who needed to multiply [latex]38 \times 99[/latex] might find it easier to think of the problem as [latex]38 \times (100 - 1) = (38 \times 100) - (38 \times 1) = 3800 - 38 = 3762[/latex].

There is another way to define multiplication using Set Theory. In order to do this, we have to remember how to take the Cartesian product of two sets, A and B.

If you are finding the Cartesian product, the answer is a set that contains ordered pairs.

In other words, to multiply two numbers, a and b, write one set A which has a elements in it and another set B that has b elements in it. To find the product, write the Cartesian
product of A and B, and count the elements in the set.
Note: There are no restrictions on what elements you choose for sets A and B. They can be disjoint or can have elements in common. It’s up to you.

The cool thing about using this geometrical approach is that any object can be used, and one only needs to be able to count to get the answer. You don't need to use addition in order to multiply! I personally advocate having lots of graph paper around for anyone just learning to multiply. To compute [latex]4 \times 7[/latex], you or the learner can mark off a rectangle on the graph paper that has 4 rows and 7 columns of boxes. The number of boxes in the rectangle is the answer to the problem. This is a useful tool for first learning a particular multiplication fact, or for anyone (adults, too) who forgot a fact and needs to figure it out again. The "objects" used on a graph paper are the individual boxes on the graph paper.

Here is how six different students might geometrically show how to multiply [latex]4 \times 7[/latex].

There is no point in making anyone stress out if they forgot, didn't memorize, or couldn't recall the answer to [latex]4 \times 7[/latex]. The point is to KNOW WHAT MULTIPLICATION MEANS so you can figure a problem out again if you have to. Of course, there is merit to using flashcards and eventually memorizing basic multiplication facts like your multiplication tables. If not, it takes longer to do problems, you aren't able to focus on more advanced issues, and you might not trust your own mind. You shouldn't have to think about multiplication every time you have to multiply two numbers together, but you should be able to think about it. Knowing how to get the answer is better than having memorized a bunch of facts if you don't know where they came from, what multiplication really means, or how to figure it out again in case you forget the answer to some fact. And, as far as I'm concerned, nothing's wrong with using your fingers, either! Let people do what is comfortable for them, if they understand and can get the right answer, that's GREAT!

Exercise 19

Write the multiplication problem represented by each geometrical representation:

a. ____	b. ____
c. ____ [latex]%begin{matrix} blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge \ blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge \ blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge \ blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge \ blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge \ blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge \ blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge \ blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge & blacklozenge end{matrix}nonumber[/latex]	d. ____
e. ____	f. ____

Exercise 19 should convince you that multiplication is commutative. In fact, the commutative property of multiplication is easiest to see using the geometrical approach. A rectangular array of objects having m rows and n columns represents the multiplication [latex]m \times n[/latex]. Once the rectangle is rotated [latex]90^{\circ}[/latex] (or looked at sideways), there are now n rows and m columns, which represents the multiplication [latex]n \times m[/latex]. But of course, it's really the same rectangle of objects, so the number of objects in [latex]m \times n[/latex] equals the number of objects in [latex]n \times m[/latex]. Therefore, [latex]m \times n = n \times m[/latex]. Okay, just for fun, one more time:

The commutative property of multiplication for whole numbers states that if a and b are any two numbers, then [latex]a \times b = b \times a[/latex].

Wow! Isn't this exciting? There are several ways to think about and define multiplication, and we can show that the commutative property still holds each time. Does it get any better than this? Well,...YES...but that's another story.

It's a little bit trickier to show on a two-dimensional page that the associative property of multiplication holds. With your work with the C-strips, you know multiplication is associative. Here's one way to geometrically model the multiplication of three numbers.

Basically, you need to make (or visualize) a 3-dimensional box to do multiplication on three numbers. As it turns out, no matter how you construct it, if at the end, one dimension is a, one is b, and the other is c, you've got the same box, and it holds the same number of cubes in it. Therefore, multiplication is associative. We won't be going through the motions of showing it in 3-dimensions, but elementary school children should have the benefit of working through it with cubes. I hope you'll model it this way if you become an elementary school teacher.

Since multiplication is both commutative and associative, any time more than two numbers are being multiplied together, the parentheses do not have to be shown, and the numbers can be multiplied together in any order.

For instance, notice how differently one might compute [latex]4 \times 10 \times 3 \times 5[/latex]

Student 1: This person just goes left to right. First, [latex]4 \times 10[/latex] is 40, then [latex]40 \times 3[/latex] is 120, then [latex]120 \times 5 = 600[/latex].

Student 2: This person first multiplies [latex]4 \times 5[/latex] to get 20, then multiplies [latex]10 \times 3[/latex] to get 30, then multiplies the two answers together, [latex]20 \times 30[/latex] to get 600.

Student 3: This person does [latex]4 \times 5[/latex] first to get 20, then multiplies by 3 to get 60, then multiplies by 10 to get 600.

Student 4: This person multiplies [latex]3 \times 5[/latex] first to get 15, then multiplies that by 4 to get 60, and then multiplies by 10 to get 600.

And there are many more possibilities. The point is that you have the option of multiplying in any order and any combination you want. The following shows how this is very convenient for certain computations:

Exercise 20 showed that [latex]P \times (Y – L) = P \times Y – P \times L[/latex]. This is one illustration that multiplication distributes over subtraction.