Chapter 5
Multiplication and Division I: Meaning
5.1 Multiplication as Repeated Addition
Multiplication is not really a basic operation. As the problems in the following activity show, it is possible to solve many “multiplication” problems by using a simpler operation.
Activity 5.1A
A. Solve the following problems using addition and appropriate units. Draw pictures if it is helpful to do so.
1. Three children are playing a game. Each child gets four cards. How many cards are in use?
2. A rectangular baby quilt is made of four strips each containing six squares. How many squares are in this quilt?
3. Rachel has two pairs of shorts and three T-shirts. Assuming she is indifferent to color coordination, how many outfits does she have?
4. A water bottle has a capacity of 11/2 liters of water. How many liters of water can five of these bottles hold?
B. Answer the following.
1a. Each of the problems in part A involved repeated ______.
b. Each of the problems in part A could have been solved more efficiently using what operation? ______
c. Thus multiplication can be defined as ______
2. Consider the following sets.
♥ ♥ ♥ ♥ ♥ ♥
a. There are _____ sets with ______hearts in each set. The union of these sets includes six ______.
b. In other words, 3 ∙ (2 ______) = 6 ______
c. In this problem, 3 refers to the ______of sets and 2 refers to the ______of a set.
3. Reconsider problem #4 in part A. Five referred to the ______of bottles and three quarters of a liter referred to the ______of a bottle.
4. In these situations, it seems that one of the numbers in a multiplication refers to the ______and the other refers to the ______.
In all of the above problems, answers can be found by using repeated addition. There are so many situations involving repeated addition that this process is called multiplication. (Be warned, however, that repeated addition is not the only meaning of multiplication. We will study another meaning in a later section of this chapter.)
Basic Definition of Multiplication as Repeated Addition
For m a whole number, the product m • B is the total number of objects in m disjoint sets, each
containing B elements. m is called the multiplier and B is called the multiplicand.
m • B = B + B + B + . . . + B
m times
The two numbers m and B play two very different roles in this basic meaning of multiplication. The multiplier m is the number of sets while the multiplicand B is the size of the set. The result of a multiplication is called a product. In situations in which multiplication is defined as repeated addition, the multiplicand can be any type of number but the multiplier must be a whole number.
Total = (Number of sets) • (Size of the set)
↓ ↓ ↓
Product = Multiplier • Multiplicand
Example 1: Melissa invited all of her running friends over for a morning run followed by brunch. She bought three dozen eggs for the occasion. How many eggs did she buy?
Total number of eggs = 3 sets of 12 eggs = 12 eggs + 12 eggs + 12 eggs = 3 · 12 eggs = 36 eggs
“Of” and “Times”
Notice that “of” is the word we often use to describe the size of a set. For instance, we might say that a platoon includes three squads of 10 soldiers. This phrasing indicates that the total number can be found by repeated addition, a.k.a., multiplication. IThus the use of the word “of” can be a signal to multiply. Conversely, “times” can often be translated as “of.” For example, “3 times 5” can be interpreted to mean “3 sets of five” or 3 fives.
Teaching Tip: Sometimes children are told that “of” means“times.” This is a misleading overgeneralization. “Of” is one of the most common words in the English language and often does not mean “times.” For example, in the following sentence, “Nine of the 12 students in the class passed the test,” it would be nonsensical to multiply 9 by 12. It actually makes more sense to say that “times” often means “of.”
Factors and Multiples
The multiplier and multiplicand are also called factors. A whole number product is called a multiple of each factor.
Example 2: Consider 3 · 2 = 2 + 2 + 2 = 6.
a. 3 is the multiplier, 2 is the multiplicand, and 6 is the product.
b. 2 is the size of the set, and 3 is the number of sets.
c. 3 and 2 are factors of 6, while 6 is a multiple of 3 and 2.
Every whole number except 0 has a finite number of whole number factor, but all whole numbers have an infinite number of whole number multiples.
Example 3: Set of factors of 6 = {1, 2, 3, 6}; set of multiples of 6 = {0, 6, 12, 18, . . .}
Teaching Tip: Students often confuse factors with multiples. For instance, a student might say that 3 is a multiple of 6 or that 12 is a factor of 6. Since these are important vocabulary words, teachers need to spend time making sure students learn which is which. Mnemonic devices such as “Factors are first” or “Multiples multiply monotonously” may be helpful to some students.
As the next examples indicate, many different notations are used to indicate multiplication.
Example 4: (a) Product of 2 and 3 = 2 times 3 = 2 threes = 2 ´ 3 = (2)(3) = 2(3) = 2 * 3 = 2 • 3
(b) Product of x and y = xy = x • y
Units in Repeated Addition
A sum has the same unit as its terms. For example, 3 feet + 3 feet is equal to 6 feet. Similarly, since the basic meaning of a product is the repeated sum of multiplicands, the product has the same unit as the multiplicand.
Example 5: Five yardsticks are placed end to end. How many feet is it from one end to another?
5 • 3 feet = 3 feet + 3 feet + 3 feet + 3 feet + 3 feet = 15 feet
Activity 5.1B
A. Fill in the blanks, representing the total as a repeated addition. Include units.
Multiplier Multiplicand Total
Ex: Three days a week Heidi walks 13/4 miles. 3 13/4 mi. 13/4 mi. + 13/4 mi. + 13/4 mi. = 51/4 mi
How many miles does she walk every week?
1. Sara has two classes of 20 students. How ______
many students does she have altogether?
2. Peter buys three ½-gallon bottles of milk. ______
How many gallons of milk has he bought?
B. Answer the following questions.
1a. Find the area of the shaded shape on the centimeter grid to the right. ______
b. What is the shape of the standard unit for measuring area? ______
2a. Suppose each cube to the right measures 1 cm by 1 cm by 1 cm.
What is the total volume of this set of cubes? ______
b. What is the shape of the standard unit for measuring volume? ______
Four Major Situations Involving Repeated Addition
1. Distinct Repeated Sets
Example 6: Consider the problem in which each of three children has four cards. How many cards are there altogether?
We have three sets of four: 3 • 4 cards = 4 cards + 4 cards + 4 cards = 12 cards.
The most obvious case of repeated sets occurs with a repeating set of physical objects. This physical set may be a hand of cards, a platoon of soldiers, a case of soft drinks, and so on.
2. Arrays
Consider the situation in which Rachel has three T-shirts and two pairs of shorts. The following diagram illustrates one way to determine that Rachel can put together a total of six different outfits.
A horizontal arrangement of objects is called a row and a vertical arrangement is called a column. The above diagram, with 2 rows and 3 columns, is an example of a 2 by 3 array. An R by C array is a set of discrete objects arranged into R rows and C columns. Because the rows of an array are the same size, the total number of elements in an array can be found by repeatedly adding the rows. Since the row size is the same as the number of columns, we have the following generalization.
The total number of elements in an R by C array is R • C.
This explains why an R by C array is also described as an “R ´ C array.”
Example 7: ۞ ۞ ۞ ۞ ۞ This is a 2 ´ 5 array, with two rows and five columns.
۞ ۞ ۞ ۞ ۞ Total number of elements = 2 • 5 = 5 + 5 = 10
3. Area and Volume
What is the total number of squares in a baby quilt made of four strips of six squares each?
This is another example of a problem that can be solved by repeated addition. The quilt
consists of four rows, each containing six squares. The total number of squares is equal
to the following: 4 sixes = 6 squares + 6 squares + 6 squares + 6 squares = 24 squares.
This quilt also illustrates why the area of a rectangle can be found by multiplying its length by its width.
Finding the number of squares in a rectangle is analogous to finding the number of elements in an array.
Rectangles as Arrays of Squares
Array with 8 elements Rectangle with an area of 8 squares
Generally speaking, we measure the area of a two-dimensional shape using squares. The squares in a rectangle form an array in which the number of rows corresponds to the length of the rectangle, while the number of columns corresponds to the width. Thus the area of a rectangle is the product of its length and width.
B
Formulas for the areas of other special shapes are derived from this basic area formula.
H
Example 8: The area of a right triangle with legs of length B and H is ½BH because
its area is half the area of a rectangle with length B and width H.
One special area is not directly derived from the area of a rectangle. The area of a circle is equal to π r2, where r is the radius of the circle.
As the following example illustrates, the area of many figures can be found by partitioning the figure.
Example 9: To find the area of the figure given below, partition it as indicated.
6 cm 6 cm Area Half-circle = 0.5 π (3.8 cm)2 ≈ 22.68 cm2
3.8 cm
7.6 cm Area Rectangle = 6 cm · 7.6 cm ≈ 45.6 cm2
16.8 cm 3.8 6.0 7.0 Area Triangles = 2 · (0.5 · 3.8 cm 7.0 cm) = 26.6 cm2
Area Total = 94.88 cm2
Volume
1″
The standard unit for measuring volume is a cube. A cube that measures one unit 1″
by one unit by one unit has a volume of one cubic unit. As the following activity
illustrates, the volume of the three-dimensional analog of a rectangle can be found 1″
by repeated addition of layers of cubes.
One Cubic Inch
Activity 5.1C
1. A solid box has a length of 4 cm, a width of 2 cm, and a height of 3 cm.
______a. What is the area or the bottom (or top) of this box?
______b. How many cubic centimeters are in the first layer of this box?
______c. How many layers does the box have?
______d. Use the above facts to determine the volume of the box.
2. What is the volume of a box that is 5'' high, 10'' long, and 3'' deep? ______
3. A cylindrical water tank is 20 feet high. It is known that when the water is one foot deep,
the volume of water in the tank is about 700 cubic feet. What is the capacity of the tank? ______
[Hint: Think about the volume of each layer.]
The formal name of a typical box is a right rectangular prism. It has rectangular faces
at right angles to each other. A right rectangular prism with length L, width W, and height H 1 1
can be partitioned into a series of identical one unit thick layers. The volume of one of these
layers has the same numerical value as L· W, the area of the “floor” or base of the prism. 1
Since the number of layers corresponds to the height of the solid, the volume of the right
rectangular prism is as follows. 1 W
L
Volume of a right rectangular solid = length • width • height
Volumes of Solids with Congruent Bases
In general, a prism is any solid with two congruent and parallel polygonal bases connected by parallel lines. This means that the other faces of a prism are parallelograms.
Various Prisms
A prism is a special type of cylinder. A cylinder is any solid with two congruent and parallel bases, not necessarily polygonal, that are connected by parallel lines.