By the end of this section, you will be able to:
Suppose you were asked to count all these pennies shown in Figure 1.11.
Would you count the pennies individually? Or would you count the number of pennies in each row and add that number 3 times.
Multiplication is a way to represent repeated addition. So instead of adding 8 three times, we could write a multiplication expression.
We call each number being multiplied a factor and the result the product. We read 3×8 as three times eight, and the result as the product of three and eight.
There are several symbols that represent multiplication. These include the symbol × as well as the dot, ⋅, and parentheses ().
There are many ways to model multiplication. Unlike in the previous sections where we used base-10base-10 blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.
In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section.
Table 1.4 shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.
What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.
What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and 11 is called the multiplicative identity.
Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that 8+9=17 is the same as 9+8=17.
Is this also true for multiplication? Let’s look at a few pairs of factors.
When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.
We will use the same strategy we used previously to solve applications of multiplication. First, we need to determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify to get the answer. Finally, we write a sentence to answer the question.
If we want to know the size of a wall that needs to be painted or a floor that needs to be carpeted, we will need to find its area. The area is a measure of the amount of surface that is covered by the shape. Area is measured in square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm.) on a side. A square inch is a square that is one inch on each side, and so on.
For a rectangular figure, the area is the product of the length and the width. Figure 1.12 shows a rectangular rug with a length of 22 feet and a width of 33 feet. Each square is 11 foot wide by 11 foot long, or 11 square foot. The rug is made of 66 squares. The area of the rug is 66 square feet.
Figure 1.12 The area of a rectangle is the product of its length and its width, or 66 square feet.
Use Multiplication Notation
In the following exercises, translate from math notation to words.
Model Multiplication of Whole Numbers
In the following exercises, model the multiplication.
Multiply Whole Numbers
In the following exercises, fill in the missing values in each chart.237.
In the following exercises, multiply.
In the following exercises, translate and simplify.
In the following exercises, simplify.
Multiply Whole Numbers in Applications
Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price rose $12$12 per share. How much money did Javier’s portfolio gain?
Salary Carlton got a $200$200 raise in each paycheck. He gets paid 24 times a year. How much higher is his new annual salary?
How confident do you feel about your knowledge of the multiplication facts? If you are not fully confident, what will you do to improve your skills?342.
How have you used models to help you learn the multiplication facts?
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?