1.4 Multiply Whole Numbers

Learning Objectives

By the end of this section, you will be able to:

  • Use multiplication notation
  • Model multiplication of whole numbers
  • Multiply whole numbers
  • Translate word phrases to math notation
  • Multiply whole numbers in applications

Use Multiplication Notation

Suppose you were asked to count all these pennies shown in Figure 1.11.

An image of 3 horizontal rows of pennies, each row containing 8 pennies.

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.

8+8+8


Multiplication is a way to represent repeated addition. So instead of adding 8 three times, we could write a multiplication expression.

3×8


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 ().

Model Multiplication of Whole Numbers

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.

Multiply Whole Numbers

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.

×0123456789
00000000000
10123456789
2024681012141618
30369121518212427
404812162024283236
5051015202530354045
6061218243036424854
7071421283542495663
8081624324048566472
9091827364554637281

Table1.4

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.

4⋅7=287⋅4=28

9⋅7=637⋅9=63

8⋅9=729⋅8=72

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

Multiply Whole Numbers in Applications

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.

An image of two squares, one larger than the other. The smaller square is 1 centimeter by 1 centimeter and has the label “1 square centimeter”. The larger square is 1 inch by 1 inch and has the label “1 square inch”.

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.

An image of a rectangle containing 6 blocks, 2 feet tall and 3 feet wide. This image has the label “2 times 3 = 6 feet squared”.

Figure 1.12 The area of a rectangle is the product of its length and its width, or 66 square feet.

Section 1.4 Exercises

Practice Makes Perfect

Use Multiplication Notation

In the following exercises, translate from math notation to words.

  1. 4×7
  2. 8×6
  3. 5⋅12
  4. 3⋅9
  5. (10)(25)
  6. (20)(15)
  7. 42(33)
  8. 39(64)

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

  1. 3×6
  2. 4×5
  3. 5×9
  4. 3×9

Multiply Whole Numbers

In the following exercises, fill in the missing values in each chart.237.

An image of a table with 11 columns and 11 rows. The cells in the first row and first column are shaded darker than the other cells. The first column has the values “x; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9”. The second column has the values “0; 0; 0; null; 0; 0; 0; 0; null; 0; 0”. The third column has the values “1; 0; 1; 2; null; 4; 5; 6; null; 8; 9”. The fourth column has the values “2; 0; 2; 4; 6; null; 10; 12; 14; null; 18”. The fifth column has the values “3; null; 3; 6; null; null; 15; null; 21; 24; null”. The sixth column has the values “4; 0; null; 8; 12; 16; null; 24; null; null; 36”. The seventh column has the values “5; 0; null; null; 15; 20; null; null; 35; null; 45”. The eighth column has the values “6; 0; 6; 12; null; null; 30; null; null; 48; null”. The ninth column has the values “7; 0; 7; null; 21; 28; null; 42; null; null; null”. The tenth column has the values “8; null; 8; null; null; 32; 40; null; 56; 64; 72”. The eleventh column has the values “9; 0; null; 18; 27; null, null; 54; 63; null; null”.

238.

An image of a table with 11 columns and 11 rows. The cells in the first row and first column are shaded darker than the other cells. The first column has the values “x; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9”. The second column has the values “0; 0; 0; 0 pink; 0; 0; 0; 0; 0; 0; 0”. The third column has the values “1; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9”. The fourth column has the values “2; 0; 2; 4; 6; 8; 10; 12; 14; 16; 18”. The fifth column has the values “3; 0; 3; 6; 9; 12; 15; 18; 21; 24; 27”. The sixth column has the values “4; 0; 4; 8; 12; 16; 20; 24; 28; 32; 36”. The seventh column has the values “5; 0; 5; 10; 15; 20; 25; 30; 35; 40; 45”. The eighth column has the values “6; 0; 6; 12; 18; 24; 30; 36; 42; 48; 54”. The ninth column has the values “7; 0; 7; 14; 21; 28; 35; 42; 49; 56; 63”. The tenth column has the values “8; 0; 8; 16; 24; 32; 40; 48; 56; 64; 72”. The eleventh column has the values “9; 0; 9; 18; 27; 36, 45; 54; 63; 72; 81”.

239.

An image of a table with 8 columns and 7 rows. The cells in the first row and first column are shaded darker than the other cells. The cells not in the first row or column are all null.  The first column has the values “x; 4; 5; 6; 7; 8; 9”. The first row has the values “x; 3; 4; 5; 6; 7; 8; 9”.

240.

PROD: An image of a table with 7 columns and 8 rows. The cells in the first row and first column are shaded darker than the other cells. The cells not in the first row or column are all null.  The first column has the values “x; 3; 4; 5; 6; 7; 8; 9”. The first row has the values “x; 4; 5; 6; 7; 8; 9”.

241.

An image of a table with 8 columns and 5 rows. The cells in the first row and first column are shaded darker than the other cells. The cells not in the first row or column are all null.  The first row has the values “x; 3; 4; 5; 6; 7; 8; 9”. The first column has the values “x;  6; 7; 8; 9”.

242.

An image of a table with 5 columns and 8 rows. The cells in the first row and first column are shaded darker than the other cells. The cells not in the first row or column are all null. The first column has the values “x; 3; 4; 5; 6; 7; 8; 9”. The first row has the values “x; 6; 7; 8; 9”.

243.

PROD: An image of a table with 6 columns and 6 rows. The cells in the first row and first column are shaded darker than the other cells. The cells not in the first row or column are all null.  The first column has the values “x; 5; 6; 7; 8; 9”. The first row has the values “x; 5; 6; 7; 8; 9”.

244.

An image of a table with 6 columns and 6 rows. The cells in the first row and first column are shaded darker than the other cells. The cells not in the first row or column are all null.  The first column has the values “x; 5; 6; 7; 8; 9”. The first row has the values “x; 5; 6; 7; 8; 9”.

In the following exercises, multiply.

  1. 0⋅15
  2. 0⋅41
  3. (99)0
  4. (77)0
  5. 1⋅43
  6. 1⋅34
  7. (28)1
  8. (65)1
  9. 1(240,055)
  10. 1(189,206)
    255.
    ⓐ 7⋅6
    ⓑ 6⋅7
    256.
    ⓐ 8×9
    ⓑ 9×8
  11. (79)(5)
  12. (58)(4)
  13. 275⋅6
  14. 638⋅5
  15. 3,421×7
  16. 9,143×3
  17. 52(38)
  18. 37(45)
  19. 96⋅73
  20. 89⋅56
  21. 27×85
  22. 53×98
  23. 23⋅10
  24. 19⋅10
  25. (100)(36)
  26. (100)(25)
  27. 1,000(88)
  28. 1,000(46)
  29. 50×1,000,000
  30. 30×1,000,000
  31. 247×139
  32. 156×328
  33. 586(721)
  34. 472(855)
  35. 915⋅879
  36. 968⋅926
  37. (104)(256)
  38. (103)(497)
  39. 348(705)
  40. 485(602)
  41. 2,719×543
  42. 3,581×724
    Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

  1. the product of 18 and 33
  2. the product of 15 and 22
  3. fifty-one times sixty-seven
  4. forty-eight times seventy-one
  5. twice 249
  6. twice 589
  7. ten times three hundred seventy-five
  8. ten times two hundred fifty-five
    Mixed Practice

In the following exercises, simplify.

  1. 38×37
  2. 86×29
  3. 415−267
  4. 341−285
  5. 6,251+4,749
  6. 3,816+8,184
  7. (56)(204)
  8. (77)(801)
  9. 947⋅0
  10. 947+0
  11. 15,382+1
  12. 15,382⋅1
    In the following exercises, translate and simplify.
  13. the difference of 50 and 18
  14. the difference of 90 and 66
  15. twice 35
  16. twice 140
  17. 20 more than 980
  18. 65 more than 325
  19. the product of 12 and 875
  20. the product of 15 and 905
  21. subtract 74 from 89
  22. subtract 45 from 99
  23. the sum of 3,075 and 95
  24. the sum of 6,308 and 724
  25. 366 less than 814
  26. 388 less than 925

Multiply Whole Numbers in Applications

Everyday Math

339

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?


340. 

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?

Writing Exercises

341

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?

Self Check

ⓐ 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?