m81255 Multiply Whole Numbers 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 9b248006-6668-446c-bcc4-bd1a9730b4ad Before you get started, take this readiness quiz. Add: 1,683+479. If you missed this problem, review . 2,162 Subtract: 605−321. If you missed this problem, review . 284

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

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 (). To describe multiplication, we can use symbols and words. Operation Notation Expression Read as Result Multiplication ×·() 3×83·83(8) three times eight the product of 3 and 8 Translate from math notation to words: ⓐ 7×6 ⓑ 12·14 ⓒ 6(13) ⓐ We read this as seven times six and the result is the product of seven and six. ⓑ We read this as twelve times fourteen and the result is the product of twelve and fourteen. ⓒ We read this as six times thirteen and the result is the product of six and thirteen. Translate from math notation to words: ⓐ 8×7ⓑ 18·11 ⓐ eight times seven ; the product of eight and seven ⓑ eighteen times eleven ; the product of eighteen and eleven Translate from math notation to words: ⓐ (13)(7)ⓑ 5(16) ⓐ thirteen times seven ; the product of thirteen and seven ⓑ five times sixteen; the product of five and sixteen

There are many ways to model multiplication. Unlike in the previous sections where we used base-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. Model: 3×8. To model the product 3×8, we’ll start with a row of 8 counters. The other factor is 3, so we’ll make 3 rows of 8 counters. Now we can count the result. There are 24 counters in all. 3×8=24 If you look at the counters sideways, you’ll see that we could have also made 8 rows of 3 counters. The product would have been the same. We’ll get back to this idea later. Model each multiplication: 4×6. Model each multiplication: 5×7.

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. 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. × 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 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. The product of any number and 0 is 0.a·0=00·a=0 Multiply:ⓐ 0·11ⓑ (42)0 ⓐ 0·11 The product of any number and zero is zero. 0 ⓑ (42)0 Multiplying by zero results in zero. 0 Find each product: ⓐ 0·19ⓑ (39)0 ⓐ 0ⓑ 0 Find each product: ⓐ 0·24ⓑ (57)0 ⓐ 0ⓑ 0 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 1 is called the multiplicative identity. The product of any number and 1 is the number.1·a=aa·1=a Multiply: ⓐ (11)1ⓑ 1·42 ⓐ (11)1 The product of any number and one is the number. 11 ⓑ 1·42 Multiplying by one does not change the value. 42 Find each product: ⓐ (19)1ⓑ 1·39 ⓐ 19ⓑ 39 Find each product: ⓐ (24)(1)ⓑ 1×57 ⓐ 24ⓑ 57 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. Changing the order of the factors does not change their product.a·b=b·a Multiply: ⓐ 8·7ⓑ 7·8 ⓐ 8·7 Multiply. 56 ⓑ 7·8 Multiply. 56 Changing the order of the factors does not change the product. Multiply: ⓐ 9·6ⓑ 6·9 54 and 54; both are the same. Multiply: ⓐ 8·6ⓑ 6·8 48 and 48; both are the same. To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction. 27×3___ We start by multiplying 3 by 7. 3×7=21 We write the 1 in the ones place of the product. We carry the 2 tens by writing 2 above the tens place. Then we multiply the 3 by the 2, and add the 2 above the tens place to the product. So 3×2=6, and 6+2=8. Write the 8 in the tens place of the product. The product is 81. When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with. Multiply: 15·4. Write the numbers so the digits 5 and 4 line up vertically. 15 ×4_____ Multiply 4 by the digit in the ones place of 15. 4⋅5=20. Write 0 in the ones place of the product and carry the 2 tens. 125 ×4_____ 0 Multiply 4 by the digit in the tens place of 15. 4⋅1=4.Add the 2 tens we carried. 4+2=6. Write the 6 in the tens place of the product. 125 ×4_____ 60 Multiply: 64·8. 512 Multiply: 57·6. 342 Multiply: 286·5. Write the numbers so the digits 5 and 6 line up vertically. 286 ×5_____ Multiply 5 by the digit in the ones place of 286. 5⋅6=30. Write the 0 in the ones place of the product and carry the 3 to the tens place.Multiply 5 by the digit in the tens place of 286. 5⋅8=40. 2836 ×5_____ 0 Add the 3 tens we carried to get 40+3=43.Write the 3 in the tens place of the product and carry the 4 to the hundreds place. 24836 ×5_____ 30 Multiply 5 by the digit in the hundreds place of 286. 5⋅2=10.Add the 4 hundreds we carried to get 10+4=14.Write the 4 in the hundreds place of the product and the 1 to the thousands place. 24836 ×5_____ 1,430 Multiply: 347·5. 1,735 Multiply: 462·7. 3,234 When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.Write the numbers so each place value lines up vertically. Multiply the digits in each place value. Work from right to left, starting with the ones place in the bottom number. Multiply the ones digit of the bottom number by the ones digit in the top number, then by the tens digit, and so on. If a product in a place value is more than 9, carry to the next place value. Write the partial products, lining up the digits in the place values with the numbers above. Repeat for the tens place in the bottom number, the hundreds place, and so on. Insert a zero as a placeholder with each additional partial product. Add the partial products. Multiply: 62(87). Write the numbers so each place lines up vertically. Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62. 7⋅2=14. Write the 4 in the ones place of the product and carry the 1 to the tens place. Multiply 7 by the digit in the tens place of 62. 7⋅6=42. Add the 1 ten we carried. 42+1=43. Write the 3 in the tens place of the product and the 4 in the hundreds place. The first partial product is 434. Now, write a 0 under the 4 in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of 87 by 62. Multiply 8 by the digit in the ones place of 62. 8⋅2=16. Write the 6 in the next place of the product, which is the tens place. Carry the 1 to the tens place. Multiply 8 by 6, the digit in the tens place of 62, then add the 1 ten we carried to get 49. Write the 9 in the hundreds place of the product and the 4 in the thousands place. The second partial product is 4960. Add the partial products. The product is 5,394. Multiply: 43(78). 3,354 Multiply: 64(59). 3,776 Multiply: ⓐ 47·10ⓑ 47·100. ⓐ 47·10. 47×10___00470___470 ⓑ 47·100 47×100_____000004700_____4,700 When we multiplied 47 times 10, the product was 470. Notice that 10 has one zero, and we put one zero after 47 to get the product. When we multiplied 47 times 100, the product was 4,700. Notice that 100 has two zeros and we put two zeros after 47 to get the product. Do you see the pattern? If we multiplied 47 times 10,000, which has four zeros, we would put four zeros after 47 to get the product 470,000. Multiply: ⓐ 54·10ⓑ 54·100. ⓐ 540 ⓑ 5,400 Multiply: ⓐ 75·10ⓑ 75·100. ⓐ 750ⓑ 7,500 Multiply: (354)(438). There are three digits in the factors so there will be 3 partial products. We do not have to write the 0 as a placeholder as long as we write each partial product in the correct place. Multiply: (265)(483). 127,995 Multiply: (823)(794). 653,462 Multiply: (896)201. There should be 3 partial products. The second partial product will be the result of multiplying 896 by 0. Notice that the second partial product of all zeros doesn’t really affect the result. We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the 2 in the hundreds place, as shown. Multiply by 10, but insert only one zero as a placeholder in the tens place. Multiply by 200, putting the 2 from the 12. 2·6=12 in the hundreds place. 896×201_____89617920__________180,096 Multiply: (718)509. 365,462 Multiply: (627)804. 504,108 When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example: to multiply 8⋅3⋅2 first multiply 8⋅3 24⋅2 then multiply 24⋅2. 48

Earlier in this section, we translated math notation into words. Now we’ll reverse the process and translate word phrases into math notation. Some of the words that indicate multiplication are given in . Operation Word Phrase Example Expression Multiplication timesproducttwice 3 times 8the product of 3 and 8twice 4 3×8,3·8,(3)(8),(3)8,or3(8)2·4 Translate and simplify: the product of 12 and 27. The word product tells us to multiply. The words of 12 and 27 tell us the two factors. the product of 12 and 27 Translate. 12⋅27 Multiply. 324 Translate and simplify: the product of 13 and 28. 13 · 28; 364 Translate and simplify: the product of 47 and 14. 47 · 14; 658 Translate and simplify: twice two hundred eleven. The word twice tells us to multiply by 2. twice two hundred eleven Translate. 2(211) Multiply. 422 Translate and simplify: twice one hundred sixty-seven. 2(167); 334 Translate and simplify: twice two hundred fifty-eight. 2(258); 516

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. Humberto bought 4 sheets of stamps. Each sheet had 20 stamps. How many stamps did Humberto buy? We are asked to find the total number of stamps. Write a phrase for the total. the product of 4 and 20 Translate to math notation. 4⋅20 Multiply. Write a sentence to answer the question.Humberto bought 80 stamps. Valia donated water for the snack bar at her son’s baseball game. She brought 6 cases of water bottles. Each case had 24 water bottles. How many water bottles did Valia donate? Valia donated 144 water bottles. Vanessa brought 8 packs of hot dogs to a family reunion. Each pack has 10 hot dogs. How many hot dogs did Vanessa bring? Vanessa brought 80 hot dogs. When Rena cooks rice, she uses twice as much water as rice. How much water does she need to cook 4 cups of rice? We are asked to find how much water Rena needs. Write as a phrase. twice as much as 4 cups Translate to math notation. 2⋅4 Multiply to simplify. 8 Write a sentence to answer the question.Rena needs 8 cups of water for 4 cups of rice. Erin is planning her flower garden. She wants to plant twice as many dahlias as sunflowers. If she plants 14 sunflowers, how many dahlias does she need? Erin needs 28 dahlias. A college choir has twice as many women as men. There are 18 men in the choir. How many women are in the choir? There are 36 women in the choir. Van is planning to build a patio. He will have 8 rows of tiles, with 14 tiles in each row. How many tiles does he need for the patio? We are asked to find the total number of tiles. Write a phrase. the product of 8 and 14 Translate to math notation. 8⋅14 Multiply to simplify. 134×8___112 Write a sentence to answer the question.Van needs 112 tiles for his patio. Jane is tiling her living room floor. She will need 16 rows of tile, with 20 tiles in each row. How many tiles does she need for the living room floor? Jane needs 320 tiles. Yousef is putting shingles on his garage roof. He will need 24 rows of shingles, with 45 shingles in each row. How many shingles does he need for the garage roof? Yousef needs 1,080 tiles. 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. shows a rectangular rug with a length of 2 feet and a width of 3 feet. Each square is 1 foot wide by 1 foot long, or 1 square foot. The rug is made of 6 squares. The area of the rug is 6 square feet.

The area of a rectangle is the product of its length and its width, or 6 square feet.

Jen’s kitchen ceiling is a rectangle that measures 9 feet long by 12 feet wide. What is the area of Jen’s kitchen ceiling? We are asked to find the area of the kitchen ceiling. Write a phrase for the area. the product of 9 and 12 Translate to math notation. 9⋅12 Multiply. 112×9___108 Answer with a sentence.The area of Jen's kitchen ceiling is 108 square feet. Zoila bought a rectangular rug. The rug is 8 feet long by 5 feet wide. What is the area of the rug? The area of the rug is 40 square feet. Rene’s driveway is a rectangle 45 feet long by 20 feet wide. What is the area of the driveway? The area of the driveway is 900 square feet Multiplying Whole Numbers Multiplication with Partial Products Example of Multiplying by Whole Numbers

Operation Notation Expression Read as Result Multiplication ×·() 3×83·83(8) three times eight the product of 3 and 8 Multiplication Property of Zero The product of any number and 0 is 0. a⋅0=0 0⋅a=0 Identity Property of Multiplication The product of any number and 1 is the number. 1⋅a=a a⋅1=a Commutative Property of Multiplication Changing the order of the factors does not change their product. a⋅b=b⋅a Multiply two whole numbers to find the product. Write the numbers so each place value lines up vertically. Multiply the digits in each place value. Work from right to left, starting with the ones place in the bottom number. Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on. If a product in a place value is more than 9, carry to the next place value. Write the partial products, lining up the digits in the place values with the numbers above. Repeat for the tens place in the bottom number, the hundreds place, and so on. Insert a zero as a placeholder with each additional partial product. Add the partial products.

Use Multiplication Notation In the following exercises, translate from math notation to words. 4×7 four times seven; the product of four and seven 8×6 5·12 five times twelve; the product of five and twelve 3·9 (10)(25) ten times twenty-five; the product of ten and twenty-five (20)(15) 42(33) forty-two times thirty-three; the product of forty-two and thirty-three 39(64) Model Multiplication of Whole Numbers In the following exercises, model the multiplication. 3×6 4×5 5×9 3×9 Multiply Whole Numbers In the following exercises, fill in the missing values in each chart. In the following exercises, multiply. 0·15 0 0·41 (99)0 0 (77)0 1·43 43 1·34 (28)1 28 (65)1 1(240,055) 240,055 1(189,206) ⓐ 7·6ⓑ 6·7 ⓐ 42 ⓑ 42 ⓐ 8×9ⓑ 9×8 (79)(5) 395 (58)(4) 275·6 1,650 638·5 3,421×7 23,947 9,143×3 52(38) 1,976 37(45) 96·73 7,008 89·56 27×85 2,295 53×98 23·10 230 19·10 (100)(36) 3,600 (100)(25) 1,000(88) 88,000 1,000(46) 50×1,000,000 50,000,000 30×1,000,000 247×139 34,333 156×328 586(721) 422,506 472(855) 915·879 804,285 968·926 (104)(256) 26,624 (103)(497) 348(705) 245,340 485(602) 2,719×543 1,476,417 3,581×724 Translate Word Phrases to Math Notation In the following exercises, translate and simplify. the product of 18 and 33 18 · 33; 594 the product of 15 and 22 fifty-one times sixty-seven 51(67); 3,417 forty-eight times seventy-one twice 249 2(249); 498 twice 589 ten times three hundred seventy-five 10(375); 3,750 ten times two hundred fifty-five Mixed Practice In the following exercises, simplify. 38×37 1,406 86×29 415−267 148 341−285 6,251+4,749 11,000 3,816+8,184 (56)(204) 11,424 (77)(801) 947·0 0 947+0 15,382+1 15,383 15,382·1 In the following exercises, translate and simplify. the difference of 50 and 18 50 − 18; 32 the difference of 90 and 66 twice 35 2(35); 70 twice 140 20 more than 980 20 + 980; 1,000 65 more than 325 the product of 12 and 875 12(875); 10,500 the product of 15 and 905 subtract 74 from 89 89 − 74; 15 subtract 45 from 99 the sum of 3,075 and 95 3,075 + 95; 3,170 the sum of 6,308 and 724 366 less than 814 814 − 366; 448 388 less than 925 Multiply Whole Numbers in Applications In the following exercises, solve. Party supplies Tim brought 9 six-packs of soda to a club party. How many cans of soda did Tim bring? Tim brought 54 cans of soda to the party. Sewing Kanisha is making a quilt. She bought 6 cards of buttons. Each card had four buttons on it. How many buttons did Kanisha buy? Field trip Seven school busses let off their students in front of a museum in Washington, DC. Each school bus had 44 students. How many students were there? There were 308 students. Gardening Kathryn bought 8 flats of impatiens for her flower bed. Each flat has 24 flowers. How many flowers did Kathryn buy? Charity Rey donated 15 twelve-packs of t-shirts to a homeless shelter. How many t-shirts did he donate? Rey donated 180 t-shirts. School There are 28 classrooms at Anna C. Scott elementary school. Each classroom has 26 student desks. What is the total number of student desks? Recipe Stephanie is making punch for a party. The recipe calls for twice as much fruit juice as club soda. If she uses 10 cups of club soda, how much fruit juice should she use? Stephanie should use 20 cups of fruit juice. Gardening Hiroko is putting in a vegetable garden. He wants to have twice as many lettuce plants as tomato plants. If he buys 12 tomato plants, how many lettuce plants should he get? Government The United States Senate has twice as many senators as there are states in the United States. There are 50 states. How many senators are there in the United States Senate? There are 100 senators in the U.S. senate. Recipe Andrea is making potato salad for a buffet luncheon. The recipe says the number of servings of potato salad will be twice the number of pounds of potatoes. If she buys 30 pounds of potatoes, how many servings of potato salad will there be? Painting Jane is painting one wall of her living room. The wall is rectangular, 13 feet wide by 9 feet high. What is the area of the wall? The area of the wall is 117 square feet. Home décor Shawnte bought a rug for the hall of her apartment. The rug is 3 feet wide by 18 feet long. What is the area of the rug? Room size The meeting room in a senior center is rectangular, with length 42 feet and width 34 feet. What is the area of the meeting room? The area of the room is 1,428 square feet. Gardening June has a vegetable garden in her yard. The garden is rectangular, with length 23 feet and width 28 feet. What is the area of the garden? NCAA basketball According to NCAA regulations, the dimensions of a rectangular basketball court must be 94 feet by 50 feet. What is the area of the basketball court? The area of the court is 4,700 square feet. NCAA football According to NCAA regulations, the dimensions of a rectangular football field must be 360 feet by 160 feet. What is the area of the football field?

Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price rose $12 per share. How much money did Javier’s portfolio gain? Javier’s portfolio gained $3,600. Salary Carlton got a $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? Answers will vary. 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?

product The product is the result of multiplying two or more numbers.