Unit 4 Number and Operations in Base Ten: Multiplying and Dividing Decimals

Transcription

1 Unit 4 Number and Operations in Base Ten: Multiplying and Dividing Decimals Introduction In this unit, students will learn how to multiply and divide decimals, and learn the algorithm for dividing whole numbers and decimals by a twodigit divisor. For multiplication of decimals, students will first multiply the decimals as if they were whole numbers. They will then find the sum of the number of decimal digits in each factor, and move the decimal point in the product to the left that many places. For division of decimals, students will learn how to multiply the divisor and the dividend by the appropriate power of 10 that will eliminate the decimal point in the divisor. They will then perform the division in the same way as they did for whole numbers, but remember to place the decimal point in the quotient in the correct position. For division by a two-digit divisor, students will learn how to estimate each digit of the quotient by rounding the divisor to the nearest ten, and counting the number of tens in the dividend. At first this method of estimation will always work, but in a later section they will have to use the guess and check method. In this method, the original estimate may be too low, too high, or just right. Students will learn how to adjust their estimate and complete the division algorithm. Number and Operations in Base Ten O-1

2 NBT5-54 Multiplying Decimals by Whole Numbers Pages STANDARDS 5.NBT.B.7, 5.NBT.B.5 Goals Students will multiply decimals up to the hundredths place by a whole number. Vocabulary associative property decimals hundredths hundredths block ones block regrouping tenths tenths block PRIOR KNOWLEDGE REQUIRED Knows how to multiply a multi-digit number by a single-digit number using the standard algorithm Can use base ten materials to model decimal arithmetic and multiplication involving regrouping Can multiply a multi-digit decimal number by multiples of 10 MATERIALS base ten blocks plastic money grid paper Review base ten materials. Review the use of base ten materials when using decimals. = 1 = 0.1 = 0.1 NOTE: In this context, we are now using the hundreds block as a ones block. One column or row of the ones block is now a tenths block. One unit of the tenths block is now a hundredths block. ASK: How many hundredths are in 1? (100) How many tenths are in 1? (10) How many hundredths are in 1 tenth? (10) (MP.4) Ask students to model the decimal 2.13 on their desks with base ten materials. (see diagram below) Model multiplying a decimal by a whole number with base ten materials and without regrouping. 3 5 ASK: What addition question can you use to find the product? Ask for a volunteer to write the answer on the board. ( ) O-2 Teacher s Guide for AP Book 5.2

3 ASK: What addition question can you use to find the product? Ask for a volunteer to write the answer on the board. ( ) Ask students to use base ten materials to add (6.39; see diagram below) What could we have done to the digits in 2.13 to get the answer 6.39? (multiply each digit separately by 3) Exercises: Write the question in your notebook and find the product mentally. a) b) c) d) e) f) g) h) Answers: a) 6.48, b) 6.93, c) 2.86, d) 4.48, e) 8.62, f) 4.86, g) 8.84, h) 4.62 (MP.4) Multiply a decimal by a whole number using place values. ASK: How can we write 2.13 using place values? (2 ones + 1 tenth + 3 hundredths) ASK: What is 3 2 ones? (6 ones) What is 3 1 tenth? (3 tenths) What is 3 3 hundredths? (9 hundredths) How can we write the answers in decimal notation? (6.39) 2.13 = 2 ones + 1 tenths + 3 hundredths ones + 3 tenths + 9 hundredths Exercise: Multiply using place values. a) b) c) Answers: a) 9.36, b) 8.24, c) 3.99 Number and Operations in Base Ten 5-54 O-3

4 (MP.4) Using base ten materials, model multiplying a decimal by a whole number with regrouping. Point out to students that none of the questions so far have involved regrouping Ask students to use base ten materials at their desks to calculate the product using addition. (see diagram below) Replace with a tenth block. ASK: How many hundredths do we have? (18) What can we use to replace 10 hundredths? (1 tenth) How many hundredths remain? (8) Ask students to replace the 10 hundredths with a tenth block, and read the answer. (6.48) (MP.4) Using money, model multiplying a decimal by a whole number with regrouping hundredths for tenths (pennies for dimes). Some students will benefit from a demonstration using plastic money. Consider the decimal 2.16 as $2.16. Write $2.16 on the board. ASK: How many dollars are there? (2) How many dimes are there? (1) How many pennies are there? (6) If your class has plastic money, ask students to represent $ using addition. If not, draw the following on the board and tell students that D and P represent dimes and pennies. $1 $1 D P P P P P P $1 $1 D P P P P P P $1 $1 D P P P P P P Replace 10 pennies with 1 dime. D ASK: What can we replace 10 pennies with? (1 dime) How many pennies are left? (8) How many dimes are there now? (4) How much money is there? ($6.48) O-4 Teacher s Guide for AP Book 5.2

5 (MP.4) Multiplying a decimal by a whole number and regrouping tenths for ones (dimes for dollars) Ask students to use base ten materials or money models to find the product using addition. ASK: What can we replace 10 tenths with? (a ones block) How many ones are there now? (3) How many tenths? (2) How many ones? (6) Ask a student to read the final answer. (3.26 or $3.26; see diagrams below) Replace 10 tenths with a one. or $1 D D D D D D P P P Replace 10 dimes with 1 dollar. $1 D D D D D D P P P $1 Exercises: Use base ten materials or plastic money to find the product. a) b) c) Answers: a) 4.74, b) 8.13, c) 9.51 (MP.7) Multiplying a decimal number involving regrouping using place values ASK: How do we write 2.63 using place values (2 ones + 6 tenths + 3 hundredths) What is 2 ones 2? (4) What is 6 tenths 2? (12 tenths) What is 3 hundredths 2? (6 hundredths) 2.63 = 2 ones + 6 tenths + 2 hundredths ones + 12 tenths + 6 hundredths ASK: We have 12 tenths. What can we use to replace 10 tenths? (a ones block) How many tenths are left? (2 tenths) How many ones are there altogether? (5) What is the answer in decimal form? (5.26) = 5 ones + 2 tenths + 6 hundredths = 5.26 Number and Operations in Base Ten 5-54 O-5

6 2.48 = 2 ones + 4 tenths + 8 hundredths ones + 8 tenths + 16 hundredths ASK: As we have 16 hundredths, what can we use to replace 10 hundredths? (a tenths block) How many hundredths are left? (6 hundredths) How many tenths are there altogether? (9) What is the answer in decimal form? (4.96) = 4 ones + 9 tenths + 6 hundredths = 4.96 Exercise: Multiply using place values. a) b) c) d) Bonus e) f) Answers: a) 7.83, b) 4.56, c) 3.84, d) 10.58, Bonus: e) 8.28, f) (MP.7) Multiplying a decimal by a whole number using a grid. Ask students to multiply using a grid and compare the answer to part a) above. ASK: What is the only difference in the answers? (the answer to part a) has a decimal point) The decimal points line up on the grid SAY: When you multiply a decimal number by a whole number, place the decimal point in the answer underneath the decimal point in the decimal number. Exercises: Find the products using grid paper. You may have to regroup more than once. a) b) c) d) e) Bonus: 9, Answers: a) 7.28, b) 15.84, c) 31.35, d) 39.44, e) 63.36, Bonus: 73, Multiplying a decimal by multiples of ASK: What is the fastest way to multiply a number by 10? (move the decimal point one place to the right) = O-6 Teacher s Guide for AP Book 5.2

7 SAY: The same rule applies to multiplying a decimal by 10. Write on the board: = 2 3. SAY: We can use the associative property to help us multiply decimals by multiples of ASK: How do we write 20 as a multiple of 10? (2 10) = (2 10) 2.3 SAY: The associative property lets us move the brackets. = 2 (10 2.3) ASK: What is ? (23) SAY: Calculate 2 23 mentally. (46) = 2 23 = 46 Exercise: Calculate using this method. a) b) c) Answers: a) 36, b) 84, c) 66 Extensions (MP.1) 1. Marty shopped at the local grocery store. This is what he bought: Product Unit Price Quantity Milk $ Bread $ Cereal $ How much did Marty spend altogether? (MP.1) Answer: $ Maria has relatives in Laos, Moldova, and Samoa. She calls them each month and keeps track of how many minutes each call lasts. Here are the calls Maria made last month. Country Called Length of Call (min) Country Called Length of Call (min) Laos 2 Moldova 1 Moldova 4 Laos 3 Samoa 3 Samoa 4 Moldova 3 Laos 3 Number and Operations in Base Ten 5-54 O-7

8 Maria s telephone service charges for long distance calls per minute according to the chart: Country Laos Moldova Samoa Cost per minute $1.49 $1.26 $1.29 Find the total cost of Maria s long distance calls last month. Answer: $31.03 (MP.1) 3. The price per gallon of gas in New York City is $3.81. Harry s motorcycle has a gas tank that holds 5 gallons. While on vacation, Harry filled his tank 7 times. Suppose he paid the same price per gallon on his trip as he did in New York City. How much did Harry spend on gas? Answer: $ O-8 Teacher s Guide for AP Book 5.2

9 NBT5-55 Multiplying Decimals by Decimals Pages (Introduction) STANDARDS 5.NBT.B.7, 5.NBT.A.3 Goals Students will multiply decimal fractions and the corresponding decimals. Vocabulary decimals denominator hundredths hundredths block ones block tenths tenths block PRIOR KNOWLEDGE REQUIRED Can convert decimal fractions into decimals MATERIALS base ten blocks overhead or digital projector BLM Ones, Tenths, Hundredths (p. O-57) (MP.4) Use base ten materials to represent decimal fractions. Use BLM Ones, Tenths, Hundredths to display the diagram of a ones block on the board (either using a digital projector or transparency). Shade in one square and ask students what fraction is represented. (1/100) Shade in one column and ask students what fraction is represented. (1/10) Shade in one row and ask students what fraction is represented. (1/10) See sample diagrams below: Using base ten materials, convert fractions with denominator 100 into decimals. Erase the shading from the board. Shade the areas shown below and ask students to name a fraction and a decimal for each. a) b) c) (MP.7) Figure 1 Figure 2 (2/10, 0.2) (3/10, 0.3) (18/100, 0.18) Multiply decimal fractions by finding overlapping shading. Erase the shading on the board. Ask a student to come to the board to shade 4/10 using columns. (see Figure 1 sample answer in margin) Without erasing the first student s shading, ask another student to come to the board and shade 3/10 using rows. (see Figure 2 sample answer in margin) Number and Operations in Base Ten 5-55 O-9

10 SAY: We can find this product by finding 3/4 of 4/10. This is the area created where the two shadings done by students overlap (see example in margin). ASK: What decimal fraction does the overlapped shading represent? (12/100) SAY: So 3/10 4/10 = 12/100. Shade in the following areas, and ask students to come to the board to write a multiplication equation for each diagram. a) b) c) = = = Multiplying decimal fractions. Remind students that, to multiply fractions, you can multiply the numerators and then multiply the denominators. Have students multiply the following fractions in their notebooks. Exercises: Multiply. 3 a) = b) = c) = d) = Answers: a) 12/100, b) 63/100, c) 27/1,000, d) 92/1,000 (MP.2) ASK: Looking at the fraction equations, how can you predict the number of zeroes in the denominator of the product? (by finding the sum of the number of zeroes in the denominators of each factor) (MP.7) Changing fraction equations into decimal equations. Remind students that, for fractions with powers of 10 as denominators, the number of digits after the decimal point is the same as the number of zeroes in the denominator. Example: 43/100 = Have students convert each of the fraction equations in the previous exercises into decimal equations. Do the first conversion with the class as an example. (a) = 0.12, b) = 0.63, c) = 0.027, d) = 0.092) Number of zeroes in denominator = + = Ask a student to fill in the bottom row of the chart O-10 Teacher s Guide for AP Book 5.2

11 = 0.12 Number of decimal digits + = Ask a student to fill in the bottom row of the chart. (MP.2) ASK: Looking at the two charts, how can you predict the number of decimal digits in the second chart? (find the number of zeroes in the denominators of the corresponding fractions) Exercises: Multiply. a) b) c) Bonus d) e) Answers: a) 0.15, b) 0.56, c) 0.015, Bonus: d) , e) Extensions (MP.1) 1. Find as many pairs of decimal fractions as you can that have the product. a) b) c) , d) , Sample answers a) , , an uncommon answer: b) , c) , , , d) , , , , Number and Operations in Base Ten 5-55 O-11

12 NBT5-56 Multiplying Decimals by Decimals Pages STANDARDS 5.NBT.B.7, 5.NBT.B.5 Goals Students will multiply decimals where the product has up to 3 decimal digits. Vocabulary decimal digits denominator hundredths tenths PRIOR KNOWLEDGE REQUIRED Can multiply multi-digit numbers by 2-digit numbers MATERIALS calculators Finding patterns in the number of decimal digits when multiplying decimals = , Number of zeroes in denominator + = Fraction as decimal = Numerator Number of decimal digits + = Ask a student to come to the board and write the number of zeroes in each decimal fraction in the first chart above. (1, 2, 3) Ask another student to come to the board and write each decimal fraction as a decimal in the second chart above. (0.3, 0.07, 0.021) (MP.2) Ask another student to come to the board and write the numerator for each fraction decimal. (3, 7, 21) Ask another student to come to the board and write the number of decimal digits in each decimal. (1, 2, 3) ASK: What was the product of the numerators? (21) ASK: What was the product of the fractions written as a decimal? (0.021) ASK: How many times did the decimal point move to the left in 21 to get 0.021? (3) ASK: What was the sum of the decimal digits? (3) O-12 Teacher s Guide for AP Book 5.2

13 Rule for placing the decimal point when multiplying decimals. Write on the board: To multiply decimals: 1. Multiply the numbers as if they were whole numbers. 2. Count the number of digits after the decimal in each factor. 3. Add the numbers from Step Shift the decimal point to the left that many places. Do the following example with the class Point out that 0.34 has 2 digits after the decimal, and 0.2 has 1 digit after the decimal = 3, so we shift the decimal point 3 places to the left. So, = Note to students that we needed to add a zero here! Have the class do the following exercises. Tell them that sometimes, to multiply the numbers, they may have to use a grid. Exercises: Multiply. a) b) c) d) Answers: a) 0.35, b) 0.048, c) 0.128, d) Multiplying multi-digit decimals Ask students to find the product using their calculators. (7.99) ASK: How many decimal digits are there in 2.35? (2) How many decimal digits are there in 3.4? (1) How many times should we move the decimal point to the left? (3) How did you get that? (2 + 1 = 3) ASK: For the answer your calculator gave, 7.99, how many places did the decimal point seem to move to the left? (2) Is there a problem here? (Students will likely say there is a problem, which will lead to finding the product without a calculator.) SAY: Let s find the product without a calculator. Draw the grid provided in the margin on the board. Ask for a student volunteer to find the product on the board. (7.990) ASK: How many times was the decimal point shifted to the left? (3) ASK: Was the calculator wrong? Why? (no, = 7.99) Number and Operations in Base Ten 5-56 O-13

14 Exercises: Use your calculator to find the product, and then check manually. a) b) c) d) Bonus: Answers: a) 4.5, b) 3.3, c) 4.335, d) 172.8, Bonus: 24.7 Extensions (MP.1) 1. The first number in each product is missing a decimal. Place the decimal point in the correct position. a) = 1.53 b) = c) = d) = 0.68 Answer: a) 0.3, b) 24.5, c) 0.08, d) 3.4 (MP.1) 2. Find eight different pairs of numbers with the product Sample answers (MP.1) 3. Put the same number in each box. = Answer: 0.2 O-14 Teacher s Guide for AP Book 5.2

15 NBT5-57 Decimal Word Problems Multiplication Page 61 STANDARDS 5.NBT.B.7, 5.NBT.B.5 Goals Students will solve word problems involving multiplying decimals by whole numbers and by decimals Vocabulary assists calories goals quire ream sales tax PRIOR KNOWLEDGE REQUIRED Knows how to multiply a multi-digit number by a two-digit number using the standard algorithm Knows how to multiply decimals by whole numbers and by decimals Dollar and cent notation. Lesson NBT5-57 provides a review of multiplying decimals involving word problems. Before having students complete the word problems, we suggest you review dollar and cent notation. In some situations involving money, amounts involving fractions of cents are used. ASK: How can we write 3 cents using dollars and cents? ($0.03) ASK: How can we write 4 cents using dollars and cents? ($0.04) ASK: How can we write 0.03 using 3 decimal digits? (0.030) ASK: How can we write 0.04 using 3 decimal digits? (0.040) SAY: 3 1 is between 3 and 4 2 ASK: What decimal is exactly in between and 0.040? (0.035) So, = $0.035 Exercises: Write the amount in dollars and cents notation. a) b) c) Bonus: Answers: a) $0.075, b) $0.095, c) $0.165, Bonus: $0.097 (MP.1) Extensions 1. In ice hockey, individual players get points by scoring goals or assisting other players in scoring goals. In the season, Wayne Gretzky scored 0.65 goals per game and earned assists per game. He played 80 games that season. a) How many goals did he score in the 80 games? b) How many assists did he earn? c) To find total points, add the goals and assists. How many points in total did Gretzky get during the season? Answers: a) 52, b) 163, c) 215 Number and Operations in Base Ten 5-57 O-15

16 (MP.4) 2. Barb s bicycle shop rents out bikes for a fee of $10.75 plus $6.80 per hour. What is the total cost of renting a bike for 4.25 hours? Answer: $39.65 (MP.4) 3. John s dad is on a diet. His diet recommends that he consume up to 800 calories at lunch. For lunch today, he ate: 125 grams of bread, 45 grams of cheese, 120 grams of broccoli, and a 200 gram apple. A gram of bread contains 2.7 calories, cheese has 4.1 calories per gram, broccoli has 0.32 calories per gram, and apples have 0.52 calories per gram. Did the lunch meet the requirements of the diet? Answer: Yes. The lunch had calories, which is less than the recommended 800 calorie limit. (MP.4) 4. John earns $10.75 per hour at a fast-food restaurant. How much does he earn if he works 8 h? 9 h? Is your answer to Question 5 on AP Book 5.2 p. 60 between these two? If not, look for a mistake. Answer: $86 for 8 h, $96.75 for 9 h; yes, the answer for Question 5 ($91.375) is between the figures for 8 h and 9 h. O-16 Teacher s Guide for AP Book 5.2

17 NBT5-58 Dividing Decimals by Whole Numbers Pages (Introduction) STANDARDS 5.NBT.B.7, 5.NBT.B.6 Goals Students will divide decimals by whole numbers using base ten materials and place values where no regrouping is required. Vocabulary decimal hundredths ones tenths whole number PRIOR KNOWLEDGE REQUIRED Knows how to multiply whole numbers using base ten materials and place values MATERIALS base ten materials plastic money Review base ten materials. Remind students that when using base ten materials to represent decimals: = 1 = 0.1 or 1 10 = 0.01 or Have students review base ten materials by having them represent the following numbers at their desks. (see diagram below for sample answer to part a)) a) 1.43 b) 1.72 c) 3.16 (MP.4) Use base ten materials to model division of decimals by whole numbers without regrouping Ask students to represent 3.69 using base ten materials. (see diagram below) Number and Operations in Base Ten 5-58 O-17

18 Ask students to divide the materials into three equal groups. (see diagram below) ASK: What is the division statement? ( = 1.23) Exercises: Use base ten materials to perform the division and then write a division statement: a) b) c) Answers: a) = 2.13, b) = 2.12, c) = 3.12 (MP.8) Use place values to model division of decimals by whole numbers without regrouping ASK: How do we write 6.82 using place values? (6 ones + 8 tenths + 2 hundredths) ASK: What is 6 ones 2? (3 ones) What is 8 tenths 2? (4 tenths) What is 2 hundredths 2? (1 hundredth) Write on the board and say: = (6 ones + 8 tenths + 2 hundredths) 2 = 3 ones + 4 tenths + 1 hundredth ASK: How do we write this in decimal notation? (3.41) Write the answer on the board. Exercises: Use place values to divide. (NOTE: Students should arrive at their answers by first noting place values, as in the example above, and then finding the decimal notation.) a) b) c) Answers: a) 2.14, b) 3.12, c) 1.21 (MP.4) Use money to model division of decimals by whole numbers without regrouping. Some students will benefit from a demonstration of division using money ASK: How can we represent $6.39 using dollar bills, dimes, and pennies? (6 dollar bills, 3 dimes, and 9 pennies) Use plastic money to model or draw the following on the board: $1 $1 $1 $1 $1 $1 D D D P P P P P P P P P O-18 Teacher s Guide for AP Book 5.2

19 ASK: If we divide the $6 among 3 friends, how many $1 bills will each friend get? (2) If we divide 3 dimes among three friends, how many dimes does each get? (1) If we divide 9 pennies among three friends, how many pennies does each get? (3) So how much money does each friend get? ($2.13) (MP.4) Exercises: Divide the money. a) $ b) $ c) $ Answers: a) $4.23, b) $2.33, c) $1.21 (MP.8) Recognize that dividing decimals by whole numbers can be done by dividing without the decimal and later placing the decimal point. Write on the board: = (6 ones + 4 tenths + 8 hundredths) Ask two students to come to the board and perform the divisions: the first using the division algorithm (see answer in margin) and the second using place values. (3 ones + 2 tenths + 4 hundredths = 3.24) ASK: What is the same about the quotients? (same digits) What is different? (when the dividend has a decimal point, the quotient has a decimal point) What do you notice about the position of the decimal points in the quotient and the dividend in the second question? (they are in the same place) SAY: To divide a decimal by a whole number, perform the division using the algorithm as if there were no decimal point and then place the decimal point in the correct place in the quotient Ask students to perform the division in their notebooks. When they have had enough time, ask a student to perform the division on the board. (see answer in margin) If = 412, then =??? Ask a student to come to the board to complete the division equation. ( = 4.12) Number and Operations in Base Ten 5-58 O-19

20 Exercises (MP.4) 1. Divide by using the division algorithm. First ignore the decimal point, and then place the decimal point in the quotient. a) b) c) d) Answers: a) 2.13, b) 2.14, c) 2.63, d) 1.41 (MP.8) 2. Use the fact that 5,284 4 = 1,321 to divide: a) b) c) d) Answers: a) 13.21, b) 132.1, c) 1.321, d) Bonus: Use the fact that 3,173,255 5 = 634,651 to divide 31, Answer: 6, Extensions (MP.1) 1. Ava earns $78.40 working for 7 hours at a part-time job. a) What is her pay per hour? NOTE: When writing numbers in dollar notation, two decimal digits are required. b) When Ava works on a holiday, she gets paid extra. She is paid 1.5 times as much per hour. What is her pay per hour on a holiday? c) How much does Ava earn for 6 hours of work on a holiday? Answers: a) $11.20, b) $16.80, c) $ (MP.1) 2. A website about fuel economy says that a particular car model will drive 29.7 miles per gallon of gas. a) The Benitez family drove that same model of car for miles using 8 gallons of gas. Was the website correct? b) How much farther would the Benitez family travel on 8 gallons if their car drove as far as the website said? Answers: a) no, they traveled 29.3 miles per gallon, so the website was not correct; b) 3.2 miles ((8 29.7) = 3.2) O-20 Teacher s Guide for AP Book 5.2

21 NBT5-59 Dividing Decimals by Whole Numbers Pages STANDARDS 5.NBT.B.7, 5.NBT.B.6 Goals Students will divide decimals by whole numbers using base ten materials, money, and the division algorithm. Vocabulary dividend divisor quotient PRIOR KNOWLEDGE REQUIRED Knows how to multiply whole numbers using base ten materials and place values MATERIALS base ten materials plastic money (MP.4) Use base ten materials to model division of decimals by whole numbers using the division algorithm where regrouping is required Ask students to model the steps of the division algorithm at their desks using base ten materials. SAY: Use base ten materials to represent (see diagram below) Ask students to follow the steps at their desks using base ten materials. Step 1: Divide the ones blocks into two equal groups. Number and Operations in Base Ten 5-59 O-21

22 Continue writing on the board as you ask the following questions. ASK: How many ones are in each group? (3) How many were placed in groups? (6) How many ones remain? (1) Step 2: SAY: Exchange the ones block for 10 tenths. ASK: How many tenths are there now? (13) Continue writing on the board: Step 3: SAY: Divide the tenths into two equal groups. Continue writing on the board as you ask the following questions. ASK: How many tenths are in each group? (6) How many tenths were placed in groups? (12) How many tenths remain? (1) O-22 Teacher s Guide for AP Book 5.2

23 Step 4: SAY: Exchange the ten block for 10 hundreths blocks. ASK: How many hundredths blocks are there now? (14) Continue writing on the board: Step 5: Divide the 14 hundreths blocks into two equal groups. SAY: Place the decimal point in the quotient directly above the decimal point in the dividend so we can line up tenths with tenths and hundredths with hundredths. Add the decimal point between the 3 and the 6 in the quotient. Continue writing on the board. ASK: How many hundredths are in each group? (7) How many hundredths were placed altogether? (14) How many hundredths are remaining? (0) Number and Operations in Base Ten 5-59 O-23

24 ASK: What decimal is represented in each group of base ten materials? (3.67) (MP.4) Use money to model division of decimals by whole numbers using the division algorithm where regrouping is required. Some students will benefit from using a money model. ACTIVITY Model using plastic money. ASK: How can we represent $7.34 using plastic money? (see diagram below) $1 $1 $1 $1 $1 $1 $1 D D D P P P P Ask students to follow these steps on their own to model the division. Step 1: Divide the dollar bills into two equal groups. Step 2: Exchange a $1 for 10 dimes. Step 3: Divide the resulting dimes into two groups. Step 4: Exchange the remaining dime for 10 pennies. Step 5: Divide the pennies into two groups. The final model should look like this: $1 $1 $1 D D D D D D P P P P P P P $1 $1 $1 D D D D D D P P P P P P P ASK: How much money is in each group? ($3.67) Exercises: Divide. NOTE: Students should notice this is exactly like dividing using whole numbers and then putting the decimal point in the correct place. 1. a) b) c) Bonus: 11, a) b) c) 11, Answers 1. a) 2.39, b) 1.236, c) 0.013, Bonus: 1, a) , b) 63.07, c) 2, O-24 Teacher s Guide for AP Book 5.2

25 Extensions (MP.1) 1. To turn a fraction into a decimal, write the numerator using at least three decimal digits and then divide the numerator by the denominator. For example, 1/4 = Find decimal representations for the fraction. a) 1 2 b) 1 4 c) 1 5 d) 1 8 Answers: a) 1 2 = 0.500, b) 1 4 = 0.250, c) 1 5 = 0.200, d) 1 8 = (MP.1) 2. In baseball, a batter s average is a decimal with three decimal digits. To find the decimal, divide the number of hits by the number of times at bat. Rewrite the number of hits using three decimal places (example: 3 = 3.000). Which batter has the highest average? Batter Number of Hits Number of Times at Bat Derek 3 8 Melky 1 4 Jose 2 5 Average Answer: Derek 0.375, Melky 0.250, Jose Jose has the highest average. (MP.3) 3. A pack of three pens costs $5.85. a) How much does each pen cost? Estimate, and then find the exact answer. b) Lina estimates that 20 pens will cost $42. Is her estimate reasonable? Explain. Answers: a) under $2 each; $1.95, b) No, because the cost of each pen is less than $2, so the cost of 20 pens should be less than $40. Number and Operations in Base Ten 5-59 O-25

26 NBT5-60 Dividing Decimals by Decimals Pages STANDARDS 5.NBT.B.7, 5.NBT.A.2 Goals Students will divide decimals by decimals by first multiplying the divisor and dividend by the power of 10 to eliminate the decimal in the divisor. Vocabulary dividend divisor quotient PRIOR KNOWLEDGE REQUIRED Knows how to divide decimals by whole numbers using the division algorithm Knows how to find equivalent fractions Knows how to multiply decimals by powers of 10 Review finding equivalent fractions. On the board, draw the diagram shown in the margin. Ask a student to come to the board to write a fraction for the shaded region. (3/4) Ask the student to explain how they got the answer. (3 is the number of shaded regions; 4 is the total number of regions in the whole) Divide each of the regions on the board into two parts so that the diagram looks like the example shown in the margin. Ask a different student to come to the board to write a fraction for the shaded region other than 3/4. (6/8) Ask the student to explain how they got the answer. (6 is the number of shaded regions; 8 is the total number of regions in the whole) SAY: We didn t change the amount of pie or pizza we divided, so what can we say about the fractions 3/4 and 6/8? (they are equal) 3? 4? = 6 8 ASK: What can we multiply the numerator and denominator by to see that 3/4 = 6/8? (multiply both by 2) SAY: Remember that you can write the fraction 3/4 as = 3 4 = SAY: How can we write this last fraction using the division symbol ( )? ((3 2) (4 2)) 3 4 = (3 2) (4 2) O-26 Teacher s Guide for AP Book 5.2

27 Ask students to find other equivalent division statements for 3 4. (Sample answers: 3 4 = (3 5) (4 5), 3 4 = (3 8) (4 8)) Review multiplying decimals by powers of Ask students to come to the board to write the answers on the board. (23, 94.8, 734, 821.9) ASK: When multiplying by 10, how many places does the decimal point move? (1) and in which direction? (right) ASK: When multiplying by 100, how many places does the decimal point move? (2), and in which direction? (right) (MP.8) Find equivalent division statements by multiplying by powers of 10. SAY: For division of decimals, we will be multiplying the numerator (dividend) and denominator (the divisor) by powers of ASK: How can we do this calculation mentally? (ignore the decimal to divide 36 4 and then place the decimal point in the quotient later) ASK: What is the difference between this question and the previous one? (in this question, the divisor has a decimal) SAY: Let s find an equivalent division statement for Write on the board and highlight the 10 in the following: = (3.6 10) (0.4 10) ASK: What is 3.6 x 10? (36) What is 0.4 x 10? (4) Continue writing on the board, again highlighting the 10: = (3.6 10) (0.4 10) = 36 4 ASK: What is 36 4? (9) ASK: What happened to the decimal in the divisor? (it is gone) ASK: Why did that happen? (because we multiplied the divisor by 10) ASK: Why do you think we multiplied by 10 and not 100 or 1,000? (there was only one decimal digit in the divisor so multiplying by 10 was enough to change the divisor to a whole number) Number and Operations in Base Ten 5-60 O-27

28 ASK: What do you think we should multiply both the divisor and dividend by this time? (100) Why? (there are two decimal places in the divisor and so multiplying by 100 will change the divisor to a whole number) Write on the board, highlighting the 100: = ( ) ( ) ASK: What is ? (2.1) ASK: What is ? (7) SAY: Calculate mentally. (0.3) Continue writing on the board: = ( ) ( ) = = 0.3 Having to add zeroes in the dividend ASK: What do you think we should multiply both the divisor and dividend by this time? (1,000) Why? (because there are three decimal digits in the divisor and so multiplying it by 1000 will change it to a whole number) ASK: What is ,000? (7) ASK: What is ,000? (140) ASK: Why did we add a zero? (there were only two digits, but we had to move the decimal point three places) SAY: Calculate mentally. (20) Continue writing on the board and highlight the 1,000: (MP.8) = (0.14 1,000) (0.07 1,000) = = 20 Exercises: Find an equivalent division statement and then find the answer. a) b) c) d) Bonus: e) f) Answers: a) 7, b) 0.9, c) 40, d) 70, Bonus: e) 90,000, f) 1,600,000,000 Finding an equivalent division question by moving the decimal point in the divisor and dividend. SAY: When it is too difficult to perform a mental calculation, we can use the division algorithm. O-28 Teacher s Guide for AP Book 5.2

29 ASK: What is the new division statement if we multiply both the divisor and dividend by 10? (4 2 8 ) Ask students to come to the board to write an equivalent division question so that the divisor has no decimals. Exercises a) b) c) d) Answers: a) 3 1 2, b) , c) , d) Dividing decimals by the division algorithm after moving the decimal point in the divisor and dividend SAY: We need to eliminate the decimal in the division. First, what power of 10 do we need to multiply by? (10) After that, what is the new division statement? ( ). Ask a student to come to the board to perform the division using the division algorithm, that is, long division. (see sample answer below) ASK: Where do we place the decimal point in the quotient? (directly above the decimal in the dividend) Have students perform the following in their notebooks and then ask for volunteers to present on the board. Tell them that, in some cases, they may have to add zeroes to the end of the dividend. Exercises: Divide. a) b) c) d) Bonus e) f) Answers: a) 15.4, b) 63.2, c) 130, d) 810, Bonus: e) 0.578, f) Number and Operations in Base Ten 5-60 O-29

30 Extensions (MP.1) 1. Sometimes we want to find the price for a unit of something, for example, the price for a pound of peanuts. To find a unit price, divide the amount of money by the unit of weight or volume. For example, if 3 pounds of peanuts cost $12.00, the unit price = $ = $4.00 per lb. Two different stores advertise their prices. Find the lower unit price: a) Dry roasted almonds: 0.6 lb for $5.37 or 0.8 lb for $7.08 b) Rolled oats: 0.5 lb for $0.55 or 0.6 lb for $0.78 Answers The lower unit price is underlined: a) almonds: $8.95/lb or $8.85/lb b) oats: $1.10 /lb or $ 1.30/lb (MP.1, MP.4) 2. Sales tax varies by state. In Alabama, the sales tax is calculated by multiplying the price by On a product with the price $80, the sales tax is 0.04 $80 = $3.20. The total price is $80 + $3.20 = $ a) Find the price of the product with the Alabama sales tax. i) $3.60 ii) $4.80 iii) $13.60 b) Find the total price if the Alabama sales tax is $ Answers a) i) $90, ii) $120, iii) $340 b) $2, (MP1, MP.8) 3. Gavin has a jar of pennies. The pennies weigh g. One penny weighs 3.1 g. Gavin estimates there are 80 pennies. Is his estimate reasonable? Explain. Answer No, because 80 3 = 240 g, but the pennies only weigh g. O-30 Teacher s Guide for AP Book 5.2

31 NBT5-61 Decimal Word Problems Division by Page 68 a Single Digit STANDARDS 5.NBT.B.7 Goals Students will solve word problems involving division of decimals by a single-digit decimal. Vocabulary dividend divisor quotient PRIOR KNOWLEDGE REQUIRED Knows how to divide a multi-digit number by a whole number using the division algorithm Knows how to divide a decimal by a whole number Knows how to divide a decimal by a single-digit decimal Word problems. Lesson NBT5-61 provides practice with decimal division using word problems. Extensions (MP.1, MP.4) 1. Mile markers are used on interstate highways to help describe locations on the highway. For example, mile marker 1 on I-65 is one mile north of the Kentucky state line. Mile marker 83.7 is 83.7 miles north of the Kentucky state line. A family traveling north on I-65 passes mile marker 83.7 at 12:00 noon. The family passes mile marker exactly 0.8 hours later. a) How far did the family travel in that time? b) To calculate a car s speed, divide the distance traveled by the amount of time in which it was traveled. How fast did the family travel measured in miles per hour? Answers: a) 48.8 miles, b) 61 miles per hour (MP.1, MP.4) 2. The price of gasoline in the United States is sometimes written using both decimals and fractions. If the price per gallon of gas is $ at a gas station, it means that a gallon costs 3 dollars and cents. a) Convert 87 6 into a decimal. 10 b) How many cents are there in 3 dollars? c) Find the price per gallon of gas written in cents. d) Find the cost of 5 gallons of gas written in cents. e) Find the cost of 5 gallons of gas written in dollars. Answers: a) 87.6, b) 300, c) 387.6, d) 1,938, e) $19.38 Number and Operations in Base Ten 5-61 O-31

32 (MP.1, MP.4) 3. At a gas station across the street, a customer spends $19.21 for 5 gallons of gas. a) How many cents are in $19.21? b) What is the cost per gallon of gas in cents? c) What is the cost per gallon of gas in dollars? d) What is the cost per gallon of gas written in decimals and fractions? Answers: a) 1,921, b) 384.2, c) $3.842, d) $3.84 2/10 O-32 Teacher s Guide for AP Book 5.2

33 NBT5-62 Division Review Pages STANDARDS 5.NBT.B.7, 5.NBT.B.6 Vocabulary dividend divisor fact family factor groups or sets items in each set quotient (MP.4) Goals Students will review how to write a division statement for items that have been grouped into sets. Students will review how to write members of a fact family for a division statement. PRIOR KNOWLEDGE REQUIRED Knows how to determine the number of sets and the number of items in each set Writing a multiplication statement for a given diagram. On the board, draw the diagram in the margin. ASK: How many groups are there? (3) How many items are in each group? (4) How many items are there altogether? (12) Ask a student to come to the board to write an addition statement for the diagram. ( = 12) Ask a different student to come to the board to write a multiplication statement for the diagram. (3 4 = 12) Although 4 3 is also correct, tell students we are going to use 3 4 = 12 because we have 3 groups of 4 items each rather than 4 groups of 3 items each. Ask another student to come to the board to write a division statement for the diagram. (12 3 = 4) Again, although 12 4 = 3 would also be correct, tell students that we are going to use the number of groups as the divisor. Writing a division statement for a diagram that uses base ten models. Draw on the board: = 1,000 = 100 = 10 = 1 ASK: Ask a student to come to the board and use the diagrams to represent 1,253. (see diagram below) Ask a student to come to the board to label the divisor, dividend, and quotient. (see answer below) 5 quotient divisor dividend Number and Operations in Base Ten 5-62 O-33

34 SAY: For now, we will use the divisor as the number of sets, the quotient as the items in each set, and the dividend as the total number of items. Draw on the board: ASK: What number is represented on the left? (320) How many groups or sets are on the right? (4) How many items are in each set? (80) How many items are there altogether? (320) Ask three different students to come to the board and fill in the divisor, the dividend, and the quotient. (see answer below) 8 0 quotient divisor dividend (MP.4, MP.7) Exercises: Write a division statement for the diagram. a) b) c) (MP.4) Answers: a) = 30, b) = 42, c) 1,224 4 = 306 Find the members of a fact family given a division question. Write on the board: ASK: How many sets are there? (6) How many items are in each set? (5) How many items are there altogether? (30) O-34 Teacher s Guide for AP Book 5.2

35 Draw the diagram shown in the margin on large grid paper, and tape to the board. 5 ASK: Why can we use this diagram to represent 6 3 0? (it has 6 rows and 5 dots in each row) What multiplication equation can this diagram represent? (6 5 = 30) Rotate the grid paper that was taped to the board by 90 o. rotate 90 o SAY: If we rotate the diagram to turn it on its side, we don t change the number of dots. ASK: In the rotated diagram, how many rows are there? (5) How many columns? (6) How many dots are there altogether (30) What multiplication equation can the rotated diagram represent? (5 6 = 30) 6 What division equation can it represent? ( or 30 5 = 6) SAY: So the dot diagrams lead to two multiplication equations and two division equations. 5 6 = = = = 5 SAY: We say these equations form a fact family. Exercises: Find all four members of the fact family. a) b) c) Answers: a) 8 3 = 24, 3 8 = 24, 24 3 = 8, 24 8 = 3; b) = 168, = 168, = 14, = 12; c) = 832, = 832, = 26, = 32 Solve a division equation by thinking of one of the multiplication equations in the fact family.? SAY: If we don t know the answer to the division equation, think of a multiplication equation in the fact family: 8? = 24 SAY: Because 8 3 = 24, we know that Number and Operations in Base Ten 5-62 O-35

36 Exercises: Find the quotient by solving an equivalent multiplication equation. a) 8 = 56 so b) 9 = 72 so c) 12 = 48 so Extensions d) 11 = 88 so (MP.1) 1. A fruit basket has fewer than 20 apples. The number of apples can be shared equally among 3, 4, or 6 people. How many apples are in the basket? Answer: 12 (MP.1) 2. The number 12 has the following factors: 1, 2, 3, 4, 6, and 12. Each of these numbers divides evenly into 12. Find all the factors of: a) 18 b) 36 c) 24 d) 45 Answers a) 1, 2, 3, 6, 9, 18 b) 1, 2, 3, 4, 6, 9, 12, 18, 36 c) 1, 2, 3, 4, 6, 8, 12, 24 d) 1, 3, 5, 9, 15, 45 (MP.3) 3. Count the total number of factors for each part in Question 2. a) Is the total number of factors even or odd? Answer: 2. a), c), d) are all even; b) is odd b) Why is the total number of factors for 2. b) odd? Answer: For 2. a), c), and d), the factors can be grouped in pairs (e.g., for 18, the factors are 1 & 18, 2 & 9, 3 & 6). When you try to group the factors for numbers like 36 (known as perfect squares), one of the pairs has the same number repeated. Because we only write this number once in the list of factors, there will be an odd number of factors: 1 & 36, 2 & 18, 3 & 12, 4 & 9, 6. O-36 Teacher s Guide for AP Book 5.2

37 NBT Digit Division (Introduction) Pages STANDARDS 5.NBT.B.7, 5.NBT.B.6 Goals Students will estimate the quotient by rounding the divisor to the nearest ten and finding the number of tens in the dividend. Vocabulary dividend divisor estimate groups multiples quotient round skip count (MP.4) PRIOR KNOWLEDGE REQUIRED Knows how to round a number to the nearest ten MATERIALS base ten materials Find the number of tens in a number using base ten materials. Write on the board: 247. Ask students to use their base ten materials to represent this number. (see diagram below) (MP.4) Ask students to exchange each hundreds block for 10 tens blocks and count the total number of tens blocks. (24) Exercises: Use base ten materials to find the number of tens. a) 318 b) 274 c) 729 Answers: a) 31, b) 27, c) 72 Find the number of tens in a number by crossing out the ones digit Ask a student to come to the board and cross out the ones digit. Ask a different student to come to the board and circle the remaining digits, as shown in the margin. SAY: The number circled is the same as the number of tens we found when using the base ten materials. Exercises: Find the number of tens by crossing out the ones digit and circling the remaining digits. a) 318 b) 274 c) 729 Answers: a) 31, b) 27, c) 72 Find the divisor and the number of tens in the dividend. Write on the board: Ask a student to write the corresponding division equation for the division question on the board. ( = 14) Number and Operations in Base Ten 5-63 O-37

38 Ask two students to come to the board and label the divisor, dividend, and quotient for each division question. (see answers below) dividend quotient 1 4 quotient = 14 divisor dividend divisor ASK: Which word represents the total number of items? (dividend) ASK: Which word represents the number of groups or sets? (divisor) ASK: Which word represents the number of items in each group? (quotient) Exercises: Find the number of groups and number of tens in the dividend. a) b) Answers: a) 13, 18; b) 17, 23; c) 18, c) (MP.4) Review rounding numbers to the nearest ten. Draw on the board: Ask a student to place a dot on the number line to represent 72. Ask: What multiples of 10 is the dot in between? (70, 80) ASK: Is the dot closer to 70 or 80? (70) SAY: So 72 rounded to the nearest ten is 70. ASK: How could we have found this answer without using a number line? (look at the last digit in 72) When do we round up to the nearest ten? (if the last digit is 5 or greater) When do we round down to the nearest ten? (if the last digit is 4 or smaller) Exercises: Round to the nearest ten. a) 84 b) 68 c) 43 d) 79 Answers: a) 80, b) 70, c) 40, d) 80 Make an initial estimate of the quotient SAY: To make an estimate of the quotient, round the divisor to the nearest ten and find the number of tens in the dividend. Number of tens in 73 = 24 rounded to nearest 10 = Ask two different students to come to the board to write the answers. If needed, provide these hints: Cross out the ones digit in 73. Is 24 closer to 20 or closer to 30? (20) SAY: To estimate the quotient, count by 20 until you pass 70. O-38 Teacher s Guide for AP Book 5.2

39 20, 40, 60, 80 too high ASK: How many multiples of 20 did we write before we passed 70? (3) SAY: So 3 is our estimate. Continue writing on the board: Exercises: Estimate the quotient by finding the number of tens in the dividend, rounding the divisor to the nearest ten, and then counting multiples of the new divisor until you pass the number of tens. a) b) c) Answers: a) 3, b) 2, c) 3 Estimate the number of tens in each group ASK: How many groups or sets are there? (18) How many items are there altogether? (612) ASK: How many tens are there in the dividend? (61) What is the divisor rounded to the nearest ten? (20) Ask a student to count by 20s until the student passes 61. Write the answer on the board. (20, 40, 60, 80) ASK: How many multiples did we write before we passed 61? (3) SAY: So there are 3 tens in each group. Exercises: Estimate the number of tens in each group for the division. a) b) c) Answers: a) 2, b) 3, c) 2 (MP.1) (MP.1) Extensions 1. Hot dog buns often come in packages of 10, while the actual hot dogs usually come in packages of 12. Find the number of hot dog bun packages needed for five packages of hot dogs. Answer: 6 2. A cashier went to the store office to exchange quarters for dimes. How many dimes should the cashier get for 324 quarters? Answer: 810 Number and Operations in Base Ten 5-63 O-39

40 NBT Digit Division Pages STANDARDS 5.NBT.B.7, 5.NBT.B.6 Vocabulary dividend divisor estimate quotient round Goals Students will divide a multi-digit number by a two-digit divisor by rounding the divisor to the nearest ten and finding the number of tens in the dividend. The questions are designed so they do not require a guess and check method. PRIOR KNOWLEDGE REQUIRED Knows how to round to the nearest ten Knows how to find the number of tens in a multi-digit number Knows how to divide a multi-digit number by a single-digit divisor MATERIALS play money (MP.4) Use money to introduce division by a two-digit divisor. Ask a student to come to your desk and hand over $714 in play money. (see diagram below) SAY: A group of 21 parents are sharing the cost of a $714 ping pong table for the school. Ask for some guesses about how much each parent should contribute. $100 $10 $1 ASK: Why can t we divide the $100 bills among the 21 parents? (there are only 7 bills, but 21 parents) SAY: We need to make change for the $100 bills. We want to find the number of $10 bills in $714. ASK: How many $10 bills are in $100? (10) SAY: Let s replace each $100 bill with ten $10 bills. Ask seven different students to come up and each replace a $100 bill with ten $10 bills. ASK: How many $10 bills are in $700? (70) How many $10 bills are in $14? (1) How many $10 bills are there altogether in $714? (71) ASK: What is the number of parents rounded to the nearest ten? (20) SAY: Count by 20 until you pass the number of $10 bills. (20, 40, 60, 80) ASK: How many multiples did you count before you passed the number of $10 bills? (3) SAY: We need to divide the money owed among 21 parents. Ask for 21 volunteers to act as the parents and come to the front of the class and each take three $10 bills back to their desks. How many $10 bills were shared? (63) How many $10 bills and $1 bills are left? (eight $10 bills, four $1 bills, as shown below) $10 $1 O-40 Teacher s Guide for AP Book 5.2