Unit 5 Ratios and Proportional Relationships: Real World Ratio and Percent Problems

Transcription

1 Unit 5 Ratios and Proportional Relationships: Real World Ratio and Percent Problems The material in this unit crosses three strands The Number System (NS), Expressions and Equations (EE), and Ratios and Proportional Relationships (RP). In this unit, students will use rounding to estimate sums and products; multiply and divide decimals by powers of 10; multiply positive and negative decimals and fractions by whole numbers; evaluate percents; compare fractions, decimals, and percents; add and subtract percents; use percents to solve real-world problems; and use tape diagrams to solve percent problems. Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-1

2 RP7-12 Rounding Pages Standards: preparation for 7.EE.B.3 Goals: Students will round whole numbers and decimals to a given place value, such as the nearest tenth. Prior Knowledge Required: Can determine which multiples of 10, 100, or 1,000 a number is between Can find which multiple of 10, 100, or 1,000 a given number is closest to Can regroup when adding whole numbers or decimals Vocabulary: approximately equal to sign ( ), estimate, round number, rounding Materials: a calculator for each student Review the meaning of rounding. Draw on the board: Circle the numbers 324, 326, 327, and 322, one at a time, and ask volunteers to draw an arrow showing which multiple of ten is closest. Remind students that when they are performing a calculation, they can use estimation to get an idea of the size of the answer or whether an answer is reasonable. To estimate the result of a calculation, replace the numbers in the calculation with numbers that are close to the original number but have one or more zeros in the rightmost place values, because these numbers are easier to work with. That process is called rounding the number. Now circle 325. SAY: 325 is equally close to 320 and 330, but the convention is to round up to 330. Rounding multi-digit numbers to any place value. Show students how numbers can be rounded in a grid. Follow the steps shown below. Example: Round 12,473 to the nearest thousand. Step 1: Underline the digit you are rounding to F-2 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

3 Step 2: Put your pencil on the digit to the right of the one you are rounding to Step 3: Write round up if the digit under your pencil is 5, 6, 7, 8, or 9, or round down if the digit is 0, 1, 2, 3, or 4. round down Step 4: Round the underlined digit up or down according to the instruction you have written round down Exercises: Round the underlined digit up or down, as indicated. a) round down b) round up c) round down Answers: a) 6, b) 1, c) 5 Step 5: Change all digits to the right of the rounded digit to zeros Step 6: Copy all digits to the left of the rounded digit as they are SAY: So 12,473 rounded to the nearest thousand is 12,000. That makes sense because the number is between 12,000 and 13,000, but is closer to 12,000 than to 13,000. Exercises: Round to the underlined place value. a) 32,623 b) 12,821 c) 12,493 d) 9,575 e) 463,511 Answers: a) 33,000; b) 12,800; c) 12,500; d) 9,600; e) 464,000 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-3

4 When students finish, draw their attention to part e) where they rounded up because 5 was the next digit. SAY: 463,511 is indeed closer to 464,000 than to 463, ,500 is equally close to both thousands but, if there are any non-zero digits to the right of the 5, the number actually is closer to 464,000. That s why the convention is to always round up. Rounding with regrouping. Write on the board: round up SAY: The 10 hundreds need to be regrouped as 1 thousand. Add that to the 7 thousands to get 8 thousands. Then copy any remaining digits to the left, as shown below: SAY: This makes sense because the number has 179 hundreds, so rounding up gives 180 hundreds. Exercises: Round to the given place value. Use grid paper. a) 39,673 to thousands b) 12,971 to hundreds c) 12,993 to tens d) 9,987 to hundreds Answers: a) 40,000, b) 13,000, c) 12,990, d) 10,000 Rounding decimals. Tell students that you use the same steps to round decimals as you use to round whole numbers. Example: Round to the nearest tenth. Step 1: Underline the digit you are rounding to Step 2: Put your pencil on the digit to the right of the digit you are rounding to F-4 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

5 Step 3: Write round up if the digit under your pencil is 5, 6, 7, 8, or 9, or round down if the digit is 0, 1, 2, 3, or round up Step 4: Round the underlined digit up or down according to the instruction you have written Step 5: Change all digits to the right of the rounded digit to zeros Step 6: Copy all digits to the left of the rounded digit as they are SAY: So rounded to the nearest tenth is 2.4. Remind students that = 2.4, so they don t need to write the zeros after the digit they rounded to. That makes sense because the number is between 2.3 and 2.4, but is closer to 2.4 than to 2.3. Exercises: Round to the underlined place value. a) b) c) d) Answers: a) 13.5, b) 38, c) , d) Rounding decimals with regrouping. Write on the board. Demonstrate rounding to the nearest tenth, as shown below: round up Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-5

6 SAY: If you have 10 in a place value, you need to regroup rounded to the nearest tenth is 3.0, or just 3. Usually, when you round to the nearest tenth, you get an answer in tenths. But, in this case, the answer is 3.0 which you can write as just 3. Exercises: Round to the given place value. Use grid paper. a) to hundredths b) to tenths c) to ones d) to thousandths Answers: a) 43.7, b) 75, c) 60, d) 84.1 NOTE: In Grade 8, students will learn that zeros at the end of a decimal can indicate precision. In this grade, the extra zeros are not necessary since they do not contribute to the value of the number. Rounding answers from calculator operations. Give students a calculator. Tell them that they will often get a decimal answer when dividing whole numbers. That never happens when they re adding, subtracting, or multiplying whole numbers, but it can when they re dividing whole numbers. Have students use a calculator to perform the division Have a volunteer write on the board the answer their calculator shows. Depending on the calculator display, students might see NOTE: Later in the year, students will learn how to do this type of division question (with repeating decimals) using long division. For now, allow students to use calculators. (MP.4) SAY: Suppose that we need to round 95 7 to the nearest hundredth. Have a volunteer underline the digit in the hundredths position, as shown below: ASK: Do you round up or down? (down) PROMPT: What is the next digit after the 7? (1) Have a volunteer do the rounding on the board, as shown below: Write on the board: Have volunteers round to the nearest tenth (2.9), then hundredth (2.85), then thousandth (2.853). SAY: It is already written as thousandths, so to round to the nearest thousandth, you don t have to do anything. Introduce the approximately equal sign. Tell students that we use a symbol that looks almost like an equal sign to show that two numbers are almost equal. Write on the board: F-6 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

7 SAY: The symbol that looks like a squiggly equal sign means almost equal. In mathematics we say approximately equal, and we call this sign the approximately equal to sign. Exercises: Use a calculator to perform the division, then round your answer to the nearest hundredth. a) b) c) d) e) Answers: a) 0.67, b) 2.78, c) 0.51, d) 1.57, e) 5.25 ASK: Which of the approximately equal signs in the previous exercises should be an equal sign? (the last one, 21 4 = 5.25) SAY: Rounding to the nearest hundredth doesn t change the number at all, so we can use the equal sign for that one. More rounding practice. Have students round the same number to different place values. Exercises: a) Round 365,257 to the nearest ten, hundred, thousand, and ten thousand. b) Round 14,581.9 to the nearest ten, hundred, thousand and ten thousand. c) Round to the nearest tenth, hundredth, thousandth, and ten thousandth. d) Round to the nearest ten, one, tenth, and hundredth. Answers: a) 365,260, 365,300, 365,000, 370,000; b) 14,580, 14,600, 15,000, 10,000; c) 0 or 0.0, 0.03, 0.027, ; d) 20, 17, 16.6, Mention to students that, in the case of part d), the number was already provided to the nearest hundredth so there s nothing to round. Extensions (MP.4) 1. a) Use a calculator to perform the division 88 7 and round your answer to the nearest tenth. Then multiply your rounded answer by 7. Is your answer more than or less than 88? Why does this make sense? b) 7 people are sharing the cost of an $88 item. How much should each person pay? Hint: How many people have to pay an extra penny? c) 9 people are sharing the cost of a $34 item. How much should each person pay? Answers: a) and = 88.2, which is greater than 88. This makes sense because we rounded up to get 12.6; b) Round to the nearest hundredth for cost = 12.57, but = 87.99, which is $0.01 short of $88.00, so one person has to pay an extra penny; c) Rounding to the nearest hundredth, So, each person should pay $3.78. Since = 34.02, two people can each pay one cent less. 2. a) Write a number that can be rounded to both 20,000 and 17,000, depending on the place value being rounded to. b) Write a number that can be rounded to 30,000, 26,000, and 26,300 Bonus: Write a number that can be rounded to 800,000, 830,000, 826,000, and 825,700. Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-7

8 Sample answers: a) 17,357 rounds to 20,000 to the nearest ten thousand and rounds to 17,000 to the nearest thousand; b) 26,310 rounds to 30,000 to the nearest ten thousand, rounds to 26,000 to the nearest thousand, and rounds to 26,300 to the nearest hundred; Bonus: 825,683 (MP.3) 3. Use the number line to round each negative number to the nearest ten How is rounding negative numbers similar to rounding positive numbers? Answers: 29 30, 22 20, 26 30, 24 20; Round as though the decimals are positive, then put back the negative sign. (MP.3, MP.4, MP.6) 4. Decide what place value it makes sense to round each of the following to. Round to the place value you selected. Justify your decisions. Height of a person: m Height of a tree: m Length of a bug: cm Distance between Washington, DC, and Hong Kong: 13, km Distance between Earth and the Moon: 384,403 km Population of Kolkata, India, in 2011: 4,486,679 people Floor area of an apartment: ft 2 Area of New York State: 141,299 km 2 Angle between two streets: Time it takes to blink: 0.33 s Speed of a car: mph Time it takes to ski a downhill course: s Answers: Answers will vary. The larger the number, the less important the smaller place values become. The way the measurement will be used is also a factor. For example, when measuring the time it takes to ski a downhill course, more accuracy might be needed at an international competition (when world records might be set) than would be needed during training. F-8 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

9 RP7-13 Upper Bounds and Lower Bounds Pages Standards: 7.EE.B.3 Goals: Students will round whole numbers and decimals to estimate sums and products. Students will find upper and lower bounds for sums and products by rounding both numbers up or both numbers down. Prior Knowledge Required: Can round decimals to a given place value Can add and multiply multi-digit whole numbers Materials: a calculator for each student BLM Upper and Lower Bounds for Subtraction (p. F-80, see Extension 4) Vocabulary: approximately equal to sign ( ), estimate, lower bound, round number, rounding, upper bound Estimating sums by rounding. Write on the board: ASK: Which of these additions is easier? (the second one) Do you think the answers will be close? (yes) Why? (because 500 is close to 475 and 300 is close to 321) Have students calculate both sums to check their prediction. (796 and 800) SAY: Sometimes you don t need an exact answer, just an answer that is close. We call this estimating. Continue writing on the board: = 800 Remind students that the squiggly equal sign means approximately equal to. Exercises: Estimate by rounding each number to the given place value. a) (tens) b) 3, ,714 (hundreds) c) 7, ,278 (thousands) d) (whole number) e) (tenths) f) (hundredths) g) (tens) h) (hundreds) i) (whole number) j) (tenths) Bonus: k) 8, ,218 (thousands) l) (hundreds) m) (tens) n) (tenths) o) (hundredths) p) (whole number) Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-9

10 Answers: a) = 580; b) 3, ,700 = 8,400; c) 8, ,000 = 9,000; d) = 5; e) = 5.7; f) = 12.24; g) = 11,000; h) = 250,000; i) = 10,000; j) = 15; Bonus: k) 9, , ,000 = 47,000; l) = 1,300; m) = 6,000; n) = 23.7; o) = 15.33; p) = 6 Rounding both numbers up to get an upper bound. Write on the board: < = 13 SAY: I know that 5.8 is less than 6 and 6.3 is less than 7, so their sum must be less than = 13. When you round both numbers up, the sum you get will be greater than the original sum. This is called an upper bound for the sum. Exercises: Round both numbers up to the next whole number to get an upper bound for the sum. a) b) c) d) e) f) Bonus: Round all numbers up to get an upper bound for the sum. g) h) Answers: a) 180, b) 11, c) 66, d) 232, e) 4, f) 874, Bonus: g) 209, h) 17 (MP.1) Have students use a calculator to check that the sum is actually less than their upper bound. If not, students should find their mistake. NOTE: Be sure students understand how to enter a decimal point on a calculator. Rounding both numbers down to get a lower bound. ASK: How would I get a lower bound for the sum? (round both numbers down) Illustrate this using the same sum as above. Write on the board: > = 11 (MP.1) Have students use a calculator to find the sum of (12.1) ASK: Is your answer between 11 and 13? (yes) If not, students should look for their mistake. Point out to students that 5.0 is equal to 5, so 5 is the best whole-number upper bound and the best whole-number lower bound for 5.0. Exercises: Find whole number upper and lower bounds, then use a calculator to find the actual sum. Is the actual sum between the upper and lower bounds? If not, find your mistake. a) b) c) d) e) f) Bonus: g) h) i) F-10 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

11 Answers: a) 3 < 3.6 < 5; b) 9 < 10 < 11; c) 100 < < 102; d) 44 < < 46; e) 29 < < 31; f) 53 < < 55; Bonus: g) 33 < 33.2 < 34; h) 20 is the actual sum, the upper bound, and the lower bound; i) 59 < < 62 (MP.5) Checking whether an answer obtained by a calculator is reasonable. Write on the board: Tell students that Kelly added these two numbers and got the answer ASK: Does the answer seem reasonable? (no) How can you tell? (the answer will be much less than 90) Point out that even rounding both numbers up to the nearest ten gives only 70 ( ), so the actual sum cannot be more than 90. SAY: You can round both numbers up and both numbers down. If the answer on the calculator isn t between those two values, then you know that you made a mistake. Exercises: A student performed the following calculations on their calculator. For each sum, find the upper and lower bounds by rounding to the higher and lower ten. Is the student s answer reasonable? a) = b) = c) = 27.9 d) = 102 e) = 8.5 f) 1, = 2,275 Answers: a) 430 < sum < 450, so is not reasonable; b) 20 < sum < 40, so is reasonable; c) 20 < sum < 40, so 27.9 is reasonable; d) 70 < sum < 90, so 102 is not reasonable; e) 10 < sum < 30, so 8.5 is not reasonable; f) 2,240 < sum < 2,270, so 2,275 is not reasonable (MP.1) Estimating products to check whether an answer is reasonable. Write on the board: = SAY: You can estimate to check whether this answer is reasonable. You can find a lower bound and an upper bound and make sure your answer is between the two numbers. Rewrite the product as shown below, underlining the whole-number part of each factor: < < SAY: Whole numbers are easier to multiply than decimals, so let s use the whole-number lower and upper bounds of each factor to check whether the answer is reasonable. ASK: What is the best whole-number lower bound for 3.14? (3) For 20.5? (20) Repeat for upper bounds. (4 and 21) Write the answers into the blanks as volunteers say them. ASK: What is 3 20? (60) What is 4 21? (84) Write on the board: 60 < < 84 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-11

12 Refer students back to the original answer on the board, ASK: Is the original answer reasonable? (no) Have students use a calculator to find the actual product. (64.37) Point out that someone might have punched the numbers into the calculator incorrectly, with the decimal point in the wrong place. So it s always a good idea to check with estimation whether the answer you calculate is reasonable. (MP.4) Importance of upper and lower bounds in real life. Write on the board: A T-shirt costs $ A sweater costs $ Tell students you want to estimate the total cost. Have students do so two ways: 1) Round both numbers to the nearest whole number and add. ($16 + $35 = $51) 2) Round both numbers up and add. ($ = $52) Have students use a calculator to find the total cost. ($51.40) ASK: Which estimate is closer to the actual sum? (rounding to the nearest whole number) Suppose you want to know how much money to bring, would $51 be enough? (no) SAY: If you want to know how much money to bring, you should round up that will be safer. Exercises: 1. A coffee costs $1.80 and a muffin costs $2.30. Round to the nearest whole number. a) Use a calculator to find the total cost of the coffee and muffin. b) Rob estimates the total cost by rounding to the nearest whole number. i) How much does Rob estimate the total cost to be? ii) If Rob brings that much money, would he have enough? c) What would the estimated cost be if Rob uses a lower bound? Is this enough to cover the total cost of the coffee and muffin? d) What would the estimated cost be if Rob uses an upper bound? Is this enough to cover the total cost of the coffee and muffin? e) Why is it better to use an upper bound in this case? Answers: a) $ $2.30 = $4.10 b) i) $4.00, ii) No, he would have $0.10 less than the actual cost. c) $ $2.30 > $ $2.00 = $3.00. This estimate is not enough to cover the total cost of $4.10 d) $ $2.30 < $ $3.00 = $5.00. This estimate is more than the total cost of $4.10, so it would be enough. e) If Rob uses a lower bound to estimate how much money to bring, then he won t have enough to cover the cost of the coffee and muffin. If Rob uses an upper bound to estimate how much money to bring, he will have more than enough to cover the cost of the coffee and muffin. In the case of costs, upper bounds are safer because they always provide an estimate that is higher than the actual cost, which is fine, since we ll get change back when making a purchase. F-12 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

13 2. a) The price of gas is $3.98 per gallon. Decide whether to use an upper or lower bound to estimate the cost of 6 gallons of gas. Explain your choice, then estimate the cost. b) 7 people go out to buy snacks costing $30 in total after taxes. Milly estimates that each person should bring $4.00 because the closest multiple of 7 to 30 is 28 and 28 7 = 4. Will Milly s estimate be enough to cover the cost? What would be a better estimate? c) A bowling alley charges $9.79 per customer for an evening of bowling. If Pedro wants to invite 14 friends to his birthday, estimate the cost for his party. Remember to include Pedro himself when estimating. Would Pedro s parents be more interested in using a lower bound or an upper bound? Why? d) A doctor is prescribing a certain medication to her patient; the amount prescribed is based on the mass of the patient. If 1.07 oz of the medication can be safely administered per pound of body mass, estimate how much medication the doctor should prescribe to a patient who weighs 120 lb. In this case, is it better to use an upper bound or a lower bound? Why? Answers: a) It is safer to use an upper bound because a lower bound would not cover the actual cost for 6 gallons of gasoline. Using an upper bound, $ gallons = $24 > $ gallons = $ b) If each person pays $4.00, then the total amount will be $ = $28.00, which is not enough to cover the $30.00 bill. Milly should use an upper bound instead: $35 7 = $5.00. c) $10 15 people = $150 > $ people = $ Pedro s parents would be more interested in using an upper bound to estimate the cost because the lower-bound estimate will be less than the actual cost. d)1.0 oz 120 lb = 120 oz < 1.07 oz 120 lb = oz. In this case, it would be safer to use a lower bound, because taking too much medication could be harmful to the patient. Extensions (MP.2) 1. Place the decimal point by estimating. You do not have to carry out the operation. a) = b) = Answers: a) , b) Without calculating the sum of , how can you tell whether the sum is greater than or less than 435? Sample answer: You can estimate an upper bound for each number. The sum is less than = 421, so it is less than 435. (MP.6) 3. Estimate by rounding both numbers to the given place value. a) tens b) ones c) tenths d) hundredths Which place value made estimating the difference the fastest? Which place value made estimating the difference the most accurate? Answers: a) = 0, b) = 1, c) = 0.6, d) = Rounding the numbers to the nearest ten or one produces an estimate quickly, but rounding the numbers to the nearest hundredth produces a more accurate estimate. Point out to students that there is always a trade-off between speed and accuracy. Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-13

14 4. On BLM Upper and Lower Bounds for Subtraction, students discover the rule for finding an upper or lower bound when calculating differences. To find an upper bound, for example, the number being subtracted from must be rounded up while the number being subtracted must be rounded down. Only give this BLM to students who can learn this rule independently, straight from the BLM. Answers: 1. b) 9 4 < < 10 3, so 5 < < 7; c) 8 3 < < 9 2, so 5 < < 7; d) 6 1 < < 7 0, so 5 < < 7 2. a) 4.81, b) 6.2, c) 6.38, d) a) A lower bound is safer so that she doesn t try to buy something worth more than the amount of money she has; b) = 420, so she has about $420 left. 4. a) He should get = 20 gallons; b) I used an upper bound to ensure that he gets enough gas for the trip. 5. Investigate how you would find an upper bound for Which number would you increase and which would you decrease? How would you find a lower bound for ? Answer: Look at each number individually. When the first number increases, so does the answer. But when you divide by a bigger number, the answer gets smaller. So, Upper bound: round up round down Lower bound: round down round up F-14 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

15 RP 7-14 Multiplying Decimals by Powers of 10 Pages Standards: preparation for 7.NS.A.2c Goals: Students will multiply decimals by 10, 100, and 1,000. Prior Knowledge Required: Knows the commutative property of multiplication Can multiply whole numbers by 10, 100, and 1,000 Understands decimal place value Can regroup ten of each place value to one of the next higher place value Can write decimals in expanded form Can read decimals in terms of smallest place value Can multiply a fraction by a whole number Vocabulary: decimal point, hundredth, tenth, thousandth Materials: 8 small cards (e.g., index cards) with a large dot drawn on each Using place value to multiply decimals by 10. Write on the board: 0.1 = 1 tenth = 1 10 so = = = = 1 Exercises: Multiply. a) b) c) Answers: a) 3, b) 8, c) 7 Draw the picture below on the board to remind students of the connection between place values: tens ones tenths hundredths thousandths Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-15

16 Write the following example on the board to illustrate: = = = 1,000 1, SAY: If 5 is in the thousandths position, then after multiplying by 10 it will be in the hundredths position. Exercises: 1. Use place value to multiply the number by 10. a) 3 hundredths 10 b) 4 tenths 10 c) 5 ones 10 d) 7 thousandths 10 Answers: a) 3 tenths, b) 4 ones, c) 5 tens, d) 7 hundredths 2. Use place value to multiply by 10. a) b) c) d) e) f) g) h) i) Answers: a) 0.1, b) 0.01, c) 0.001, d) 0.3, e) 0.03, f) 0.003, g) 5, h) 0.2, i) 0.06 Using expanded form to multiply decimals by 10. Remind students how to use expanded form to represent what they are doing. Write on the board: 4.36 = SAY: To multiply by 10, you can multiply each place value by 10. Write on the board: = = 43.6 Exercises: Use expanded form to multiply by 10. a) b) c) d) e) f) g) h) Sample solution: f) = , so = = Answers: a) 54, b) 603, c) 30.04, d) 58.1, e) 31.2, f) 840.6, g) 32.94, h) Move the decimal point to multiply decimals by 10. Ahead of time, draw a large decimal point on eight different cards. Write on the board the numbers from the Exercises above and tape each card to the board so it acts as a decimal point. The first two numbers are shown below F-16 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

17 Ask volunteers to move the decimal point to show the answer for all 8 questions. The first two answers are shown below Ask the rest of the class to look for a pattern in how the decimal point is being moved. (it always moved one place to the right) Point out that multiplying a number by 10 will always shift the decimal point one position to the right, because multiplying by 10 makes the number 10 times as big. Exercises: Move the decimal point one place to the right to multiply by 10. a) b) c) d) Bonus: 98, Answers: a) 32, b) 5.8, c) 2.16, d) 74.6, Bonus: 987, (MP.1) Move the decimal point to multiply decimals by 100 and by 1,000. Write on the board: = SAY: Move the decimal point once to multiply by 10, and again to multiply by another 10. Show this on the board: So = SAY: To multiply by 100, move the decimal point two places to the right. Exercises: Move the decimal point two places to the right to multiply by 100. a) b) c) d) Answers: a) 362, b) 72.5, c) 167.3, d) 8.5 (MP.8) SAY: We move the decimal point one place to multiply by 10 and two places to multiply by 100. ASK: How many places do we move the decimal point to multiply by 1,000? (3) Point out that the number of places to move the decimal point matches the number of zeros in 10, 100, or 1,000. Show the decimal point movement on the board: So ,000 = 2,467. Exercises: Move the decimal point to multiply by 1,000. a) ,000 b) ,000 c) ,000 Answers: a) 462, b) 11,241, c) 9, Using zero as a placeholder when multiplying decimals. Write 3.42 in a grid on the board as shown below. Use the card with a large dot for the decimal point so it can be moved Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-17

18 ASK: How many places do I have to move the decimal point if I am going to multiply 3.42 by 1,000? (3) Move the decimal point three times, as shown below: ASK: Are we finished writing the number? (no) Why not? What s missing? (a zero) Add the zero, as shown below: SAY: Each digit is worth 1,000 times as much as it was. The number was 3 ones, 4 tenths, and 2 hundredths. Point to each digit in the second grid and SAY: Now it is 3 thousands, 4 hundreds, and 2 tens. So the number is 3,420. Encourage struggling students to write each place value in its own cell of the grid for the Exercises below, and to draw arrows to show how they moved the decimal point. Students can also use expanded form to check their work if they are struggling. Exercises: 1. Multiply by 1,000. a) 0.4 1,000 b) ,000 c) ,000 d) ,000 Sample solution: b) So ,000 = 5,240. Answers: a) 400; b) 5,240; c) 23,600; d) Multiply by 10, 100, or 1,000. a) b) c) ,000 d) e) ,000 f ) ,000 g) Bonus: ,000 Answers: a) 60; b) 72.8; c) 25,600; d) 180; e) 21,900; f) 326,300; g) 0.02; Bonus: 23,000 (MP.1) Connect multiplying whole numbers by 10 to multiplying decimals by 10. Write on the board: 3 4 Leave enough space between the digits to place the card with the decimal point. ASK: What is 34 10? (340) SAY: We can also multiply by moving the decimal point. Write on the board: F-18 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

19 Place the card in the position of the decimal point. Move it one place to the right and point out that this is the same answer you get the other way, as shown below: SAY: Multiplying whole numbers is really the same method we use to multiply decimals. (MP.4) Word problems practice. Exercises: a) Nancy earns $12.50 an hour mowing lawns. How much does she earn in 10 hours? b) A clothing-store owner wants to buy 100 coats for $32.69 each. How much will the coats cost? c) A dime is cm thick. How tall would a stack of 100 dimes be? d) A necklace has 100 beads. Each bead has a diameter of 1.32 mm. How long is the necklace? Answers: a) $125.00, b) $3,269.00, c) 13.5 cm, d) 132 mm Extensions 1. Fill in the blanks. a) 10 = 38.2 b) 100 = 6.74 c) 42.3 = 4,230 d) 0.08 = 0.8 Answers: a) 3.82, b) , c) 100, d) 10 (MP.7) 2. Write the next two numbers in the pattern. a) , 0.007, 0.07, b) 3.895, 38.95, 389.5, Answers: a) 0.7, 7; b) 3,895, 38, (MP.2) a) Switch the numbers around to make the product easier to find. Then find the product. i) (3.2 5) 20 ii) (6.73 2) 50 iii) ( ) (25 8) b) In part a), what property did you use? Answers: a) i) 320, ii) 673, iii) 7,836; b) the associative property (MP.4) 4. Create a word problem that requires multiplying by 1,000. Have a partner solve it. (MP.4) 5. One marble weighs 3.5 g. The marble bag weighs 10.6 g. How much does the bag weigh with 100 marbles in it? Answer: g Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-19

20 RP7-15 Multiplying and Dividing by Powers of 10 Pages Standards: preparation for 7.NS.A.2c Goals: Students will shift the decimal point to multiply and divide decimals by 10, 100, and 1,000. Prior Knowledge Required: Can multiply whole numbers and decimals by 10, 100, and 1,000 Knows that multiplication and division are opposite operations Understands decimal place value Can write decimals in expanded form Can multiply a fraction by a whole number Vocabulary: decimal point, hundredth, tenth, thousandth Materials: a card with a large dot drawn on it lots of beans and a scale (see Extension 4) Divide by 10 by inverting the rule for multiplying by 10. SAY: We know how to multiply by powers of 10. Remember that if you know multiplication facts, then you can write division facts, too. Write on the board: 2 3 = = so so 6 2 = = 6 3 = 2 Have a volunteer do the division. (34.25) SAY: I moved the decimal point one place to the right to multiply by 10. ASK: How did the volunteer move the decimal point when dividing by 10? (one place to the left) Emphasize that division does the opposite of multiplication. SAY: To multiply by 10, move the decimal point one place to the right. And, to divide by 10, move the decimal point one place to the left. Another way to look at this is that dividing a number by 10 makes the number one tenth as big (for example, = 3), so the decimal point shifts one place to the left. Exercises: Divide by 10. a) b) c) d) Answers: a) 1.45, b) 6.48, c) 0.922, d) F-20 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

21 Dividing by 100. Write on the board: = 583.1, so = Ask a volunteer to fill in the blank. (5.831) Point out that the equations are in the same fact family, so knowing how to multiply by 100 also tells us how to divide by 100. ASK: How do we move the decimal point to divide by 100? (two places to the left) Point out that you had to move it two places to the right to multiply by 100. Then, to get back, you need to move it to the left two places. Exercises: Divide by 100. a) b) c) d) Answers: a) 0.145, b) 4.648, c) , d) Dividing whole numbers by 10 and 100. Write on the board: 6 7 Leave room between the digits for the card with the large dot. Tell students you want to know the answer for SAY: I would do the division by moving the decimal point, but I don t see any decimal point here. What should I do? (add the decimal point to the right of the ones, because 67 = 67.0; the decimal point always goes after the ones place) Do so, using the card. Then invite a volunteer to move the decimal point one place to the left to get = 6.7. Repeat the process with and 1, (0.18, 19.87) Exercises: Divide by 10 or 100. a) b) c) 1, d) 14, Answers: a) 23.6, b) 5.73, c) or 12.3, d) 1,488.9 Dividing by 1,000. ASK: How would you shift the decimal point to divide by 1,000? (3 places to the left) Show this example done on a grid on the board: So 45 1,000 = Exercises: Divide. a) 2,934 1,000 b) 423 1,000 c) ,000 d) ,000 Bonus: ,000 Answers: a) 2.934, b) 0.423, c) , d) , Bonus: Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-21

22 (MP.1) Strategies for remembering which way to move the decimal point. SAY: Remember: Multiplying by 10, 100, or 1,000 makes the number bigger, so the decimal point moves to the right. Dividing makes the number smaller, so the decimal point moves to the left. If students have trouble deciding which direction to move the decimal point when multiplying and dividing by 10, 100, or 1,000, one hint that some students might find helpful is to use the case of whole numbers as an example. Which way does the decimal point move when multiplying = 340? (to the right) Exercises: Multiply or divide. a) 78,678 1,000 b) c) d) ,000 e) f) g) h) i).31 1,000 Bonus: j) 31, ,000,000 k) 31, ,000,000 Answers: a) ; b) 242.3; c) 1.89; d) 1,310; e) 0.06; f) 0.82; g) 0.002; h) 510; i) 310; Bonus: j) ; k) 31,498,765,320 Remind students who are struggling to write each place value in its own cell on grid paper when multiplying or dividing decimals by powers of 10. (MP.1) Connect multiplying decimals by 10 to multiplying fractions by 10. Write on the board: ASK: What is the numerator of the answer? (the whole number times the numerator, or 70) SAY: 10 7 is 70. Write 70 in the numerator position. ASK: What is the denominator of the answer? (1,000, the denominator of the fraction) Write the answer on the board, as shown below: Have a volunteer write the answer in lowest terms. (7/100) Write on the board: = Have a volunteer tell you the answer as a decimal. (0.07) SAY: Whether you write 7 thousandths as a decimal or a fraction, 10 times it is still 7 hundredths. F-22 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

23 Exercises: 1. Multiply. Write your answer in lowest terms. 1 7 a) 10 = b) = 7 c) = d) , 000 = e) , 000 = f) , 000 = Answers: a) 1/10, b) 7, c) 7/10, d) 3/100, e) 37/100, f) 143/ Multiply. a) = b) = c) = d) = e) = f) = Bonus: Do you notice any similarities in your answers to Exercise 1 (multiplying fractions by 10) and Exercise 2 (multiplying decimals by 10)? Why do you think this is? Answers: a) 0.1, b) 7, c) 0.7, d) 0.03, e) 0.37, f) 1.43, Bonus: The answers are the same. The reason is that each fraction in Exercise 1 has the same numerical value as the decimal in Exercise 2. We re multiplying both numbers by 10, so we should get the same result. (MP.4) Word problems practice. Exercises: a) In 10 months of fundraising, a charity has raised $26, How much did they raise each month on average? b) A stack of 100 cardboard sheets is 13 cm high. How thick is one sheet of the cardboard? c) A thousand people attended a pay what you can event. The total money paid was $5,750. Kim paid $0.60. Did she pay more or less than average? d) A hundred walruses weigh metric tonnes (1 metric tonne = 1,000 kg). How much does one walrus weigh on average, in kilograms? e) A box of 1,000 nails costs $ i) How much did each nail cost, to the nearest cent? ii) One hundred of the nails have been used. What is the cost for the nails that are left, to the nearest cent? Hint: Use the actual cost of a nail in your calculations, not the rounded cost from part a). Answers: a) $2,657.58; b) 0.13 cm; c) less, the average was $5.75; d) 1,215 kg; e) i) 1, ii) $11.65 or $11.66 (the calculation can be done as = which rounds to 11.66, or as = by rounding to 1.30) Extensions 1. A penny has a width of mm. How long would a line of 10,000 pennies, laid end-to-end, be in millimeters, centimeters, meters, and kilometers? Answers: 190,500 mm, 19,050 cm, m, km Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-23

24 2. (MP.2) a) Ten of an object laid end-to-end have a length of 48 cm. How long is the object? What might the object be? b) One hundred of an object laid end-to-end have a length of 2.38 m. How long is the object, in centimeters? What might the object be? c) One thousand of an object laid end-to-end have a length of 274 m. How long is the object, in centimeters? What might the object be? Answers: a) 4.8 cm, sample answer: an eraser; b) 2.38 cm, sample answer: a quarter (coin); c) 27.4 cm, sample answers: a shoe, a sheet of paper (MP.4) 3. Create your own word problems that require multiplying and/or dividing decimals by powers of 10. Then trade with a partner and solve the problems. (MP.4) 4. Find the mass of one bean by weighing 100 or 1,000 beans. Use a calculator to determine how many beans are in a 2 lb (908 g) package. (MP.3) 5. How would you shift the decimal point to divide by 10,000,000? Explain. Answer: Move the decimal 7 places (because there are 7 zeros in 10,000,000) to the left (because I am dividing). F-24 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

25 RP7-16 Multiplying Decimals by Whole Numbers Pages Standards: 7.NS.A.2c Goals: Students will multiply decimals by whole numbers. Prior Knowledge Required: Can multiply and divide whole numbers and decimals by powers of 10 Can use the standard algorithms to multiply and divide whole numbers by 1-digit numbers Understands decimal place value Can regroup decimals Can write decimals in expanded form Vocabulary: decimal point, hundredth, round number, rounding, tenth, thousandth Multiplying decimals without regrouping. Draw on the board: Ask a volunteer to draw on the board a model for Then extend the model yourself to show on the board: ASK: What number is this? (2.46) Write on the board: = 2.46 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-25

26 SAY: This is 2 ones, 4 tenths, and 6 hundredths. Exercises: Draw models to multiply. a) b) Answers: a) 8.02, b) 9.36 Point out that what students did is the same as writing the decimal in expanded form and multiplying each place value separately. Write on the board: 1.23 = 1 one + 2 tenths + 3 hundredths = 2 ones + 4 tenths + 6 hundredths SAY: Each digit is multiplied by 2. Exercises: Multiply mentally. a) b) c) Bonus: d) e) Answers: a) 8.44, b) 6.93, c) , Bonus: d) , e) Using expanded form to multiply decimals with regrouping. Write on the board: Number Tens Ones Tenths Hundredths =? 6 16 Ask a volunteer to regroup the ones and tenths to find the number that equals (7 ones and 6 tenths = 7.6) Have students copy a blank chart with the same headings into their notebook to do the Exercises below. Exercises: Multiply by regrouping when necessary. a) b) c) d) e) f) Answers: a) 7.2, b) 12.8, c) 6.12, d) 13.26, e) 13.28, f) (MP.1) Compare multiplying decimals to multiplying whole numbers. Have a volunteer use the chart above to multiply (6 tens + 16 ones = 7 tens + 6 ones = 76) Discuss with students the similarities and differences between the two problems and solutions. (The digits are the same, the regrouping is the same, but the place values are now 10 times as big tens instead of ones, and ones instead of tenths.) SAY: 2 38 is 10 times more than because 38 is 10 times more than 3.8. F-26 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

27 Using the standard method to record multiplication. Use grids to multiply 38 5 and on the board: Emphasize that none of the digits in the answer changes when the question has a decimal point; only the place values of the digits change. Even the regrouping looks the same. In placing the decimal point, point out that we want the same number of digits after the decimal point in the answer as there are in the decimal in the question. SAY: 3.8 has one digit after the decimal point, so put one digit after the decimal point in You can remove any final zeros that occur after the decimal point. In this case, 19.0 becomes 19. Exercises: Multiply. a) b) c) d) e) f) g) h) Bonus: i) 834, j) 5, Answers: a) or 20.1; b) 82.62; c) 2,617.0 or 2,617; d) ; e) 275.4; f) 15.12; g) 604.8; h) ; Bonus: i) 1,669,559.36; j) 38, (MP.5) Explain that you can use estimation to check the answer. SAY: Suppose you ve done the multiplication and got SAY: If I multiply 332 4, I can check if the digits in my answer are correct. Have a volunteer do this. (332 4 = 1,328) SAY: So I know the digits are correct, but is my answer to the original multiplication reasonable? (no) Why not? (3.32 is close to 3, so should be close to 3 4 = 12.) SAY: It s easier to estimate if you round each number to its highest non-zero place value so you only have one non-zero digit. So 332 rounds to 300, and 3.32 rounds to 3, and rounds to Exercises: 1. Estimate to make sure your answers to the previous Exercises are reasonable. Round each number to its highest non-zero place value. Answers: a) 3 6 = 18; b) 40 2 = 80; c) = 2,500; d) 10 3 = 30; e) 3 80 = 240; f) = 12; g) 8 70 = 560; h) = 14; Bonus: i) 800,000 2 = 1,600,000; j) 5,000 7 = 35, David thinks = What mistake did he make? How can you use estimation to know that he made a mistake? Answer: David multiplied 4 3 and put both digits in the answer after the decimal point. The answer should be more than 0.4, but 0.12 is less than 0.4, so you can tell he made a mistake. Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-27

28 (MP.3) Multiplying negative decimals by whole numbers. Remind students that multiplying by a whole number is repeated addition. Write on the board: 3 4 = = 3 ( 4) = = Have a volunteer fill in the first answer (12), then SAY: You can add negative numbers by first adding the positive numbers, then changing the sign. Show this on a sketch of a number line, as shown below: Write 12 in the remaining blank. Write on the board: = = 3 ( 1.2) = = Have a volunteer fill in the first blank. (3.6) ASK: So what is 3 times 1.2? ( 3.6) Exercises: Use repeated addition to multiply. a) b) 2 ( 0.1) c) 3 ( 0.3) d) 4 ( 0.25) e) 4 ( 1.25) f) 7 ( 0.15) g) 1 ( 0.67) h) 9 ( 1.11) Bonus: 3 ( ) Answers: a) 0.2, b) 0.2, c) 0.9, d) 1, e) 5, f) 1.05, g) 0.67, h) 9.99, Bonus: Writing simple fractions as decimals. SAY: If you can write a fraction with the numerator 1 as a decimal, then you can write as a decimal any fraction with that same denominator. Write on the board: 1 5 = = + = ASK: What is one fifth written as a decimal? (0.2) Continue writing on the board: 1 2 = = = + = = F-28 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

29 ASK: What is ? (0.4) Demonstrate how decimals can be added by lining up the decimal points, as shown below: Now continue with 3/5 and 4/5. Continue writing on the board: 1 5 = = + = = = + + = = = = = Have a volunteer fill in the sums. (0.6 and 0.8) (MP.7) Exercises: a) Use = to write 7 as a decimal. 2 b) Use = to write 5 as a decimal. 4 (MP.1) Bonus: Use 1 4 = 0.05 to write as a decimal. What reduced fraction is 4 20 equivalent to? Does it have the same decimal? Answers: a) 3.5; b) 1.25; Bonus: 0.20, and 4/20 is equivalent to 1/5 or 0.2, which is equivalent to 0.20 Using multiplication to write decimal fractions as decimals. SAY: You can use multiplication instead of repeated addition. Write on the board: = + + = = = 3 = = Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-29

30 (MP.1) Exercises: 1. Do the previous Exercises using multiplication instead of repeated addition. Make sure you get the same answer. (MP.7) 2. a) Multiply both numbers by 20 to check that 1 20 = b) Multiply 0.05 by 3 to write 3 20 as a decimal. c) Multiply 0.05 by 13 to write 13 as a decimal. 20 d) How much bigger is than 3? Write your answer as a reduced fraction. 20 e) How much bigger is the decimal for than the decimal for 3 20? f) Are your answers to d) and e) equivalent? If not, find your mistake. Answers: a) both numbers multiplied by 20 get 1, b) 0.15, c) 0.65, d) 1/2 bigger, e) 0.50 or 0.5 bigger, f) yes NOTE: Students will learn how to write any fraction as a decimal when they learn repeating decimals later in the year. Extensions 1. Do you expect the product to be greater than 10? Check your prediction. a) b) c) d) Answers: a) no, check: = 9; b) yes, check: = 13.2; c) yes, check: = 14; d) no, check: = 0.76 (MP.2) 2. Multiply (0.8 5) (0.2 3) ( ). Answer: = (or 4 60) = 240 (MP.5) 3. a) Multiply. i) ii) iii) iv) 3, b) Which multiplication is easiest? Why? Answers: a) all answers are 712.6, b) Students can give different answers as to which one is easiest and why. (MP.1, MP.3) 4. Compare the answers to and What do you notice? Why is this the case? Answer: They are both the same, 9. This makes sense because 0.5 = 1/2 and multiplying by 1/2 is the same as dividing by 2. F-30 Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships

31 RP7-17 Percents Pages Standards: preparation for 7.EE.A.2, 7.EE.B.3 Goals: Students will write given fractions as percents, where the given fractions have a denominator that divides evenly into 100. Prior Knowledge Required: Can find equivalent fractions Can reduce fractions to lowest terms Can convert terminating decimals to fractions Can find a decimal equivalent to a fraction that is equivalent to a decimal fraction Vocabulary: percent (MP.6) Percents as ratios. Write on the board: Megan can type 60 words per minute. Raj scores 3 goals per game. Bev makes $10 per hour. The car travels at a speed of up to 140 kilometers per hour. Ask students what the word per means in these sentences. Emphasize that per means for each or for every or in every. Ask volunteers to read the sentences with for every or in every or for each replacing per. Then write percent on the board. ASK: What is a cent (it s an amount of money; 100 cents is a dollar) What is a century? (100 years) Tell students that per cent is short for a Latin phrase that means for each hundred. For example, a score of 84% on a test would mean that you got 84 out of every 100 marks or points. Another example: If a survey reports that 72% of people who commute by public transportation read the newspaper, that means 72 out of every 100 people who commute by public transportation read the newspaper. Exercises: Have students rephrase the percent using the phrase for every 100 or out of 100. a) 52% of students in the school are girls. b) 40% of tickets sold were on sale. c) Eddy scored 95% on the test. d) About 60% of your body weight is water. Answers: a) For every 100 students, 52 are girls OR 52 out of every 100 students in the school are girls. b) For every 100 tickets sold, 40 were on sale OR 40 out of every 100 tickets were on sale. c) For every 100 possible points, Eddy scored 95 points on the test OR Eddy got 95 out of every 100 points on the test. d) For every 100 kg of body weight, about 60 kg is water OR about 60 kg out of every 100 kg of body weight is water. Teacher s Guide for AP Book 7.1 Unit 5 Ratios and Proportional Relationships F-31