unit 5 ratios and proportional relationships: real world ... for cc edition... · teacher’s guide...

Teacher’s Guide for AP Book 7.1 — Unit 5 Ratios and Proportional Relationships F-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.

F-2 Teacher’s Guide for AP Book 7.1 — Unit 5 Ratios and Proportional Relationships

RP7-12 Rounding Pages 119–121 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.

1 2 4 7 3

320 321 322 323 324 325 326 327 328 329 330


Step 2: Put your pencil on the digit to the right of the one you are rounding to.

1 2 4 7 3

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.

1 2 4 7 3

Step 4: Round the underlined digit up or down according to the instruction you have written.

1 2 4 7 3 2

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

5 1 2 1 5

1 6 4 7 3

2 0 7 5 2

1 2 4 7 3 2 0 0 0

1 2 4 7 3 1 2 0 0 0

round down

round down


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 463,000. 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: 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 2.365 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.

1 7 9 6 8 10

1 7 9 6 8 1 8 0 0 0

round up

2 3 6 5

2 3 6 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 4.

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 2.365 rounded to the nearest tenth is 2.4. Remind students that 2.400 = 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) 13.451 b) 38.479 c) 612.389 d) 804.749 Answers: a) 13.5, b) 38, c) 612.39, d) 804.7 Rounding decimals with regrouping. Write 2.965 on the board. Demonstrate rounding 2.965 to the nearest tenth, as shown below:

round up

round up

2 3 6 5

4

2 3 6 5

0 0 4

2 3 6 5

2 0 0 4

2 3 6 5

2 9 6 5

2 10

3 0


SAY: If you have 10 in a place value, you need to regroup. 2.965 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) 43.698 to hundredths b) 74.952 to tenths c) 59.517 to ones d) 84.09971 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 95 ÷ 7. Have a volunteer write on the board the answer their calculator shows. Depending on the calculator display, students might see 13.57142857. 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: 13.57142857 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: 13.57 Write on the board:

2.853 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:

2.853 ≈ 2.9 2.853 ≈ 2.85


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) 2 ÷ 3 ≈ 0.___ ___ b) 25 ÷ 9 ≈ 2.___ ___ c) 33 ÷ 65 ≈ 0.___ ___ d) 11 ÷ 7 ≈ 1.___ ___ e) 21 ÷ 4 ≈ 5.___ ___ 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 0.02749 to the nearest tenth, hundredth, thousandth, and ten thousandth. d) Round 16.57 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, 0.0275; d) 20, 17, 16.6, 16.57 Mention to students that, in the case of part d), the number 16.57 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) 88 ÷ 7 ≈ 12.571 ≈ 12.6 and 12.6 × 7 = 88.2, which is greater than 88. This makes sense because we rounded 12.571 up to get 12.6; b) Round to the nearest hundredth for cost. 88 ÷ 7 = 12.57, but 12.57 × 7 = 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, 34 ÷ 9 ≈ 3.78. So, each person should pay $3.78. Since 3.78 × 9 = 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.


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.

−29 ≈ ___ −22 ≈ ___ −26 ≈ ___ −24 ≈ ___

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: 1.524 m Height of a tree: 13.1064 m Length of a bug: 1.267 cm Distance between Washington, DC, and Hong Kong: 13,116.275 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: 973.91 ft2

Area of New York State: 141,299 km2

Angle between two streets: 82.469° Time it takes to blink: 0.33 s Speed of a car: 66.560639 mph Time it takes to ski a downhill course: 233.81 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.

−27 −21−23−24−25 −26 −28 −29 −20 −30 −22


RP7-13 Upper Bounds and Lower Bounds Pages 122–123 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:

475 + 321 500 + 300

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:

475 + 321 ≈ 500 + 300 = 800

Remind students that the squiggly equal sign means “approximately equal to.” Exercises: Estimate by rounding each number to the given place value. a) 421 + 159 (tens) b) 3,652 + 4,714 (hundreds) c) 7,980 + 1,278 (thousands) d) 1.1 + 3.8 (whole number) e) 3.14 + 2.62 (tenths) f) 5.985 + 6.251 (hundredths) g) 102.3 × 110.1 (tens) h) 489.8 × 512.2 (hundreds) i) 99.8 × 100.39 (whole number) j) 1.49 × 9.99 (tenths) Bonus: k) 8,541 + 972 + 37,218 (thousands) l) 479 + 612 + 162 (hundreds) m) 13 × 21 × 33 (tens) n) 10.10 + 3.67 + 9.87 (tenths) o) 1.678 + 9.254 + 4.398 (hundredths) p) 0.97 × 2.35 × 3.14 (whole number)


Answers: a) 420 + 160 = 580; b) 3,700 + 4,700 = 8,400; c) 8,000 + 1,000 = 9,000; d) 1 + 4 = 5; e) 3.1 + 2.6 = 5.7; f) 5.99 + 6.25 = 12.24; g) 100 × 110 = 11,000; h) 500 × 500 = 250,000; i) 100 × 100 = 10,000; j) 1.5 × 10.0 = 15; Bonus: k) 9,000 + 1,000 + 37,000 = 47,000; l) 500 + 600 + 200 = 1,300; m) 10 × 20 × 30 = 6,000; n) 10.1 + 3.7 + 9.9 = 23.7; o) 1.68 + 9.25 + 4.40 = 15.33; p) 1 × 2 × 3 = 6 Rounding both numbers up to get an upper bound. Write on the board: 5.8 + 6.3 < 6 + 7 = 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 6 + 7 = 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) 162.34 + 16.234 b) 6.3 + 3.147 c) 52.4 + 12.5 d) 20.12 + 210.2 e) 2.6 + 0.68 f) 655.001 + 217.91 Bonus: Round all numbers up to get an upper bound for the sum. g) 103.5 + 19.6 + 84.73 h) 1.320 + 4.14 + 5.99 + 3.6 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: 5.8 + 6.3 > 5 + 6 = 11 (MP.1) Have students use a calculator to find the sum of 5.8 + 6.3. (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) 1.1 + 2.5 b) 0.1 + 9.9 c) 67.6 + 33.3 d) 13.12 + 31.32 e) 16.24 + 13.3 f) 2.95 + 51.21 Bonus: g) 7.0 + 26.2 h) 16.0 + 4.0 i) 19.7 + 26.33 + 14.012


Answers: a) 3 < 3.6 < 5; b) 9 < 10 < 11; c) 100 < 100.9 < 102; d) 44 < 44.44 < 46; e) 29 < 29.54 < 31; f) 53 < 54.16 < 55; Bonus: g) 33 < 33.2 < 34; h) 20 is the actual sum, the upper bound, and the lower bound; i) 59 < 60.042 < 62 (MP.5) Checking whether an answer obtained by a calculator is reasonable. Write on the board:

27.3 + 38.5

Tell students that Kelly added these two numbers and got the answer 95.8. 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 (30 + 40), 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) 387.52 + 53.31 = 392.83 b) 14.67 + 10.50 = 25.17 c) 1.2 + 26.7 = 27.9 d) 62 + 14 = 102 e) 0.15 + 17.2 = 8.5 f) 1,523.4 + 214.3 + 510.3 = 2,275 Answers: a) 430 < sum < 450, so 392.83 is not reasonable; b) 20 < sum < 40, so 25.17 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:

3.14 × 20.5 = 6.437

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:

_____ × _____ < 3.14 × 20.5 < _____ × _____

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 < 3.14 × 20.5 < 84


Refer students back to the original answer on the board, 6.437. 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 $15.95. A sweater costs $35.45. 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. ($16 + 36 = $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) $1.80 + $2.30 = $4.10 b) i) $4.00, ii) No, he would have $0.10 less than the actual cost. c) $1.80 + $2.30 > $1.00 + $2.00 = $3.00. This estimate is not enough to cover the total cost of $4.10 d) $1.80 + $2.30 < $2.00 + $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.


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, $4.00 × 6 gallons = $24 > $3.98 × 6 gallons = $23.88. b) If each person pays $4.00, then the total amount will be $4.00 × 7 = $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 > $9.79 × 15 people = $146.85. 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 = 128.4 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) 27.21 + 832.5 = 8 5 9 7 1 b) 57.23 + 2.5 = 5 9 7 3 Answers: a) 859.71, b) 59.73 2. Without calculating the sum of 9.5 + .37 + 407.63, 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 10 + 1 + 410 = 421, so it is less than 435. (MP.6) 3. Estimate 14.502 − 13.921 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) 10 − 10 = 0, b) 15 − 14 = 1, c) 14.5 − 13.9 = 0.6, d) 14.50 − 13.92 = 0.58. 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.


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 < 9.4 − 3.2 < 10 − 3, so 5 < 9.4 − 3.2 < 7; c) 8 − 3 < 8.57 − 2.19 < 9 − 2, so 5 < 8.57 − 2.19 < 7; d) 6 − 1 < 6.38 − 0.57 < 7 − 0, so 5 < 6.38 − 0.57 < 7 2. a) 4.81, b) 6.2, c) 6.38, d) 5.81 3. 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) 750 − 330 = 420, so she has about $420 left. 4. a) He should get 3 + 8 + 9 = 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 14.3 ÷ 2.8. Which number would you increase and which would you decrease? How would you find a lower bound for 14.3 ÷ 2.8? 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


RP 7-14 Multiplying Decimals by Powers of 10 Pages 124–125 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

0.1 × 10 = 1

10× 10 =

1 1 1 1 1 1 1 1 1 1

10 10 10 10 10 10 10 10 10 10+ + + + + + + + +

= 10

10 = 1

Exercises: Multiply. a) 0.3 × 10 b) 0.8 × 10 c) 0.7 × 10 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

× 10 × 10 × 10 × 10


Write the following example on the board to illustrate: 0.005 × 10 = 0.05

5 50 510

1,000 1,000 100´ = =

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) 0.01 × 10 b) 0.001 × 10 c) 0.0001 × 10 d) 0.03 × 10 e) 0.003 × 10 f) 0.0003 × 10 g) 0.5 × 10 h) 0.02 × 10 i) 0.006 × 10 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 = 4 + 0.3 + 0.06

SAY: To multiply by 10, you can multiply each place value by 10. Write on the board: 4.36 × 10 = 40 + 3 + 0.6 = 43.6 Exercises: Use expanded form to multiply by 10. a) 5.4 × 10 b) 60.3 × 10 c) 3.004 × 10 d) 5.81 × 10 e) 3.12 × 10 f) 84.06 × 10 g) 3.294 × 10 h) 7.806 × 10 Sample solution: f) 84.06 = 80 + 4 + .06, so 84.06 × 10 = 800 + 40 + .6 = 840.6 Answers: a) 54, b) 603, c) 30.04, d) 58.1, e) 31.2, f) 840.6, g) 32.94, h) 78.06 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. 5 4 6 0 3


Ask volunteers to move the decimal point to show the answer for all 8 questions. The first two answers are shown below. 5 4 6 0 3 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) 3.2 × 10 b) 0.58 × 10 c) 10 × 0.216 d) 10 × 7.46 Bonus: 98,763.60789 × 10 Answers: a) 32, b) 5.8, c) 2.16, d) 74.6, Bonus: 987,636.0789 (MP.1) Move the decimal point to multiply decimals by 100 and by 1,000. Write on the board:

3.462 × 100 = 3.462 × 10 × 10

SAY: Move the decimal point once to multiply by 10, and again to multiply by another 10. Show this on the board: 3 . 4 6 2 So 3.462 × 100 = 346.2. 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) 3.62 × 100 b) 0.725 × 100 c) 1.673 × 100 d) 0.085 × 100 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:

2 . 4 6 7 So 2.467 × 1,000 = 2,467.

Exercises: Move the decimal point to multiply by 1,000. a) 0.462 × 1,000 b) 11.241 × 1,000 c) 9.32416 × 1,000 Answers: a) 462, b) 11,241, c) 9,324.16 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.

2 3 4


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) 5.24 × 1,000 c) 23.6 × 1,000 d) 0.01 × 1,000 Sample solution: b) So 5.24 × 1,000 = 5,240. Answers: a) 400; b) 5,240; c) 23,600; d) 10 2. Multiply by 10, 100, or 1,000. a) 0.6 × 100 b) 7.28 × 10 c) 25.6 × 1,000 d) 1.8 × 100 e) 21.9 × 1,000 f ) 326.3 × 1,000 g) 0.002 × 10 Bonus: 2.3 × 10,000 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 34 × 10 by moving the decimal point. Write on the board: 3 4 . 0

5 2 4

2 3 4

2 3 4 0


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: 3 4 0 3 4 0 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 0.135 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) 0.0674, c) 100, d) 10 (MP.7) 2. Write the next two numbers in the pattern. a) 0.0007, 0.007, 0.07, … b) 3.895, 38.95, 389.5, … Answers: a) 0.7, 7; b) 3,895, 38,950 3. (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) (7.836 × 5) × (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: 360.6 g


RP7-15 Multiplying and Dividing by Powers of 10 Pages 126–127 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 = 6 34.25 × 10 = 342.5

so so 6 ÷ 2 = 3 342.5 ÷ 10 = ______ 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, 30 ÷ 10 = 3), so the decimal point shifts one place to the left. Exercises: Divide by 10. a) 14.5 ÷ 10 b) 64.8 ÷ 10 c) 9.22 ÷ 10 d) 0.16 ÷ 10 Answers: a) 1.45, b) 6.48, c) 0.922, d) 0.016


Dividing by 100. Write on the board:

5.831 × 100 = 583.1, so 583.1 ÷ 100 = ______

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 5.831 by 100. Then, to get 5.831 back, you need to move it to the left two places. Exercises: Divide by 100. a) 14.5 ÷ 100 b) 464.8 ÷ 100 c) 9.22 ÷ 100 d) 0.6 ÷ 100 Answers: a) 0.145, b) 4.648, c) 0.0922, d) 0.006 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 67 ÷ 10. 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 67 ÷ 10 = 6.7. Repeat the process with 18 ÷ 100 and 1,987 ÷ 100. (0.18, 19.87) Exercises: Divide by 10 or 100. a) 236 ÷ 10 b) 573 ÷ 100 c) 1,230 ÷ 100 d) 14,889 ÷ 10 Answers: a) 23.6, b) 5.73, c) 12.30 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 = 0.045. Exercises: Divide. a) 2,934 ÷ 1,000 b) 423 ÷ 1,000 c) 18.9 ÷ 1,000 d) 1.31 ÷ 1,000 Bonus: 423 ÷ 100,000 Answers: a) 2.934, b) 0.423, c) 0.0189, d) 0.00131, Bonus: 0.00423

4 5

0 4 5


(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 34 × 10 = 340? (to the right) Exercises: Multiply or divide. a) 78,678 ÷ 1,000 b) 2.423 × 100 c) 18.9 ÷ 10 d) 1.31 × 1,000 e) 6 ÷ 100 f) .082 × 10 g) 0.2 ÷ 100 h) 5.1 × 100 i) .31 × 1,000 Bonus: j) 31,498.76532 ÷ 1,000,000 k) 31,498.76532 × 1,000,000 Answers: a) 78.678; b) 242.3; c) 1.89; d) 1,310; e) 0.06; f) 0.82; g) 0.002; h) 510; i) 310; Bonus: j) 0.03149876532; 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: 10 × 0.007 = ______ 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.


Exercises: 1. Multiply. Write your answer in lowest terms.

a) 10 × 1

100 = _____ b) 10 ×

7

10= _____

c) 10 × 7

100 = _____ d) 10 ×

3

1,000 = _____

e) 10 ×37

1,000 = _____ f) 10 ×

143

1,000 = _____

Answers: a) 1/10, b) 7, c) 7/10, d) 3/100, e) 37/100, f) 143/100 2. Multiply. a) 10 × 0.01 = _____ b) 10 × 0.7 = _____ c) 10 × 0.07 = _____ d) 10 × 0.003 = _____ e) 10 × 0.037 = _____ f) 10 × 0.143 = _____ 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 $26,575.80. 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 121.5 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 $12.95. 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 12.95 − 1.295 can be done as 12.950 − 1.295 = 11.655 which rounds to 11.66, or as 12.95 − 1.30 = 11.65 by rounding 1.295 to 1.30) Extensions 1. A penny has a width of 19.05 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, 190.5 m, 0.1905 km


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).


RP7-16 Multiplying Decimals by Whole Numbers Pages 128–130 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 1.23. Then extend the model yourself to show 2 × 1.23 on the board: ASK: What number is this? (2.46) Write on the board:

2 × 1.23 = 2.46

1.0 0.1 0.01


SAY: This is 2 ones, 4 tenths, and 6 hundredths. Exercises: Draw models to multiply. a) 2 × 4.01 b) 3 × 3.12 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 × 1.23 = 2 ones + 4 tenths + 6 hundredths SAY: Each digit is multiplied by 2. Exercises: Multiply mentally. a) 4 × 2.11 b) 3 × 2.31 c) 3 × 1.1213 Bonus: d) 2 × 1.114312 e) 3 × 1.1212031 Answers: a) 8.44, b) 6.93, c) 3.3639, Bonus: d) 2.228624, e) 3.3636093 Using expanded form to multiply decimals with regrouping. Write on the board:

Number Tens Ones Tenths Hundredths3.8 3 8

2 × 3.8 = ? 6 16

Ask a volunteer to regroup the ones and tenths to find the number that equals 2 × 3.8. (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) 3 × 2.4 b) 4 × 3.2 c) 3 × 2.04 d) 3 × 4.42 e) 4 × 3.32 f) 3 × 3.45 Answers: a) 7.2, b) 12.8, c) 6.12, d) 13.26, e) 13.28, f) 10.35 (MP.1) Compare multiplying decimals to multiplying whole numbers. Have a volunteer use the chart above to multiply 2 × 38. (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 2 × 3.8 because 38 is 10 times more than 3.8.


Using the standard method to record multiplication. Use grids to multiply 38 × 5 and 3.8 × 5 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 19.0. You can remove any final zeros that occur after the decimal point. In this case, 19.0 becomes 19. Exercises: Multiply. a) 3.35 × 6 b) 41.31 × 2 c) 523.4 × 5 d) 9.801 × 3 e) 3.4 × 81 f) 0.24 × 63 g) 8.4 × 72 h) 0.734 × 18 Bonus: i) 834,779.68 × 2 j) 5,480.63 × 7 Answers: a) 20.10 or 20.1; b) 82.62; c) 2,617.0 or 2,617; d) 29.403; e) 275.4; f) 15.12; g) 604.8; h) 13.212; Bonus: i) 1,669,559.36; j) 38,364.41 (MP.5) Explain that you can use estimation to check the answer. SAY: Suppose you’ve done the multiplication 3.32 × 4 and got 132.8. 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 3.32 × 4 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 0.00332 rounds to 0.003. 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) 500 × 5 = 2,500; d) 10 × 3 = 30; e) 3 × 80 = 240; f) 0.2 × 60 = 12; g) 8 × 70 = 560; h) 0.7 × 20 = 14; Bonus: i) 800,000 × 2 = 1,600,000; j) 5,000 × 7 = 35,000 2. David thinks 0.4 × 3 = 0.12. 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.

×

4

8

5

0 9 1

3

1

4

×

3 8

5

9 0


(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 = 4 + 4 + 4 = ______ 3 × (−4) = −4 − 4 − 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: −12 −8 −4 0 4 8 12 Write “−12” in the remaining blank. Write on the board:

3 × 1.2 = 1.2 + 1.2 + 1.2 = _____ 3 × (−1.2) = −1.2 − 1.2 − 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) 2 × 0.1 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 × (−21.31223) Answers: a) 0.2, b) −0.2, c) −0.9, d) −1, e) −5, f) −1.05, g) −0.67, h) −9.99, Bonus: −63.93669 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=

2 1 1

5 5 5= + =

ASK: What is one fifth written as a decimal? (0.2) Continue writing on the board:

1 2

0.25 10= =

2 1 1

5 5 5= + = 0.2 + 0.2 =


ASK: What is 0.2 + 0.2? (0.4) Demonstrate how decimals can be added by lining up the decimal points, as shown below: 0.2

+ 0.2 0.4 Now continue with 3/5 and 4/5. Continue writing on the board:

1

5= 0.2

2 1 1

5 5 5= + = 0.2 + 0.2 = 0.4

3 1 1 1

5 5 5 5= + + = 0.2 + 0.2 + 0.2 =

4 1 1 1 1

5 5 5 5 5= + + + = 0.2 + 0.2 + 0.2 + 0.2 =

Have a volunteer fill in the sums. (0.6 and 0.8) (MP.7) Exercises:

a) Use 1

0.52= to write

7

2 as a decimal.

b) Use 1

0.254= to write

5

4 as a decimal.

(MP.1) Bonus: Use 1

0.0520

= to write 4

20 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 1 1 1

5 5 5 5= + + = 0.2 + 0.2 + 0.2 = 0.6

3 1

35 5= ´ = 3 × 0.2 = 0.6


(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 = 0.05.

b) Multiply 0.05 by 3 to write 3

20 as a decimal.

c) Multiply 0.05 by 13 to write 13

20 as a decimal.

d) How much bigger is 13

20 than

3

20? Write your answer as a reduced fraction.

e) How much bigger is the decimal for 13

20 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) 0.3 × 30 b) 1.2 × 11 c) 2.8 × 5 d) 0.04 × 19 Answers: a) no, check: 0.3 × 30 = 9; b) yes, check: 1.2 × 11 = 13.2; c) yes, check: 2.8 × 5 = 14; d) no, check: 0.04 × 19 = 0.76 (MP.2) 2. Multiply (0.8 × 5) × (0.2 × 3) × (0.5 × 200). Answer: 4 × 0.6 × 100 = 2.4 × 100 (or 4 × 60) = 240 (MP.5) 3. a) Multiply. i) 3.563 × 200 ii) 35.63 × 20 iii) 356.3 × 2 iv) 3,563 × 0.2 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 18 × 0.5 and 18 ÷ 2. 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.


RP7-17 Percents Pages 131–132 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.


Percents as fractions. Explain to students that a percent is just a short way of writing a fraction with denominator 100. For example, you can write the fraction 84/100 as 84%. Exercises: 1. Write the fraction as a percent.

a) 28

100 b)

9

100 c)

34

100 d)

67

100 e)

81

100 f)

3

100

Answers: a) 28%, b) 9%, c) 34%, d) 67%, e) 81%, f) 3% 2. Write the percent as a fraction. a) 6% b) 19% c) 8% d) 54% e) 79% f) 97% Answers: a) 6/100, b) 19/100, c) 8/100, d) 54/100, e) 79/100, f) 97/100 Writing decimal hundredths as percents. Write on the board:

0.35 = 100

= _____% 0.04 =

ASK: How would you write 0.35 as a fraction with denominator 100? (35/100) Write “35” as the numerator, then ASK: How would you write 35/100 as a percent? (35%) Write “35” as the percent. Repeat for 0.04. (4/100 = 4%) Exercises: Write the decimal as a fraction with denominator 100, then as a percent. a) 0.74 b) 0.03 c) 0.12 d) 0.83 e) 0.91 f) 0.09 Answers: a) 74/100 = 74%, b) 3/100 = 3%, c) 12/100 = 12%, d) 83/100 = 83%, e) 91/100 = 91%, f) 9/100 = 9% Changing fractions to percents when the denominator divides evenly into 100. Write the fraction 3/5 on the board and have a volunteer find an equivalent fraction with denominator 100 (60/100). ASK: If 3 out of every 5 students at a school are girls, how many out of every 100 students are girls? (60) What percent of the students are girls? (60%) Write on the board:

3

5 =

60

100 = 60%

Exercises: Find an equivalent fraction with denominator 100.

a) 4

5 b)

9

10 c)

7

20 d)

1

2 e)

4

25 f)

1

20

Answers: a) 80/100, b) 90/100, c) 35/100, d) 50/100, e) 16/100, f) 5/100 SAY: Once you change the fraction to have denominator 100, it is easy to write it as a percent.


Exercises: Write the fraction as a percent.

a) 2

5 b)

4

5 c)

1

5 d)

5

5 e)

3

5

Answers: a) 40%, b) 80%, c) 20%, d) 100%, e) 60% Using percents to order fractions. Point out that percents are easily ordered because they are all fractions with denominator 100. Exercises: a) Write the fraction as a percent.

i) 4

10 ii)

9

20 iii)

3

4 iv)

29

50 v)

21

25 vi)

1

2

b) Use the equivalent percents that you found in part a) to put the fractions in order from least to greatest. Answers: a) i) 40%, ii) 45%, iii) 75%, iv) 58%, v) 84%, vi) 50%; b) 4/10 < 9/20 < 1/2 < 29/50 < 3/4 < 21/25 Writing decimal tenths as percents. Remind students how to write decimal tenths as percents by first changing the decimal to a fraction with denominator 100, as shown below:

2 20

0.2 20%10 100

= = =

Exercises: Write the decimal as a percent. a) 0.3 b) .9 c) 0.7 d) 0.5 Answers: a) 30%, b) 90%, c) 70%, d) 50% When students finish, point out a faster way: Write the decimal as a decimal hundredth, then write the hundredth as a percent. For example: 0.7 = 0.70 = 70% (MP.4) Percent of a figure. Explain to students that they can find a percent of a figure just as they can find a fraction of a figure. Exercises: Decide first what fraction, then what percent of each figure below is shaded. a) b) c) Answers: a) 4/10 or 40%, b) 1/4 or 25%, c) 7/20 or 35%


Fractions that need to be reduced before changing the denominator to 100. Write on the board:

9

15

Tell students that you want to find an equivalent fraction with denominator 100 so that you can turn it into a percent. ASK: How is this fraction different from previous fractions you have changed to percents? (the denominator does not divide evenly into 100) Is there any way to find an equivalent fraction whose denominator does divide evenly into 100? (reduce the fraction by dividing both the numerator and the denominator by 3) Write on the board:

9 3 60

60%15 5 100

= = =

Summarize the steps for finding the equivalent percent of a fraction, pointing to the relevant part of the sequence of equations on the board as you say each step. Step 1: Reduce the fraction so that the denominator is a factor of 100. Step 2: Find an equivalent fraction with denominator 100. Step 3: Write as a percent the fraction with denominator 100. (MP.7) Exercises: Write the fraction as a percent.

a) 3

12 b)

6

30 c)

24

30 d)

3

75 e)

6

15 f)

36

48 g)

60

75

Answers: a) 25%, b) 20%, c) 80%, d) 4%, e) 40%, f) 75%, g) 80% NOTE: Some fractions cannot be reduced so that the denominator is a factor of 100. Students will learn about these fractions later in the year. Extensions (MP.2) 1. ASK: How many degrees are in a circle? (360) If I rotate an object 90° counterclockwise, what fraction and what percent of a complete 360° turn has the object made? PROMPT: 90 out of 360 is what out of 100? (90/360 = 1/4 = 25/100 = 25%) What percent of a complete 360° turn is the degree rotation? a) 180° b) 18° c) 126° d) 270° e) 72° f) 216°.


Answers: a) 180/360 = 1/2 = 50/100 = 50%, b) 18/360 = 1/20 = 5/100 = 5%, c) 126/360 = 7/20 = 35%, d) 270/360 = 3/4 = 75/100 = 75%, e) 72/360 = 1/5 = 20/100 = 20%, f) 216/360 = 3/5 = 60%

2. Write fractions greater than 1 as percents. Example: Change 7

5 to a percent.

Answer: 7/5 = 140/100 = 140% 3. a) What percent of an hour is 3 minutes? b) 4.8 hours is what percent of a day? c) You spend 7.2 hours per day at school. What percent of the day is that? d) If you get 13 weeks off in a school year, when percent of the year (52 weeks) is that? Answers: a) 5%, b) 20%, c) 30%, d) 25%


RP7-18 Decimals, Fractions, and Percents Pages 133–134 Standards: preparation for 7.EE.A.2, 7.EE.B.3 Goals: Students will visualize various percentages of different shapes, including rectangles, squares, and number lines. Students will compare and order fractions, percents, and decimals. Prior Knowledge Required: Can write equivalent fractions Understands the relationship between decimal tenths and hundredths and fractions with denominator 100 Understands percents as fractions with denominator 100 Can order fractions Can find equivalent fractions Knows the signs for less than (<) and greater than (>) Vocabulary: decimal, fraction, percent Materials: the 0% to 100% strip from BLM Percents Strips, one for each student (p. F-81) Percent of a shape. Draw on the board: Have students use a fraction, a decimal, and a percent to write what part of the block is shaded. (39/100, 0.39, 39%) Then, draw and shade the shapes below on the board: ASK: What fraction of each shape is shaded? Have students change each fraction to an equivalent fraction with denominator 100, then to a decimal, and a percent. (7/10 = 70/100 = 0.70 = 70%, 1/5 = 20/100 = 0.20 = 20%, 9/25 = 36/100 = 0.36 = 36%, 14/20 = 70/100 = 0.70 = 70%, 11/20 = 55/100 = 0.55 = 55%)


Percent of a line. Draw on the board: 0% 100% 0 1 Tell students that you want to write percents on the top of this number line and fractions on the bottom. Pointing to the one-fifth mark, ASK: What fraction do I write here? (1/5) PROMPT: How many equal parts are in 1? (5) SAY: One out of 5 equal parts is one fifth. ASK: What is 2 out of 5 equal parts? (2/5) Have volunteers write the remaining fractions. (3/5 and 4/5) Then move to the percents on top. ASK: If we divide 100% into 5 equal parts, what is one part? (20%) What would two parts be? (40%) Write those answers in, then have volunteers write the rest. (60%, 80%) (MP.7) Exercises: Write the fractions and percents for each mark on the number line. a) 0 100% b) 0% 100% 0 1 0 1 Bonus: 0% 25%

0 1

4

Answers: a) 10% to 90% and 1/10 to 9/10; b) 25%, 50%, 75% and 1/4, 2/4, 3/4; Bonus: 5%, 10%, 15%, 20%, and 1/20, 2/20 or 1/10, 3/20, 4/20 or 1/5 (MP.1) Estimating percents. Provide each student with the 0% to 100% strip from BLM Percent Strips. (The other four strips are not used in this lesson.) Have students trace a line from the 0% mark to the 100% mark, label only the 0% and 100% marks, and put an X at any percent they choose. Have students trade with a partner and estimate to the nearest 10% which percent their partner picked. Students can then check their estimate with their own percent strip. Review comparing fractions with the same denominator. SAY: 7 tenths is greater than 4 tenths because all the tenths are the same size. Draw the following picture on the board to illustrate:

(ten parts)


Exercises: Which fraction is greater?

a) 3

5 or

1

5 b)

1

9 or

7

9 c)

6

20 or

11

20

Bonus: Order the fractions from greatest to least.

d) 1

15,

3

15,

2

15 e)

3

4,

2

4,

4

4 f)

59

100,

20

100,

50

100

Answers: a) 3/5, b) 7/9, c) 11/20, Bonus: d) 3/15 > 2/15 > 1/15, e) 4/4 > 3/4 > 2/4, f) 59/100 > 50/100 > 20/100 Review comparing percents. Point out that because percents are fractions with denominator 100, they are easy to compare. Write on the board:

30 24

100 100> so 30% > 24%

Exercises: Which percent is greater? a) 7% or 39% b) 16% or 56% c) 21% or 81% Bonus: Order the following percents from greatest to least. d) 85%, 76%, 67% e) 16.5%, 96.5%, 49.5% f) 9%, 100%, 75% Answers: a) 39%, b) 56%, c) 81%, Bonus: d) 85% > 76% > 67%, e) 96.5% > 49.5% > 16.5%, f) 100% > 75% > 9% Review comparing fractions with different denominators. Write on the board:

2

3

3

4

SAY: These fractions have different denominators so, to compare them, we have to make them have the same denominator. ASK: What is the smallest denominator we can use? PROMPT: It has to be a multiple of 3 and 4. (12) Write on the board, underneath 2/3 and 3/4:

12

12

Have volunteers write the numerators. (8, 9) Then write the less than symbol (<) between the fractions. The final picture should look like this:

8 9

12 12<

Exercises: Write < or >.

a) 1

5

7

10 b)

3

8

3

4 c)

1

3

3

12 d)

2

3

1

2

Answers: a) <, b) <, c) >, d) >


(MP.7) Comparing fractions and percents. Remind students that they can compare fractions and decimals by changing them both to fractions with denominator 100. Write on the board:

3

0.525> because

60 52

100 100>

SAY: You can compare fractions and percents the same way. Write on the board:

352%

5> because

60 52

100 100>

Exercises: 1. Which is larger?

a) 1

2 or 38% b)

3

5 or 70% c)

9

10 or 84% d)

7

25 or 30% e)

9

20 or 46%

Answers: a) 1/2, b) 70%, c) 9/10, d) 30%, e) 46% 2. Which is closer to 50%? Hint: Change the fractions to percents first.

a) 1

4 or

2

5 b)

3

10 or

4

5 c)

3

5 or

1

4 Bonus:

2

5 or

3

5

Answers: a) 2/5, b) 3/10, c) 3/5, Bonus: they’re the same distance from 50% Comparing decimals and percents. SAY: You can compare decimals and percents by changing both to equivalent fractions with denominator 100. Write on the board: 0.7 and 52% Have volunteers change each number to a fraction with denominator 100. (70/100, 52/100) SAY: 70 is greater than 52, so 0.7 is greater than 52%. Students can also compare decimals to percents by changing them to decimal hundredths. (0.70 and 0.52) This method is faster but also more prone to error. Students should know both methods. Exercises: 1. Which is larger? a) 0.9 or 10% b) 0.09 or 10% c) 28% or 0.34 d) 4% or 0.3 Answers: a) 0.9, b) 10%, c) 0.34, d) 0.3 2. Use “is greater than” for > and “is less than” for < when you read the sentence. Is the sentence true? a) 0.8 > 53% b) 0.7 > 92% c) 0.3 < 26% d) 95% < 0.97 Answers: a) true, b) false, c) false, d) true


3. Change each number to a fraction with denominator 100, then put the numbers in order from least to greatest.

a) 0.28 42% 3

10 b)

14

50 23% 0.3

c) 19

25 0.72 7% d)

1

4 4% 0.4

Bonus: 13

20 0.6 66% 0.7 7%

16

25

3

50

Answers: a) 0.28, 3/10, 42%; b) 23%, 14/50, 0.3; c) 7%, 0.72, 19/25; d) 4%, 1/4, 0.4; Bonus: 3/50, 7%, 0.6, 16/25, 13/20, 66%, 0.7 (MP.3) Comparing fractions and percents when a denominator does not divide evenly into 100. ASK: How can we compare 35% to 1/3? If we changed 35% to a fraction, what would it be? (35/100) Do we have a way to compare 1/3 to 35/100, or are we stuck? PROMPT: We have two fractions with different denominators, but 3 doesn’t divide evenly into 100. How can we give both fractions the same denominator? (use denominator 300) Have volunteers change both fractions to equivalent fractions with denominator 300. Ask the class to identify which is greater, 35% or 1/3, and to explain how they know. (35% = 105/300 and 1/3 = 100/300, so 35% is greater) Exercises: 1. Compare.

a) 5

6 and 85% b)

3

7 and 42% c)

2

9 and 21%

Bonus: Make up your own question and have a partner solve it. Answers: a) 5/6 < 85%, b) 3/7 > 42%, c) 2/9 > 21% 2. Is the statement true?

a) 1

18%6< b)

570%

7< c)

557%

9> d)

888%

9<

Answers: a) yes, b) no, c) no, d) no Have students order lists of numbers (fractions, percents, and decimals) in which the fractions do not have denominators that divide evenly into 100. Exercises: Order the numbers from least to greatest.

a) 1

6, 0.17, 13% b) 0.37,

1

3, 28% c)

5

7, 71%, 0.68

Bonus: 7

9, .8,

4

7, 51%, .78, 62%

Answers: a) 13%, 1/6, 0.17; b) 28%, 1/3, 0.37; c) 0.68, 71%, 5/7; Bonus: 51%, 4/7, 62%, 7/9, .78, .8


(MP.4) Word problems practice. Exercises: a) A pizza has a total of 10 slices. If 3 of the slices have pepperoni and the other 7 slices have just cheese, what fraction and percent of the pizza have pepperoni? b) A baseball stadium can seat 60,000 people. If the stadium sold 75% of its seats, how many tickets were sold? c) Jen has run 4 km of a 20 km cross-country race. What fraction and what percent of the race has she completed? What percent of the race is left to run?

d) In a gumball machine, 30% of the gumballs are red, 45% of the gumballs are blue, and 1

4 of

the gumballs are green. Which gumball color is there fewest of? Answers: a) 3/10, 30%; b) 45,000; c) 1/5 or 20% is completed and 80% is left; d) 30% of the gumballs are red, 45% are blue, and 25% are green, so there are fewer green gumballs than any other color Extensions 1. Have students shade 25% of a triangle like this one: Answer: (MP.1) 2. a) Draw a 12 cm line segment. This line segment is 80% of a larger line segment. Draw the whole line segment. b) Draw a 5 cm line segment. This line segment is 40% of a larger line segment. Draw the whole line segment. Selected solution: a) If 12 cm is 4/5 of the line segment, it is 4 equal parts of 5. Since 12 cm is 4 equal parts, each equal part is 3 cm. Since the whole is 5 equal parts, it is 15 cm long. Answer: b) 12.5 cm

(MP.7) 3. Multiply both by 7 to decide which is larger: 4

7 or 55%. Which is greater,

4 or 7 × 55%? Hint: What is 4 as a percent? Use 4 = 4 ?

1 100= .

Answer: 7 × 55% = 385%, which is less than 400% = 4, so 4/7 > 55% (MP.2) 4. Name percents that indicate … a) almost all of something b) very little of something c) a little less than half of something Explain your thinking. Sample answers: a) 98%, b) 3%, c) 48%

(MP.4) 5. Ask students to look for fractions, decimals, and percents in newspapers, flyers, magazines, cookbooks, and on other printed materials, such as food packaging, trading cards, and order forms. What kind of information is expressed as a fraction? as a decimal? as a percent? Ask students to clip examples and to make a collage for a class display.


(MP.1, MP.4) 6. Students who remember how to multiply fractions can solve the following problem:

Grace spent her birthday money this way: She donated 1

10 to charity and placed

1

3 of the

remainder in a savings account. One sixth of what was left was a gift card to an ice-cream parlor. She used the rest of the money to buy three books for $7.50 each, a T-shirt for $4.25, and a basketball for $23.25. How much birthday money was she given? Solution: She gave 10% to charity and put 1/3 of 90%, or 30% of the total, into a savings account. She had 60% left. 1/6 of that, or 10% of the total, was a gift certificate. She spent the rest, or 50% of the total, on her purchases. The total amount she spent on her purchases was $50, so her birthday money was $100.


RP7-19 Finding Percents (Introduction) Pages 135–136 Standards: 7.EE.B.3 Goals: Students will use 10% of a number to find multiples of 10% of the number and will use the multiples of 10% of the number to find multiples of 5% of the number. Prior Knowledge Required: Can convert fractions to decimals and decimals to fractions Understands the relationship between percents and fractions Vocabulary: percent Materials: base ten materials (MP.2) Percents and base ten representations. Draw on the board: Tell students that in an earlier lesson, they used a hundreds block to represent one whole, but now they will use a thousands block to represent one whole. ASK: What fraction of a thousands block does a hundreds block represent? (one tenth) SAY: A hundreds block is one tenth of a thousands block. ASK: What is one hundredth of a thousands block? (a tens block) What is one thousandth? (a ones block) SAY: You can use base ten blocks to represent decimal place values just like you can with whole-number place values. Under the diagram, write the decimal represented: 1.34 Now tell students that you want to use base ten blocks to find 1/10 of the number 1.34. ASK: What is one tenth of the thousands block? (one hundreds block) Draw the block on the board: ASK: What is one tenth of the three hundreds blocks? (3 tens blocks) Draw the tens blocks on the board, as shown below:


ASK: What is one tenth of the four tens blocks? (4 ones blocks) Draw the ones blocks on the board, as shown below: ASK: What number does this represent? (0.134) SAY: Each place value becomes one tenth of what it was, so you are just moving the decimal point one place left.

Exercises: Find 1

10 of the number by moving the decimal point.

a) 10 b) 100 c) 500 d) 1,000 Answers: a) 1.0, b) 10.0, c) 50.0, d) 100.0 ASK: When you find one tenth of a number, what are you dividing by? (10) Point out that when you find one tenth of a number, you are dividing the number into 10 equal parts, and that’s what you do when you divide the number by 10. So, it makes sense that you move the decimal point one place left, the same as you do when dividing a number by 10. Exercises: Find one tenth of the number. a) 0.75 b) 16.1 c) 1,500 d) 0.02 e) 15.6 f) 902.1 Answers: a) 0.075, b) 1.61, c) 150.0, d) 0.002, e) 1.56, f) 90.21 Finding 10% of a number. ASK: What percent is the same as one tenth? (10%) Write on the board:

1 10

10%10 100

= =

SAY: So, to find 10% of a number, you can divide the number by 10 or simply move the decimal point one place to the left. Exercises: Find 10% of the number. a) 40 b) 4 c) 7.3 d) 500 e) 408 f) 3.07 Bonus: 432.5609 Answers: a) 4, b) 0.4, c) 0.73, d) 50.0 or 50, e) 40.8, f) 0.307, Bonus: 43.25609 ASK: If you know 10% of a number, how can you find 30% of that number? (multiply 10% of the number by 3) PROMPT: 30% is how many times 10%? (3) Tell students that you would like to find 70% of 12. ASK: What is 10% of 12? (1.2) Write on the board: 1.2


ASK: If I know that 10% of 12 is 1.2, how can I find 70% of 12? (multiply 1.2 × 7). Remind students that multiplying 1.2 × 7 is just like multiplying 12 × 7, but they have to put the decimal point in the right place, as shown below: 1

1.2 × 7 8.4 SAY: When you multiply a decimal by a whole number, you put the decimal point in the answer right below the decimal point in the decimal. Exercises: Find the percent of the number. a) 60% of 15 b) 40% of 40 c) 60% of 4 d) 20% of 1.5 e) 90% of 8.2 f) 70% of 4.3 g) 80% of 5.5 h) 30% of 3.1 Answers: a) 6 × 1.5 = 9, b) 4 × 4 = 16, c) 6 × 0.4 = 2.4, d) 2 × 0.15 = 0.30, e) 9 × 0.82 = 7.38, f) 7 × 0.43 = 3.01, g) 8 × 0.55 = 4.4, h) 3 × 0.31 = 0.93 (MP.1) Finding percents with a number line. Draw on the board: 0 30 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Have a volunteer fill in the missing numbers on the top of the number line. (3, 6, 9, 12, 15, 18, 21, 24, 27) Point out that once students know that 10% of 30 is 3, they can skip count by 3s to finish the number line. Write on the board:

a) 20% of 30 b) 40% of 30 c) 90% of 30 d) 70% of 30

Ask volunteers to look at the completed number line and identify the percent of each number. (a) 6, b) 12, c) 27, d) 21) ASK: What is 10% of 21? (2.1) What is 20% of 21? (4.2) Exercise: Create a number line divided into tenths that goes from 0 to 21. Use it to find 70% of 21. Answer: The 10% intervals will be: 0, 2.1, 4.2, 6.3, 8.4, 10.5, 12.6, 14.7, 16.8, 18.9, 21, so the 70% mark is at 14.7. (MP.7) Finding multiples of 5% of a number. Refer students to the number line from 0 to 30 above: 0 3 6 9 12 15 18 21 24 27 30 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%


Tell students that 5% is halfway between 0% and 10%. Have a volunteer mark where this is on the number line. ASK: What is 5% of 30? (1.5) Have a volunteer find 45% of 30 on the number line. ASK: What is 45% of 30? (13.5) Point out that 45% is halfway between 40% and 50%, so 45% of 30 is halfway between 12 and 15. Exercises: 1. What is the percent? a) 25% of 30 b) 35% of 30 c) 85% of 30? d) 95% of 30? Answers: a) 7.5, b) 10.5, c) 25.5, d) 28.5 2. Make a number line from 0 to 20 divided into 10% intervals. Then find … a) 15% of 20 b) 45% of 20 c) 55% of 20 d) 85% of 20 Answers: a) 3, b) 9, c) 11, d) 17 3. Make a number line from 0 to 40 divided into 10% intervals. Then find … a) 15% of 40 b) 45% of 40 c) 55% of 40 d) 85% of 40 Answers: a) 6, b) 18, c) 22, d) 34 (MP.1) When students finish, ASK: How does 15% of 40 compare to 15% of 20? (it’s double) PROMPT: What do you multiply by? (2) Tell students that because 40 is double 20, any percent of 40 should be double the same percent of 20. Have students make sure that is the case for all their answers for Exercises 2 and 3. If it isn’t, students can check with a partner to find their mistake. (MP.4) Word problems practice. Exercises: a) A bag has 20 marbles. If 40% of the marbles are green, how many green marbles are there? b) An airport has 40 flights traveling from the United States to Europe each day. If 15% of those flights travel to France, how many airplanes will go to France each day? c) Mike works at a furniture store, where he receives 20% of the sale price on every sale that he makes. If Mike sold a living room set for $6,000, how much would he make? Answers: a) 8 green marbles, b) 6 airplanes will visit France each day, c) Mike would make $1,200 Extensions (MP.7) 1. a) Find 1% of each number by shifting the decimal point two places to the left. i) 27 ii) 3.2 iii) 773 iv) 12.3 v) 68.2 vi) 0.041 b) Use your answers to part a) to find 7% of the numbers. i) 27 ii) 3.2 iii) 773 iv) 12.3 v) 68.2 vi) 0.041 Answers: a) i) 0.27, ii) 0.032, iii) 7.73, iv) 0.123, v) 0.682, vi) 0.00041; b) i) 0.27 × 7 = 1.89, ii) 0.224, iii) 54.11, iv) 0.861, v) 4.774, vi) 0.00287


NOTE: Students who remember how to multiply fractions can do Extensions 2 and 3. 2. Randi took 20% of the candies. She gave 30% of them to a friend. What percent of the candies did her friend get? Answer: 30% of 20% is 3/10 × 2/10 = 6/100 = 6% 3. Peter took 40% of a pie. Tess took 80% of the remainder. Who took more pie? Answer: Tess took 80% of 60% which is 4/5 of 3/5, or 12/25 = 48/100 = 48%. Tess took more than Peter.


RP7-20 More Percents Pages 137–138 Standards: 7.EE.B.3 Goals: Students will find any whole-number percent of a whole number. Students will use estimation to determine if the percent they calculated is reasonable. Students will discover and explain the commutativity of percents (e.g., 30% of 10 is 10% of 30). Prior Knowledge Required: Can reduce fractions Knows the standard algorithm for multiplying whole numbers Can multiply a decimal by a whole number Can multiply a fraction by a whole number Can convert between fractions, percents, and decimals Knows that multiplication commutes Vocabulary: percent, rounding Materials: BLM Percent Strips (p. F-81) Finding fractions of whole numbers. Review finding fractions of whole numbers. Write on the board:

2

3 of 12 = 2 ×

1

3 of 12

= 2 × (12 ÷ 3) = 2 × 4 = 8 Exercises: Find the fraction of the number.

a) 2

3 of 15 b)

3

4 of 20 c)

1

6 of 30 d)

2

5 of 35

Answers: a) 10, b) 15, c) 5, d) 14 Remind students that they can multiply first, then divide, and get the same answer both ways. Demonstrate with the example above. Write on the board: 2 × (12 ÷ 3) = (2 × 12) ÷ 3 = 24 ÷ 3 = 8 Exercises: Find the same fractions in the Exercise above by multiplying first, then dividing. If you don’t get the same answer, find your mistake.


(MP.8) Using multiplication to find percents. Write on the board: 53% of 12 SAY: I want to find 53% of 12, but I don’t know how to find a percent of a number; I only know how to find fractions of numbers. ASK: How can I use what I know to solve this problem? (change the percent to a fraction) Continue writing on the board:

53% of 12 = 53

100 × 12 = (53 × 12) ÷ 100

Remind students that they can use long multiplication (or mental multiplication, if it is an easy product) to find products such as 53 × 12. Have a volunteer do so. (636) Also, remind students that dividing by 100 shifts the decimal point two places to the left. Write on the board: So (53 × 12) ÷ 100 = 636 ÷ 100 = 6.36 Students should use grid paper to do the long multiplication required in the Exercises below. Exercises: Find the percent. a) 68% of 33 b) 5% of 42 c) 76% of 85 d) 55% of 21 Answers: a) 22.44, b) 2.1, c) 64.6, d) 11.55 Have students use BLM Percent Strips to check their answers. Each of the four numbers in the Exercise has been placed on a number line (all of the same length but each with a different scale). A fifth number line, divided into 100 parts to represent percents, is also included on the BLM. Students should cut out the percent strip so they can line it up with the other strips. To estimate 68% of 33, students can find 68% on the percent number line, then locate the number that is in the same position on the number line for 33. (MP.1) Using estimation to check answers. Tell students that they won’t usually have a percent strip to check their answers, so they will have to use estimation to check whether their answers are reasonable. For instance, they can round a given percent to the nearest multiple of ten and use the rounded percent to estimate the answer. ASK: How can we tell if 6.36 is a reasonable answer to 53% of 12? Is there a percentage of 12 that is close to 53% and easy to calculate? (yes, 50%) What is 50% of 12? (6) Should 53% be more or less than 50%? (more) Is our answer a little bit more than 6? (yes) SAY: For an even quicker estimate, you can round the number as well as the percent. Write on the board: 53% of 12 is about 50% of 10 = 5 Exercises: Use estimation to check if your answers to the above Exercises are reasonable.


(MP.3) Ask students what method they used to estimate 68% of 33 to see if their answer is reasonable. Encourage them to give a variety of answers. For example, “68% is close to 70% and 33 is close to 30. I found 70 × 30 mentally (2,100), then shifted the decimal point two places to the left. So my estimate is 21.” Or, “68% is close to 75%, which is the same as 3/4 (75/100 = 3/4). And 33 is close to 32. I know 1/4 of 32 is 8, so 3/4 of 32 is 24. So my estimate is 24.” (MP.1) Changing percents to decimals instead of fractions gets the same answer. Write on the board:

53% of 12 = 53

12100

´ = 0.53 × 12

SAY: Percents are equivalent to fractions or decimals, so you could also find a percent of a number by finding the equivalent decimal of the number. ASK: How would you find 0.53 × 12? (multiply the whole numbers and then move the decimal point) Write on the board: SAY: This is the same thing we did when changing 53% to a fraction instead of a decimal. We multiplied 53 × 12, then we moved the decimal point to divide by 100. Now, we are multiplying 0.53 × 12 and doing the long multiplication as though it was 53 × 12. Then we put the decimal point in the answer in the correct place. Exercises: Change the percent to a decimal, then find that percent of the number. a) 26% of 36 b) 11% of 40 c) 19% of 45 d) 96% of 57 Answers: a) 9.36, b) 4.4, c) 8.55, d) 54.72 Tell students to look at their answer to part a). ASK: What easy percent is 26% close to? (25%) What is 25% as a fraction? (1/4) What is 25% of 36? PROMPT: What is 1/4 of 36? (9) Is your answer to 26% a little bit more than your answer to 25%? (yes, 9.36 is a little bit more than 9) (MP.1, MP.5) Exercises: Find the percent, then estimate to make sure your answers are reasonable. a) 76% of 24 b) 19% of 25 c) 48% of 76 d) 11% of 32

0 5 3

1 0 6

5 3 0

× 1 2

6 3 6


Answers: a) 18.24, should be slightly more than 3/4 of 24, which is 6 × 3 = 18; b) 4.75, should be slightly less than 1/5 of 25, which is 25 ÷ 5 = 5; c) 36.48, should be slightly less than 1/2 of 76, which is 38; d) 3.52, should be slightly more than 1/10 of 32, which is 3.2 Commutativity of percents. Have students predict which is greater: 20% of 60 or 60% of 20. Tell students that the first is a smaller percentage of a larger number, so you’re not sure which is bigger. Then have students calculate both. (both are 12) SAY: These are both the same. ASK: How does 20% of 60 compare to 20% of 20? (it is 3 times greater) How does 60% of 20 compare to 20% of 20? (it is also 3 times greater) So they are both equal. Exercises: Calculate and compare. a) 30% of 50 and 50% of 30 b) 40% of 20 and 20% of 40 c) 70% of 90 and 90% of 70 d) 80% of 60 and 60% of 80 e) 50% of 40 and 40% of 50 f) 36% of 24 and 24% of 36 g) 17% of 35 and 35% of 17 h) 29% of 78 and 78% of 29 i) 48% of 52 and 52% of 48 Answers: a) 15 and 15, b) 8 and 8, c) 63 and 63, d) 48 and 48, e) 20 and 20, f) 8.64 and 8.64, g) 5.95 and 5.95, h) 22.62 and 22.62, i) 24.96 and 24.96 (MP.3, MP.8) ASK: What pattern do you see? (both answers are the same) Challenge students to figure out why this pattern holds. Students can write an explanation individually, then work with a partner to refine their explanation. Guide pairs who need help by asking what calculations they need to do to get 24% of 36 (24 × 36 ÷ 100) and 36% of 24 (36 × 24 ÷ 100). ASK: What rule can you use to explain why these calculations will get the same answer? (multiplication is commutative) Extensions (MP.3) 1. Tony says that to find 10% of a number, he can divide the number by 10. So, to find 5% of a number, he can divide the number by 5. Is he right? Explain. Answer: No. 5% of a number is 5/100 or 1/20 of the number. So, to find 5%, or 1/20, of the number, Tony should divide it by 20. 2. a) Calculate: i) 48% of 300 ii) 50% of 300 iii) 52% of 300 b) How much more is 52% of 300 than 50% of 300? c) How much more is 50% of 300 than 48% of 300? d) How do your answers to b) and c) compare? Why is that the case? Answers: a) i) 144, ii) 150, iii) 156; b) 6; c) 6; d) they are the same, because the difference between 48% of 300 and 50% of 300 is 2% of 300, which is also the difference between 50% of 300 and 52% of 300.


RP7-21 Mental Math and Percents Pages 139–140 Standards: 7.EE.A.2, 7.EE.B.3 Goals: Students will add and subtract percents and will use mental math to compute easy percents of numbers. Prior Knowledge Required: Can add and subtract fractions Can change percents to fractions Knows that multiplication commutes Vocabulary: fraction, decimal, lowest terms, percent Materials: BLM Percent Strips (p. F-81) base ten blocks Review finding 25% of a number by using 1/4. Remind students that 25% is equal to 1/4, so they can find 25% of a number by finding 1/4 of the number. Exercises: Calculate. a) 25% of 33 b) 25% of 42 c) 25% of 85 d) 25% of 21 Answers: a) 8.25, b) 10.5, c) 21.25, d) 5.25 Have students use BLM Percent Strips (including their cut-out percent strip) to check their answers. (see answer to part a) below)

So 25% of 33 is a little more than 8, as calculated. (MP.7) Finding 5% of a number using 10% of the number. ASK: If I know 10% of 42 is 4.2, how can I find 5% of 42? (5% is half of 10%, so if 10% is 4.2, then 5% is 2.1) Have students find 5% of the following numbers by first finding 10%, then dividing by 2. a) 80 b) 16 c) 72 d) 50 e) 3.2 f) 2.34 (a) 4, b) 0.8, c) 3.6, d) 2.5, e) 0.16, f) 0.117)


(MP.3) Adding and subtracting percents. SAY: You can add and subtract percents the same way you add and subtract fractions, because percents are fractions. Write on the board:

20% + 5% = 20 5

100 100+

= 25

100

= 25% Ask a volunteer to show how to subtract 20% − 5%. (15%) Exercises: Add or subtract the following percents. a) 100% − 40% b) 16% − 10% c) 25% + 35% d) 0% + 4% e) 19% − 16% f) 45% + 30% Answers: a) 60%, b) 6%, c) 60%, d) 4%, e) 3%, f) 75% Finding 15% from 10% and 5%. SAY: I know that 10% of 42 is 4.2, and 5% of 42 is 2.1. ASK: What is 15% of 42? (15% of a number is 10% of the number plus 5% of the number, so 15% of 42 is 4.2 + 2.1 = 6.3) Have students use BLM Percent Strips to verify this answer by lining up the percent strip with the “42” strip. ASK: Does 10% look as though it lines up with 4.2? (yes) Does 15% look as though it lines up with 6.3? (yes) Exercises: Calculate 15% by finding 10% and 5%, then adding. a) 60 b) 240 c) 12 d) 7.2 e) 3.8 f) 6.10 Answers: a) 6 + 3 = 9, b) 24 + 12 = 36, c) 1.2 + 0.6 = 1.8, d) 0.72 + 0.36 = 1.08, e) 0.38 + 0.19 = 0.57, f) 0.61 + 0.305 = 0.915 (MP.4) Tell students that when going out to eat at a restaurant, people are expected to leave a tip. Depending on how good the service is, the tip could be anywhere from 10% to 20% of the bill, before taxes. Before giving the Exercises below, write on the board: 15% = 10% + 5% Exercises: Calculate mentally what tip you should leave for each meal price if you want to leave a 15% tip. a) $30 b) $22 c) $18 d) $35 e) $47 Answers: a) $4.50, b) $3.30, c) $2.70, d) $5.25, e) $7.05 (MP.1) As a check on their calculations, have students order the meal costs from least to greatest and the tips they calculated from least to greatest—these should be in the same order! (c, b, a, d, e) Exercises: Calculate 15% of the same numbers as above, using 15 × (the number) ÷ 100. Make sure you get the same answer as before, or find your mistake.


(MP.7) Finding 100% from 10%. Tell students that 10% of a number is 4 and you want to know what the number is. Point out that the number is 100% of itself. ASK: How can I get 100% from 10%? (multiply by 10) So, if 10% is 4, what is 100%? (40) SAY: So the number is 40. Exercises: Find 100% if 10% is … a) 9 b) 1.5 c) 150 d) 0.30 Answers: a) 90, b) 15, c) 1,500, d) 3.0 (MP.4) The total is always 100%. Tell students that 30% of students in an after-school club are girls. ASK: What percent are boys? (70%) How did you get that? (100% − 30% = 70%) Tell students that in a classroom, 70% of students have brown eyes and 20% have blue eyes. ASK: What percent have a different color of eyes? (10%) Point out that the total is always 100%. Exercises: Answer the question by changing the fraction to a percent. a) If one fourth of Blanca’s children are boys, then what percent of her children are girls? b) Alex received acceptance letters from 3 of the 5 universities that he applied to. What percent of the universities did not accept Alex? c) A recent survey stated that 7 out of every 10 commuters prefer taking the train over the bus. The rest prefer the bus. What percent of commuters prefer taking the bus? Answers: a) 75%, b) 40%, c) 30% ASK: What is 10% of 70? (7) What is 20% of 70? (14) How did you get that from 10%? (multiply by 2) What is 80% of 70? (56) Write on the board: 20% of 70 + 80% of 70 (MP.3) Have students add the two numbers (14 + 56 = 70) ASK: Why does this answer make sense? (because 20% + 80% is 100% and 100% of 70 is 70) Tell students that they can use this to check their answers. Exercises: Calculate each percent mentally, then add them. a) 25% of 100 is ______ and 75% of 100 is ______, so 25% of 100 + 75% of 100 is ______ b) 30% of 80 is ______ and 70% of 80 is ______, so 30% of 80 + 70% of 80 is _______ c) 20% of 150 is ______ and 80% of 150 is ______, so 20% of 150 + 80% of 150 is ______ d) 50% of 1 is ______ and 50% of 1 is ______, so 50% of 1 + 50% of 1 is ______ Answers: a) 25, 75, 100; b) 24, 56, 80; c) 30, 120, 150; d) 0.50, 0.50, 1.00 SAY: Is 100% of each number in the previous Exercises equal to the number? If not, you may have made a mistake somewhere. This is a good way to check your work because you know 100% of a number should equal the number. Another way to check answers. Have students find 15% of 80. (12) Ask various students how they solved it. Now tell students you want to make sure the answer is reasonable. ASK: What is 25% of 80? (1/4 of 80 is 20) Is 15% of a number closer to 0% or 25%? (25%) Is 12 closer to 0 or 20? (20) SAY: This is another way to make sure that your answer makes sense. Since 15% is closer to 25% than to 0, 15% of the number should be a bit closer to 25% than to 0%.


Exercises: a) Find 15% of the number. i) 20 ii) 60 iii) 100 iv) 40 (MP.1) b) Is your answer closer to 0 or to 25%? Is your answer reasonable? Answers: a) i) 3, ii) 9, iii) 15, iv) 6; b) Since 15 is closer to 25 than to 0, 15% of the number should always be closer to 25% (or 1/4) than to 0. (Students should check that this is the case by dividing each number in part a) by 4 to get 25% of the number and then comparing it to their answer.) Extensions 1. (MP.1, MP.3) a) Liz said that 25% of 60 is 15, so 15% of 60 is 25. Is she right? How do you know? b) What mistake do you think she made? Answers: a) No, 15% of 60 should be less than 25% of 60; b) She was thinking of the commutative property, but she was interchanging the wrong numbers. 25% of 60 is the same as 60% of 25. 2. a) How can you get 1% of a number from the number? b) How can you get 34% of a number from 1% of the number? (MP.1) c) How is this method the same as finding 34/100 of a number? Answers: a) divide by 100; b) multiply by 34; c) Instead of multiplying by 34, then dividing by 100, you are now dividing by 100 first, then multiplying by 34. The answers are still the same. (MP.7) 3. Without using a calculator, mentally calculate 37% of 32 by following the steps below: a) First find 30% of 32 (by adding 30% of 30 and 30% of 2). b) Then find 7% of 32 (by adding 7% of 30 and 7% of 2). c) Add the results of a) and b) to get 37% of 32. Hint: 30% of a number can be obtained by multiplying 10% of the number by 3, and 7% of a number can be obtained by multiplying 1% of the number by 7. Answers: a) 10% of 30 is 3, so 30% of 30 is 9; 10% of 2 is 0.2, so 30% of 2 is 0.6; altogether, 30% of 32 is 9.6; b) 1% of 30 is 0.3, so 7% of 30 is 2.1; 1% of 2 is 0.02, so 7% of 2 is 0.14, so 7% of 32 is 2.24; c) 9.6 + 2.24 = 11.84 (MP.3) 4. You can add percents by adding the numbers: 10% + 5% = 15%. Can you multiply percents the same way? Does 10% × 5% = 50%? Answer: No, 10 × 5% is 50%, but 10% × 5% is 10% of 5%, which is less than 5%, so it cannot be 50%. In fact, 10% × 5% is 1/10 × 1/20 = 1/200 = 0.005 = 0.5%. (Students will review multiplying fractions in a later unit.) (MP.1) 5. 4% of a number is 32. Find the number two ways (make sure you get the same answer both ways): a) Find 1% of the number and multiply it by 100. b) Multiply 4% of the number by 25. Answers: a) if 4% is 32, then 1% is 8, so 100% is 800; b) 32 × 25 = 8 × 4 × 25 = 800.


(MP.1) 6. Sales tax is 8%. Estimate the taxes on a $60 item by adding 5% + 1

2 of 5%. Will your

estimate be lower or higher than the actual answer? Answer: 5% of $60 = $3 so 1/2 of 5% of $60 is $1.50. So estimate the taxes as $4.50. This estimate is for a sales tax of 7 1/2%, which is lower than the actual sales tax of 8%.


RP7-22 Word Problems

Pages 141–142 Standards: 7.EE.A.2, 7.EE.B.3, 7.RP.A.3 Goals: Students will solve real-world percent problems in contexts involving taxes, commissions, discounts, and percent increases. Prior Knowledge Required: Can find the percent of a number Can add percents Vocabulary: commission, discount, percent increase Materials: BLM Use an Easier Expression to Solve Problems (pp. F-82–84) Review how to find the percent of a number. Write on the board:

7% of 18 = 7 × 18 ÷ 100 Have a volunteer calculate 7 × 18 (126), then finish demonstrating how to find the percent. Write on the board:

= 126 ÷ 100 = 1.26

ASK: What is 10% of 18? (1.8) Is 7% more or less than 10%? (less) Point out that we can use 10% of 18 to check our answer. ASK: Is our answer less than 1.8? (yes) SAY: So our answer is reasonable. Exercises: Find the percent of the number. Check if your answer is reasonable. a) 11% of 90 b) 21% of 3 c) 9% of 20 d) 6% of 60 Answers: a) 9.9, b) 0.63, c) 1.8, d) 3.6 (MP.4) Word problems practice. Have students warm up with some simple word problems. Exercises: a) Greg earned a 15% tip on a $120 dinner bill. How much money did Greg earn? b) Helen bought a new computer for $1,200 before taxes at Electronics ‘R’ Us. If the tax is 15%, how much money did Helen spend just in tax?


Bonus: Carlos decided to take his family out for dinner. The dinner for his family cost $250 before tax, which is 8%, and Carlos tipped 20% of the cost before taxes. How much money did Carlos spend altogether? Answers: a) $18, b) $180, Bonus: $320 Introduce commission. Tell students that sales people often get paid commission. That means, on top of a salary, they get paid a percent of the sales they make. Write on the board: shirts cost $20 Tell students to pretend that they are working at a store that sells shirts for $20, and they get 2% commission. That means that they get 2% of the $20 every time they sell a shirt. ASK: How much do you get for each shirt? PROMPT: What is 1% of $20? (20 cents) What is 2% of $20? (40 cents) SAY: So you get 40¢ for each shirt you sell. (MP.4) Exercises: For each sale, a salesperson makes 10% in commission. How much would the salesperson receive on each sale? a) $350 bike b) $499 cell phone c) $36 pair of shoes d) $42,199 car Answers: a) $35, b) $49.90, c) $3.60, d) $4,219.90 (MP.2) Introduce discounts. Tell students that a sweater that costs $35 is on sale for 20% off. ASK: What is 20% of $35? PROMPT: What fraction is 20% equivalent to? (1/5) What is 1/5 of 35? (7) ASK: If the discount is $7, what is the sale price? ($28) Write on the board: $35 − $7 = $28 original price discount sale price (MP.4) Exercises: Determine the sale price for each of the following items after subtracting the discount. a) 5% off a $44 pair of jeans b) 60% off a $2,499 piece of jewelry c) 15% off a $33 book d) 35% off a $160 coat Answers: a) $44.00 − $2.20 = $41.80, b) $2,499.00 − $1,499.40 = $999.60, c) $33.00 − $4.95 = $28.05, d) $160 − $56 = $104 (MP.7) Simplifying percent decreases. After students finish, point out that we have the same structure in every problem. Write on the board: $44 − 5% of $44 original − decrease amount


SAY: You can do each calculation separately, or you can simplify by factoring. Write on the board: 44 − 0.05 × 44 SAY: 44 is multiplied by 0.05 in the second expression. ASK: What is it being multiplied by in the first expression? (1) PROMPT: 44 is 44 multiplied by what number? (1) Write on the board: = 1 × 44 − 0.05 × 44 SAY: Now we can use the distributive property to make an easier expression. Continue writing on the board:

= (1 − 0.05)(44)

= (0.95)(44)

= 41.8

(MP.8) SAY: By keeping the 44 in the expression, it’s easier to remember what we did to get the 41.8. This could come in handy. When doing similar examples, we can see that we just have to multiply the original price by 0.95 to get the price after a 5% discount. Exercises: What would you multiply the original price p by to get the price after each discount? a) 10% b) 5% c) 40% d) 60% Answers: a) 0.9, b) 0.95, c) 0.6, d) 0.4 Have volunteers write the equations that show their answers. (p − 0.1p = 0.9p, p − 0.05p = 0.95p, p − 0.4p = 0.6p, p − 0.6p = 0.4p) (MP.1) Exercises: Do the same discount Exercises as before, but use the simplified calculation instead. Make sure you get the same final price as before. a) 5% off a $44 pair of jeans b) 60% off a $2,499 piece of jewelry c) 15% off a $33 book d) 35% off a $160 coat Selected solution: b) 0.4 × $2,499 = $999.60 (MP.4) Car sales commission. Tell students that commission for a car salesperson is different from how most commissions are done. Instead of getting a small percentage of the price, they get a larger percentage of the profit. Write on the board: Profit = Selling Price of Car − Buying Price of Car Tell students that the company selling the car has to either make the car themselves or buy it from someone; either way, it costs them money, so they take that into account when they give


commission to their salesperson. A typical salesperson might get 30% of the profit while the company gets 70% of the profit. Write on the board:

Buying Price of Car Selling Price of Car Profit 30% Commission a) $30,000 $35,000 b) $35,000 $39,000 c) $40,000 $49,000 d) $46,000 $56,000 e) $79,000 $91,000

Have students copy the chart into their notebook. Then go through the first one together as a class. (the profit is $5,000, so the commission is $1,500) Exercises: Complete the chart. Answers: b) $4,000; $1,200, c) $9,000; $2,700, d) $10,000; $3,000, e) $12,000; $3,600 (MP.2, MP.4) Percent increases. Tell students that it is important that taxes increase the cost by a percentage of the price rather than by a set amount. It wouldn’t make sense for people to pay $5 in taxes on a 20 cent item, nor would it make sense for people to pay $5 in taxes on a $30,000 car. If that was the case, nobody would want to buy things that cost 20 cents! Tell students to assume the taxes are 5%. ASK: What is the amount of tax on the 20 cent item? PROMPTS: What is 10% of 20 cents? (2 cents) So what is 5%? (1 cent) ASK: If the amount of tax is 1¢, and the original price before taxes is 20¢, what is the total price after taxes? (21¢) Write on the board: 20¢ + 1¢ = 21¢ original price amount of tax price after taxes Exercises: Calculate the price after 5% in taxes for each price before taxes. a) $10 b) $30 c) $8 d) $17 e) $30,000 Answers: a) $10.50, b) $31.50, c) $8.40, d) $17.85, e) $31,500 (MP.7) Simplifying percent increases. After students finish, point out that we have the same structure in every problem. Write on the board: $10 + 5% of $10 original + increase amount


SAY: You can do each calculation separately, or you can simplify by factoring. Continue writing on the board: 10 + 0.05 × 10 = 1 × 10 + 0.05 × 10

= (1 + 0.05)(10)

= (1.05)(10)

= 10.5 (MP.8) SAY: You can multiply the original price by 1.05 to get the price after 5% in taxes. Exercises: The amount of tax is 5%. Multiply the original price by 1.05 to calculate the price after taxes. a) a $30 sweater b) a $12 CD c) a $200 jacket d) an $800 computer e) a 20¢ toy f) a $40,000 car Answers: a) $31.50, b) $12.60, c) $210, d) $840, e) 21¢, f) $42,000 Have volunteers demonstrate their calculations for some of the Exercises above, keeping the original price in the expression. (sample answers: a) 1.05 × 30, b) 1.05 × 12, c) 1.05 × 200) Exercises: (MP.7, MP.8) 1. What would you multiply the original price by to get the price after taxes if the tax is … a) 8% b) 30% c) 15% d) 4% Bonus: What would you multiply the original price by to get the price after a 15% tip and a 6% tax? Answers: a) 1.08, b) 1.3, c) 1.15, d) 1.04, Bonus: 1.21 2. a) A $20,000 car is taxed at 8%. What is the cost after taxes? b) A $40 meal is taxed at 6%. What is the cost after taxes? Bonus: A $20 haircut is taxed at 8% and you decide to leave a 10% tip. What is the cost after taxes and tip? Answers: a) $21,600, b) $42.40, Bonus: $23.60 (MP.4) SAY: You can use percent increases in contexts other than taxes and tips. For example, a price increases over time because a store’s costs increase over time, a person’s salary can increase over time, and the value of a house can increase over time. Exercises: a) Every morning, Marta buys a coffee that costs $1.80. If the price of the coffee is going to increase by 10%, what will be the new price of the coffee?


b) Jay’s salary increased by 8% this year. If his salary was $30,000 last year, what will his salary be this year? c) Beth wrote a test, but complained to her teacher that she had a headache. Her teacher let her

write another test later, and her mark increased by 40%. Beth’s original mark was 13

20.

i) What was her original mark as a percent? ii) What was her new mark as a percent? (Hint: The answer is not her original percent mark + 40) Bonus: Ethan bought a house for $80,000. He spent $5,000 renovating it. Two years after he bought the house, the value increased by 20%. If he sells the house, what would his profit be, per year? Answers: a) 1.1 × $1.8 = $1.98; b) 1.08 × $30,000 = $32,400; c) i) 65%, ii) 1.4 × 65% = 91%; Bonus: $5,500 The following exercises include percent increases and decreases. Exercises: Complete BLM Use an Easier Expression to Solve Problems. Answers: 1. 36 + 84 = 120, 3 × 40 = 120 2. a) 1,080¢ − 1,044¢ = 36¢; b) 3¢ × 12 = 36¢; c) i) part a), ii) part b) 3. a) $420.00, b) $23.00, c) $1,200.00, d) $55.00 4. She made $400, but paid $160, so her profit was $240. Her profit per hour was $12.00. 5. a) 0.09 × 24, b) 0.09 × 9, c) 0.09 × 300, d) 0.09n 6. b) 0.05 × 6, 6 + 0.05 × 6; c) 0.05 × 10, 10 + 0.05 × 10; d) 0.05n, n + 0.05n 7. a) ii), 0.06 × 6, 6 + 0.06 × 6, 1.06 × 6; iii) 0.06 × 10, 10 + 0.06 × 10, 1.06 × 10; iv) 0.06n, n + 0.06n, 1.06n; b) $13,780 8. 0.85 × $45,000 = $38,250 9. $25,000 × 0.7 = $17,500 10. $540 × 0.75 = $405 11. 299 accidents 12. 28 minutes 13. 1.08 14. $42.80 15. Increase, because 5% of 15,000 is more than 5% of 12,000. Another solution is 15,000 + 12,000 = 27,000 apples and pears one year, and 15,750 + 11,400 = 27,150 the next year. 16. A and C are both the best deal with the price being $977.50, whereas B has the price of $1,000. Extensions (MP.3) 1. Introduce decimal percents. In the Exercises below, students compute 2% and 3% to decide what 2.5% would be. a) Jake makes 2% commission on the sale of a $30 item. How much does he make on each sale? b) Micky makes 3% commission on the sale of a $30 item. How much does she make on each sale?


c) Peter makes 2.5% commission on the sale of a $30 item. How much does he make on each sale? Hint: Use your answers to a) and b). Sample solution: c) 2.5 is halfway between 2 and 3, so 2.5% is halfway between 2% and 3%. Halfway between 0.6 and 0.9 is 0.75, so Peter makes $0.75 on each sale. Answers: a) $0.60, b) $0.90, c) $0.75 (MP.4) 2. Alice is a car salesperson. She makes a 30% commission on the company’s profit. All the cars Alice sells cost the company $22,000. She is allowed to negotiate a price to sell the cars, as long as the company does not lose money. a) When Alice tries to negotiate a deal for $27,000, she needs to work an average of 60 hours before making a deal. How much does she make per hour? b) When Alice tries to negotiate a deal for $25,000, she needs to work an average of 30 hours before making a deal. How much does she make per hour? c) If you were Alice, would you try to negotiate a deal for $27,000 or $25,000? Why? Answers: a) $25; b) $30; c) $25,000, because I would get more money per hour of work 3. Tell students that computer programmers try to find the most efficient way for a computer to do a calculation. The amount of time it takes a computer to do a calculation depends on the number of operators. Often, addition and subtraction each count as 1 operator, but multiplication and division each count as 2 operators. A computer programmer is debating what is the best way to calculate the price of a $9 item after a 15% tax. For each method below, write how many operators the computer would do and explain why the method works. a) 9 + 0.15 × 9 b) 1.15 × 9 c) 9 + 9 ÷ 10 + [answer to 9 ÷ 10] ÷ 2 Answers: a) 3. This method works because you are just adding 15% of the price to the price. b) 2. This method gives the same answer as a) by the distributive property: 1 × 9 + 0.15 × 9 = 1.15 × 9. c) 6. This method works because it is adding 10% and 5% to find 15% and adding that to the original price


RP7-23 Tape Diagrams, Fractions, and Percents Problems Pages 143–145 Standards: 7.RP.A.3, 7.EE.B.3 Goals: Students will use tape diagrams to solve problems in which one quantity is expressed as a percent of another. Prior Knowledge Required: Can write percents as reduced fractions Can add percents Can multiply a percent by a whole number Can use division to find a unit fraction of a whole number Knows that a fraction can be greater than 1 Vocabulary: percent, tape diagram Review fractions and percents. Write on the board: For each picture, ASK: What fraction of the whole rectangle is each block? (1/2 and 1/4) What percent of the whole rectangle is each block? (50% and 25%) Write on the board:

1 50

50%2 100= =

1 2525%

4 100= =

Exercises: What fraction and what percent of the whole rectangle is each block? a) b) 10 blocks Answers: a) 1/5; 20%; b) 1/10; 10% Review tape diagrams. Remind students that a tape diagram uses blocks of the same size to represent a situation. Tape diagrams can be used when comparing two or more quantities. Draw on the board:

A:

B:

20%


SAY: Each block is 20% of A. ASK: What percent of A is B? (60%) Write 20% in each block of B to emphasize the point. Exercises: What percent of A is each block? What percent of A is B? a) A: b) A: B: B: c) A: B: Answers: a) 25%, 75%; b) 20%, 40%; c) 10%, 30% (MP.1, MP.4) Using tape diagrams to model one quantity as a percent of another. Write on the board:

The number of girls is 80% of the number of boys. g: b:

ASK: What fraction of the number of boys is the number of girls? (4/5) SAY: So if you draw 4 blocks for girls, then you have to draw 5 blocks for boys. Add the tape diagrams to the board, as shown below:

g: b:

(MP.6) Emphasize that it is important to label the bars so, when they answer questions about the situation, they will know which is which. Exercises: Draw a tape diagram to show the given information. a) The number of girls is 40% of the number of boys. b) The number of boys is 60% of the number of girls. c) The number of adults on a school trip is 10% of the number of children. d) The number of trucks on a highway is 50% of the number of buses.

10 blocks


Answers: a) g: b) b: b: g: c) a: d) t: c: b: More than 100% of a number. ASK: Does it make sense to talk about 140% of a number? What does 140% of a number mean? Discuss what 100% and 40% of a number mean separately. Remind students that 50% of a number can be obtained by adding 20% and 30% of that number. Point out that 140% can be obtained by adding 100% and 40% of the number. Write on the board: 10% of 70 = _______

so 30% of 70 = _______ so 130% of 70 = _______

Have volunteers finish the equations. (7, 21, 91) SAY: Since fractions can be greater than 1, and percents are just fractions with denominator 100, percents can also be greater than 1. That means that they can be greater than 100%. Exercises: Find the percents of the number. a) 30% of 20 and 130% of 20 b) 50% of 60 and 150% of 60 c) 70% of 80 and 170% of 80 d) 60% of 40 and 160% of 40 Answers: a) 6, 26; b) 30, 90; c) 56, 136; d) 24, 64 Using tape diagrams to find more than 100%. Draw on the board: A:

B: ASK: What percent of A is each block? (25%) PROMPT: A has four blocks, so each block is one fourth. ASK: How many blocks are in B? (7) SAY: So B is 175% of A. Write 25% in each block of B, then write on the board: 7 × 25% = 175%


Exercises: (MP.1) 1. What percent of A is B? a) A: b) A: B: B: c) A: B: Answers: a) 150%, b) 250%, c) 1,000% (MP.1, MP.4) 2. Model the situation with a tape diagram. a) The number of girls is 200% of the number of boys. b) The number of boys is 150% of the number of girls. c) The number of girls is 160% of the number of boys. d) The price after taxes is 120% of the price before taxes. Answers: a) g: b) b: c) g: b: g: b: d) price after taxes:

price before taxes: Using tape diagrams to find percents of the total. Write on the board:

A: total

B: SAY: I want to know what percent of the total A is. I know that A is 3 out of the 5 blocks. ASK: What percent is that? (60%) Write on the board:

3 60

5 100=

Exercises: What percent of the total is A? a) A: b) A: c) A: B: B: B: Bonus: A: B:

C:


Answers: a) 50%, b) 40%, c) 40%, Bonus: 30% (MP.1, MP.4) Drawing tape diagrams to model percent of a total. Write on the board:

The number of girls is 80% of the total number of children. SAY: The number of girls is 4/5 of the total. ASK: If I draw 4 blocks for girls, how many do I have to draw for boys? (1) SAY: I need 5 blocks in total and, if I have 4 already, then I need only 1 more for boys. Draw on the board:

g: b:

Exercises: Draw a tape diagram to show the situation. a) The number of girls is 25% of the total number of children. b) The number of boys is 50% of the total number of children. c) The number of girls is 60% of the total number of children. d) The number of boys is 75% of the total number of children. Answers: a) g: b) b: c) g: d) b: b: g: b: g: Using tape diagrams to solve problems. Write on the board:

The number of girls is 80% of the number of children. There are 24 girls. How many children are there?

Have a volunteer draw a tape diagram that shows the information in the first sentence, as shown below:

g:

b: Read the second sentence in the problem, then ASK: Which part of the tape diagram shows 24? (the 4 squares for girls) Show this on the tape diagram, as shown below: 24

g:

b:


ASK: If 4 squares represent 24 children, how many children does 1 square represent? (6) Write 6 in each square of the tape diagram:

g:

b: ASK: How many children are there altogether? (30) Exercises: Use a tape diagram to solve the problem. a) The number of boys is 30% of the total number of children. If there are 40 children in total, how many boys and girls are there? b) The number of girls is 80% of the number of boys. If there are 24 girls, how many boys are there? How many children are there in total? c) The number of boys is 25% the number of girls. If there are 50 boys, how many girls are there? How many children are there in total? Answers: a) 12 boys, 28 girls; b) 30 boys, 54 children; c) 200 girls, 250 children (MP.1, MP.4) Word problems practice. Exercises: Use a tape diagram to solve the problem. a) Jennifer has 3 times as many pairs of black jeans as she does blue jeans. If Jennifer has 8 pairs of jeans in total, how many of them are blue? b) The number of striped ties that Carl owns is 75% of the solid ties. If he owns 12 solid ties, how many striped ties does Carl own? c) The price after taxes and tip at a restaurant is 130% of the original price. If the price after taxes and tip is $52, what was the original price? Answers: a) 2 pairs of blue jeans, b) 9 striped ties, c) $40 NOTE: To do the Bonus in Question 8 on AP Book 7.1 p. 145, students should use a tape diagram to represent the given information (the number of girls is 40% of the number of boys), then interpret the tape diagram to fill in the needed information (the number of boys is what percent of the number of girls?). Extensions (MP.5) 1. What makes it difficult to draw a tape diagram to show that the number of girls is 43% of the total number of children? Answer: You would need to draw 43 individual squares for the girls and 57 individual squares for the boys because 43 is prime, so no smaller number of squares can show 43%.

2. A box of chocolates has milk chocolate, dark chocolate, and white chocolate. There is 3

5 as

much dark chocolate as milk chocolate. There is 2

3 as much white chocolate as dark chocolate.

What percent of the total is dark chocolate?

6 6 6 6

6


Answer: 30%. The tape diagram is Dark: Milk: White: 3. A is 125% of B. What percent of A is B? Answer: 80%

4. a) One number is 3

5 of another number. The two numbers add to 32. What are the numbers?

(MP.1) b) Three numbers A, B, and C satisfy the following conditions:

A is 2

3 of B.

C is 2 more than B. A + B + C = 42

What are the numbers? Hint: Use a different color square to represent the 2 in “C is 2 more than B.” Answers: a) The tape diagram is There are 8 squares in total, so each square represents 4, so the numbers are 12 and 20. b) One possible tape diagram is A: B: 2 C: The 8 white squares represent 40, so each white square represents 5, so the numbers are 10, 15, and 17.


RP7-24 Percent More Than Pages 146–148 Standards: 7.RP.A.3, 7.EE.B.3 Goals: Students will use tape diagrams to solve problems in which one quantity is expressed as a certain percent more than another. Prior Knowledge Required: Can use tape diagrams to solve problems Vocabulary: [percent] more than Comparing the extra part of A when A is more than B. Draw on the board: A: B: extra part of A Tell students that in some contexts, you want to know how much more the bigger part is than the smaller part, so sometimes you want to compare the extra part of A to B. If A is bigger than B, then there is a part of A that is equal to B, and there is an extra part. ASK: What percent of B is the extra part of A? (75%) SAY: B is 4 blocks, and the extra part of A is 3 blocks, and 3 out of 4 is 75%, so the extra part of A is 75% of B. (MP.1) Exercises: What percent of B is the extra part of A? a) A: b) A: B: B: c) A: d) A: B: B: Answers: a) 20%, b) 50%, c) 25%, d) 40% Tell students that sometimes the extra part of A is more than or equal to B. In that case, the extra part of A will be equal to or more than 100% of B. Draw on the board:

A:

B: extra part of A


SAY: B has two parts. ASK: What percent of B is each part of B? (50%) How many parts does the extra part of A have? (4) So what percent of B is the extra part of A? (200%) SAY: If each part is 50% of B, and the extra part of A has 4 parts, then it is 200% of B. (MP.1) Exercises: What percent of B is the extra part of A? a) A: b) A: B: B: c) A: d) A: B: B: Answers: a) 100%, b) 200%, c) 500%, d) 300% (MP.1, MP.4) Drawing a tape diagram to represent a given situation. Write on the board: The extra part of A is 60% of B. SAY: Let’s start by drawing B. We need to find out how many parts to divide B into, and the easiest way to do that is to write 60% as a fraction. ASK: What is 60% as a fraction? (3/5) If students say 60/100, tell them that you want to use smaller numbers because you don’t want to divide B into 100 parts. If students say 6/10, ask if there are even smaller numbers they can use. When students say 3/5, SAY: So B has 5 parts and the extra part of A has 3 parts. Draw B on the board and have a volunteer draw A. The final picture should look like this: A: B: Exercises: Draw a tape diagram so that the extra part of A is a) 40% of B b) 80% of B c) 110% of B d) 1,000% of B Answers: a) A: 7 squares, B: 5 squares; b) A: 9 squares, B: 5 squares; c) A: 21 squares, B: 10 squares; d) A: 11 squares, B: 1 square Introduce “percent more than.” Write on the board: There are 5 more girls than boys in a class. There are 15 boys in the class.


ASK: Are there more girls or more boys? (girls) How many more girls? (5) Write on the board: g: b: 15 ASK: How many girls are in the class? (20) How do you know? (20 is 5 more than 15) Now erase the 5 in the word problem and write 20% instead, as shown below:

There are 20% more girls than boys in a class. There are 15 boys in the class. g: b: 15 Tell students we are still given the number of boys in the class, and there are still more girls than boys, but we don’t know how many more. Tell students that the phrase “20% more than” means that the number of extra girls is 20% of the number of boys. Exercises: Fill in the blanks. a) g: b) b: b: g: There are ____% more girls than boys. There are ____% more boys than girls. c) g: d) b: b: g: There are ____% more girls than boys. There are ____% more boys than girls. Answers: a) 25, b) 40, c) 50, d) 100 Refer to the example above (There are 20% more girls than boys…). Have a volunteer draw the tape diagram. Remind students that, when drawing the tape diagram, they have to start with the boys—they can only draw the “20% more than the boys” after they have the picture for the boys. ASK: How did the volunteer know to draw 5 parts for the boys? (because the 20% is telling us the extra part of girls is 1 part for every 5 parts representing the boys) SAY: The first step is to write the fraction, 20%, in lowest terms. Since we need to represent 20% of B, we need to represent 1/5 of B, and the easiest way to do that is make B have 5 equal parts. (see below) g: b: 15


(MP.1) Exercises: Draw a tape diagram to show the situation. a) There are 25% more boys than girls. b) There are 5% more girls than boys. Answers: a) g: 4 squares, b: 5 squares; b) g: 21 squares, b: 20 squares SAY: Once you have the tape diagram, it is easy to solve the problem. Refer students to the example above. ASK: If there are 15 boys and the bar for the boys consists of 5 parts, how much is each part? (3) Write 3 in each square of the tape diagram. SAY: So there are 18 girls. Exercises: Draw a tape diagram to answer the question. a) There are 50% more boys than girls in a class. There are 25 students in the class altogether. How many boys and how many girls are in the class? b) There are 200% more girls than boys in a class. There are 20 students in the class altogether. How many boys and how many girls are in the class? Answers: a) g: 2 squares each representing 5 girls, b: 3 squares each representing 5 boys, 15 boys and 10 girls; b) g: 3 squares each representing 5 girls, b: 1 square representing 5 boys, 15 girls and 5 boys. NOTE: In the Extensions below, students can use a calculator to multiply and divide decimals. Extensions 1. The value of a house increased by 20% in one year and 50% the next year. Show the percent it increased by in two years, in two ways. a) Use a tape diagram. b) Show what you multiply the price by each year. Then find the amount you multiply the original price by over two years. Answers: a) From the tape diagram, the value increased from 5 squares to 9 squares in the two years, so that is an increase of 80% (4 squares is 80% of the original 5 squares).

Year 0:

Year 1:

Year 2: b) Over two years, the price becomes (original price) × 1.2 × 1.5 = (original price) × 1.8, so the price increased by 80% over the two years. 2. Over two years, the price of a house doubled. In the first year, the price increased by 60%. By what percent did the price increase in the second year? Solution: Using a tape diagram, the first price can be represented by 5 squares, the second price by 8, and the third price by 10 (because it is double of 5), so that is a 25% increase from the second price. Alternatively: (original price) × 1.6 × ? = (original price) × 2, so the answer is 2 ÷ 1.6 = 1.25. 3. A newspaper article claimed that the price of bread increased by 300% in the past 10 years, from 80¢ to $2.40. What mistake did they make? Answer: The amount tripled (80¢ × 3 = $2.40), so the price became 300% of the original price, but the increase in the price was only $1.60, or 200% of the original price, so the price increased by 200%, not 300%.


RP7-25 Ratios, Fractions, and Percents Problems Pages 149–151 Standards: 7.EE.B.3 Goals: Students will solve problems involving percents, ratios, and fractions, and will translate the given information from one form into another. Prior Knowledge Required: Can draw a tape diagram to represent a given situation involving ratios, fractions, percents, or percent more than Can calculate a percent or fraction of a number Vocabulary: fraction, percent, percent more than, ratio, tape diagram Exercises: a) Determine the numbers of boys, girls, and children in the class. i) There are 3 boys and 4 girls in a class. ii) There are 7 boys and 13 children in a class. iii) There are 8 girls and 19 children in a class. b) Write the fraction of girls and boys in each class from part a). Answers: a) i) 3 boys, 4 girls, 7 children altogether; ii) 7 boys, 6 girls, 13 children altogether; iii) 8 girls, 11 boys, 19 children altogether; b) i) b: 3/7, g: 4/7; ii) b: 7/13, g: 6/13; iii) g: 8/19, b: 11/19 Write on the board:

The ratio of girls to boys is 1 : 3. g: b: SAY: If the ratio of girls to boys is 1 : 3, then for every 1 part that are girls (point to the one square for girls in the tape diagram), there are 3 parts that are boys (point to the three squares for boys in the tape diagram). ASK: What fraction of the class is girls? (1/4) SAY: There are 4 parts in total and one out of the four parts is girls, so 1/4 of the class is girls. In the Exercises below, students who are struggling can draw a tape diagram. Exercises: Write the fraction of girls and boys in each class. a) The ratio of boys to girls is 1 : 2. b) The ratio of girls to boys is 2 : 3. c) The ratio of boys to girls is 12 : 11. d) The ratio of boys to girls is 11 : 12. e) The ratio of girls to boys is 11 : 12.


Answers: a) b: 1/3, g: 2/3; b) g: 2/5, b: 3/5; c) b: 12/23, g: 11/23; d) b: 11/23, g: 12/23; e) g: 11/23, b: 12/23 ASK: Which two of the last three questions have the same answer? (parts c) and e)) What is another question that would have the same answer as part a)? (The ratio of girls to boys is 2 : 1.) Remind students that if they know the fraction of a class that is girls, and the number of students, then they can find the number of girls and the number of boys. Write on the board: 24 students

3

4 are girls

How many girls and how many boys? SAY: The number of girls is 3/4 of 24. ASK: How many girls are there? (18) Write on the board:

3 3

of 24 244 4

= ´

= 3 × 24 ÷ 4 ASK: What is easier to do first, the 3 × 24 or the 24 ÷ 4? (24 ÷ 4) Continue evaluating the expression on the board: = 3 × 6 = 18 SAY: There are 24 students and 18 of them are girls. ASK: How many boys are there? (6) Exercises: Determine the number of girls and boys in the class.

a) There are 30 children and 3

5 are girls.

b) There are 36 children and 4

9 are girls.

c) There are 21 children and 4

7 are boys.

d) There are 18 children. The ratio of boys to girls is 7 : 2. e) There are 18 children. The ratio of girls to boys is 2 : 7. f) There are 30 children and 60% are girls. g) There are 45 children and 40% are girls. h) There are 36 children and 75% are boys. Answers: a) 18 girls and 12 boys, b) 16 girls and 20 boys, c) 12 boys and 9 girls, d) 14 boys and 4 girls, e) 4 girls and 14 boys, f) 18 girls and 12 boys, g) 18 girls and 27 boys, h) 27 boys and 9 girls


Translating between percent of, fraction of, ratio, and percent more than. Draw on the board: Girls: Boys: Fraction of students that are girls: ______ Ratio of girls to boys: ______ Percent of students that are girls: ______ _____ % more girls than boys Have volunteers fill in the blanks. (3/5, 3 : 2, 60%, 50) SAY: If you can draw the tape diagram for any of the ways of describing the situation, then you can describe the situation given any of the other ways. (MP.1) Exercises: Show your answer with a tape diagram. a) The number of boys is 80% of the number of girls. The ratio of boys to girls is ____ : ____. There are ____% more girls than boys. b) The ratio of girls to boys is 3 : 2. The fraction of students that are girls is ______. There are _____% more girls than boys.

c) 5

12 of the students are boys.

The ratio of girls to boys is _____ to _____. There are _____% more _____ than _____.

d) The number of girls is 5

8 of the number of boys.

The ratio of girls to boys is _____ to _____. There are _____% more _____ than _____. Answers: a) 4 : 5, 25% more girls than boys; b) 3/5, 50% more girls than boys; c) 7 to 5, 40% more girls than boys; d) 5 to 8, 60% more boys than girls SAY: If you can draw a tape diagram for any way of describing a situation, then you can solve all sorts of problems. (MP.4) Exercises: 1. A baseball player got a hit 2 out of every 3 times at bat. She was at bat 9 times. How many hits did she have?


2. a) The ratio of girls to boys in a school is 12 : 13. If the school has 200 students, how many girls are there? b) A jar has red and green marbles. The number of red marbles is 25% of the number of green marbles in a jar. If there are 60 marbles in the jar, how many green marbles are there? c) In a fruit salad, 40% of the fruit pieces are apples and 30% are oranges. There are 6 more apple pieces than orange pieces. How many fruit pieces are there altogether? d) There are 2 apples in a bowl for every 3 oranges. If there are 15 fruits in the bowl, how many apples are there? e) There are 60% more girls than boys in a class. There are 24 girls in the class. How many boys are in the class?

f) A recipe for lemonade calls for 4

3 cup lemon juice,

1

2 cup maple syrup, and 4 cups water.

Marla used the juice from 6 lemons and got 3

15

cups lemon juice. How much of the other

ingredients should she use? Answers: 1. 6; 2. a) 96, b) 48, c) 60, d) 6, e) 15, f) 3/5 cup maple syrup and 4 4/5 cups water Extensions 1. To estimate a fraction or ratio, you can change one or both parts slightly.

Example A: 5 out of 11 is close to 5 out of 10, which is close to 1

2 or 50%.

Example B: 9 out of 23 is close to 10 out of 25, which is 2

5 (

10

25 =

2

5), which is 40%.

The chart shows the lengths of calves and adult whales (in feet). Approximately what fraction and what percent of each adult’s length is the calf’s length? Do you need to know how long a foot is to answer this question? (no) Type Orca Humpback Narwhal Fin-backed Sei Calf length (ft) 7 16 5 22 16 Adult length (ft) 15 50 19 70 60

Sample answers: Orca: 1/2 and 50%; Humpback: 1/3 and 33%; Narwhal: 1/4 and 25%; Fin-backed: 21/70 = 3/10 or 30%; Sei: 1/4 and 25% (MP.1) 2. There are red and blue marbles in a bag with the ratio red : blue = 3 : 5. Some marbles of each color were added to increase the number of marbles in the bag by 50%. Now there are the same number of red marbles as blue marbles in the bag. What percent did the red marbles increase by? What percent did the blue marbles increase by? Answer: R: B: The total increased by 50%, and there were 8 squares to start, so there must be 12 after the increase. Since there are the same number of each color, each color has 6 squares at the end, so the red marbles increased by 100% and the blue marbles increased by 20%.


(MP.3) 3. A newspaper article said that among the children playing soccer at the park, the ratio of girls to boys was 1 to 2. Han said that means that half the children playing soccer were girls. Is he right? Explain. Answer: No, there are twice as many boys as girls, so only one third of the children playing soccer are girls. (MP.4) 4. A company notices that Eva and Andy are being paid different amounts for the same job. To improve pay equity, the company increases Eva’s pay by 20% and decreases Andy’s pay by 25%. Now they are paid the same amount. What percent more was Andy’s original pay than Eva’s? Answer: 60%. To see this, draw a tape diagram: Eva’s original pay: Final pay: Andy’s original pay:

unit 5 ratios and proportional relationships: real world ... for cc edition... · teacher’s guide...

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