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Objectives To demonstrate how to estimate the area of a surface having a curved boundary; and to provide practice converting from one square unit to another. 1 materials Teaching the Lesson Key Activities Students estimate the area of the front of their hand by tracing it on a 1-inch grid and counting squares. Then they use a rule of thumb to estimate their total skin area. Students convert measurements from square inches to square feet and from square feet to square yards. Math Journal 2, pp. 230 and 231 Study Link 8 3 Key Concepts and Skills Teaching Aid Masters (Math Masters, p. 388 or 389; and p. 444) calculator • Use the terms estimate and guess. [Operations and Computation Goal 6] • Use an estimate to judge the reasonableness of a solution. scissors tape [Operations and Computation Goal 6] • Count squares and partial squares or use a formula to estimate area. See Advance Preparation [Measurement and Reference Frames Goal 2] • Convert between square inches and square feet. [Measurement and Reference Frames Goal 3] Ongoing Assessment: Recognizing Student Achievement Use a Math Log or Exit Slip. [Measurement and Reference Frames Goal 2] 2 Ongoing Learning & Practice Students resume the World Tour in South America. Students practice and maintain skills through Math Boxes and Study Link activities. materials Math Journal 2, pp. 229, 329 (optional), 330, 331, 334, and 335 Student Reference Book Study Link Master (Math Masters, p. 255) Teaching Aid Masters (Math Masters, pp. 419 and 420, optional) 3 materials Differentiation Options READINESS Students use square stick-on notes to determine the area of an irregular region. EXTRA PRACTICE Students place objects on a grid, trace their outlines, and count grid squares to estimate their areas. Additional Information Advance Preparation For Part 1, make 3 copies of 1-inch grid paper (Math Masters, page 444) for each student. Place copies near the Math Message. Teaching Aid Masters (Math Masters, p. 388 or 389; and p. 444) objects with flat irregular surfaces masking tape; square stick-on notes Technology Assessment Management System Math Log or Exit Slip See the iTLG. Lesson 8 4 675 Getting Started Mental Math and Reflexes Math Message Pose mental subtraction problems. Suggestions: Take 3 sheets of grid paper. Cut and tape the grids to make a square with sides that measure 1 foot (12 inches). How many square inches are there in 1 square foot? 80 – 50 30 86 – 41 45 64 – 45 19 120 – 30 90 149 – 42 107 203 – 150 53 250 – 60 190 278 – 73 205 671 – 80 591 370 – 90 280 357 – 122 235 417 – 60 357 Study Link 8 3 Follow-Up Have students share their conclusions about the area of their new shapes. Confirm with the class that shapes that look different can have the same area. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS ACTIVITY (Math Journal 2, p. 230) Each student should have made a square consisting of 12 rows with 12 one-inch squares in each row. Since 12 12 144, there are 144 square inches in 1 square foot. Students record this equivalency in Problem 1 on journal page 230. Tell students that in this lesson they will use the 1-foot square to estimate the area of their skin. Estimating the Area Student Page Date 8 4 of Your Skin Time LESSON What Is the Total Area of My Skin? Follow your teacher’s directions to complete this page. 1. There are 144 square inches in 1 square foot. (Math Journal 2, pp. 230 and 231) Answers vary for Problems 2–5 and 6b. 1 ft 2. My guess is that the total area of my skin is about PARTNER ACTIVITY square feet. Rule of Thumb: The total area of your skin is about 100 times the area of the outline of your hand. 1 ft Ask students to guess the total area of their skin in square feet and record it in Problem 2 on journal page 230. Have them refer to their 1-foot square to help them with their guess. 1 square foot Follow these steps to estimate the total area of your skin: Ask your partner to trace the outline of your hand on the grid on page 231. Estimate the area of the outline of your hand by counting squares on the grid. Record your estimate in Problem 3 below. Use the rule of thumb to estimate the total area of your skin (area of skin 100 area of hand). Record your estimate in Problem 4 below. 3. I estimate that the area of the outline of my hand is about square inches. 5. I estimate that the total area of my skin is about square feet. 9 Rule of Thumb: The total area of your skin is about 100 times the area of the outline of your hand. square inches. 4. I estimate that the total area of my skin is about 6. a. There are Tell them to use the following to check their guess. Then have partners follow these steps: 1. Trace the outline of your partner’s hand on the grid on your partner’s journal page 231. square feet in 1 square yard. b. I estimate that the total area of my 2. Estimate the area of your own hand by counting grid squares (square inches) inside the tracing of your hand. square yards. 3 ft skin is about 3 ft 1 square yard 230 Math Journal 2, p. 230 676 Unit 8 Perimeter and Area 3. Use the rule of thumb to estimate the total area of your skin (area of skin 100 area of hand). Students record the results in Problems 3 and 4. A student’s fingers should be closed when his or her partner traces the hand outline on journal page 231. Ongoing Assessment: Recognizing Student Achievement Math Log or Exit Slip Use a Math Log or an Exit Slip to assess students’ ability to count unit squares and fractions of unit squares to estimate the area of an irregular figure. Have students explain how they found the area of their hands. Students are making adequate progress if their strategy includes counting whole squares and combining partial squares to approximate whole squares. Some students may describe a strategy in which they find the area of a rectangular region within the hand tracing by multiplying its length and width. [Measurement and Reference Frames Goal 2] Sharing the Results of WHOLE-CLASS ACTIVITY the Experiment (Math Journal 2, pp. 230 and 231) Review the information students have obtained so far. In Problem 2, they guessed the total area of their skin in square feet. In Problem 4, they estimated the total area of their skin in square inches. Remind students about the difference between a guess and an estimate—an estimate is more systematic and usually more accurate than a guess. Ask students to compare how close the guesses they made in Problem 2 are to their estimates in Problem 4. The guess and the estimate cannot be immediately compared because the guess is in square feet and the estimate is in square inches. Lesson 8 4 677 Help students convert their square-inch estimates to square feet. Have them record the square-foot estimates in Problem 5. Example: Guess is 15 square feet. Estimate is 1,512 square inches. Convert 1,512 square inches to square feet. Use the fact that 144 square inches equals 1 square foot. You want to know how many 144s there are in 1,512. Since 1,512 144 10.5, it follows that 1,512 square inches equals 10.5 square feet. ELL Adjusting the Activity Have students use their 1-foot squares to model the conversion of a volunteer’s skin estimate in square inches to square feet. Ask students to come to the board, one at a time, with 1-foot squares and tape them onto the board. Have a student with a calculator add 144 each time a 1-foot square is added. Students should tape the first nine 1-foot squares in three rows with three squares in each row. Call attention to the fact that 9 square feet is equal to 1 square yard. Have students tape 1-foot squares until the total is equal to or greater than the estimated area. Discuss how to use the taped squares to find the approximate area in square feet. For example, 1,500 square inches is between 10 square feet (10 144 1,440) and 11 square feet (11 144 1,584). A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Now ask students to convert their estimates to an approximate number of square yards (probably between 1 and 2) and record it in Problem 6. To convert to square yards, students divide the number of square feet by 9. 2 Ongoing Learning & Practice World Tour Option: SMALL-GROUP ACTIVITY Visiting South America (Math Journal 2, pp. 329–331, 334, and 335; Student Reference Book ) Social Studies Link If you have chosen to extend the scope of the World Tour for your class, have small groups visit a second country in South America. You may let them choose which countries to visit, or you may assign a country to each group. Remind students to update their Route Log (if they are keeping one) and Route Map and to complete Country Notes for the country they visit. Adjusting the Activity Have students determine the latitude and longitude of the capital city of the country they visit. A U D I T O R Y 678 Unit 8 Perimeter and Area K I N E S T H E T I C T A C T I L E V I S U A L Student Page Math Boxes 8 4 INDEPENDENT ACTIVITY (Math Journal 2, p. 229) Date Time LESSON Math Boxes 8 4 1. Measure the sides of the figure to the 2. If you tossed a coin onto the grid below, nearest centimeter. Then find its perimeter. 3 cm Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-2. The skill in Problem 6 previews Unit 9 content. Writing/Reasoning Have students write a response to the following: Describe the strategy you used to estimate the product in Problem 4. Sample answer: I used 200 instead of 192, and 50 instead of 49, because they are “close-but-easier” numbers. 200 50 10,000 1 2 4 cm 8 , 16 2 cm 15 Perimeter cm 2 INDEPENDENT ACTIVITY P E P E R O P E R E R O P 45 84 cm Fraction 1 10 20 100 0.1 0.20 0.8 0 4 5 0 3 4. Which number is closest to the product 100 B 1,000 C 10,000 D 100,000 181 18 6. Shade more than but less than 100 3 of the grid. 10 notation. 105 a. 100,000 A 61 5. Write each number in exponential O of 192 and 49? Circle the best answer. Decimal c. R O 131 whole number. b. 1 2 cm 3. Write an equivalent fraction, decimal, or a. or cm cm 1 d. Study Link 8 4 about what fraction of the time would you expect it to land on a vowel? Sample answer: 1 (Math Masters, p. 255) b. 10 10 107 109 1,000,000,000 c. 10,000,000 Home Connection Students estimate the area of São Paulo State and Rio de Janeiro State in Brazil. d. 5 27 229 Math Journal 2, p. 229 Study Link Master Name Date STUDY LINK 8 4 1. Below is a map of São Paulo State in Brazil. Each grid square represents 2,500 square miles. Estimate the area of São Paulo State. I counted about The area is about 2. Time Areas of Irregular Figures 133 35 grid squares. 87,500 square miles. To the right is a map of Rio de Janeiro State in Brazil. Each grid square represents 2,500 square miles. Estimate the area of Rio de Janeiro State. I counted about The area is about 7 grid squares. 17,500 square miles. Practice 3. 88.71 5. 386.174 73.04 15.67 4. 86.05 27.97 312.11 74.064 6. 57.1 39.002 58.08 18.098 Math Masters, p. 255 Lesson 8 4 679 3 Differentiation Options READINESS Counting Squares to Find Area PARTNER ACTIVITY 5–15 Min (Math Masters, p. 388 or 389) To explore the concept of area as the number of unit squares and fractions of unit squares needed to cover a surface without gaps or overlaps, have students use square stick-on notes to cover an irregular region. Using masking tape, create (or have students create) a large figure on the floor. Students first estimate the area and then cover the figure with stick-on notes. They count stick-on notes and partial areas not covered by stick-on notes to refine their estimates. In a Math Log or on an Exit Slip, have students describe how they determined the area of the irregular region. Students use square stick-on notes to estimate the area of an irregular region. EXTRA PRACTICE Estimating Areas of INDEPENDENT ACTIVITY 5–15 Min Irregular Regions (Math Masters, p. 444) To practice finding the area of irregular regions, have students place objects that have flat, irregular surfaces on a grid and trace their outlines. Then have students count grid squares and partial grid squares to estimate the areas. Students estimate areas of irregular regions on Math Masters, page 444. 680 Unit 8 Perimeter and Area