Surface Area. Assessment Management


 Emerald Johnston
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1 Surface Area Objective To introduce finding the surface area of prisms, cylinders, and pyramids. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Measure the dimensions of a cylinder in inches and centimeters. [Measurement and Reference Frames Goal ] Use rectangle and triangle area formulas to find the surface area of prisms and cylinders. Apply a formula to calculate the area of a circle. [Measurement and Reference Frames Goal 2] Identify and use the properties of prisms, pyramids, and cylinders in calculations. [Geometry Goal 2] Key Activities Students find the surface areas of prisms, cylinders, and pyramids by calculating the area of each surface of a solid and then finding the sum. Ongoing Assessment: Informing Instruction See page 892. Ongoing Assessment: Recognizing Student Achievement Use journal pages 389 and 390. [Measurement and Reference Frames Goals and 2] Key Vocabulary surface area Materials Math Journal 2, pp. 389 and 390 Study Link 6 slate calculator cardboard box per workstation: 2 cans, tape measure, ruler, triangular prism, square pyramid Plotting and Analyzing Insect Data Math Journal 2, pp. 390A and 390B Students plot insect lengths in fractional units on a line plot. They use the line plot to analyze the data. Math Boxes Math Journal 2, p. 39 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. 34 Students practice and maintain skills through Study Link activities. ENRICHMENT Finding the Smallest Surface Area for a Given Volume Math Masters, p. 342 per partnership: 2 cubes Students find the rectangular prism with the smallest surface area for a given volume. ETRA PRACTICE Finding Area, Surface Area, and Volume Math Masters, p. 343 Student Reference Book, pp. 96 and 97 Students practice calculating area, surface area, and volume. Advance Preparation For Part, organize workstations for groups of 4 students (2 partnerships each). Each group will need 2 cans (see Lesson 3) and each of the triangular prism and square pyramid models (Math Masters, pages 324 and 326) constructed in Lesson . Place a cardboard box with the top taped shut near the Math Message. Teacher s Reference Manual, Grades 4 6 pp Unit Volume
2 Getting Started Mental Math and Reflexes Have students write numbers from dictation and mark the indicated digits. Suggestions:,024,039 Circle the digit in the thousands place, and put an through the digit in the hundredthousands place. 28,794,02 Circle the digit in the tenthousands place, and put an through the digit in the hundreds place Circle the digit in the thousandths place, and put an through the digit in the hundredths place. 0, Circle the digit in the thousandths place, and put an through the digit in the thousands place Circle the digit in the tens place, and put an through the digit in the hundredths place. 94,679,424 Circle each digit in the millions period, and put an through each digit in the ones period. Math Message If you were to wrap this box as a gift, how would you calculate the least amount of wrapping paper needed? Study Link 6 FollowUp Briefly review the answers. Ask: How would you explain the difference between capacity and volume? Volume is the measure of how much space a solid object occupies. Capacity is the measure of how much a container can hold. Teaching the Lesson Math Message FollowUp (Math Journal 2, p. 389) WHOLECLASS Ask volunteers to use geometry vocabulary to describe the box. The box is a rectangular prism that has six rectangular faces. Then have students explain their solution strategies. If wrapping paper covers each face of the box exactly, the least amount of wrapping paper needed is the sum of the area of each of the six faces. Tell students that the sum of the areas of the faces or the curved surface of a geometric solid is called surface area. If wrapping paper covers each face of the box exactly, the area of paper required is the surface area of the box. If the area of the available wrapping paper is less than the surface area of the box, the paper will not completely cover the box. Ask a pair of students to measure the length, width, and height of the box to the nearest inch. Have students record these dimensions on the figure in Problem on journal page 389, find and record the areas of the six sides of the box, and add these to find the total surface area. Remind students that opposite sides (top and bottom, left and right, front and back) of the box have the same area. This means that students need to calculate only three different areas. Finding the Surface Area of a Can (Math Journal 2, p. 389) PARTNER PROBLEM SOLVING Algebraic Thinking Distribute one can to each partnership. Ask students to imagine that the top lids of their cans have not been removed. Ask: How would you find the surface area of your can? 7 Surface Area The surface area of a box is the sum of the areas of all 6 sides (faces) of the box.. Your class will find the dimensions of a cardboard box. a. Fill in the dimensions on the figure below. b. Find the area of each side of the box. Then find the total surface area. Area of front 3 in Area of back in 2 Area of right side in 2 Area of left side in 2 Area of top in 2 Area of bottom in Total surface area in 2 2. Think: How would you find the area of the metal used to manufacture a can? a. How would you find the area of the top or bottom of the can? b. How would you find the area of the curved surface between the top and bottom of the can? c. Choose a can. Find the total area of the metal used to manufacture the can. Remember to include a unit for each area. Area of top Area of bottom Area of curved side surface Total surface area Sample answers: First measure the diameter of the can, and divide by 2 to find the radius. Then calculate the area (A π º r 2 ). Calculate the circumference of the base (c π º d ), and multiply that by the height of the can Math Journal 2, p in. front 7 top right side Sample answers for a can with a diameter and 3 cm height: in. in. Lesson 89
3 3. Use your model of a triangular prism. a. Find the dimensions of the triangular and rectangular faces. Then find the areas of these faces. Measure lengths to the nearest _ 4 inch. base = 2 in. length = 4 _ 2 in. height = 3_ 4 in. width = 2 in. Area = 3_ 4 in 2 Area = 9 in 2 b. Add the areas of the faces to find the total surface area. Area of 2 triangular bases = 3 _ 2 in 2 Analyzing Fraction Data Kerry measured lifesize photos of common insects to the nearest _ 8 of an inch. His measurements are given in the table below. Insect Length Length (nearest _ 8 in.) Insect (nearest _ 8 in.) American Cockroach 3_ 8 Heelfly _ 3 8 Ant _ 4 Honeybee 3_ 4 Aphid _ 8 House Centipede 3_ 8 Bumblebee _ 2 Housefly _ 4 Cabbage Butterfly _ 4 June Bug Cutworm _ 2 Ladybug _ 8 Deerfly 3_ 8 Silverfish 3_ 8 Field Cricket 7_ 8. Make a line plot of the insect lengths on the number line below. Be sure to label the axes and provide a title for the graph. Number of Insects Surface Area continued Area of 3 rectangular sides = in 2 Total surface area = 30 _ 2 in 2 4. Use your model of a square pyramid. a. Find the dimensions of the square and triangular faces. Then find the areas of these faces. Measure lengths to the nearest tenth of a centimeter. length = 6.3 cm base = 6.3 cm width = 6.3 cm height =. Area = cm 2 Area = 7.32cm 2 b. Add the areas of the faces to find the total surface area. Area of square base = cm 2 Area of 4 triangular sides = cm 2 Total surface area = cm _EMCS_S_MJ2_U_76434.indd 390 Formula for the Area of a Triangle A = _ 2 º b º h where A is the area of the triangle, b is the length of its base, and h is its height Math Journal 2, p. 390 Insect Lengths Length (inches) /4/ 7:04 PM Remind students that in this problem they are finding the area of the surface of their can, not the volume. Give students plenty of time to explore solution strategies among themselves, without your help. Add the areas of the top, bottom, and curved surface to find the total surface area. Ask a volunteer to explain how to find the area of the circular top and bottom. Use the formula A = π r 2. Ask another volunteer to explain how to find the area of the curved surface between the top and bottom of the can. Calculate the circumference of the base using the formula c = π d, and then multiply the circumference by the height of the can. If students are unable to suggest an approach for finding this area, remind or show them that the label on a food can is shaped like a rectangle if it is cut off and unrolled. Draw a rectangle on the board: If you cut a can label in a straight line perpendicular to the bottom, it will have a rectangular shape when it is unrolled and flattened. How would you measure the can to find the length of the base of the rectangle? Guide students to recognize the following two possibilities:. Measure the circumference of the base of the can with a tape measure. 2. Measure the diameter of the base of the can. Calculate the circumference of the base, using the formula c = π d. How would you find the other dimension (the width) of the rectangle? Measure the height of the can. Have partners measure the dimensions of their cans and complete Part 2 on journal page 389. Circulate and assist. Ongoing Assessment: Informing Instruction Watch for students who seem to have difficulty with visualizing the cutting and unrolling of the can label. Have them use the following procedure to cut a sheet of paper to use as a label for the side of the can:. Mark the top and bottom rims of the can at the can s seam. 2. Lay the can on the sheet of paper with the marks touching the paper. Mark these points on the paper. 3. Roll the can one complete revolution, until the marks touch the paper again. Mark these points on the paper. 4. Connect the four marks on the paper to form a rectangle. Cut out the rectangle. Math Journal 2, p. 390A _EMCS_S_MJ2_U_76434.indd 390A 3/3/ 9:23 AM 892 Unit Volume
4 Finding the Surface Area of a Prism and a Pyramid (Math Journal 2, p. 390) PARTNER Algebraic Thinking Have partners measure each solid constructed in Lesson  from Math Masters, pages 324 and 326 and record the dimensions on journal page 390. Then have them calculate and record the face areas and total surface area for each solid. Circulate and assist. Ongoing Assessment: Recognizing Student Achievement Journal pages 389 and 390 Use journal pages 389 and 390 to assess students ability to measure to the nearest _ 4 inch and 0. cm and to find the area of circles, triangles, and rectangles. Students are making adequate progress if they correctly answer Problems 2 and 3 and use rectangle, circle, and triangle area formulas to find the surface area of prisms and cylinders. [Measurement and Reference Frames Goals and 2] Analyzing Fraction Data continued Use the data and the line plot you made on journal page 390A to solve these problems. Give all answers in simplest form. 2. What is the maximum length of the insects measured? 3. What is the minimum length? 4. What is the range of insect lengths? _ 8 in.. What is (are) the mode length(s) of the insects? 6. What is the median length of the insects? 7. How much longer is the American cockroach that Kerry measured than the field cricket? 3_ 8 in. 3_ 8 in. 3_ 4 in. _ 2 in. _ 2 in. 8. Suppose ladybugs were placed end to end. How long would they be? 3 _ 8 in. 9. How many times as long is the bumblebee than the housefly? 6 times as long 0. a. Suppose all of the insects less than 3_ 4 inch long were placed endtoend. How long would they be? 2 3_ 8 in. b. Suppose all of the insects greater than _ 8 inch long were placed endtoend. How long would they be? 9 _ 8 in. c. What is the total length of all insects placed end to end? 2 in.. What is the mean length of the insects that were measured? Math Journal 2, p. 390B 4_ in _EMCS_S_MJ2_G_U_76434.indd 390B 4/7/ 3:8 PM 2 Ongoing Learning & Practice Plotting and Analyzing Insect Data (Math Journal 2, pp. 390A and 390B) INDEPENDENT Students plot insect lengths in fractional units on a line plot. They use the line plot to analyze data and solve problems involving addition, subtraction, multiplication, and division with fractions. As needed, review how to make a line plot based on fractional units and how to perform operations with fractions. Math Boxes Math Boxes (Math Journal 2, p. 39) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson . The skill in Problem previews Unit 2 content.. Write each number in simplest form. a. b. 80_ = 6 3_ 6 8 _ = 6 8 _ 2 c. 8 24_ 48 = 9_ d. 38_ 4 = Find the volume of the rectangular prism. Volume of rectangular prism V = B * h 2. How are cylinders and cones alike? Both have a circular base and one curved surface Find the area and perimeter of the rectangle. 3 2 cm Study Link (Math Masters, p. 34) INDEPENDENT Home Connection Students practice using formulas to calculate volume and surface area. If students use calculators, remind them to enter 3.4 for π if necessary. V = 2 cm 3. Solve the panbalance problems below. a. b. 3 cm cm Area = Perimeter = (unit) 2 cm (unit) One weighs as much as 3 s. One weighs as much as 6 s. One weighs as much as One weighs as much as paper clips. paper clips Math Journal 2, p _EMCS_S_MJ2_U_76434.indd 39 3/4/ 7:04 PM Lesson 893
5 Name STUDY LINK Volume and Surface Area Area of rectangle: A = l º w Volume of rectangular prism: V = l º w º h. Kesia wants to give her best friend a box of chocolates. Figure out the least number of square inches of wrapping paper Kesia needs to wrap the box. (To simplify the problem, assume that she will cover the box completely with no overlaps.) Amount of paper needed: 88 in 2 Explain how you found the answer. Sample answer: I found the area of each of the 6 sides and then added them together. Yes Explain. A 4 in. 4 in. 3 _ 2 in. box has a volume of 6 in 3 and a surface area of 88 in Could Kesia use the same amount of wrapping paper to cover a box with a larger volume than the box in Problem? Find the volume and the surface area of the two figures in Problems 3 and Volume: 4. Volume: Surface area: 3. 2 Study Link Master 0 cm Circumference of circle: c = π º d Area of circle: A = π º r 2 Volume of cylinder: V = π º r 2 º h 26 in 3 Surface area: 26 in 2 2 in. 6 in. 6 in. cube in. 3 Differentiation Options ENRICHMENT PARTNER Finding the Smallest Surface Area for a Given Volume (Math Masters, p. 342) 30 Min Algebraic Thinking To apply students understanding of volume and surface area, have them use given volumes to find the dimensions of rectangular prisms. Partners use the patterns in the dimensions to identify the dimensions of a prism that would have the smallest surface area. When students have finished, discuss any difficulties or curiosities they encountered _EMCS_B_MM_G_U_76973.indd 34 Math Masters, p. 34 3/9/ 8:44 AM ETRA PRACTICE INDEPENDENT Finding Area, Surface Area, and Volume (Math Masters, p. 343; Student Reference Book, pp. 96 and 97) Min Students practice calculating area, surface area, and volume of various geometric figures. Name Teaching Master A SurfaceArea Investigation Name Teaching Master Area, Surface Area, and Volume In each problem below, the volume of a rectangular prism is given. Your task is to find the dimensions of the rectangular prism (with the given volume) that has the smallest surface area. To help you, use centimeter cubes to build as many different prisms as possible having the given volume. Record the dimensions and surface area of each prism you build in the table. Do not record different prisms with the same surface area. Put a star next to the prism with the smallest surface area.. 2. Dimensions (cm) Surface Area (cm 2 ) Volume (cm 3 ) Dimensions (cm) Surface Area (cm 2 ) Volume (cm 3 ) If the volume of a prism is 36 cm 3, predict the dimensions that will result in the smallest surface area. Explain. 3 º 3 º 4; I used the smallest 3 dimensions that would result in the given volume. 4. Describe a general rule for finding the surface area of a rectangular prism in words or with a number sentence. Find the area of each face. Then find the sum of these areas. Math Masters, p. 342 Area of rectangle: A = l w Volume of rectangular prism: V = B h or V = l w h. Record the dimensions and find the area. Length = Width = _ 2 ft 2 _ 8 ft Area = 4 7_ 6 ft2 How many square tiles, each ft long on a side, would be needed to fill the rectangle? 4 7_ 6 Circumference of circle: c = π d Area of circle: A = π r 2 Volume of cylinder: V = π r 2 h Record the dimensions, and find the volume and surface area for each figure below. Round results to the nearest hundredth. 3. Rectangular prism 2 ft Volume = 280 cm ft Surface area = 262 cm 2 Math Masters, p Record the dimensions and find the volume. Area of base = 20 cm 2 2 cm Number of centimeter unit cubes needed to fill the box: 40 Volume = 40 cm 3 4. Cylinder Diameter = Height = 9 ft 7 ft 7 ft 9 ft Volume = ft 3 2 cm Surface area = ft _EMCS_B_MM_G_U_76973.indd 342 3/9/ 8:44 AM _EMCS_B_MM_G_U_76973.indd 343 4/7/ 9:43 AM 894 Unit Volume
6 Name Area, Surface Area, and Volume Area of rectangle: A = l w Volume of rectangular prism: V = B h or V = l w h Circumference of circle: c = π d Area of circle: A = π r 2 Volume of cylinder: V = π r 2 h. Record the dimensions and find the area. 2. Record the dimensions and find the volume. 2 8 ft 2 cm 2 ft Length = Width = Area = How many square tiles, each ft long on a side, would be needed to fill the rectangle? Area of base = Number of centimeter unit cubes needed to fill the box: Volume = Copyright Wright Group/McGrawHill Record the dimensions, and find the volume and surface area for each figure below. Round results to the nearest hundredth. 3. Rectangular prism 4. Cylinder Diameter = Height = 9 ft 7 ft Volume = Volume = Surface area = Surface area = 343
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