Assessment Management


 Adele Owens
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1 Areas of Rectangles Objective To reinforce students understanding of area concepts and units of area. 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 Multiply fractions and mixed numbers to find the area of a rectangle. [Operations and Computation Goal 5] Use a formula to calculate the areas of rectangles. [Measurement and Reference Frames Goal 2] Compare inch and centimeter measures for length and area. [Measurement and Reference Frames Goal 3] Key Activities Students review area concepts and the names and notations for common area units. They find areas of rectangles by counting and by applying an area formula. Ongoing Assessment: Informing Instruction See page 725. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 414). [Measurement and Reference Frames Goal 2] Key Vocabulary area square units base height formula variable Materials Math Journal 2, pp. 304 and 305 Student Reference Book, p. 188 Study Link 9 3 Math Masters, p. 414 transparency of Math Masters, p. 436 Class Data Pad inch ruler slate roll of paper towels, wax paper, or aluminum foil (optional) Fraction Division Review Math Journal 2, p. 306 Student Reference Book, pp B Students use visual models and number stories to solve fraction problems. Math Boxes Math Journal 2, p. 307 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. 265 Students practice and maintain skills through Study Link activities. READINESS Comparing Perimeter and Area Math Masters, p. 266 per partnership: 2 sixsided dice, 36 centimeter cubes Students use centimeter grids to compare the perimeters and areas of rectangles. ENRICHMENT Comparing Perimeter and Area for Irregular Figures Math Masters, pp. 267 and differentcolored pencils or markers scissors Students compare the perimeters and areas of irregular polygons. EXTRA PRACTICE 5Minute Math 5Minute Math, p. 212 Students calculate the areas of rectangles. EXTRA PRACTICE Area: Tiling and Using a Formula Math Masters, pp. 293A and 436 Students find the areas of rectangles with fractional units by tiling and using a formula. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 142 Students define and illustrate the terms length, height, base, and width. Advance Preparation For Part 1, display a set of unit squares. (See Planning Ahead, Lesson 92.) Use a roll of paper or foil to demonstrate carpet rolls for Math Journal 2, page 305, Problem 2. Teacher s Reference Manual, Grades 4 6 pp , 233, 234, Unit 9 Coordinates, Area, Volume, and Capacity
2 Getting Started Mental Math and Reflexes Have students write fractions as equivalent decimals and percents. Suggestions: 2_ ; 66 2_ 3 % 4_ 0.8; 80% 5 8_ 0.32; 32% 25 19_ 0.95; 95% 20 24_ 0.48; 48% 50 3_ ; 37.5% Math Message Read page 188 of the Student Reference Book, and write two important facts about area. Study Link 9 3 FollowUp Have partners compare answers and resolve any differences. 1 Teaching the Lesson Math Message FollowUp (Student Reference Book, p. 188) WHOLECLASS DISCUSSION Ask volunteers to share what they wrote about area. Use their responses and the display of common units of area to review basic area concepts. Emphasize the following points: Area is a measure of the surface, or region, inside a closed boundary. It is the number of whole and partial unit squares needed to cover the region without gaps or overlaps. NOTE It is more precise to talk about the area of a rectangular region, the area of a triangular region, and so on. However, it is customary to refer to area in terms of the figure that is the boundary: the area of a rectangle, the area of a triangle, and so on. Measurement Student Page Area is measured in square units. There are many units to choose from, and some choices make more sense than others. Call students attention to the classroom display of unit squares and to alternative ways of writing the units: square inch, sq in., or in 2 ; square meter, sq m, or m 2 ; and so on. Ask students to share the relationships they observe among the units for example, a square meter is larger than a square yard. There are 9 square feet in a square yard and 144 square inches in a square foot. A square inch is larger than a square centimeter. Area Area is a measure of the amount of surface inside a closed boundary. You can find the area by counting the number of squares of a certain size that cover the region inside the boundary. The squares must cover the entire region. They must not overlap, have any gaps, or extend outside the boundary. Sometimes a region cannot be covered by an exact number of squares. In that case, count the number of whole squares and fractions of squares that cover the region. Area is reported in square units. Units of area for small regions are square inches (in. 2 ), square feet (ft 2 ), square yards (yd 2 ), square centimeters (cm 2 ), and square meters (m 2 ). For large regions, square miles (mi 2 ) are used in the United States, while square kilometers (km 2 ) are used in other countries. You may report area using any of the square units. But you should choose a square unit that makes sense for the region being measured. Examples Area of the field is 6,000 square yards. Area 6,000 yd 2 The area of a fieldhockey field is reported below in three different ways. 100 yd Area of the field is 54,000 square feet. Area 54,000 ft ft 1 in. 1 cm 1 cm 1 square centimeter (actual size) 1 in. 1 square inch (actual size) Area of the field is 7,776,000 square inches. Area 7,776,000 in. 2 3,600 in. 60 yd 180 ft 2,160 in. Although each of the measurements above is correct, reporting the area in square inches really doesn t give a good idea about the size of the field. It is hard to imagine 7,776,000 of anything! The International Space Station (ISS) orbits the Earth at an altitude of 250 miles. It is 356 feet wide and 290 feet long, and has an area of over 100,000 square feet. Student Reference Book, p. 188 Lesson 723
3 Date 1 cm 2 1 cm 1 cm A Student Page Time Areas of Rectangles base (or length) C height (or width) 1. Fill in the table. Draw rectangles D, E, and F on the grid. B Rectangle Base (length) Height (width) Area A 2 cm 5 cm 10 cm 2 B 4 cm 4 cm 16 cm 2 E D F Finding the Area of a Rectangle (Math Journal 2, p. 304) Ask volunteers to define the terms base and height. The term base is often used to mean both a side of a figure and the length of that side. The height of a rectangle is the length of a side adjacent to the base. Ask students to decide upon the phrasing of a common definition for these vocabulary terms. Record the student definitions on the Class Data Pad. Ask a volunteer to draw a rectangle on the board and label the base and height. C 2.5 cm 2.5 cm 6.25 cm 2 D 6 cm 2 cm 12 cm 2 E F 3.5 cm 4 cm 3 cm 3.5 cm 14 cm cm 2 height 2. Write a formula for finding the area of a rectangle. Area = base height (b h), or length width (l w) Math Journal 2, p _EMCS_S_G5_MJ2_U09_ indd 304 2/22/11 5:18 PM base In Fourth Grade Everyday Mathematics, students found the area of a rectangle by counting unit squares. Then they developed a formula for finding the area of a rectangle. Expect that students might use either method formula or counting squares to find the areas of the rectangles on journal page 304. With the counting method, some rectangles enclose partial grid squares, and students must count and add the full and partial squares to find areas. For example, rectangle C encloses 4 full squares (4 cm 2 ), 4 halfsquares (4 1_ 2 = 2 cm2 ), and 1 quartersquare ( 1_ 4 cm2 ). Its total area is _ 4 = 61_ 4 cm2. 4 full squares 4 cm 2 4 halfsquares 2 cm 2 Each halfsquare has an area of 1_ 2 cm2. +1 quartersquare 1_ 4 cm2 The quartersquare is 1_ 2 cm total area 6 1_ 4 cm2 long and 1_ 2 cm wide. 1 cm 2 1 cm 1 cm C Assign journal page 304, Problem 1. Circulate and assist. 724 Unit 9 Coordinates, Area, Volume, and Capacity
4 Discussing Formulas for the Area of a Rectangle (Math Journal 2, p. 304; Math Masters, p. 436) WHOLECLASS DISCUSSION Algebraic Thinking Ask volunteers to give the dimensions of rectangles A F as other volunteers draw the rectangles on a transparency of Math Masters, page 436. Ask: What do you notice about the relationship between the base and height and the actual area of each figure? The base multiplied by the height is equal to the area. Reinforce this rule: If the length of the base and the height of a rectangle are known, the area can be found by multiplying the length of the base by the height. Such a rule is called a formula. The formula can be written in abbreviated form as: A = b h, where A stands for the area, b stands for the length of the base, and h stands for the height. Ask students to complete Problem 2 on journal page 304. Remind students that letters used in this way are called variables. Add the abbreviated formula to the definitions on the Class Data Pad, and have students write the abbreviated formula after their answers for Problem 2 on journal page 304. Refer students to the rectangles drawn on the transparency. Have students apply the formula for the rectangles in Problem 1. Ask volunteers to record a number model for the area of each rectangle on the transparency. For example, 2 cm 5 cm = 10 cm 2. Have students check their total count of the squares with the product from the number model. For rectangles C and E, ask students to think about the decimals as fractions (2 1_ cm 21_ cm = 61_ cm2, and 4 cm 3 1_ 2 cm = 14 cm2 ). Ask partners to estimate, in inches, the length of the sides of the rectangles on journal page 304. Then have students measure the sides of rectangle C using their inch rulers. (Each side of rectangle C is about 1 inch long.) Ask students what the area of rectangle C is when the unit is inches. 1 square inch Point out that there are about 2.5, or 2 1_ centimeters in 1 inch, and about 6.25, or 61_ 2 4 square centimeters in 1 square inch. Applying the Area Formulas (Math Journal 2, p. 305) INDEPENDENT PROBLEM SOLVING NOTE An alternative formula for the area of a rectangle is A = l w, where l stands for the length and w stands for the width of the rectangle. Students are familiar with both versions of the formula from Fourth Grade Everyday Mathematics. Date Area Problems 1. A bedroom floor is 12 feet by 15 feet (4 yards by 5 yards). Floor area = Floor area = Student Page square feet square yards 2. Imagine that you want to buy carpet for the bedroom in Problem 1. The carpet comes on a roll that is 6 feet (2 yards) wide. The carpet salesperson unrolls the carpet to the length you want and cuts off your piece. What length of carpet will you need to cover the bedroom floor? Time 3. Calculate the areas for the 4. Fill in the missing lengths for figures below. the figures below. a. 9 yd a. 12 ft 12 yd 3 yd 6 yd 72 6 yd Area = yd 2 6 yd 12 ft (4 yd) 6 ft (2 yd) 30 ft, or 10 yd 15 ft (5 yd) 360 ft 30 2 ft 30 ft 12 ft Have students complete journal page 305. Circulate and assist. b ft b. 15 yd 8 ft 2 ft 4 ft ft 25 yd 375 yd 2 25 yd 76 Area = ft 2 Math Journal 2, p yd _EMCS_S_MJ2_G5_U09_ indd 305 3/22/11 12:42 PM Lesson 725
5 Student Page Date Time Fraction Division Review 1. Liz, Juan, and Michael equally share 9_ 12 of a pizza. a. To show how the 9 pieces can be distributed, write the student s initial on each piece that he or she is getting. b. Each student will get 3 pieces. c. Write a number model to show what fraction of the whole pizza each student gets. 2. A football team orders a 6footlong submarine sandwich. Each player will eat 1_ 3 of a foot of sandwich. Draw tick marks on the ruler to help you find how many players the sandwich will serve. Math Journal 2, p. 306 J M L L J M L M J 9_ 12 3 = 3_ 12, or 1_ 4 0 FEET The sandwich will serve players. So, 6 1_ 3 =. 3. When Mr. Showers won a large amount of money, he donated half to a local college. The funds were divided equally among 5 departments. a. Write an open number model to show how much of Mr. Shower s prize went to each department. b. Each department received of the prize. 4. The stone is a unit of weight used in the United Kingdom and Ireland. One pound equals 1_ 14 stone. Anna s pen pal in Ireland weighs 6 stones. How much is this in pounds? 1_ 2 5 = p 1_ pounds 5. Write a number story that can be solved by dividing 3 by 1_ 4. Solve your problem. Ongoing Assessment: Informing Instruction Watch for students who hesitate with Problem 2. Ask: Which unit makes more sense to work with, feet or yards? yards Refer students to Problem 1 and ask how many yards of carpet are needed. 20 square yards Illustrate Problem 2 using a roll of paper towels, waxed paper, or aluminum foil to demonstrate how a piece of carpet is cut from a roll. As the carpet unrolls, the area increases. For example, in Problem 2, each foot unrolled adds an additional 6 square feet or 2 square yards. As you unroll the model, have students shade the grid in Problem 1 to show how much floor is covered. Prompt students to tell you where to cut the model. When most students have completed the journal page, ask volunteers to share their solution strategies. Ask questions like the following: In Problem 2, how will you cut the carpet to make it fit the bedroom? If students determine that they will need a piece of carpet 30 feet or 10 yards long, they will need to cut it in half to fit the bedroom. Each of the two pieces will be 2 yards by 5 yards _EMCS_S_MJ2_G5_U09_ indd 306 4/6/11 11:01 AM 2 yd 10 yd 2 yd 2 yd 4 yd 5 yd Date Math Boxes Student Page Time In Problem 4, how did you find the missing lengths? Encourage students to use multiplication/division relationships and open number sentences. For example, = h, or 25 b = A rope 3_ 4 meter long is cut into 6 equal pieces meters a. Draw lines on the rope to show how long each piece will be. b. Write a number model to describe the problem. 3_ 4 6 = 1_ 8 3. Compare. Use <, >, or =. a. 8 * 10 5 > 80,000 b million = 12,400,000 c. 7,000,000 > 7 * 10 5 d. 8 2 < 28 e. 5.4 * 10 2 < 5,400 80A 80B 2. What is volume of the prism? Choose the best answer. 4. Solve. 240 units 3 90 units 2 30 units 3 90 units 3 a. 429 b. 134 * 15 * 82 6,435 10,988 c. 706 * , Ongoing Assessment: Recognizing Student Achievement Exit Slip Use an Exit Slip (Math Masters, page 414) to assess students ability to calculate area. Have students write a response to the following: Explain how you found the area for the figure in Problem 3b on journal page 305. Students are making adequate progress if they multiply the side lengths to find the areas of the rectangles in order to find the total area. [Measurement and Reference Frames Goal 2] Write true or false. a. 5,278 is divisible by 3. false b. 79,002 is divisible by 6. true c. 86,076 is divisible by 9. true d. 908,321 is divisible by 2. false 6. Complete the What s My Rule? table, and state the rule. Rule: in out Math Journal 2, p _EMCS_S_MJ2_G5_U09_ indd 307 4/6/11 11:01 AM 726 Unit 9 Coordinates, Area, Volume, and Capacity
6 2 Ongoing Learning & Practice Fraction Division Review (Math Journal 2, p. 306; Student Reference Book, pp B) Study Link Master Name Date Time STUDY LINK More Area Problems 1. Rashid can paint 2 square feet of fence in 10 minutes. Fill in the missing parts to tell how long it will take him to paint a fence that is 6 feet high by 25 feet long. Rashid will be able to paint 150 sq ft of fence in. 12 hr 30 min (area) (hours/minutes) 2. Regina wants to cover one wall of her room with wallpaper. The wall is 9 feet high and 15 feet wide. There is a doorway in the wall that is 3 feet wide and 7 feet tall. How many square feet of wallpaper will she need to buy? 114 square feet Have students review information about fraction division problems on pages 80 80B of the Student Reference Book. Discuss the use of visual models and number stories. Then have students complete the journal page. Calculate the areas for the figures below yd yd 4 yd 4 yd 4 ft 2 ft 1 ft 9 ft 10 yd Math Boxes (Math Journal 2, p. 307) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 92. The skill in Problem 6 previews Unit 10 content. Writing/Reasoning Have students write a response to the following: Explain how you could determine the volume of the rectangular prism in Problem 2 by counting the unit cubes. Then write a number model for the formula you could use to find the volume. Sample answers: I could count the number of unit cubes in the bottom layer first. The width is 6 and the length of the base is 5. So there are 30 unit cubes in the bottom layer. The height is 3, so there are three layers of 30. The volume is 90 units 3. The formula is B h = V. The number model is (5 6) 3 = 90 units Area = yd 2 Area = ft 2 Fill in the missing lengths for the figures below cm cm _497_EMCS_B_MM_G5_U09_ indd 265 3,000 cm 2 50 cm 6 m 60 cm Math Masters, p ft 33 5 ft 33 m 198 m 2 33 m 6 m 3/23/11 1:09 PM Study Link (Math Masters, p. 265) Home Connection Students solve area problems. INDEPENDENT Teaching Master Name Date Time Comparing Perimeter and Area 3 Differentiation Options READINESS Comparing Perimeter and Area (Math Masters, p. 266) 5 15 Min Roll 2 sixsided dice. The numbers on top are the lengths of 2 sides of a rectangle. Draw the rectangle in the grid below. Record the perimeter and the area of the rectangle in the table. Use centimeter cubes to find other rectangles that have the same area, but different perimeters. Draw the rectangles and record their perimeters and areas in the table. Repeat until you have filled the table. You might need to roll the dice several times. Rectangle Perimeter Area A B C D E F To support students understanding of perimeter and area, have partners roll 2 sixsided dice to determine the dimensions of a rectangle. They draw a rectangle with those dimensions and find the perimeter and area of the rectangle. Partners then find other rectangles with the same area, but different perimeters. They repeat this process until the table on page 266 is completed. Discuss what conclusions can be drawn from the table. Sample answers: Different perimeters can have the same area; the dimensions are factor pairs for the area. Math Masters, p _497_EMCS_B_MM_G5_U09_ indd 266 3/23/11 1:09 PM Lesson 727
7 Teaching Master py g g p Name Date Time Perimeter and Area of Irregular Figures Cut 6 rectangles that are 6 columns by 7 rows from the centimeter grid paper. Record the area and the perimeter of one of these rectangles in Problem 1. Divide each rectangle by using 3 different colored pencils to shade three connected parts with the same number of boxes. The parts must follow the grid, and the squares must be connected by sides. Divide each rectangle in a different way. 1. For a rectangle that is 6 cm by 7 cm: Area = 42 cm 2 Perimeter = _497_EMCS_B_MM_G5_U09_ indd Record the perimeters for the divisions of the 6 rectangles in the table. Rectangle What is the area for each of the parts? Perimeters Part 1 Part 2 Part 3 4. What is the range of the perimeters for each of the parts? 5. a. Describe one relationship between perimeter and area. 26 cm 14 square centimeters Sample answer: Shapes with the same area can have different perimeters. b. Is the relationship the same for rectangles and irregular figures? Explain. Math Masters, p /23/11 1:09 PM ENRICHMENT Comparing Perimeter and Area for Irregular Figures (Math Masters, pp. 267 and 436) Min To apply students understanding of perimeter and area to irregular figures, have partners divide rectangles and compare the relationship between perimeter and area. When students have finished, have them share the parts they cut from their rectangles and discuss questions such as the following: Can you use the area of a figure to predict the perimeter of that figure? No Can you use the perimeter of a figure to predict the area of that figure? No Guide students to conclude that perimeter and area are independent measures. EXTRA PRACTICE 5Minute Math SMALLGROUP 5 15 Min To offer students more experience with calculating the area of rectangles, see 5Minute Math, page 212. Name Date Time Area: Tiling and Using a Formula For each rectangle below, cut out a rectangle from the centimeter grid paper (Math Masters, page 436) that has the same dimensions. Follow the directions for each problem. 1. The length of the base of the rectangle is 6 cm and the height is 2 1_ 2 cm. a. Tape the centimeter grid over the rectangle, and then use the counting method to find the area of the rectangle. 15 cm 2 b. Use the formula to write an open number model that can be used to find the area. c. Area = 15 cm2 Teaching Master 6 2 1_ 2 = A EXTRA PRACTICE Area: Tiling and Using a Formula (Math Masters, pp. 293A and 436) Min To find the area, students use grid paper to tile rectangles that have fractional side lengths. They also use the formula for the area of a rectangle to show that both methods produce the same result. ELL SUPPORT SMALLGROUP Building a Math Word Bank (Differentiation Handbook, p. 142) 5 15 Min 2. The length of the base of the rectangle below is 12 1_ 2 cm and the height is 2 1_ 2 cm. a. Tape the centimeter grid over the rectangle, and then use the counting method to find the area of the rectangle. 31 1_ 4 cm 2 b. Use the formula to find the area. 31 1_ 4 cm 2 3. a. Use the formula to find the area of the rectangle below. 15 3_ 4 cm 2 b. Tape the centimeter grid over the rectangle, and then use the counting method to find the area of the rectangle. 15 3_ 4 cm 2 1 1_ 2 cm py g g p To provide language support for area, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the terms length, height, base, and width; draw pictures relating to each term; and write other related words. See the Differentiation Handbook for more information. 10 1_ 2 cm c. Explain why the formula and the counting method produce the same area. Sample answer: Each row has the same number of squares. So, multiplying the base by the height gives you the number of squares there are in all. Counting the squares gives you the same area. Math Masters, p. 293A 293A_EMCS_B_MM_G5_U09_ indd 293A 3/22/11 9:31 AM 728 Unit 9 Coordinates, Area, Volume, and Capacity
8 Name Date Time Area: Tiling and Using a Formula For each rectangle below, cut out a rectangle from the centimeter grid paper (Math Masters, page 436) that has the same dimensions. Follow the directions for each problem. 1. The length of the base of the rectangle is 6 cm and the height is 2 1 _ 2 cm. a. Tape the centimeter grid over the rectangle, and then use the counting method to find the area of the rectangle. cm 2 b. Use the formula to write an open number model that can be used to find the area. c. Area = cm 2 2. The length of the base of the rectangle below is 12 1 _ 2 cm and the height is 2 1 _ 2 cm. a. Tape the centimeter grid over the rectangle, and then use the counting method to find the area of the rectangle. cm 2 b. Use the formula to find the area. cm 2 3. a. Use the formula to find the area of the rectangle below. cm 2 b. Tape the centimeter grid over the rectangle, and then use the counting method to find the area of the rectangle. cm _ 2 cm 1 1 _ 2 cm c. Explain why the formula and the counting method produce the same area. Copyright Wright Group/McGrawHill 293A
Assessment Management
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