Family Letters. Assessment Management. Playing Fraction Of


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1 Formula for the Area of a Parallelogram Objectives To review the properties of parallelograms; and to guide the development and use of a formula for the area of a parallelogram. 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 Find the area of a rectangle. [Measurement and Reference Frames Goal 2] Develop a formula for calculating the area of a parallelogram. [Measurement and Reference Frames Goal 2] Calculate perimeter. [Measurement and Reference Frames Goal 2] Identify perpendicular line segments and right angles. [Geometry Goal 1] Describe properties of parallelograms. [Geometry Goal 2] Key Activities Students construct models of parallelograms and use them to review properties of parallelograms. Students cut apart and rearrange parallelogram shapes; they develop and use a formula for the area of a parallelogram. Ongoing Assessment: Informing Instruction See page Playing Fraction Of Student Reference Book, pp. 244 and 245 Math Masters, pp counters (optional) Students practice finding fractions of collections. Playing Angle AddUp Math Masters pp per partnership: 4 of each of number cards 1 8 and 1 of each of number cards 0 and 9 (from the Everything Math Deck, if available) fullcircle protractor (transparency of Math Masters, p. 439) dryerase markers straightedge Students draw angles and then use addition and subtraction to find the measures of unknown angles. Math Boxes 8 6 Math Journal 2, p. 239 Students practice and maintain skills through Math Box problems. Constructing Figures with a Compass and Straightedge Student Reference Book, pp. 114, 117, and 118 compass straightedge Students construct figures with a compass and straightedge. Solving Area and Perimeter Problems Math Masters, pp. 263, 264, and 437 scissors tape Students explore ways of combining various twodimensional shapes to form new shapes. Key Vocabulary base perpendicular Materials Math Journal 2, pp Study Link 8 5 Math Masters, p. 260 centimeter ruler straws and twistties scissors tape index card or other squarecor ner object slate Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 4. [Operations and Computation Goal 5] Study Link 8 6 Math Masters, pp. 261 and 262 Students practice and maintain skills through Study Link activities. Advance Preparation For Part 1, each student needs 2 short straws, 2 long straws, and 4 twistties. Pairs of straws should be the same length. Place them near the Math Message. Teacher s Reference Manual, Grades 4 6 pp , 221, 222 Lesson
2 Getting Started Mental Math and Reflexes Pose multiplication facts and problems. Suggestions: Mathematical Practices SMP1, SMP2, SMP3, SMP5, SMP6, SMP7, SMP8 Content Standards 4.NF.4c, 4.MD.3, 4.MD.7, 4.G.1 Math Message Take 2 short straws, 2 long straws, and 4 twistties. Use them to construct a parallelogram. 3 7 = = = = = = = 4, = 2, = = = = 440 Study Link 8 5 FollowUp Have partners compare answers and discuss how they found the missing side measure in Problems 5 and 6. 1 Teaching the Lesson Math Message FollowUp WHOLECLASS Ask students to tell what they know about parallelograms, using their straw constructions as models, while you list the properties they name on the board. The list should include: A parallelogram is a foursided polygon called a quadrangle or quadrilateral. Opposite sides of a parallelogram are parallel. Opposite sides of a parallelogram are the same length. Rectangles and squares are special kinds of parallelograms. Have students form a rectangle with their straw constructions, and then ask them to pull gently on the opposite corners. They should get a parallelogram that is not a rectangle. Ask the following questions: NOTE Height is the distance perpendicular to the base of a figure. Any side of a parallelogram can be the base. The choice of the base determines the. base base Does the perimeter remain the same? yes Does the area remain the same? No, because although the length of the base stays the same, the decreases, so the area decreases. Draw a parallelogram on the board. Choose one of the sides, for example, the side on which the parallelogram sits, and label it the base. Explain that base is also used to mean the length of the base. The shortest distance between the base and the side opposite the base is called the of the parallelogram. Draw and label a dashed line to show the. Include a rightangle symbol. Point out that the dashed line can be drawn anywhere between the two sides as long as it is perpendicular to (forms a right angle with) the base. Remind students that rectangles are parallelograms whose sides form right angles. If you think of one side of a rectangle as its base, then the length of an adjacent side is its. 688 Unit 8 Perimeter and Area
3 Tell students that in this lesson they will use the formula for the area of a rectangle to develop a formula for the area of a parallelogram. 8 6 Areas of Parallelograms 1. Cut out Parallelogram A on Math Masters, page 260. DO NOT CUT OUT THE ONE BELOW. Cut it into 2 pieces so that it can be made into a rectangle Links to the Future Parallelogram A The use of a formula to calculate the area of a parallelogram is a Grade 5 Goal. base = 6 length of base = 6 Developing a Formula for the Area of a Parallelogram (Math Journal 2, pp. 236 and 237; Math Masters, p. 260) PROBLEM SOLVING WHOLECLASS = 2 width () = 2 Area of parallelogram = 12 2 Area of rectangle = Do the same with Parallelogram B on Math Masters, page 260. Parallelogram B Point out that Parallelogram A on journal page 236 is the same as Parallelogram A on Math Masters, page 260. Guide students through the following activity: 1. Cut out Parallelogram A from the master. 2. Cut the parallelogram into two pieces along one of the vertical grid lines. 3. Tape the pieces together to form a rectangle. cut base = 4 length of base = 4 = 4 width () = 4 Area of parallelogram = 16 2 Area of rectangle = 16 2 Math Journal 2, p Tape this rectangle in the space next to the parallelogram in the journal. Discuss the relationship between the parallelogram and the rectangle formed from the parallelogram. Why must the parallelogram and the rectangle both have the same area? The rectangle was constructed from the parallelogram. Nothing was lost or added. 5. Record the dimensions and area of the parallelogram and the rectangle. Length of base of parallelogram and length of base of rectangle = 6 ; of parallelogram and width () of rectangle = 2 ; area of each figure = 12 2 Have students repeat these steps with Parallelograms B, C, and D, working on their own or with a partner. Bring students together to develop a formula for the area of a parallelogram. These are three possible lines of reasoning: The area of each parallelogram is the same as the area of the rectangle that was made from it. The area of the rectangle is equal to the length of its base times its width (also called the ). 8 6 Areas of Parallelograms continued 3. Do the same with Parallelogram C. Parallelogram C base = 4 length of base = 4 = 3 width () = 3 Area of parallelogram = 12 2 Area of rectangle = Do the same with Parallelogram D. Parallelogram D base = 3 length of base = 3 = 4 width () = 4 Area of parallelogram = 12 2 Area of rectangle = Write a formula for the area of a parallelogram. A = b h base Math Journal 2, p. 237 Lesson
4 about Perimeter = about 16.4 Area = 12 2 The length of the base of the parallelogram is equal to the length of the base of the rectangle. The of that parallelogram is equal to the width () of that rectangle. Therefore, the area of the parallelogram is equal to the length of its base times its. Using variables: A = b h where b is the length of the base and h is the. Have students record the formula at the bottom of journal page Perimeter = 16 Area = 12 2 Ongoing Assessment: Informing Instruction Watch for students who think that the perimeter of each parallelogram and rectangle pair is also the same. Point out that although the and base are the same measure, the of a parallelogram is only used in computing its perimeter when the parallelogram is a rectangle or square. (See margin.) Solving Area Problems (Math Journal 2, p. 238) Algebraic Thinking Work with the whole class on Problem 6, journal page 238. Students can place an index card (or other squarecorner object) on top of the shape, align the bottom edge of the card with the base, and then use the edge of the card to draw a line for the. They will need a centimeter ruler to measure the length of the base and the. index card 8 6 Areas of Parallelograms continued 6. Draw a line segment to show the of Parallelogram DORA. D Use your ruler to measure the base and. Then find the area. base 5 4 Area Draw the following shapes on the grid below: a. A rectangle whose area is 12 square centimeters b. A parallelogram, not a rectangle, whose area is 12 square centimeters c. A different parallelogram whose area is also 12 square centimeters a. c. Sample answers: O b. R A Drawing the of a parallelogram Have partnerships complete Problems 7 and 8. Problem 7 illustrates the fact that shapes that do not look the same can have the same area. Problem 8b lends itself to a variety of solution strategies. Some students may have partitioned the trapezoid into a rectangle flanked by two triangles. The rectangle covers 12 grid squares. If one triangle were cut apart and placed next to the other triangle to form a rectangle, the pair would cover 6 squares. The rectangle and two triangles cover = What is the area of: a. Parallelogram 24 ABCD? b. Trapezoid 18 EBCD? c. Triangle 6 ABE? E D A E D B C B C Math Journal 2, p. 238 Problem 8b 690 Unit 8 Perimeter and Area
5 Problem 8c can be solved without using a formula for the area of a triangle. The parallelogram area minus the trapezoid area is the triangle area = Ongoing Learning & Practice Playing Fraction Of (Student Reference Book, pp. 244 and 245; Math Masters, pp ) Students play Fraction Of to practice finding fractions of collections. See Lesson 73 for additional information. Playing Angle AddUp (Math Masters, pp. 439 and ) 8 6 Math Boxes 1. Dimensions for actual rectangles are given. Make scale drawings of each rectangle described below. Scale: 1 represents 20 meters. a. Length of rectangle: 80 meters Width of rectangle: 30 meters b. Length of rectangle: 90 meters Width of rectangle: 50 meters 2. What is the area of the parallelogram? 7 3 = 21 Area = 21 in 2 4. Add or subtract. _ 10 a. 3_ _ 16 = b. 2_ 3 + 1_ 6 = 3" 6_ 3_ 7" 16, 5_ or 5_ 8 6 c. 10, or 3_ 5 = 9_ 103_ 10 d. 8 = 3_ 43_ 8 a. b. 3. A jar contains 8 blue blocks, 4 red blocks, 9 orange blocks, and 4 green blocks. You put your hand in the jar and without looking pull out a block. About what fraction of the time would you expect to get a blue block? 8_ Multiply. Use a paperandpencil algorithm. 6,142 = To further explore the idea that angle measures are additive, have students play Angle AddUp. See Lesson 79 for more information. Math Journal 2, p. 239 Math Boxes 8 6 (Math Journal 2, p. 239) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 88. The skill in Problem 5 previews Unit 9 content. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 4 Use Math Boxes, Problem 4 to assess students ability to solve fraction addition and subtraction problems. Students are making adequate progress if they are able to solve Problems 4a and 4c, which involve fractions with like denominators. Some students may be able to solve Problems 4b and 4d by using equivalent fractions with like denominators, using manipulatives, or drawing pictures. [Operations and Computation Goal 5] Name STUDY LINK 8 6 Areas of Parallelograms Find the area of each parallelogram. Study Link Master ' 9' Study Link 8 6 (Math Masters, pp. 261 and 262) INDEPENDENT 4 9 = Area = square feet Area = square centimeters 3. 4 ft ft 8 3 = Home Connection Students calculate the areas of parallelograms on Math Masters, page 261. NOTE Math Masters, page 262 should be completed before Lesson 91, in which students share and discuss examples of percents they have collected. 6 4 = ,680 Area = square feet Area = square centimeters Try This The area of each parallelogram is given. Find the length of the base in = 4, m? Area = 26 square inches 13 Area = 5,015 square meters 85 base = inches base = meters? Math Masters, p. 261 Lesson
6 Teaching Master 8 6 Perimeter and Area Differentiation Options Constructing Figures with a Compass and Straightedge (Student Reference Book, pp. 114, 117, and 118) INDEPENDENT 30+ Min To apply students understanding of the properties of parallelograms, have them construct parallelograms and perpendicular line segments as described on pages 114, 117, and 118 of the Student Reference Book. Math Masters, p Solving Area and Perimeter Problems (Math Masters, pp. 263, 264, and 437) 30+ Min To apply students understanding of area and perimeter, have them explore different ways of combining various 2dimensional shapes to form new shapes. Possible solutions to Problem 6: Name 8 6 Perimeter and Area continued Cut out and use only the shapes in the top half of Math Masters, page 263 to complete Problems Make a square out of 4 of the shapes. Draw the square on the centimeter dot grid on Math Masters, page 437. Your picture should show how you put the square together. 2. Make a triangle out of 3 of the shapes. One of the shapes should be the shape you did not use to make the square in Problem 1. Draw the triangle on Math Masters, page Find the area of the following: Teaching Master a. the small triangle 2 b. the square 2 c. the parallelogram 2 4. a. What is the perimeter of the large square you made in Problem 1? b. What is the area of that square? What is the area of the large triangle you made in Problem 2? Name Dot Paper Teaching Aid Master Try This 6. Cut out the 5 shapes in the bottom half of Math Masters, page 263 and add them to the other shapes. Use at least 6 pieces each to make the following shapes. Answers vary. a. a square b. a rectangle c. a trapezoid d. any shape you choose Tape your favorite shape onto the back of this sheet. Next to the shape, write its perimeter and area. py g g p Math Masters, p. 264 Math Masters, p Unit 8 Perimeter and Area
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Formula for the rea of a Triangle Objective To guide the development and use of a formula for the area of a triangle. www.everydaymathonline.com epresentations etoolkit lgorithms Practice EM Facts Workshop
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Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game
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