Formula for the Area of a Parallelogram

Transcription

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 ] Develop a formula for calculating the area of a parallelogram. [Measurement and Reference Frames Goal ] Calculate perimeter. [Measurement and Reference Frames Goal ] Identify perpendicular line segments and right angles. [Geometry Goal ] Describe properties of parallelograms. [Geometry Goal ] 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 690. Key Vocabulary base height perpendicular Materials Math Journal, pp. 6 8 Study Link 8 5 Math Masters, p. 60 centimeter ruler straws and twist-ties scissors tape index card or other square-cor ner object slate Playing Fraction Of Student Reference Book, pp. and 5 Fraction Of Cards (Math Masters, pp. 77, 78, and 80) Math Masters, p. 79 Students practice finding fractions of collections. Playing Angle Add-Up Math Masters pp per partnership: of each of number cards 8 and of each of number cards 0 and 9 (from the Everything Math Deck, if available) full-circle protractor (transparency of Math Masters, p. 9) dry-erase markers straightedge Students draw angles and then use addition and subtraction to find the measures of unknown angles. Math Boxes 8 6 Math Journal, p. 9 Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem. [Operations and Computation Goal 5] Study Link 8 6 Math Masters, pp. 6 and 6 Students practice and maintain skills through Study Link activities. ENRICHMENT Constructing Figures with a Compass and Straightedge Student Reference Book, pp., 7, and 8 compass straightedge Students construct figures with a compass and straightedge. ENRICHMENT Solving Area and Perimeter Problems Math Masters, pp. 6, 6, and 7 scissors tape Students explore ways of combining various two-dimensional shapes to form new shapes. Advance Preparation For Part, each student needs short straws, long straws, and twist-ties. Pairs of straws should be the same length. Place them near the Math Message. Teacher s Reference Manual, Grades 6 pp ,, Lesson

2 Getting Started Mental Math and Reflexes Pose multiplication facts and problems. Suggestions: Math Message Take short straws, long straws, and twist-ties. Use them to construct a parallelogram. 7 = 9 = = = = = =, =, = 6 6 = = = 0 Study Link 8 5 Follow-Up Have partners compare answers and discuss how they found the missing side measure in Problems 5 and 6. Teaching the Lesson Math Message Follow-Up WHOLE-CLASS 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 four-sided 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 height. base height height 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 height 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 height of the parallelogram. Draw and label a dashed line to show the height. Include a right-angle 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 height. 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. Date 8 6 Student Page Time Areas of Parallelograms. Cut out Parallelogram A on Math Masters, page 60. DO NOT CUT OUT THE ONE BELOW. Cut it into pieces so that it can be made into a rectangle. 5 Links to the Future Parallelogram A Tape your rectangle in the space below. Sample answer: 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, pp. 6 and 7; Math Masters, p. 60) PROBLEM SOLVING WHOLE-CLASS height = width (height) = Area of parallelogram = Area of rectangle =. Do the same with Parallelogram B on Math Masters, page 60. Parallelogram B Tape your rectangle in the space below. Sample answer: Point out that Parallelogram A on journal page 6 is the same as Parallelogram A on Math Masters, page 60. Guide students through the following activity:. Cut out Parallelogram A from the master.. Cut the parallelogram into two pieces along one of the vertical grid lines.. Tape the pieces together to form a rectangle. cut base = length of base = height = width (height) = Area of parallelogram = 6 Area of rectangle = 6 Math Journal, p. 6. 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 ; height of parallelogram and width (height) of rectangle = ; area of each figure = 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 height). Date 8 6 Areas of Parallelograms continued. Do the same with Parallelogram C. Parallelogram C Tape your rectangle in the space below. base = length of base = height = width (height) = Area of parallelogram = Area of rectangle =. Do the same with Parallelogram D. Parallelogram D Student Page Tape your rectangle in the space below. base = length of base = height = width (height) = Area of parallelogram = Area of rectangle = 5. Write a formula for the area of a parallelogram. A = b h Time Sample answer: Sample answer: height base Math Journal, p. 7 Lesson

4 about. 6 Perimeter = about 6. Area = The length of the base of the parallelogram is equal to the length of the base of the rectangle. The height of that parallelogram is equal to the width (height) of that rectangle. Therefore, the area of the parallelogram is equal to the length of its base times its height. Using variables: A = b h where b is the length of the base and h is the height. Have students record the formula at the bottom of journal page 7. 6 Perimeter = 6 Area = 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 height and base are the same measure, the height 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, p. 8) PARTNER Algebraic Thinking Work with the whole class on Problem 6, journal page 8. Students can place an index card (or other square-corner 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 height. They will need a centimeter ruler to measure the length of the base and the height. index card Date Student Page Time height 8 6 Areas of Parallelograms continued 6. Draw a line segment to show the height of Parallelogram DORA. D Use your ruler to measure the base and height. Then find the area. base 5 height Area 0 7. Draw the following shapes on the grid below: a. A rectangle whose area is square centimeters b. A parallelogram, not a rectangle, whose area is square centimeters c. A different parallelogram whose area is also square centimeters a. c. Sample answers: O b. R A Drawing the height 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 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 + 6 = What is the area of: a. Parallelogram ABCD? b. Trapezoid 8 EBCD? c. Triangle 6 ABE? E D A E D B C B C Math Journal, p. 8 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. - 8 = 6 Ongoing Learning & Practice Playing Fraction Of PARTNER (Student Reference Book, pp. and 5; Math Masters, pp ) Students play Fraction Of to practice finding fractions of collections. See Lesson 7- for additional information. Playing Angle Add-Up (Math Masters, pp. 9 and ) PARTNER Date 8 6 Math Boxes. Dimensions for actual rectangles are given. Make scale drawings of each rectangle described below. Scale: represents 0 meters. a. Length of rectangle: 80 meters Width of rectangle: 0 meters b. Length of rectangle: 90 meters Width of rectangle: 50 meters. What is the area of the parallelogram? 7 = Area = in. Add or subtract. _ 0 a. _ 6 + 7_ 6 = b. _ + _ 6 = " 6 7" 6, 5_ or 5_ 8 6 c. 0, or _ 5 = 9_ 0 - _ 0 d. 8 = _ - _ 8 Student Page Time a. b.. A jar contains 8 blue blocks, red blocks, 9 orange blocks, and 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 paper-and-pencil algorithm. 6, = To further explore the idea that angle measures are additive, have students play Angle Add-Up. See Lesson 7-9 for more information. Math Journal, p. 9 Math Boxes 8 6 (Math Journal, p. 9) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-8. The skill in Problem 5 previews Unit 9 content. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem Use Math Boxes, Problem to assess students ability to solve fraction addition and subtraction problems. Students are making adequate progress if they are able to solve Problems a and c, which involve fractions with like denominators. Some students may be able to solve Problems b and d by using equivalent fractions with like denominators, using manipulatives, or drawing pictures. [Operations and Computation Goal 5] Name Date Time STUDY LINK 8 6 Areas of Parallelograms Find the area of each parallelogram. Study Link Master.. ' 9' 8 5 Study Link 8 6 (Math Masters, pp. 6 and 6) INDEPENDENT 9 = 6 6 Area = square feet Area = square centimeters. ft ft 8 = 7 Home Connection Students calculate the areas of parallelograms on Math Masters, page 6. NOTE Math Masters, page 6 should be completed before Lesson 9-, in which students share and discuss examples of percents they have collected. 6 =,680 Area = square feet Area = square centimeters Try This The area of each parallelogram is given. Find the length of the base in =, m? Area = 6 square inches Area = 5,05 square meters 85 base = inches base = meters? Math Masters, p. 6 Lesson

6 Teaching Master 8 6 Perimeter and Area 50 Differentiation Options ENRICHMENT Constructing Figures with a Compass and Straightedge (Student Reference Book, pp., 7, and 8) INDEPENDENT 0+ Min To apply students understanding of the properties of parallelograms, have them construct parallelograms and perpendicular line segments as described on pages, 7, and 8 of the Student Reference Book. Math Masters, p. 6 6 ENRICHMENT Solving Area and Perimeter Problems (Math Masters, pp. 6, 6, and 7) PARTNER 0+ Min To apply students understanding of area and perimeter, have them explore different ways of combining various -dimensional shapes to form new shapes. Possible solutions to Problem 6: Name Date Time 8 6 Perimeter and Area continued Cut out and use only the shapes in the top half of Math Masters, page 6 to complete Problems 5.. Make a square out of of the shapes. Draw the square on the centimeter dot grid on Math Masters, page 7. Your picture should show how you put the square together.. Make a triangle out of of the shapes. One of the shapes should be the shape you did not use to make the square in Problem. Draw the triangle on Math Masters, page 7.. Find the area of the following: Teaching Master a. the small triangle b. the square c. the parallelogram. a. What is the perimeter of the large square you made in Problem? b. What is the area of that square? 5. What is the area of the large triangle you made in Problem? 6 Name Date Time Dot Paper Teaching Aid Master Try This 6. Cut out the 5 shapes in the bottom half of Math Masters, page 6 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. 6 Math Masters, p Unit 8 Perimeter and Area

7 Name Date Time Angle Add-Up Materials number cards 8 ( of each) number cards 0 and 9 ( of each) dry-erase marker straightedge full-circle protractor (transparency of Math Masters, p. 9) Angle Add-Up Record Sheet (Math Masters, p. 509) Players Skills Drawing angles of a given measure Recognizing angle measures as additive Solving addition and subtraction problems to find the measures of unknown angles Objective To score the most points in rounds. Directions. Shuffle the cards and place the deck number-side down on the table.. In each round, each player draws the number of cards indicated on the Record Sheet. Copyright Wright Group/McGraw-Hill. Each player uses the number cards to fill in the blanks and form angle measures so the unknown angle measure is as large as possible.. Players add or subtract to find the measure of the unknown angle and record it in the circle on the Record Sheet. The measure of the unknown angle is the player s score for the round. 5. Each player uses a full-circle protractor, straightedge, and marker to show that the angle measure of the whole is the sum of the angle measures of the parts. 6. Players play rounds for a game. The player with the largest total number of points at the end of the rounds wins the game. 507

8 Name Date Time Angle Add-Up Example Example: In Round, Suma draws a, 7,, and 5. She creates the angle measures 5 and 7 and records them on her record sheet. Round : Draw cards = Using addition, Suma finds the sum of the measures of angles ABD and DBC. She records the measure of angle ABC on her record sheet and scores points for the round. Round : Draw cards = Suma uses her full-circle protractor to show that m ABD + m DBC = m ABC. A degrees 0 9 B D Copyright Wright Group/McGraw-Hill C 508

9 Name Date Time Angle Add-Up Record Sheet Game Round : Draw cards. + = Round : Draw cards. Round : Draw cards. + = 90 + = 80 Total Points = Copyright Wright Group/McGraw-Hill Game Round : Draw cards. Round : Draw cards. Round : Draw cards. + = + = 90 + = 80 Total Points = 509