LESSON 10.1 Algebra Area of Parallelograms FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards Learning Objective 6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Find the area of parallelograms. Also 6.EE.A., 6.EE.B.7 MATHEMATICAL PRACTICES (See ematical Practices in GO! in the Planning Guide for full text.) MP Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP8 Look for and express regularity in repeated reasoning. F C R Coherence: Language Objective Students draw a four-panel picture poster showing how to find the area of parallelograms. Materials Board, Grid Paper (see Teacher Resources), scissors, straightedge, number cubes Standards Across the Grades Before Grade 6 After 5.NF.B.b 6.G.A.1 7.G.A.1 7.G.B.6 F C R Rigor: Level 1: Understand Concepts...Share and Show ( Checked Items) Level : Procedural Skills and Fluency...On Your Own, Practice and Homework Level 3: Applications...Think Smarter and Go Deeper F C R For more about how GO! fosters Coherence within the Content Standards and ematical Progressions for this chapter, see page 531J. About the Professional Development If Students Ask Students may not understand how a parallelogram and rectangle can have the same area when the parallelogram extends farther in one direction than the rectangle. Students may not understand that the height is perpendicular to the base. They may use the side lengths in the formula for area rather than the base and height. In this lesson, students cut out a parallelogram and examine its relationship to a rectangle with the same base and height. This will reinforce that the height of a parallelogram is different from a side length. Professional Development Videos 533A Chapter 10 Interactive Student Edition Personal Trainer on the Spot Video itools: Geometry HMH Mega
Daily Routines Common Core Problem of the Day 10.1 Mrs. Hanson bought 1.6 pounds of grapes. The grapes cost $.50 per pound. How much did she pay for the grapes? $.00 Vocabulary area, parallelogram Interactive Student Edition Multimedia Glossary e 1 3 Fluency Builder Materials eteacher Resources pages TR13 TR135 Operations Find the quotient. Write the remainder with an r. 1. 1,358 5 5 r8. 3,897 3 98 r83 3. 90,65 903 100 r35 Common Core Fluency Standard 6.NS.B. 1 ENGAGE with the Interactive Student Edition Essential Question How can you find the area of a parallelogram? Making Connections Draw a rectangle and review how to find its perimeter. What formula do you use? P = l + w, where l and w are the length and width of the rectangle Learning Activity Discuss the relationship between an equation and a formula. Focus on the area formula for a rectangle. What two measures of a rectangle are needed to calculate the area? length and width Is it possible to compute the length if you know the area and the width? yes What operation would you use? division Literacy and ematics Visit the lesson opener with the students. Have students brainstorm common applications of how area of a figure is used in their everyday lives. Have students write a real-world problem that involves finding the area of a rectangle.. 56,087 30 185 r17 5. 1,603 5 163 r01 1 3 Pages 18 19 in Strategies and Practice for Skills and Facts Fluency provide additional fluency support for this lesson. Literature Connection From the Grab-and-Go TM Differentiated Centers Kit Students read about using area to find the amount of paint and the tile needed for a room. How can you find the area of a parallelogram? Room Makeover: Serving the Community 533B
LESSON 10.1 EXPLORE Discuss the definition of area. Have students count the number of grid squares, or unit squares, inside the -by-3 rectangle. Students should connect the number of grid squares with the product of the dimensions. Students may be familiar with the formula A = bh rather than A = lw. Explain that these formulas are equivalent for rectangles. Unlock the Problem MATHEMATICAL PRACTICES Explain that the area of the quilt piece is measured in square centimeters because it is the product of two dimensions that are measured in centimeters. MP Reason abstractly and quantitatively. What characteristics affect the area of a parallelogram? the length of a base and the height Talk Use Talk to expand on students understanding of parallelograms. DEEPER Remind students that the height of a parallelogram is the length of a segment that forms a 90º angle with the base and extends to the opposite side. MP8 Look for and express regularity in repeated reasoning. Why is the area of the rectangle the same as the area of the parallelogram? Possible answer: Since the number of grid squares covered by the two shapes is the same, the area of the shapes must be the same. ELL Strategy: Restate Restate the steps of the process for finding the area of a parallelogram using familiar language. Be sure to use simplified language to explain what are parallelograms, rectangles, and quadrilaterals. Use real objects or drawings to demonstrate area and height. 533 Chapter 10 6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Image Credits: (r) Roman Soumar/Corbis Area of Parallelograms Essential Question How can you find the area of parallelograms? connect The area of a figure is the measure of the number of unit squares needed to cover it without any gaps or overlaps. The area of a rectangle is the product of the length and the width. The rectangle shown has an area of 1 square units. For a rectangle with length l and width w, A = l w, or A = lw. Recall that a rectangle is a special type of parallelogram. A parallelogram is a quadrilateral with two pairs of parallel sides. Algebra Area of Parallelograms The formula for the area of a parallelogram is the product of the base and height. height 5 h base 5 b Find the area. A 5 bh A 5 s Step 1 Identify the figure. 1 mi 13 m 1 mi yd 5 1 yd side 5 s The figure is a parallelogram, so use the formula A 5 bh. Step Substitute 5 1_ for b and for h. A 5 5 1_ 3 Step 3 Multiply. A 5 1_ 5 3 5 11 3 _ 1 5 11 So, the area of the parallelogram is 11 yd. Find the area. 1. Figure:.7 m Formula: A 5. 3. 1 The formula for the area of a square is the square of one of its sides. parallelogram bh 13.7 61.1 A 5 3 5 m 3 yd 7 1 5 yd 1 3_ 5 mi yd Chapter Resources Unlock the Problem Victoria is making a quilt. She is using material in the shape of parallelograms to form the pattern. The base of each parallelogram measures 9 cm and the height measures cm. What is the area of each parallelogram? See below. Activity Use the area of a rectangle to find the area of the parallelogram. Materials grid paper scissors Draw the parallelogram on grid paper and cut it out. Cut along the dashed line to remove a triangle. Move the triangle to the right side of the figure to form a rectangle. What is the area of the rectangle? 36 sq cm What is the area of the parallelogram? 36 sq cm base of parallelogram = length of rectangle height of parallelogram = width of rectangle area of parallelogram = area of rectangle For a parallelogram with base b and height h, A = bh or b h Area of parallelogram = b h = 9 cm cm = _ 36 sq cm So, the area of each parallelogram in the quilt is _ 36 sq cm. Problem Type: Area Unknown Product Reteach 10.1 3 Differentiated Enrich 10.1 1 Instruction Reteach 10-5 Reteach All Aboard Parallelogram and Square Express Find the areas of the train cars below. Write the shape of the car and the formula you used to find the area. (Do not include the wheels as part of the area.) 73 yd Figure: Car D 13 1 yd Train Car A Formula for the Area: Area: Figure: Train Car C Formula for the Area: Area: Car C 36. ft 3.3 ft parallelogram A 5 bh,318 ft parallelogram A 5 bh 83.6 ft 1. Stretch Your Thinking Add another car to the train. Draw and label the car. Then find the area. Check students drawings and answers. Chapter Resources width (w) cm ALGEBRA Geometry 6.G.A.1 Also 6.EE.A.c, 6.EE.B.7 MATHEMATICAL PRACTICES MP, MP5, MP6 Talk 3 Car B 0.6 ft Figure: 0.6 ft Car A 61 ft 38 ft Train Car B Formula for the Area: Area: Figure: Train Car D Formula for the Area: Area: height (h) cm base (b) 9 cm length (l) 9 cm The height of a parallelogram forms a 90º angle with the base. The parallelogram can be cut along the height and put back together to form the rectangle, so the areas are the same. MATHEMATICAL PRACTICES 6 Explain how you know that the area of the parallelogram is the same as the area of the rectangle. Chapter 10 533 Enrich square A 5 s 1,68.36 ft parallelogram A 5 bh 10 11 16 yd. Stretch Your Thinking If the unshaded triangles in Train Car C are removed, the figure is a square. What is the area of the shaded section of Train Car C? 5.89 ft 10-6 Enrich
Example 1 of the parallelogram. Write the formula. Replace b and h with their values. Multiply. A = bh A = 6.3.1 A = 13.3 So, the area of the parallelogram is _ 13.3 square meters. A square is a special rectangle in which the length and width are equal. For a square with side length s, A = l w = s s = s, or A = s. Example Find the area of a square with sides measuring 9.5 cm. Write the formula. Substitute 9.5 for s. Simplify. Use the formula A = bh to find the area A = s So, the area of the square is 90.5 cm. A = ( ) 9.5 = 90.5 Example 3 A parallelogram has an area of 98 square feet and a base of 1 feet. What is the height of the parallelogram? Write the formula. Replace A and b with their values. Use the Division Property of Equality. 98 Solve for h. A = bh 98 1 = h = 1h 1 1 7 = h So, the height of the parallelogram is 7 feet..1 m 6.3 m? 9.5 cm 9.5 cm Area 5 98 ft 1 ft Problem Type: Area Unknown Product Example 1 What are the dimensions of a rectangle with the same area as this parallelogram? Possible answer: 6.3 meters long by.1 meters wide Example Why is the area measured in square centimeters? Possible answer: because each side is measured in centimeters and area is measured in square units. Example 3 Why did you use division to find the width? Possible answer: Since multiplication and division are inverse operations, and area is the product of length and width, you can divide the area by the length to find the width. DEEPER MP3 Construct viable arguments and critique the reasoning of others. Can you use the formula for the area of a parallelogram to find the area of a rectangle? Explain. Yes; a rectangle is a special type of parallelogram. The length of the rectangle is the base, and the width of the rectangle is the height, because the sides of a rectangle are perpendicular. MATHEMATICAL PRACTICE 6 Compare Explain the difference between the height of a rectangle and the height of a parallelogram. Possible answer: The height of a rectangle is the length of one side of the rectangle. The height of a parallelogram is the length of a perpendicular line segment drawn from the base to an opposite side or to a line segment extended from the opposite side. If the parallelogram is a rectangle, the height will be the length of a side of the parallelogram. 53 Advanced Learners Logical / ematical Partners Materials Grid Paper (see eteacher Resources), number cubes labeled 1 to 6 Give each pair of students grid paper, two number cubes, and the following instructions. Take turns rolling the number cubes and finding the product of the numbers rolled. The product represents the area of a parallelogram. Find the dimensions of a figure with this area. On the grid paper, draw the figure with the area and dimensions found above. Figures may share sides but may not overlap. Make an identifying mark, such as initials, inside the figure and record the figure s area. When there is no more room to draw parallelograms, the game is over. The player with the greatest total area wins. COMMON ERRORS Error When finding the area of a parallelogram, students may incorrectly identify the height from a drawing. Example A = bh 1 in. 1 A = 15 in. 1 A = 195 in. 15 in. Springboard to Learning Remind students that the height, h, of a parallelogram forms a right angle with its base, b. 53
3 If Then EXPLAIN Share and Show The first problem connects to the learning model. Have students use the Board to explain their thinking. Talk Quick Check MATH MATH BOARD Use Talk to focus on students understanding of area of parallelograms. MP8 Look for and express regularity in repeated reasoning. If you know the area and height of a parallelogram, how can you calculate the base? Divide the area by the height. Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the Board. 3 1 Rt I a student misses the checked exercises Differentiate Instruction with Reteach 10.1 Personal Trainer 6.G.A.1, 6.EE.A.c RtI Tier 1 Activity (online) Share and Show 1. A = bh 1. m. 8.3 m 6 ft A = 8.3 1. 15 ft 9.96 90 3. A = m.5 mm 9.1 m.5 mm 5. Area = 11 yd yd On Your Own? yd 1 5 yd Find the area of the parallelogram. 7. 6. m MATH BOARD Find the area of the parallelogram or square. 6.5 mm Find the unknown measurement for the parallelogram.. ft 3 3 ft 8 ft ft ft yd 1 ft 58. 168 m Find the unknown measurement for the figure. 9. square A = 5 ft s = 15 ft 10. parallelogram A = 3 m b = m h = 8 m 6. Area = 3 yd Possible answer: The area of a parallelogram with no right angles is the same as the area of a rectangle that has the same base (length) and height (width). 13. The height of a parallelogram is four times the base. The base measures 3 1_ ft. Find the area of the parallelogram. 9 f t 8. 8 yd 11. parallelogram Talk A = 51 1_ in. b = 8 1_ 5 in. 6 h = 1_ in.? yd MATHEMATICAL PRACTICES Reasoning Explain how the areas of some parallelograms and rectangles are related. ft 1. parallelogram A = 11 mm b = 11 mm h = 11 mm Chapter 10 Lesson 1 535 1_ On Your Own You may wish to provide grid paper to students to help them solve Exercises 10 13. Ask for volunteers to share their solutions for Exercise 13. As a class, compare the various methods the volunteers used. MP6 Attend to precision. Describe a method you can use to determine if your answer is reasonable. Possible answer: Estimation. I can round the base and the height and multiply. Additional Example Kelly made an enclosed square garden with a side length of 5.5 feet. Write and evaluate an expression for the area of the garden using an exponent. A = 5.5 ; 30.5 square feet PROBLEM TYPE SITUATIONS Addition and Subtraction Put Together/Take Apart Total Unknown Exercise: 15 Multiplication and Division Area Unknown Product Exercises: 13, 1, 15, 16 Area Unknown Factor Exercise: 16 535 Chapter 10
MATHEMATICAL PRACTICES M 1. Jane s backyard is shaped like a parallelogram. The base of the parallelogram is 90 feet, and the height is 5 feet. What is the area of Jane s backyard? ELABORATE,50 ft 15. Jack made a parallelogram by putting together two congruent triangles and a square, like the figures shown at the right. The triangles have the same height as the square. What is the area of Jack s parallelogram? 10 cm 16. DEEPER The base of a parallelogram is times the parallelogram s height. If the base is 1 inches, what is the area? 7 in. 5 cm 8 cm 5 cm Problem Solving Applications MATHEMATICAL PRACTICES Exercise 15 requires students to use higher order thinking skills. Students must infer that the area of the parallelogram is the sum of the areas of two congruent triangles and a square. 17. MATHEMATICAL PRACTICE 3 Verify the Reasoning of Others Li Ping says that a square with 3-inch sides has a greater area than a parallelogram that is not a square but has sides that have the same length. Does Li Ping s statement make sense? Explain. Yes; possible explanation: A parallelogram with sides that are each 3 inches does not have a on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. height of 3 inches. Its height will be less than 3 inches, and its area will be less than that of the on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com. square, which is 9 square inches. 18. Find the area of the parallelogram. 536 The area is _ 60 in. 5 in. 1 in. DIFFERENTIATED INSTRUCTION 6 in. INDEPENDENT ACTIVITIES Exercise 18 assesses a student s ability to use the formula to find the area of a parallelogram. Students must find the area by multiplying the base and the height. Students who answer 7 square inches have used the side dimension, 6 inches, rather than the 5-inch height. Students who multiply all three numbers may not have known the formula. 5 EVALUATE Formative Assessment Activities Risky Rectangles Differentiated Centers Kit Activities Areas of Parallelograms and Trapezoids Literature Room Makeover: Serving the Community Essential Question Using the Language Objective Reflect Have students draw a four-panel picture poster to answer the essential question. How can you find the area of a parallelogram? Possible answer: To find the area of a parallelogram, multiply its base by its height. Students complete orange Activity Card 3 by drawing and measuring rectangles. Students complete online blue Activity Card 19 by finding the area of parallelograms and trapezoids. Students read about using area to find the amount of paint and tile needed for a room. Journal WRITE Copy the two triangles and the square in Exercise 15. Show how you found the area of each piece. Draw the parallelogram formed when the three figures are put together. Calculate its area using the formula for the area of a parallelogram. 536
Area of Parallelograms Practice and Homework Practice and Homework Use the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write section to determine student s understanding of content for this lesson. Encourage students to use their Journals to record their answers. Find the area of the figure. 1. A = bh 7 ft A = 18 3 7 A = 16 ft 18 ft Find the unknown measurement for the figure. 3. parallelogram A = 9.18 m b =.7 m h = 3. m. parallelogram 9 A = 3 100 m b = 3 10 m h = 1 10 m. 5 cm 7 cm 35 cm 5. square A = 1,5 cm s = 35 cm COMMON CORE STANDARD 6.G.A.1 Solve real-world and mathematical problems involving area, surface area, and volume. 6. parallelogram A = 6.3 mm b = 7 mm h = 0.9 mm Problem Solving 7. Ronna has a sticker in the shape of a parallelogram. The sticker has a base of 6.5 cm and a height of 10.1 cm. What is the area of the sticker? 8. A parallelogram-shaped tile has an area of 8 in.. The base of the tile measures 1 in. What is the measure of its height? 65.65 cm 9. WRITE Copy the two triangles and the square in Exercise 15 on page 536. Show how you found the area of each piece. Draw the parallelogram formed when the three figures are put together. Calculate its area using the formula for the area of a parallelogram. Check students work. in. Chapter 10 537 Cross-Curricular Cartographers, or mapmakers, make maps and globes. In the past, they used pen and paper, but today, with the help of computers, they can make quicker work of mapmaking. They often begin with an estimate of the area of a continent, country, state, or town. Then they apply more precise computer techniques to make the map a true model of the actual area. Use a map of your town to make an estimate of the area of your town. Explain your reasoning. Check students work. SOCIAL STUDIES The game of chess has its ancestry in a game from sixth-century India called chaturanga, which literally means having four parts. This refers to the four-part army of elephant brigades, chariots, cavalry, and foot soldiers, corresponding to the modern-day chess pieces bishops, rooks, knights, and pawns, respectively. The square chessboard contains 6 smaller squares. If each of those squares on a particular chessboard has sides that are 1 1_ inches long, what is the area of the chessboard? 1 i n Image Credits: PhotoDisc/Getty Images 537 Chapter 10
Lesson Check (6.G.A.1, 6.EE.A.c, 6.EE.B.7) 1. Cougar Park is shaped like a parallelogram and has an area of 1 16 square mile. Its base is 3_ 8 mile. What is its height? 1_ 6 mile. Square County is a square-shaped county divided into 16 equal-sized square districts. If the side length of each district is miles, what is the area of Square County? 56 square miles Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section. Spiral Review (6.EE.B.5, 6.EE.B.8, 6.EE.C.9) 3. Which of the following values of y make the inequality y < true? y = y = 6 y = 0 y = 8 y =. On a winter s day, 9 F is the highest temperature recorded. Write an inequality that represents the temperature t in degrees Fahrenheit at any time on this day. y = 6 and y = 8 t 9 5. In seconds, an elevator travels 0 feet. In 3 seconds, the elevator travels 60 feet. In seconds, the elevator travels 80 feet. Write an equation that gives the relationship between the number of seconds x and the distance y the elevator travels. 538 y = 0x 6. The linear equation y = x represents the number of bracelets y that Jolene can make in x hours. Which ordered pair lies on the graph of the equation? Students answers will vary, but the y-value of the ordered pair should be times as great as the x-value. Possible answers are (1, ), (, 8), and (3, 1). FOR MORE PRACTICE GO TO THE Personal Trainer Connecting and Science Mean, Median, Mode, and Range You can analyze both the measures of central tendency and the variability of data using mean, median, mode, and range. Tutorial Orbit eccentricity measures how oval-shaped the elliptical orbit is. The closer a value is to 0, the closer the orbit is to a circle. Examine the eccentricity values below. Orbit Eccentricities of Planets in the Solar System Mercury 0.05 Jupiter 0.09 Venus 0.007 Saturn 0.057 Earth 0.017 Uranus 0.06 Mars 0.09 Neptune 0.011 Mean The mean is the sum of all of the values in a data set divided by the total number of values in the data set. The mean is also called the average. Median The median is the value of the middle item when data are arranged in numerical order. If there is an odd number of values, the median is the middle value. If there is an even number of values, the median is the mean of the two middle values. Mode The mode is the value or values that occur most frequently in a data set. Order the values to find the mode. If all values occur with the same frequency, the data set is said to have no mode. Range The range is the difference between the greatest value and the least value of a data set. 66 Unit 8 The Solar System 0.007 + 0.011 + 0.017 + 0.06 + 0.09 + 0.057 + 0.09 + 0.05 8 1 Add up all of the values. Divide the sum by the number of values. mean = 0.061 0.007 0.011 0.017 0.06 0.09 0.057 0.09 0.05 1 Order the values. The median is the middle value if there is an odd number of values. If there is an even number of values, calculate the mean of the two middle values. median = 0.075 0.007 0.011 0.017 0.06 0.09 0.057 0.09 0.05 1 Order the values. Find the value or values that occur most frequently. mode = none 0.05 0.007 1 Subtract the least value from the greatest value. range = 0.198 Do the Mean, Median, Mode, and Range Develop Vocabulary S.T.E.M. Activity Chapter 10 1. Define the following terms in your own words. Mean (Average): Possible answer: The mean is the sum of all of the values in a data set divided by the total Do the! number of values in the data set. 3. The data table below shows the masses and densities of the planets. Develop Concepts Use with ScienceFusion pages 66 67. g Planets Mass ( 10 kg) Density ( cm 3 ) Median: Mercury 0.33 5.3 Possible answer: The median is the value of the middle item when data are arranged in Venus.87 5. numerical order. If there is an odd number of values, the median is the middle value. If there Earth 5.97 5.5 is an even number of values, the median is the mean of the two middle values. Mars 0.6 3.3 Mode: Jupiter 1,899 1.33 Possible answer: The mode is the value or values that occur most frequently Saturn in a data set. 568 0.69 Uranus 87 1.7 Order the values to find the mode. If all values occur with the same frequency, the data set Neptune 10 1.6 is said to have no mode.. Find the mean, median, mode, and range for the density of the planets. Range: Show your work. Possible answer: The range is the difference between the greatest value and the least value Mean: of a data set. (5.3 + 5. + 5.5 + 3.3 + 1.33 + 0.69 + 1.7 + 1.6) 8 = 3.06 cm 3. A planet has an orbit eccentricity of 0.99. Is the orbit circular? Mode: Possible answer: No. A planet s orbit has to have an eccentricity close to 0 for No density appears more than once. The data has no mode. it to be circular. Range: Range is 5.5 g cm 3 0.69 g cm 3 =.83 g cm 3. S.T.E.M. Activity 67 Summarize 68 Median: from smallest to largest: 0.69, 1.7, 1.33, 1.6, 3.3, 5., 5.3, 5.5. Since there is an even (1.6 + 3.3) number of values, I take the average of 1.6 and 3.3. =.9 g cm 3. 5. Why is it important to know the mean, median, mode, and range of data? Possible answer: so that scientists can identify the central tendency of the data and how much the data varies In Chapter 10, students extend their understanding of area, by performing calculations with multiple values. These same topics are used often in the development of various science concepts and process skills. Help students make the connection between math and science through the S.T.E.M. activities and activity worksheets found at www.thinkcentral.com. In Chapter 10, students connect math and science with the S.T.E.M. Activity Mean, Median, Mode, and Range and the accompanying worksheets (pages 67 and 68). Through this S.T.E.M. Activity, students will connect the GO! Chapter 10 concepts and skills with various problems about mass and density, including finding the mean density of the planets in our solar system. It is recommended that this S.T.E.M. Activity be used after. 538