Instructional Strategies and Activities. Support/Connections/Resources Writing for the Lesson... 7

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1 SCALE CITY The Road to Proportional Reasoning: Louisville Slugger Lesson TABLE OF CONTENTS Click on a title to go directly to the page. You also can click on web addresses to link to external web sites. Overview of Lesson Including Kentucky Standards Addressed... 2 Instructional Strategies and Activities Day One: Introduction to Proportional Reasoning Problems Day Two: Using Similar Triangles and Shadows To Determine Unknown Heights Day Three: Open Response Assessment and Game... 6 Day Four: Extension Activity and Multiple Choice Assessment... 6 Support/Connections/Resources... 6 Writing for the Lesson... 7 Adaptations for Diverse Learners/Lesson Extensions Comparing Baseball Fields... 7 The Rule... 7 Using Chalk To Measure Shadows... 8 Advanced Extension: Independent Research on the Pythagorean Theorem... 8 Applications Across the Curriculum... 9 Open Response Assessment Multiple Choice Assessment LOUISVILLE SLUGGER: Similar Triangles

2 Grades 6-8 Essential Question How can I use the geometry of similar triangles, measurement, and proportional reasoning to find an unknown height? Materials whiteboards markers Technology computer Internet connection computer projector Vocabulary angle congruent equivalent fractions proportion ratio right triangle similar LOUISVILLE SLUGGER: SIMILAR TRIANGLES Length: 1-4 days Concept/Objectives: Students will learn about similar triangles and proportional reasoning. Students will apply proportional reasoning to solve everyday problems using shadows to determine the height of objects. Activity: Using the Similar Shadows interactive, students will explore and calculate how measurement, right triangles, and shadows can be used to determine an unknown height. Students also will practice skills in proportional reasoning by solving problems related to batting average, scale sizes, and shadows. Students will construct formulas to solve problems using words and numbers. Instructional Strategies and Activities Resources Used in This Lesson Plan: Scale City Video: Greetings from the Louisville Slugger Museum Online Interactive: Similar Shadows Assessments (included in this lesson) Classroom Handouts (PDFs) All resources are available at Note to teacher: This lesson plan applies students understanding of geometry and proportional reasoning to real world problems. You could use only Day 2. Day 1 uses the subject of baseball to provide practice using percents, decimals, proportional reasoning, and scale. Day 3 includes an Open Response and a game and Day 4 an Extension Activity and a Multiple Choice Assessment. DAY ONE: INTRODUCTION TO PROPORTIONAL REASONING AND REVIEW OF TRIANGLES 1. If you have one, show students a Louisville Slugger Bat, the official bat of major league baseball. Babe Ruth, one of baseball s greatest players, used a Louisville Slugger. Ruth s career batting average was.342, an amazing statistic when you consider that a batting average of.300 is excellent for major league players. Batting averages are computed by dividing base hits by the number of times at bat and rounding off the answer to three decimal places. So an average of.342 means that in 1000 at bats, Ruth averaged 342 base hits. Go to to see all of Ruth s statistics. Not only was he a great hitter, he was a great pitcher as well. And his slugging average a more complicated statistic that takes into account whether the player hits a single, a double, a triple, or a homerun and thus measures the power of his swing is even more amazing. 2. Discuss batting averages as a way of evaluating skill and performance. Model and solve together: If someone is at bat 50 times and gets 10 base hits, what would the batting average be? Answer:.200 LOUISVILLE SLUGGER: Similar Triangles 2

3 If you think students need additional practice in the computation, give them a few more problems to practice. 3. Ask questions that require the students to reason proportionally. For example, you might ask, Ten hits out of 50 at bats is the equivalent of how many hits out of 25 at bats? 100 at bats? You can use equivalent fractions to help the students see the proportional relationships. Follow up with some comparison questions: Which batting average is better 10 out of 50 or 10 out of 40? 10 out of 50 or 5 out of 30? 4. Distribute Handout 1: Batting Practice. This activity helps students prepare to use calculators and practice proportional reasoning. After students complete the problems, a good approach would be to encourage them to discuss their reasoning with partners and then share it with the whole class. Then other students can respond with alternative ways of thinking about and solving the problems. 5. Introduce the video. Babe Ruth s statistics demonstrate that he was a great baseball player many argue the greatest that has ever played. Mathematics backs up this argument. One of the Louisville Slugger bats that Babe Ruth used in 1927, the year he set the record for most homeruns, is on display at the Louisville Slugger Museum and Factory in Louisville, Kentucky. Louisville Sluggers are created using math, science, and engineering. 6. Watch the video. Call attention to the large model bat and glove at the end. Students will be using proportional reasoning to calculate model sizes. Students will also watch the video again on Day Distribute Handout 2: Sizing Up the Team. In this activity, students practice using proportional reasoning to determine the heights of different scale models. Before you discuss the method outlined below for solving problem one, ask the students to think about the problem. Will the statue be taller or shorter than the bat? How did Babe s height compare to the height of the 34-inch bat? Can you come up with a rough estimate for the height of the statue? Thinking about the problem in concrete terms will help the students see the meaning behind their calculations. Solve problem one as a demonstration. Solve problem two as guided practice. Students should solve problem three on their own. TEACHING TIP: The five-step method described here gives students a way to set up equivalent ratios and introduces cross multiplication as a possible means of solving proportional reasoning problems. However, it s good practice to encourage students to think about the concepts and come up with their own solutions. Let students know there s no single correct approach to these problems. Kentucky Academic Expectations Kentucky Program of Studies Grade 6 MA-6-NPO-U-3 MA-6-NPO-S-NO2 MA-6-NPO-U-4 MA-6-NPO-S-RP2 MA-6-NPO-S-RP3 MA-6-G-S-SR6 Grade 7 MA-7-NPO-U-4 MA-7-NPO-S-RP2 MA-7-NPO-S-RP3 MA-7-G-S-SR2 MA-7-G-S-SR3 MA-7-G-S-SR6 Grade 8 MA-8-NPO-S-NO1 MA-8-NPO-U-4 MA-8-G-S-SR4 MA-8-G-S-SR5 MA-8-G-S-TS1 MA-8-G-U-5 Allowing students to puzzle over problems and discuss various solutions with partners, small groups, and/or whole class will help them see the concepts that underlie cross-multiplication and other methods of solving proportional problems. Unless students have a conceptual grasp of proportionality, many won t remember how to cross-multiply when they need to apply proportional reasoning in subsequent math and science classes or in everyday life. LOUISVILLE SLUGGER: Similar Triangles 3

4 Step 1: Use words to describe the ratios, and use words to describe what the equivalent ratios or proportion would be. Step 2: Convert the numbers to the same unit of measurement. Step 3: Put the numbers into the ratio formulas. Write the equivalent ratios or proportion using numbers. Use x for the unknown number. Kentucky Core Content for Assessment 4.1 Grade 6 MA MA MA MA MA MA Grade 7 MA MA MA MA Grade 8 MA MA MA MA KET, 2009 NOTE TO TEACHER: Before you get to step 4, ask students to propose ways of solving for x. These might include finding the equivalent fraction or seeing that the value of a known ratio (1:2 = 0.5, for example) will equal the value of the equivalent ratio in the proportion (1:2 = x:50; 50 x 0.5 = 25; x = 25). You also might ask them to explain why cross multiplication works as a way of solving these problems (1:2 = 25:50; 1 50 = 50; 2 25 = 50). They could work as pairs or small groups to come up with an explanation and then share it with the rest of the class. Encourage students to use alternative methods when they are solving the problems. Step 4: Solve for x by cross multiplying the equivalent ratios and dividing or by using some other method. Step 5: Convert the answer to the appropriate unit of measurement. 7. If desired, use Handout 3: Do You Remember Triangles? as a review and checkup. This handout also could be homework. The students answers will inform you about their understanding of the properties of triangles and whether they have the computation skills needed to work with similar triangles. You may decide this review is not necessary based on your knowledge of student ability and knowledge. One commercial product that helps students explore and understand triangles and similarity is AngLegs, a set of plastic pieces of different lengths that snap together to form geometric shapes (see 8. The writing for this lesson is another homework option or a possible math journal entry. Have students choose a professional athlete and research the statistics that prove that the athlete is a great player. Then ask the students to use the numbers to write a persuasive piece on why this player is among the best. DAY TWO: SIMILAR TRIANGLES 1. Distribute Handout 4: Batter Up! Pay Attention viewer s guide for students to complete while watching Greetings from the Louisville Slugger Museum. If students watched the video during Session One, reviewing it with the viewer s guide will add variation as well as introduce today s lesson. 2. Check answers to the viewer s guide. 3. Allow students some time to come up with theories on how to measure the world s largest bat using shadows, a regular bat, and proportional reasoning. 4. Tell students that we use proportional reasoning to calculate and compare. Today we will compare the dimensions of triangles using proportional reasoning. Use a projector to show students the Similar Shadows interactive at LOUISVILLE SLUGGER: Similar Triangles 4

5 5. As you explore the interactive, ask students the following questions: What is similar about the objects? How would these bats and shadows form triangles? (Once they have offered possible answers, click on the triangle button to view.) In what ways are these triangles alike? (The smaller triangle can be moved to fit into the larger triangle. They both have right angles. The lower lefthand angles are the same size.) If two angles are the same, what do we know about the third angle? Why is the ratio of each bat s height to its shadow the same? 6. As a class exercise, complete the questions in the interactive. Select a different object or time of day for additional practice. This page offers ample exploration and computation practice. Additional practice at home or in a computer lab would be beneficial to students. (See sample message to parents below.) In addition to the interactive computations, students also could determine the size of the third angle in the triangles at dawn, midmorning, and late afternoon. 7. Discuss congruency and similarity through exploring the bat examples in the interactive. Are these triangles congruent? (No, the sides are not the same.) Are their angles congruent? (Yes, the corresponding angles are congruent.) Define similarity. Discuss angle-angle as a means of proving triangles to be similar. How do these triangles meet the criteria for similar triangles? 8. If desired, use Handout 3: Do You Remember Triangles? as a review and check-up (if you didn t use it on Day One). This handout also could be homework. The students answers will inform you about their understanding of the properties of triangles and whether they have the computation skills needed to work with similar triangles. You may decide this review is not necessary based on your knowledge of student ability and knowledge. One commercial product that helps students explore and understand triangles and similarity is AngLegs, a set of plastic pieces of different lengths that snap together to form geometric shapes (see 9. Handout 6: Similar Shapes is provided for extra practice. Sample to parents Dear Parents, We are working on a study of proportional reasoning in math class using the following web site: You may access this web site at home for additional practice and interaction with your child. KET has designed this web site for exploration and practice of mathematical concepts. The Check Answer button allows students to monitor their own understanding as they work the problems. Sincerely, Teacher LOUISVILLE SLUGGER: Similar Triangles 5

6 DAY THREE: OPEN RESPONSE AND GAME 1. Students complete the Open Response. 2. As time allows, divide the class into two baseball teams for Game Time. Use Handout 7: Game Time Questions to quiz the students. You may want to put the questions up on the overhead one by one rather than distributing them to the teams. If the teams have the questions ahead of time, it might give them unfair advantage. Directions: For the batter s boxes, place two seats at the front of the class beside whiteboards. Different students from each team will go to the batter s box to answer questions. Alternately, you could have two students from each team go to the batter s box so they could talk to each other about the problems. Draw two baseball diamonds, one for each team, on the classroom whiteboard or blackboard, to mark what base students are on when they answer correctly and to advance any players already on base to second, third, or home as appropriate. You might designate one or two students as umpires to make sure this is done accurately. Teams could be at bat until they have three outs, just as in the real game. Practically speaking, however, it would probably work better to have them alternate. Allow students one minute to answer each question using calculators or computation. After completing the problem, the batter shows the team his or her answer and gets a thumbs-up or thumbs-down. At this point, the batter may change the answer or stick with the original response. The team with the most runs wins the game. Regardless of who is batting, all students are expected to work each problem. Easy questions are singles. Difficult questions are home runs. The teacher may choose the difficulty level for individual students. Since they are working problems in front of the class, all the batters should have a good chance to make a hit. Students could also choose which level question they want to answer. DAY FOUR: EXTENSION ACTIVITY AND MULTIPLE CHOICE ASSESSMENT 1. Choose one of the Extension Activities described on pages Complete the Multiple Choice Assessment (see pages 11-12). Multiple Choice Assessment Key 1. D, 2. C, 3. B, 4. D, 5. A, 6. C, 7. A, 8. D, 9. C, 10. A Support/Connections/Resources Finding Shadow Length This short article from a regional newspaper provides a clear explanation of how the position of the sun affects the length of shadows. Louisville Slugger, the Official Bat of Major League Baseball This web site includes a history of Louisville Slugger bats and information about the science and technology of their manufacturing process as well as tips for improving baseball skills. Pyramids, Olives, and Donkeys ecr.lausd.k12.ca.us/staff/llocurto/algebra/thalesprofile.pdf This story from Mathematicians Are People Too is a fictionalized account of how the Greek mathematician Thales used proportional reasoning to measure the Great Pyramid. LOUISVILLE SLUGGER: Similar Triangles 6

7 Writing for the Lesson For journal or persuasive writing, chose a professional athlete and research the statistics that prove the athlete to be a great player. Use the numbers to write a persuasive piece on why this player is among the best. Adaptations for Diverse Learners/ Lesson Extensions Extension: Comparing Baseball Fields For additional experience, students could compare baseball field dimensions using Handout 8: Little League and Major League Baseball Fields. This handout could serve as regular class work, extra credit, or independent work. Baseball Field Dimensions This site includes information about the dimensions of high school, college, and professional baseball fields. Extension: The Rule To make sure corners have perfectly square angles, carpenters often use the rule. A triangle will have a perfect 90 angle across from its hypotenuse if it is similar to a triangle with sides measuring 3, 4, and Explain the rule, and have students test it with centimeter graph paper and rulers. What happens when you make a triangle with legs of 6 and 8 cm? Of 9 and 12 cm? Then divide students into teams and give them chalk, pencils, tape measures and/or rulers, and lengths of string and ask How can you use some of these tools and the rule to determine if apparent right angles in our classroom (door and window frames, desks, corners, whiteboards, bookshelves) are perfectly square? Students would need to decide an appropriate scale for their triangles (i.e., inches for a bookshelf, feet for the corner of the room, etc.). They could measure and use the chalk to mark three units down one side of the supposed right angle and four units down the other and then measure the hypotenuse to see if it is five units. Others might realize that if they knot the string into 12 evenly spaced sections using the ruler or measuring tape, it can form a perfect triangle. In either case, the groups should chart their findings, report back to the class, and compare their results with those of other students. To extend this lesson further, students could research could research how string is used for measurement. The following web sites provide background information. Building the Great Pyramid This is a fascinating illustrated discussion of the methods used by builders in ancient Egypt. The 47th Problem of Euclid This site provides instructions for using string to create perfect squares. LOUISVILLE SLUGGER: Similar Triangles 7

8 Activity for Students with Learning Disabilities Using chalk to record shadows throughout the day is a good way to engage students with cognitive disabilities in the phenomena and mathematics of shadows. This could also be a nice family project. Students should be instructed to pick a sunny day and have a partner draw their shadow at different times of day while they stand or sit in the same location. Primary Activity Using Chalk To Measure Shadows For students with cognitive disabilities, the numbers in this lesson may interfere withunderstanding. Begin by discussing shadows that are the same size as the object. Then discuss shadows that are twice the size. TEACHING TIP: All your students might benefit from seeing one or two of these simple proportions before they move on to the decimals that are used in these lessons. Increase the complexity to match student understanding. To make the lesson more engaging, use photographs of students and their shadows. For example, the student s image could be cut out of copy of a photograph and compared to the photographic image of his or her shadow to show that the shadow is twice student s height. This lesson also can offer practice in creating and reading charts. Students could chart the lengths of their own shadows at various times of day. They also could make a chart listing other objects and the lengths of their shadows. More advanced students might predict the length of the objects shadows and test their predictions by measuring the actual shadows at different times of day. Advanced Extension: Independent Research Advanced students can explore the basic idea behind the Pythagorean theorem. Using actual squares in the pattern of a 2 + b 2 = c 2 and experimenting with different dimensions will add meaning to the computation. Students could conduct an investigation of the Pythagorean Theorem by exploring the following questions: Who was Pythagoras? What is the Pythagorean Theorem? What does the Pythagorean Theorem have to do with right triangles? How could you visually represent the Pythagorean Theorem with squares? How does the Pythagorean Theorem show that it is impossible for the side opposite the right angle to be the shortest side? How is the Pythagorean Theorem used today? How could you use the Pythagorean Theorem to calculate the length of a throw from first to third base or from second to home? (Students may need prompting to see that the baseball diamond is really a square.) What some other Pythagorean theorems involving triangles? LOUISVILLE SLUGGER: Similar Triangles 8

9 Applications Across the Curriculum Science A companion study on shadows relative to time of day and time of year fits nicely with this lesson. Students could photograph and record data about their shadows at various points in the day from the same location. Challenge students to measure shadows of objects of unknown height and use their skills with similar triangles to determine the heights. The Angle of the Sun s Rays istp.gsfc.nasa.gov/stargaze/sunangle.htm A diagram and explanation of the way in which the sun s rays strike the Earth s northern hemisphere A Walk Through Time: Early Clocks physics.nist.gov/genint/time/early.html A brief history of the evolution of clocks, from the earliest sundials to mechanized clocks Practical Living Students can investigate the use of triangles and triangulation in the professions of surveyor, architect, or carpenter. Visual Arts Students can explore tessellations, triangles, and art integrating measurement, principles of geometry, elements of art, and principles of design. After looking at examples, students could create their own designs. Totally Tessellated: Escher s Tessellations library.thinkquest.org/16661/escher/tessellations.8.html This site has examples of Escher s tessellations as well as hands-on activities to help students create their own tessellating designs. Social Studies An exploration of Egyptian pyramids and what is known of their construction would go very nicely with this lesson. As a topic for enrichment or even extra credit in math, students could explore the following questions: What is known about the age and purpose of the pyramids? How have the pyramids endured over time? How are triangles involved in the construction of the pyramids of Ancient Egypt? How are right angles important in the construction of the pyramids? What is the story of how Thales measured the height of the pyramids? LOUISVILLE SLUGGER: Similar Triangles 9

10 OPEN RESPONSE ASSESSMENT SCALE CITY Prompt: You re waiting for the bus with a friend. There s a flagpole nearby. Use shadows to determine the height of the flagpole. Directions: You have a measuring tape that is long enough to measure the length of your friend s shadow and the length of the flagpole s shadow. How could you calculate the actual height of the flagpole with the tools and information you have? Identify two triangles that will aid you in your computation. What information confirms that the properties of similar triangles apply to this problem? Illustrate your understanding by labeling a drawing. Write out an equation with words and symbols that show how you would use the information to compute the height of the flagpole. OPEN RESPONSE SCORING GUIDE The student correctly identifies the property of angle-angle similarity. The student draws and correctly labels an illustration depicting two similar triangles. The writing indicates excellent understanding of triangles and their similarity. The student identifies that the length of the shadows and the height of the friend would need to be measured to determine the height of the flagpole. The details and explanation indicate outstanding comprehension of geometric The student correctly identifies the property of angle-angle similarity. The student draws an illustration depicting two similar triangles. The writing indicates good understanding of triangles and their similarity. The student identifies that the length of the shadows and the height of the friend would need to be measured to determine the height of the flagpole. The details and explanation have few errors. The student draws an illustration depicting two similar triangles. The writing indicates basic understanding of triangles and their similarity. The details and explanation reveal a few concepts in need of review. LOUISVILLE SLUGGER: Similar Triangles 10 The student draws an illustration depicting two triangles. The writing indicates limited understanding of triangles and their similar ity. The details and explanation reveal minimal effort or understanding. Blank or no response

11 MULTIPLE CHOICE ASSESSMENT Name: Date: 1. One triangle has sides measuring 6, 8 and 10 centimeters. Another triangle is similar to this triangle with sides measuring A. 8, 10, and 12 B. 2, 4, and 6 C. 3, 9, and 12 D. 3, 4, and 5 2. The angles of one triangle are congruent to angles in another triangle. So it must be true that A. all the sides are equal B. the triangles are congruent C. the triangles are similar D. one angle is 90 degrees 3. A fashion designer known for her use of triangles is coming to your school. As a welcoming poster, you d like to use a design she has created. The triangles on the design have sides measuring three inches, three inches, and two inches. If you re making a poster design with one triangle side measuring eight inches, then the other sides should measure A. 12 and 24 inches B. 12 and 12 inches C. 14 and 14 inches D. 6 and 8 inches 4. If two triangles are similar then you know that they A. have corresponding sides that are the same length B. can be used to measure shadows C. have two corresponding sides the same length D. have corresponding congruent angles 5. The team mascot s shadow was seven feet long or 84 inches. Tim s shadow was six feet long or 72 inches. Tim is 64 inches tall. The mascot s height in inches would be A inches B inches C inches D inches 6. The shadow from the school flagpole was 16 feet long. Mia s shadow was three feet long. Mia is 5 feet, 7 and 1/2 inches or feet. The flagpole is A. 20 feet tall B. 25 feet tall C. 30 feet tall D. 18 feet tall LOUISVILLE SLUGGER: Similar Triangles 11

12 Multiple Choice Assessment 7. Triangle ABC is similar to triangle MNO. AB/MN = BC/NO = A. AC/MO B. CB/OM C. AC/BC D. AB/OM 8. Triangle EFG is similar to triangle XYZ. The sides of triangle EFG measure 5, 6, and 14. Two sides of triangle XYZ measure 15 and 18. The third side measures A. 24 B. 26 C. 38 D Legend holds that the Greek mathematician Thales estimated the height of the Great Pyramid using shadows and principles of similar triangles. The formula he would have used could have been A. Thale s height/pyramid shadow length = Pyramid s height/thale s shadow length B. Thale s height/thale s arm span = Pyramid s height/pyramid s width C. Thale s height/thale s shadow length = Pyramid height/pyramid shadow length D. Thale s height/thale s head circumference = Pyramid height/pyramid perimeter 10. The World s Largest Bat is 120 feet tall. It s a replica of a 34-inch bat. One of the greatest players in baseball history was Ted Williams, who stood 6 feet 3 inches tall. A statue of Williams to match the bat would be A. 265 feet B. 186 feet C. 320 feet D. 173 feet LOUISVILLE SLUGGER: Similar Triangles 12