SCALE CITY The Road to Proportional Reasoning: Kentucky Horse Park 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: Mathematics and Horses... 2-3 Day Two: Direct and Inverse Proportions... 3-4 Day Three: Performance Assessment, Open Response, and Multiple Choice Assessment... 4-6 Support/Connections/Resources... 6-7 Adaptations for Diverse Learners/Lesson Extensions Suggestions for Use with Students with Special Needs... 7 Applications Across the Curriculum... 8 Performance Assessment... 9-10 Open Response Assessment... 11 Multiple Choice Assessment... 12-13 KENTUCKY HORSE PARK: Time, Distance, and Speed
Grades 6-8 Essential Question: How are time, distance, and speed related mathematically? Materials measuring tape rulers or yardsticks (one for each student or pair of students) graphing paper Technology computer computer projector Internet connection computer lab for individual or paired exploration overhead projector Vocabulary: direct proportion x/y = k where x and y are two related variables and k is a constant inverse proportion x y = k where x and y are two related variables and k is a constant equivalent fractions proportion rate ratio scale standard and nonstandard measurement KENTUCKY HORSE PARK: TIME, DISTANCE, AND SPEED Length: 1-3 days Concept/Objectives: Students will learn to calculate rate of speed. Students will learn that speed and distance are directly proportional and speed and time are inversely proportional. Activity: Students will explore the concept of rate through hands-on activities, an online interactive activity, video, and word problems. Students will use skills in computation and problem solving to determine answers to problems related to rate. Students will apply understanding of inverse and direct proportions to solve problems related to speed and other rate problems. Students will analyze and create graphs of inverse and direct proportions. Instructional Strategies and Activities NOTE TO TEACHER: The core lesson covering the objectives is Day Two. Day One is provided as an exploration of standard and non-standard measurement. Day Three can be used for follow-up, Open Response, Performance, and Multiple Choice Assessments. DAY ONE: MATHEMATICS AND HORSES Resources Used in This Lesson Plan: Scale City Video: Greetings from the Kentucky Horse Park Online Interactive: At the Track Assessments (included in this lesson) Classroom Handouts (PDFs) All resources are available at www.scalecity.org 1. Distribute Handout 1: Measurement Origins as a hands-on exploration of standard and non-standard measurement. Students with recent perspective on measurement may not need this review. 2. Post the following question on the board or overhead: How would you measure the height of a horse? 3. Discuss student answers. If some students have equine experience, a demonstration may even be possible. Answer: A special measuring device is used that resembles a 90-degree angle. Horses are measured in hands. A hand is 4 inches tall, or about 10 centimeters. Horses are measured from the ground to the withers. The withers is the highest point of the horse s back, just in front of the saddle. When a horse is said to be 16.2 hands, that means 16 hands 2 inches, not 16 and two-tenths hands. KENTUCKY HORSE PARK: Time, Distance, and Speed 2
4. Students complete Handout 2: Horse Height as an exercise in proportional reasoning and problem solving. Students use basic understanding of ratios and proportion to solve these problems. The proportions are direct proportions. 5. Use an Internet projector to watch the Greetings from the Kentucky Horse Park video at www.scalecity.org. Or download the video to a DVD to show to your class. 6. Have students complete Handout 3: Greetings from the Kentucky Horse Park Review as an exit slip. DAY TWO: DIRECT AND INVERSE PROPORTIONS 1. Watch or review Greetings from the Kentucky Horse Park. After viewing, ask students the following questions: What were some ways that math was used to make comparisons? Weight, height (hands), winning record, speed, and stride What is the name of the legendary horse buried at the Kentucky Horse Park? Man o War. 2. Using an Internet projector, go to the At the Track page at www.scalecity.org and race Man o War against other opponents. Background Information for the Online Interactive At the Track Man o War s size, speed, and record (20 wins in 21 races) are still celebrated today. Many think he is the greatest racehorse of all time. Others argue that Secretariat or Seabiscuit could have beaten Man o War in a race. Ironically, the only horse to beat Man o War was named Upset, although it s often argued that the race was unfair. There were no starting gates at the time; instead, someone dropped a flag when all the horses were lined up and ready to run. The flag was dropped while Man o War was facing backward. He still managed to finish a close second, and Upset never beat him again. The cheetah is the fastest land animal, clocking speeds at 60 mph and higher. However, cheetahs can t run 60 mph for more than several hundred yards, so they sneak up on their prey and then attack in a burst of speed. Other exceptionally fast runners include pronghorn antelopes, wildebeests, lions, gazelles, and horses. For a chart comparing the speeds of different species, see www.homeworkspot.com/ask/fastestanimals.htm. The adult runner in the activity, Roger Bannister, is a hero of track and field. Before he broke the four-minute record for the mile in 1954, many people thought it was impossible for a man to run a mile faster than four minutes. Bannister was a 25- year-old British medical student who believed that better training would make the difference. Since his record-breaking run, Bannister has had a distinguished career in medicine. The record for the mile today is around 3:43, around 17 seconds faster than in 1954. The metric system is now used in American high school track and field the closest distance to the mile race is the 1600-meter race, which is 0.994 miles. Seabiscuit was a grandson of Man o War. Kentucky Academic Expectations 2.7 2.8 Kentucky Program of Studies Grade 6 MA-6-NPO-S-NS1 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 Grade 7 MA-7-NPO-U-1 MA-7-NPO-S-NS6 MA-7-NPO-U-2 MA-7-NPO-S-NO3 MA-7-NPO-U-4 MA-7-NPO-S-RP2 MA-7-NPO-S-RP3 Grade 8 MA-8-NPO-U-1 MA-8-NPO-S-NS3 MA-8-NPO-NO1 MA-8-NPO-U-2 MA-8-NPO-U-4 MA-8-NPO-S-RP1 KENTUCKY HORSE PARK: Time, Distance, and Speed 3
3. Distribute Handout 4: At the Track for students to complete while participating in the online interactive. Ask students questions as they complete the chart and analyze the graphs: What can you observe about the times of the girl, man, and cheetah? (The girl runs half as fast as the man. The man is 1/4 as fast as the cheetah.) How might the ratio of their times predict the ratio of the speeds for the girl and man? Or the man and cheetah? Kentucky Core Content for Assessment 4.1 Grade 6 MA-06-1.3.1 MA-06-1.4.1 Grade 7 MA-07-1.3.1 MA-07-1.4.1 Grade 8 MA-08-1.3.1 MA-08-1.4.1 KET, 2009 How do you calculate speed? Hint: mph (distance time = speed). What is a direct proportion? What examples of direct proportion have we used? What is an inverse proportion? How might speed time = distance be an inverse proportion? 4. Review the answers to At the Track as you progress through the activity. 5. Distribute Handout 5: Practice At the Track, and work through the problems together in class. Depending on student experience, you may want to use the entire sheet as guided practice. 6. Distribute Handout 6: Everyday Proportions as homework. DAY THREE: PERFORMANCE, OPEN RESPONSE, AND MULTIPLE CHOICE ASSESSMENT Use a combination of open response, performance assessment, and/or multiple choice to assess student understanding. Open Response Assessment (see page 11) Students compare the speed of a variety of modes of transportation by completing a data chart and creating a graphical representation of the data. Provide rulers and graphing paper for their graphs. Key to Open Response Mode of Transportation Distance to Travel (miles) Speed (mph) Time (hours) Newer car 300 60 5 hours Older car 300 50 6 hours Moped 300 30 10 hours Bicycle 300 10 30 hours Foot 300 3 100 hours Wagon train 300 2 150 hours The graph that students create should reflect the inversely proportional relationship between speed and time with a curving L-shaped graph line similar to the speed vs. time graph found in the At the Track interactive. It should be correctly titled and labeled and reasonably scaled, with consistent spacing on the x- and y-axes. KENTUCKY HORSE PARK: Time, Distance, and Speed 4
Performance Assessment (see page 9) Students use the concept of speed to complete data charts, create graphical representations, and explain direct and inverse proportions. Provide rulers and graphing paper for their graphs. Key to Performance Assessment Country Road Runs Name Distance (miles) Time (minutes) Speed (mph) Sam 10 miles 90 6.6667 or 6 and 2/3 Felix 9 miles 90 6 Connie 8.5 miles 90 5.6667 or 5 and 2/3 Red 8 miles 90 5.333 or 5 1/3 Dora 7 miles 90 4.6667 or 4 and 2/3 Chip 6 miles 90 4 Kyla 5 miles 90 3.333 or 3 1/3 The graph that students create should reflect the directly proportional relationship between speed and distance with a straight line starting at the origin similar to the speed vs. distance graph in the At the Track interactive. It should be correctly titled and labeled and reasonably scaled, with consistent spacing on the x- and y-axes. TEACHING TIP: You might want to ask the students how many hours 90 minutes represents, and point out that since 90 minutes is 1.5 hours, one hour is two-thirds of 90 minutes. Ask students to discuss why this is so, and how it can help them solve the problems. Remind them what mph means. Some students might see they can multiple the distance by two (how far the runner would go in three hours) and then divide the answer by three. 5K Race Name Distance (miles) Time (minutes) Speed (mph) George 3.1 20 9.3 Nick 3.1 22 8.5 Ginny 3.1 24 7.8 James 3.1 27 6.9 Kara 3.1 28 6.6 Bob 3.1 31 6.0 Minnie 3.1 33 5.6 The graph that students create should reflect the inversely proportional relationship between speed and time with a curving L-shaped graph similar to the speed vs. time graph found in the At the Track interactive. It should be correctly titled and labeled and reasonably scaled, with consistent spacing on the x- and y-axes. KENTUCKY HORSE PARK: Time, Distance, and Speed 5
TEACHING TIP: This table and graph presents an opportunity to review how to determine intervals on a graph as well as how to use a space to show that you are not starting at zero. You might suggest using speed in mph (perhaps 5.0 mph to 10.0 mph in intervals of 0.5) on the x-axis and time in minutes (perhaps 18 minutes to 34 minutes with intervals of 1 or 2) on the y-axis. Why would bigger intervals not work as well? You also might discuss how students know from the graphs or tables whether these are direct or inverse proportions. Direct proportions will have a constant increase in the y value when there is a constant increase in the x value. This translates to a graph that is a straight line. You will not see the same thing for an inverse proportion since the two are related by multiplication (xy = k): as x gets larger or smaller, y must get proportionally smaller or larger to keep the product constant. This graph will be a smooth L-shaped curve with no straight lines. If you would like to extend this problem, you could ask students to convert mph to kilometers per hour, and compare the graph of that table to the one for mph. (The shape should be the same.) Students are asked to round the figures in this table to the nearest tenth due to rules governing significant digits and measurement. See the Vocabulary list at the Teacher s Diner for more information about this concept. Multiple Choice Assessment (see page 12) Key to Multiple Choice Assessment 1. D, 2. C, 3. A, 4. D, 5. B, 6. A, 7. D, 8. D, 9. A,10. C Support/Connections/Resources An Electronic Field Trip to a Horse Farm: A KET Production www.ket.org/trips/farm/ Students learn what life is like on a working horse farm. The United Kingdom s National Horseracing Museum www.horseracinghistory.co.uk/hrho/jsp/education/measure.jsp A diagram and exercise illustrate measuring horses in hands versus meters and centimeters. A Dictionary of Units of Measurement www.unc.edu/~rowlett/units/custom.html The history of the English measurement system units is described on this web site. Thumbelina: World s Smallest Horse www.worldssmallesthorse.com/index.php?option=com_content&task=view&id=22&itemid=35 Thumbelina is a miniature horse. And she s extra tiny because she is also a dwarf that makes her a mini mini. This web site provides information about Thumbelina. American Miniature Horse Museum www.amha.org/index.asp?keyname=515 This site provides a description of how to measure a miniature horse. KENTUCKY HORSE PARK: Time, Distance, and Speed 6
Man o War Came Close to Perfection espn.go.com/sportscentury/features/00016132.html ESPN offers a history of Man o War, whose strength and speed made him a national hero in the 1920s and energized a flagging racing industry. The Legacy of the Horse: The Story of Humans and Their Relationship with the Horse imh.org/museum/history.php?pageid=9 This site, created by the International Museum of the Horse at the Kentucky Horse Park, provides chronological information about horses from their earliest origins to the modern era. Academy of Achievement www.achievement.org/autodoc/page/ban0bio-1 This site features an interview and profile of Roger Bannister, the first man to break the 4-minute mile. Adaptations for Diverse Learners/Lesson Extensions Suggestions for Use with Students with Special Needs Discuss how standard measurement units influence choices we make every day. Ask students to fill in the following blanks: A football field is measured in yards. The size of farms is measured in acres. We buy soda in 2 liter bottles. We buy milk by the gallon. The distance to the next town is measured in miles or kilometers. The doctor s office scale records your weight in pounds. Two common systems of measurement are the English and metric systems. Most of the world uses the metric system. U.S. Metric Association lamar.colostate.edu/~hillger/internat.htm This site explores the use of the metric system in the U.S. and other nations. Increased global trade is making the metric system more widely accepted in the U.S., since most global measurement standards are metric, and nearly every country in the world has taken steps to replace traditional measurements. Students with developing basic math skills may benefit from extended experience with the measurement of hands for horses. Handout 1: Measurement Origins provides exploration of standard and non-standard measurement. Activities could be extended to measure more items. Related activities: Discuss ways people may still use non-standard measurement in real life. Some examples include measuring with arm span, fingers, and pencils. Use a copier to create multiple images of a clenched fist to try measuring items in hands. Compare four inches with the height of the fist. A Handy Measure oklahoma4h.okstate.edu/aitc/lessons/primary Look under Farm Animals for A Handy Measure, a PDF describing methods for measuring horses that was produced by the Oklahoma Extension Service for primary grades. KENTUCKY HORSE PARK: Time, Distance, and Speed 7
Applications Across the Curriculum Practical Living: Health and Physical Education Learn how to set a pedometer using stride. Instruct students to walk ten steps from a specified point. Mark and measure the length of the 10 steps. Divide the total distance by 10 to get the average stride. There are 5,280 feet in a mile. Calculate how many steps students take to walk a mile. Many health organizations establish 10,000 steps a day as a target for healthy living. If possible, have students wear pedometers over a period of several days and average the steps per day. If a school track coach is available, discuss ways that students can begin a running or walking program and increase speed, strength, and fitness. The Active Lifestyle Program www.presidentschallenge.org/the_challenge/active_lifestyle.aspx The Active Lifestyle program for people under 18 helps set realistic goals to encourage fitness for a lifetime. Social Studies The speed limit for U.S. highways is often a controversial issue. Explore how commuters, long haul truck drivers, emergency medical staff, state police, and environmentalists may have differing perspectives on the national speed limit. Divide the class into groups to examine possible perspectives. Each group will use mathematics to support their arguments. For example, students might research the mathematical relationship between speed limits and accident rates to argue that speed limits should be lower or they might examine how higher speed limits affect the profit margins of trucking companies to argue for higher speed limits. They also might explore how speed affects gas mileage. Science Terms such as speed, friction, gravity, and force can be introduced through classroom experiments. Newton s Laws also can be discussed as they relate to distance, speed, and time. KENTUCKY HORSE PARK: Time, Distance, and Speed 8
PERFORMANCE ASSESSMENT SCALE CITY Prompt: How does the concept of speed explain both direct and inverse proportions? Examine these examples from running events. Directions: Complete the charts by calculating speeds, and then create one graph showing speed and distance using data from the first chart and a second graph showing speed and time using data from the second chart. Decide on a reasonable scale for depicting the data and label and title your graphs. Determine which depicts a direct proportion and which depicts an inverse proportion and be prepared to explain how you decided. Present your findings to the class and explain the terms inverse proportion and direct proportion. Country Road Runs A fitness club has one Saturday each month for a Road Run. The club selects a country road and meets at the starting point. The club member to go the farthest in a 90-minute period is considered the winner. Complete the chart, rounding your decimal answers to the nearest 10th place and providing the equivalent fractions, and then create a graph using the data from the chart to depict distance vs. speed with speed on the x-axis and distance on the y-axis. Name Distance (miles) Time (min) Speed (mph) Sam 10 miles 90 6.7 or 6 and 2/3 Felix 9 miles 90 Connie 8.5 miles 90 5.7 or 5 and 2/3 Red 8 miles 90 5.3 or 5 1/3 Dora 7 miles 90 Chip 6 miles 90 Kyla 5 miles 90 5K Race Here are the results of a 5K race. Complete the chart, rounding your answers to the nearest 10th place. Graph the information with speed on the x-axis and time on the y-axis. Name Distance (miles) Time (min) Speed (mph) George 3.1 20 9.3 Nick 3.1 22 8.5 Ginny 3.1 24 James 3.1 27 Kara 3.1 28 Bob 3.1 31 Minnie 3.1 33 KENTUCKY HORSE PARK: Time, Distance, and Speed 9
Performance Assessment PERFORMANCE SCORING GUIDE 4 3 2 1 0 The student demonstrates excellent understanding of direct and inverse proportions. The mathematical information is correct and complete. The graphs are correctly labeled and drawn. The graphs clearly represent inverse and direct proportions. The work indicates outstanding effort, understanding, and application of proportional reasoning. The student s presentation and product are exemplary. The student demonstrates good understanding of direct and inverse proportions. The mathematical information has few errors. The graphs clearly represent inverse and direct proportions. The work indicates adequate application of proportional reasoning. The student s presentation and product are acceptable. The student demonstrates basic understanding of direct and inverse proportions. The mathematical information and graphs indicate necessary review of concepts or data. The work indicates application of proportional reasoning. The student s presentation and product are generally acceptable. The student demonstrates minimal understanding of direct and inverse proportions. The mathematical information and graphs are incorrect and/or incomplete. The student s presentation and product indicate a need for review and/or limited effort. Blank or no response. KENTUCKY HORSE PARK: Time, Distance, and Speed 10
OPEN RESPONSE ASSESSMENT SCALE CITY Prompt: A family s vacation destination is 300 miles away. The parents wanted to drive their newer car, which easily travels an average speed of 60 mph. However, that car is at the repair shop and won t be fixed in time. So they ll have to take the older car, which travels at an average speed of 50 mph. The father says, Cheer up, kids! At least we won t be traveling by wagon train, bicycle, or on foot. Directions: Calculate the time it will take the family to travel in the newer car and the older car. Compare these calculations with the time it would take to travel by wagon train, moped, bicycle, or on foot. Explain how the relationship of time versus speed is an inverse proportion, and use this completed chart to create a simple graph showing this relationship. Make sure the scale of your graph is reasonable and that it s correctly labeled and titled. Mode of Transportation Distance to Travel Speed (mph) Time (hours) (miles) Newer car 300 60 Older car 300 50 Moped 300 30 10 hours Bicycle 300 10 Foot 300 3 Wagon train 300 2 OPEN RESPONSE SCORING GUIDE 4 3 2 1 0 The writing clearly and accurately explains the inverse proportion of speed to time. The calculations and graph are correct and clear. The student uses appropriate mathematical terminology reflecting excellent understanding of proportional reasoning. The writing adequately explains the inverse proportion of speed to time. The calculations and graph reflect good understanding with few errors. The written response indicates a good understanding of inverse proportions and the mathematical basis for proportional reasoning. The writing generally explains the inverse proportion of speed to time. The calculations and graph reflect general understanding with some incomplete or incorrect responses. The written response indicates a basic under standing of proportional reasoning. The explanations, calculations, and graph reflect minimal under standing of inverse proportions and proportional reasoning. The written response indicates a limited effort or a need to review basic concepts. Blank, or no response. KENTUCKY HORSE PARK: Time, Distance, and Speed 11
MULTIPLE CHOICE ASSESSMENT Name: Date: 1. The school band is traveling 120 miles to a competition. If the school bus travels at 60 mph, travel time will be A. 4 hours B. 3.5 hours C. 2.5 hours D. 2 hours 2. There are about 3.1 miles in a 5 kilometer race. Mike Hart finished the 5K race in 20 minutes. His speed in mph is A. under 6 mph B. almost 8 mph C. around 9 mph D. over 10 mph 3. Speed is A. distance time B. distance time C. time distance D. distance + time 4. A common unit to measure speed of automobiles is A. friction B. gallons C. Hz D. mph 5. Cory finished five pages of homework in 90 minutes. If he works at the same pace he should complete three pages in A. 75 minutes B. 54 minutes C. 30 minutes D. 25 minutes 6. Grandma always drives 65 mph on the highway. If she drives for 3 hours, she will travel A. 195 miles B. 145 miles C. 130 miles D. 65 miles KENTUCKY HORSE PARK: Time, Distance, and Speed 12
7. Chip s father runs half the speed of his son. Chip will finish a 10 K race in 36 minutes. His father will finish in A. 18 minutes B. 48 minutes C. 56 minutes D. 72 minutes 8. A farmer s car travels at 45 mph on the back road. A tractor travels at 9 mph. On a three-mile stretch of road, the family car would travel A. 45 minutes and the tractor would travel 9 minutes B. 4 minutes and the tractor would travel 45 minutes C. 10 minutes and the tractor would travel 20 minutes D. 4 minutes and the tractor would travel 20 minutes 9. The relationship of speed to time is an A. inverse proportion because speed time = distance B. inverse proportion because speed time = distance C. direct proportion because distance speed = time D. direct proportion because distance + time = speed 10. In an inverse proportion, as one variable increases A. another variable increases proportionally B. there is no effect on the other variable C. another variable decreases proportionally D. another variable equals zero Multiple Choice Assessment KENTUCKY HORSE PARK: Time, Distance, and Speed 13