W illie the Wheel Man Common Core State Standards for Mathematics* • Make sense of problems and persevere in solving them. (MP.1) • Reason abstractly and quantitatively. (MP.2) • Model with mathematics. (MP.4) • Attend to precision. (MP.6) • Look for and make use of structure. (MP.7) • Look for and express regularity in repeated reasoning. (MP.8) • Write and interpret numerical expressions. (5.OA.A) • Analyze patterns and relationships. (5.OA.B) • Graph points on the coordinate plane to solve real-world and mathematical problems. (5.G.A) You Need Wheel-shaped pasta (see Before You Begin 1) Student pages Colored pencils, 4 different colors Straight edge, such as a ruler Before You Begin 1. One box of rotelle pasta (wagon wheel shape) wheel-shaped cereal, or other circular cereal/candy will provide plenty of manipulatives, as students will find that the pattern is easy to discern and will soon discontinue its use. 5th GR MATHEMATICAL PRACTICES Patterns and Relationships Students will use tables and graphs to analyze the relationship between various vehicles and the number of wheels they need. Do This 1. Ask students how many wheels his/her bicycle has. Discuss how many wheels are on a bike and how many bicycle wheels they would have if every member in their family had a bicycle. Record these responses on the board. Have the class play a guessing game of “In the family.” For example: If a student says his or her family would need 10 wheels, the class would respond by saying that there are five in the family, because five bicycles have 10 wheels. Record this as an ordered pair, (5,10). Do several of these examples, leaving the ordered pairs randomly recorded on the board. 2. Tell the class that you have a friend, Willie the Wheel Man, and that he deals in wheels. Explain that his customers tell him what kind of vehicle they are building and how many vehicles they want to build, and he figures out how many wheels they will need. 3. Show the students the wheel-shaped pasta and explain that they may use it to help them model the bicycles. 4. Distribute the pasta and first student page. Explain that when Willie gets the information from a customer, he makes a t-table. Ask students to fill in the t-table with the number of wheels needed for the various numbers of bicycles, using the pasta as necessary. 1 5. Once the table has been filled out, ask students how they determined the number of wheels. [E.g., used the pasta to model the bikes and counted, skip counted (2, 4, 6, …) the wheels per bicycle, multiplied the number of bicycles by two to find the number of wheels, etc.] Emphasize the importance of understanding multiple methods to solve a problem. (MP.4, MP.7) 6. Ask the students to share the patterns they see in the t-table. Have them look at the horizontal pattern and decide as a group what rule they used to find the number of wheels on a certain number of bikes and write the rule using the letter B on the first student page. Share out as a class. [Any number of bikes times two is the number of wheels needed, 2B.] Invite the students to use the expression to answer how many wheels are needed for 27 bicycles, 50 bicycles, etc. (MP.1, MP.2, MP.4) 7. Distribute the second student page and colored pencils. Inform students that they are going to look at the information in a different way by building a graph on a coordinate plane. On the first student page, help students write each row in the t-table as an ordered pair. On the second page, draw students’ attention to the two axes. The horizontal axis is labeled number of vehicles because they will be graphing lines for three types of vehicles—a bicycles, tricycles, and wagons. Now have them graph © 2012 AIMS Education Foundation each ordered pair on the coordinate plane with one of their colored pencils. Assist as needed. Ask: From the graph, how can you tell how many wheels are on four bicycles? …five bicycles? Which model is easier for you to find the number of wheels needed? Why? (MP.4, MP.6) 8. Ask groups: How can you use the graph to determine the number of wheels on seven bicycles? After groups answer the question, have a few groups share answers. If necessary, steer students to using their straight edge to extend the line. Ask: How many wheels do you need for eight bicycles? …10 bicycles? (MP.6) 9. Tell the class that now they are going to apply their new skills, looking at the number of wheels they need to order from Willie for tricycles. Students will fill out a t-table, write the ordered pairs, find the rule, and add the line to the graph on the student page in a different color. Be sure they color the key. 10. Ask: How does the tricycle line compare to the bicycle line? [It’s above the bicycle line. Both are straight.] 11. Ask: Using your experience with the patterns from bicycles and tricycles, where on the graph do you think the line for wagons will be? What patterns will help you make your predictions? Justify your answer. (MP.3, MP.6, MP.8) 12. Now have students fill out the t-table, write the ordered pairs, find the rule, and add the line to the graph on the second student page in a different color for wagons. 5th GR MATHEMATICAL PRACTICES Ask These 1. How many wheels will be needed for 15 bicycles? …for 13 tricycles? Describe two ways you could find out. [Students can use the expression or the graph to answer the question. If students use the graph, they will need to make it larger.] 2. If someone orders 24 wheels, how many bicycles could they make? [12] How many tricycles? [8] How many wagons? [6] How do you know? [When I find 24 wheels on the number of wheels axis, I follow the line until it intersects with the line graph of the bicycle, tricycle, or wagon. Then I use the coordinates to give me the answer to how many bicycles, tricycles, or wagons I could make.] Journal Prompt Make a table, write the ordered pairs, and graph the possibilities if you needed to find the number of wheels for a unicycle. Digging Deeper Have students design a vehicle with multiple wheels. On a blank piece of paper, have students write a prediction of where the line will be on the graph. Then, have them draw a t-table, find the rule, and add the line to the graph on the student page in a different color. * © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. 3. How is the t-table helpful? …ordered pairs? …rule? …graph? [The t-table helps us to see numerical patterns. Ordered pairs help us to make our graph. The rule helps me represent the math procedure I use to find any number of wheels for a certain vehicle. The graph is a visual representation that allows me to see patterns and make generalizations.] 4. Explain in your own words how you can use a graph to know how many wheels are on three wagons. …on eight bicycles. 5. Which method did you find easiest for determining the number of wheels on a vehicle: using the rule, using the graph, adding wheels, or skip counting? Explain why you thought it was the easiest. 2 © 2012 AIMS Education Foundation W illie the Wheel Man Bicycles 1. Fill in the t-table column Number of Wheels. 2. Look for the horizontal pattern. 3. Write the rule. 4. Use the information from the t-table to write ordered pairs. Number of Bicycles Number of Wheels Ordered Pairs 0 0 (0,0) 1 ( , ) 2 ( , ) 3 ( , ) 4 ( , ) 5 ( , ) Rule: Tricycles Wagons Number of Tricycles Number of Wheels Ordered Pairs Number of Wagons Number of Wheels Ordered Pairs 0 0 (0,0) 0 0 (0,0) 1 ( , ) 1 ( , ) 2 ( , ) 2 ( , ) 3 ( , ) 3 ( , ) 4 ( , ) 4 ( , ) 5 ( , ) 5 ( , ) Rule: 5th GR MATHEMATICAL PRACTICES Rule: 3 © 2012 AIMS Education Foundation W illie the Wheel Man 1. Graph the data. 2. Color the graph key to correspond to the graph. Number of Wheels W Willie Key Bicycles Tricycles Wagons 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Number of Vehicles 5th GR MATHEMATICAL PRACTICES 4 © 2012 AIMS Education Foundation W illie the Wheel Man CON N EC Ask These T I NG AR LE NI N G 1. How many wheels will be needed for 15 bicycles? …for 13 tricycles? Describe two ways you could find out. 2. If someone orders 24 wheels, how many bicycles could they make? How many tricycles? How many wagons? How do you know? 3. How is the t-table helpful? …ordered pairs? …rule? …graph? 5th GR MATHEMATICAL PRACTICES 5 © 2012 AIMS Education Foundation W illie the Wheel Man CON N EC T I NG AR LE NI N G 4. Explain in your own words how you can use a graph to know how many wheels are on three wagons. …on eight bicycles. 5. Which method did you find easiest for determining the number of wheels on a vehicle: using the rule, using the graph, adding wheels, or skip counting? Explain why you thought it was the easiest. 5th GR MATHEMATICAL PRACTICES 6 © 2012 AIMS Education Foundation
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