# Comparing Value for Money: Baseball Jerseys

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1 PROBLEM SOLVING Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Comparing Value for Money: Baseball Jerseys Mathematics Assessment Resource Service University of Nottingham & UC Berkeley For more details, visit: MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Comparing Value for Money: Baseball Jerseys MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to: Interpret a situation and represent the variables mathematically. Select appropriate mathematical methods to use. Explore the effects of systematically varying the constraints. Interpret and evaluate the data generated and identify the break-even point, checking it for confirmation. Communicate their reasoning clearly. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practices 1, 2, 3, and 4: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 8. Look for and express regularity in repeated reasoning. This lesson gives students the opportunity to apply their knowledge of the following Standards for Mathematical Content: 8.EE: 8.F: INTRODUCTION Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and pairs of simultaneous linear equations. Use functions to model relationships between quantities. Before the lesson students attempt the task individually. After reviewing responses, you formulate questions for students to consider when planning improvements to their work. At the start of the lesson, students think individually about the questions posed. Next, students work in small groups to combine their thinking and work together to produce a collaborative solution to the Baseball Jerseys task, in the form of a poster. In the same small groups, students evaluate and comment on sample responses, identifying the strengths and weaknesses in these responses and comparing them with their own work. In a whole-class discussion students compare and evaluate the strategies they have seen and used. In a follow-up lesson, students reflect for 10 minutes on their work and what they have learned. MATERIALS REQUIRED Each individual student will need a copy of the assessment task: Baseball Jerseys, some plain paper, a mini-whiteboard, pen, and eraser, and a copy of the How Did You Work? questionnaire. Each small group of students will need a large sheet of paper, some felt tipped pens, and copies of Sample Responses to Discuss. Provide calculators and graph paper only when requested. There is a projector resource to support whole-class discussions. TIME NEEDED 15 minutes before the lesson, a 70-minute lesson (or split into two shorter ones), and 10 minutes in a follow-up lesson. Timings given are only approximate. Teacher guide Comparing Value for Money: Baseball Jerseys T-1

9 For this problem, what are the advantages and disadvantages of using a graph? You may also want to use Slides P-3, P-4, P-5, and P-6 of the projector resource and the questions in the Common issues table to support this whole-class discussion. Follow-up lesson: individual reflection (10 minutes) Once students have had a chance to discuss the sample responses as a whole-class, distribute the How Did You Work? questionnaire. Ask students to spend a couple of minutes, working individually, to answer the questions. Think carefully about your work this lesson and the different methods you have seen and used. On your own, answer the review questions as carefully as you can. Some teachers give this as a homework task. Teacher guide Comparing Value for Money: Baseball Jerseys T-8

10 SOLUTIONS 1. For one or two jerseys, the one-off set-up cost means that each jersey is much more expensive from Top Print. As the number of jerseys increases, the difference in the costs becomes smaller. For 15 jerseys the costs are equal and for more than 15 the cost at Top Print becomes less than at Print It. Tabular method: Number of jerseys Cost at Print It Cost at Top Print 1 \$21 \$63 2 \$42 \$81 5 \$105 \$ \$210 \$ \$315 \$ \$420 \$ \$525 \$ \$336 \$333 Algebraic method: Top Print jerseys cost less than Print It jerseys when: 18n n n is the number of jerseys. 3n 45 n 15 Number of jerseys Print It Top Print Graphical method: By plotting the 16 lines C = 18n and 333 C = 21n where C is the cost of jerseys and n is the number of jerseys it can 20be seen that 420the two lines 405 intersect at the point (15, 315). This shows that when buying 15 jerseys it costs the same from both companies.!"#\$%&'(%!!"#!&\$#!&"#!%\$#!%"#!"\$#!""# %&# %!# %'# %\$# %(# %)# %*# )*+,-.%"/%0-.#-1#% +,-./#0/# 123#+,-./# Teacher guide Comparing Value for Money: Baseball Jerseys T-9

13 Sample Responses to Discuss: Danny What mistakes has Danny made? In what ways would Danny s mistakes affect his advice to Bill? Why has Danny calculated the cost of 16 jerseys? How could Danny s work be improved? Student materials Comparing Value for Money: Baseball Jerseys S MARS, Shell Center, University of Nottingham

14 Sample Responses to Discuss: Jeremiah Where does Jeremiah get the equation 21n = n from? Is Jeremiah s solution complete? In what ways could Jeremiah s work be improved? To help you to understand Jeremiah s work, what question(s) could you ask him? Student materials Comparing Value for Money: Baseball Jerseys S MARS, Shell Center, University of Nottingham

15 Sample Responses to Discuss: Bella Is Bella s reasoning correct? Explain your answer. What does Bella mean by \$3 cheaper? In what ways could Bella s work be improved? To help you to understand Bella s work, what question(s) could you ask her? Student materials Comparing Value for Money: Baseball Jerseys S MARS, Shell Center, University of Nottingham

16 Sample Responses to Discuss: Tanya What is not clear about Tanya s work? In what ways could Tanya s work be improved? To help you to understand Tanya s work, what question(s) could you ask her? Student materials Comparing Value for Money: Baseball Jerseys S MARS, Shell Center, University of Nottingham

17 Mark the boxes and complete the sentences that apply to your work. How Did You Work? 1 Our group solution was mathematically better than my individual solution This is because OR My individual solution was mathematically better than our group solution 2 We checked our method We checked our method by We could check our method by 3 Our solution is similar to one of the sample responses OR Our solution is different from all the sample responses Our solution is similar to Add name of sample response I prefer our method / the sample response method (circle) This is because This is because 4 The solution most useful to Bill is This is because Student materials Comparing Value for Money: Baseball Jerseys S MARS, Shell Center, University of Nottingham

18 Baseball Jerseys 1. Give Bill some advice. When should he choose PRINT IT? When should he choose TOP PRINT? 2. VALUE PRINTING never wants to be the most expensive and never wants to be the cheapest. Projector resources Comparing Value for Money: Baseball Jerseys P-1

19 Sharing Individual Solutions 1. Take turns to share your work. 2. Describe how you did the task and how you think it could be improved. 3. If explanations are unclear, ask questions until everyone in the group understands the individual solutions. Projector resources Comparing Value for Money: Baseball Jerseys P-2

20 Sample Responses to Discuss: Danny Projector resources Comparing Value for Money: Baseball Jerseys P-3

21 Sample Responses to Discuss: Jeremiah Projector resources Comparing Value for Money: Baseball Jerseys P-4

22 Sample Responses to Discuss: Bella Projector resources Comparing Value for Money: Baseball Jerseys P-5

23 Sample Responses to Discuss: Tanya Projector resources Comparing Value for Money: Baseball Jerseys P-6

24 Graph Showing Cost of Jerseys Cost (\$) Print It Top Prints Number of Jerseys Projector resources Comparing Value for Money: Baseball Jerseys P-7

25 Mathematics Assessment Project Classroom Challenges These materials were designed and developed by the Shell Center Team at the Center for Research in Mathematical Education University of Nottingham, England: Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead We are grateful to the many teachers and students, in the UK and the US, who took part in the classroom trials that played a critical role in developing these materials The classroom observation teams in the US were led by David Foster, Mary Bouck, and Diane Schaefer This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) by Alan Schoenfeld at the University of California, Berkeley, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the University of Nottingham Thanks also to Mat Crosier, Anne Floyde, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularly grateful to Carina Wong, Melissa Chabran, and Jamie McKee The full collection of Mathematics Assessment Project materials is available from MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact if this license does not meet your needs.

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