# Interpreting Distance-Time Graphs

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1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Interpreting Distance- Graphs 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

5 SUGGESTED LESSON OUTLINE If you have a short lesson or you find the lesson is progressing at a slower pace than anticipated, we suggest you break the lesson after the first sharing of posters and continue it at a later time. Whole-class introduction: interpreting and sketching graphs (15 minutes) Throughout this activity, encourage students to articulate their reasoning, justify their choices mathematically, and question the choices put forward by others. This introduction will provide students with a model of how they should work with their partners in the first small-group activity. Show the class the projector resource Matching a Graph to a Story: Matching a Graph to a Story A. Tom took his dog for a walk to the park. He set off slowly and then increased his pace. At the park Tom turned around and walked slowly back. B. Tom rode his bike east from his up a steep hill. After a while the slope eased off. At the top he raced down the other side.!"#\$%&'() *+,-).,-( C. Tom went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tom left his friend he walked quickly back. /"-( Projector Resources Interpreting Distance- Graphs Ask students to match the correct story to the graph. They are to write down at least two reasons to support their decision. After two or three minutes ask students who selected option A to raise their hands. Ask one or two to justify their choice. You may wish to use some of the questions on the sheet Suggested questions and prompts to encourage students to justify their choices and others to challenge their reasoning. Repeat this with options B and C. Even if explanations are incorrect or only partially correct, write them next to the appropriate section of the graph. Encourage students to challenge these interpretations. Slide P-2 of the projector resource allows you to write three different student explanations on the board at the same time. P-1 Teacher guide Interpreting Distance- Graphs T-4

6 A graph may end up looking like this: Line not too steep - this means Tom slows down. Furthest Tom gets from. Distance from Negative slope means Tom is walking back to his. Tom starts from Tom returns. This is how students should annotate their graphs when working on the collaborative task. Collaborative activity: matching Card sets A and B (20 minutes) Ask students to work in small groups of two or three students. Give each group the Card Set A: Distance Graphs, and Card Set B: Interpretations together with a large sheet of paper, and a glue stick for making a poster. You are now going to continue exploring matching graphs with a story, but as a group. You will be given ten graph cards and ten story cards. In your group take a graph and find a story that matches it. Alternatively, you may want to take a story and find a graph that matches it. Take turns at matching pairs of cards. Each time you do this, explain your thinking clearly and carefully. If you think there is no suitable card that matches, write one of your own. Place your cards side by side on your large sheet of paper, not on top of one another, so that everyone can see them. Write your reasons for the match on the cards or the poster just as we did with the example in class. Give explanations for each line segment. Make sure you leave plenty of space around the cards as, eventually, you will be adding another card to each matched pair. The purpose of this structured group work is to encourage students to engage with each other's explanations and take responsibility for each other s understanding. Slide P-3 of the projector resource summarizes these instructions. Teacher guide Interpreting Distance- Graphs T-5

8 When you return to your own desk, you need to consider as a group whether to make any changes to your own poster. Students may now want to make changes to their poster. At this stage there is no need for students to glue the cards onto their posters as they may decide to make further changes. If you need to extend the lesson over two days: Once students have finished sharing posters, organize a whole-class discussion. Invite pairs of students to describe one pair of cards that they think they have matched correctly and the reasoning they employed. Encourage other students to challenge their explanations. Finally, ask students to note their matches on the back of their poster and to use a paperclip to attach all cards to the poster. At the start of the second lesson, spend a few minutes reminding the class about the activity: Can you remember what we were working on in the last lesson? Return the posters to each group. The whole-class discussion on interpreting tables can serve as an introduction to the lesson. Whole-class discussion: Interpreting tables (15 minutes) Bring the class together and give each student a mini-whiteboard, a pen, and an eraser. Display Slide P-5 of the projector resource: Making Up Data for a Graph Distance 0!"#\$%&'() *+,-).,-(! " # \$ % &! /"-( Projector Resources On your whiteboard, create a table that shows possible times and distances for Tom s journey. After a few minutes, ask students to show you their whiteboards. Ask some students to explain how they created their tables. Write their figures on the board. Ask the rest of the class to check these figures. Is Tom s speed slower or faster in this section compared to that section? How do you know from the graph? From the table? Is this speed constant? How can you tell? Do the figures in the table show a constant speed for this section of the journey? What units might these be measured in? Are these figures realistic? Interpreting Distance- Graphs P-5 Teacher guide Interpreting Distance- Graphs T-7

9 Collaborative activity: matching Card Set C (20 minutes) Hand out Card Set C: Tables of Data and ask students to match these cards with the cards already on their poster. You are now going to match tables with the cards already on your desk. In your group take a graph and try to find a table that matches it, or take a table and find a graph that matches it. Again take turns at matching cards you think belong together. Each time you do this, explain your thinking clearly and carefully. Write your reasons for the match on the poster. Students may also wish to suggest suitable units for the distances and times on the cards. The tables should help students confirm or modify existing matches. As they work on the matching, support the students as in the previous matching activity. In the past, some students have had difficulty understanding the repetition in Table R. The table is intended to show the impossibility of Graph H. R Distance H Some teachers have found that it helps students to look at the average speeds between consecutive times, by calculating differences. For example, average speeds for Table of Data Q would look like this. A Q Distance Average speed B This may help students to understand C that the table on Card Q matches Tom s hill walk, and that the correct distance time graph should therefore be Card D. D E If some students finish quickly, encourage them to devise their own pairs of cards. F Teacher guide Interpreting Distance- Graphs T-8

11 SOLUTIONS Assessment task: Journey to the Bus Stop 1. The straight lines indicate that Tom moves at a constant but different speed in each section. Tom walks a total of 280 yards ( ). He is walking for 100 seconds. The graph shows Tom s journey is split into four sections. A In this section of the journey Tom walks away from at a speed of 2 meters per second (100 50) for 50 seconds. B The negative slope here means a change in direction. At 100 meters from Tom starts to walk towards. He walks for 60 meters at a speed of 3 meters per second (60 20). C At the start of this section Tom changes direction. He is now walking away from at a fast pace. His speed is 4 meters per second (120 30). He moves at this speed for 30 seconds and covers 120 meters. D Here the slope is zero. This means at 160 meters from Tom stops. It has taken him 100 seconds to get to this point. 2. The speeds provided in the answer to question 1 are realistic. A speed of 2 meters per second is a brisk walk. A speed of 4 meters per second means Tom is running. Post Assessment task: Journey Home 1. The straight lines indicate that Sylvia moves at a constant but different speed in each section. The overall journey takes 90 minutes and Sylvia bikes a total of 10 miles ( ). The graph shows Sylvia s journey is split into four sections. A In this section of the journey Sylvia bikes towards at a speed of 4 miles per hour (2 0.5) for 30 minutes. B Here the slope is zero. Six miles from Sylvia has stopped for 20 minutes. C The positive slope here means a change in direction. After 50 minutes Sylvia bikes away from for 1 mile. She bikes at a speed of 6 miles per hour (1 10/60). in miles D In this section Sylvia spends 30 minutes biking 7 miles. Her speed is 17 miles per hour (7 1/2). Teacher guide Interpreting Distance- Graphs T A A B C B Minutes C D D

12 2. Although unrealistic for anyone to bike at a constant speed the average speeds for each section are realistic, although for the first 60 minutes Sylvia is not biking very fast. Maybe the wind was against her! Collaborative activity Graph Interpretation Table Graph Interpretation Table A 5 W B 10 S C 4 V D 2 Q E 6 T F 3 G 1 P H 8 R I 7 U J 9 X Teacher guide Interpreting Distance- Graphs T-11

13 Journey to the Bus Stop Every morning Tom walks along a straight road from his to a bus stop, a distance of 160 meters. The graph shows his journey on one particular day. 1. Describe what may have happened. You should include details like how fast he walked. 2. Are all sections of the graph realistic? Fully explain your answer. Student materials Interpreting Distance- Graphs S MARS, Shell Center, University of Nottingham

14 Card Set A: Distance Graphs A B C D E F Student materials Interpreting Distance- Graphs S MARS, Shell Center, University of Nottingham

15 Card Set A: Distance Graphs (continued) G H I J Student materials Interpreting Distance- Graphs S MARS, Shell Center, University of Nottingham

16 Card Set B: Interpretations 1 Tom ran from his to the bus stop and waited. He realized that he had missed the bus so he walked. 2 Opposite Tom s is a hill. Tom climbed slowly up the hill, walked across the top, and then ran quickly down the other side. 3 Tom skateboarded from his house, gradually building up speed. He slowed down to avoid some rough ground, but then speeded up again. 4 Tom walked slowly along the road, stopped to look at his watch, realized he was late, and then started running. 5 Tom left his for a run, but he was unfit and gradually came to a stop! 6 Tom walked to the store at the end of his street, bought a newspaper, and then ran all the way back. 7 Tom went out for a walk with some friends. He suddenly realized he had left his wallet behind. He ran to get it and then had to run to catch up with the others. 8 This graph is just plain wrong. How can Tom be in two places at once? 9 After the party, Tom walked slowly all the way. 10 Make up your own story! Student materials Interpreting Distance- Graphs S MARS, Shell Center, University of Nottingham

17 Card Set C: Tables of Data P Distance S Distance V Distance Q Distance T Distance W Distance R Distance U Distance X Distance Y Make this one up! Distance Z Make this one up! Distance Student materials Interpreting Distance- Graphs S MARS, Shell Center, University of Nottingham

18 Journey Home Sylvia bikes along a straight road from her friend s house, a distance of 7 miles. The graph shows her journey in miles Minutes 1. Describe what may have happened. You should include details like how fast she bikes. 2. Are all sections of the graph realistic? Fully explain your answer. Student materials Interpreting Distance- Graphs S MARS, Shell Center, University of Nottingham

19 Matching a Graph to a Story A. Tom took his dog for a walk to the park. He set off slowly and then increased his pace. At the park Tom turned around and walked slowly back. B. Tom rode his bike east from his up a steep hill. After a while the slope eased off. At the top he raced down the other side.!"#\$%&'() *+,-).,-( C. Tom went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tom left his friend he walked quickly back. /"-( Projector resources Interpreting Distance- Graphs P-1

20 A. Tom took his dog for a walk Explaining to the park. He set off the match slowly and then increased his pace. At the park Tom turned around and walked slowly back.!"#\$%&'() *+,-).,-( /"-( B. Tom rode his bike east from his up a steep hill. After a while the slope eased off. At the top he raced down the other side.!"#\$%&'() *+,-).,-( C. Tom went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tom left his friend he walked quickly back.!"#\$%&'() *+,-).,-( /"-( /"-( Projector resources Interpreting Distance- Graphs P-2

21 Matching Cards Take turns at matching pairs of cards. You may want to take a graph and find a story that matches it. Alternatively, you may prefer to take a story and find a graph that matches it. Each time you do this, explain your thinking clearly and carefully. If you think there is no suitable card that matches, write one of your own. Place your cards side by side on your large sheet of paper, not on top of one another, so that everyone can see them. Write your reasons for the match on the cards or the poster just as we did with the example in class. Give explanations for each line segment. Make sure you leave plenty of space around the cards as, eventually, you will be adding another card to each matched pair. Projector resources Interpreting Distance- Graphs P-3

23 Making Up Data for a Graph Distance 0!"#\$%&'() *+,-).,-(! " # \$ % &! /"-( Projector resources Interpreting Distance- Graphs P-5

24 Mathematics Assessment Project CLASSROOM CHALLENGES This lesson was designed and developed by the Shell Center Team University of Nottingham, England Malcolm Swan, Clare Dawson, Sheila Evans, Colin Foster and Marie Joubert with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead It was refined on the basis of reports from teams of observers led by David Foster, Mary Bouck, and Diane Schaefer based on their observation of trials in US classrooms along with comments from teachers and other users. This project was conceived and directed for MARS: Mathematics Assessment Resource Service by Alan Schoenfeld, Hugh Burkhardt, Daniel Pead, and Malcolm Swan and based at the University of California, Berkeley We are grateful to the many teachers, in the UK and the US, who trialed earlier versions of these materials in their classrooms, to their students, and to Judith Mills, Mat Crosier, Anne Floyde, Michael Galan, Nick Orchard, and Alvaro Villanueva who contributed to the design. 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 2015 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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