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Interpreting Distance-Time Graphs
During the
  • Whole-Class Introduction: Interpreting and Sketching Graphs (10 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:
      • 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.

      A graph may end up looking like this:

      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-Time 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 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.

      You have two tasks during the small-group work: to make a note of student approaches to the task, and to support student reasoning.

      Make a note of student approaches to the task:

      Listen and watch students carefully.

      Note different student approaches to the task and any common mistakes. For example, students may interpret the graph as a picture, or students may read the graph from right to left.

      Also notice the ways students check to see if their match is correct and how they explain and justify a match to each other. You can use this information to focus a whole-class discussion.

      Support student reasoning:

      Try not to make suggestions that move students towards a particular match. Instead, ask questions to help students to reason together.

      If you find one student has produced a solution for a particular match, challenge another student in the group to provide an explanation:

        • John matched these cards. Sharon, why do you think John matched these two cards?

      If you find students have difficulty articulating their decisions, then use the sheet Suggested Questions and Prompts to support your own questioning of students.

      In trials of this lesson, some students had difficulty stating where home is on the graph.

        • For this graph, where does the journey start? Is that home?
        • Give me a graph that shows a journey starting away from home.
        • For this graph, does the journey end at home? How do you know?

      If the whole class is struggling on the same issue, you could write a couple of questions on the board and hold an interim, whole-class discussion. You could ask students who performed well in the assessment to help struggling students.

      Some of the cards are deliberate distracters. For example, a student who matches Card 2 and E indicates that they think that graphs are pictures of the situation.

      Allow students time to match all the cards they can.

  • Sharing Posters (5 minutes)
      • As students finish matching the cards, ask one student from each group to visit another group's poster.
      • You may want to use Slide P-4 of the projector resource to display the following instructions.

        • If you are staying at your desk, be ready to explain the reasons for your group's matches.
        • If you are visiting another group, write your card placements on a piece of paper. Go to another group's desk, and check to see which matches are different from your own.
        • If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
        • 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, below, 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:

        • 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?
        • In what units might these be measured in?
        • Are these figures realistic?
  • Collaborative Activity: Matching Card Set C (20 minutes)
      • Hand out Card Set C: Tables of Data and ask students to match these cards in Card Set C 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 that 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.

      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:

      This may help students to understand that the table on Card Q matches Tom's hill walk, and that the correct distance-time graph should, therefore, be Card D.

      • If some students finish quickly, encourage them to devise their own pairs of cards.
  • Sharing Posters (5 minutes)
      • When students have completed the task, the student who has not already visited another pair should share their work with another pair of students. Students are to share their reasoning as they did earlier in the lesson unit.
      • Students may now want to make final changes to their poster.
      • When they are completely satisfied, ask them to glue their cards onto the large sheet of paper.
  • Whole-Class Discussion (10 minutes)
      • Using mini-whiteboards, make up some journeys and ask the class to show you the corresponding graphs.
        • On your whiteboards, draw a distance-time graph to show each of the following stories:
          • Sam ran out of his front door, then slipped and fell. He got up and walked the rest of the way to school.
          • Sara walked from home up the steep hill opposite her house. She stopped at the top to put her skates on, then skated quickly down the hill, back home again.
          • Chris cycled rapidly down the hill that starts at his house. He then slowed down as he climbed up the other side.
      • Ask students to show their whiteboards to the whole class.
      • Select some to explain their graph to the class.
      • Encourage others in the class to challenge their reasoning.
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