Time-Distance Graphs

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Objective

SWBAT interpret and graph functions relating time and distance traveled.

Big Idea

Each point or interval in a time-distance graph reveals a part of a story.

Launch

15 minutes

As they arrive for class today, each student receives a copy of Homeward Bound Entrance Slip. The task asks students to analyze the graph. The x-axis represents time in minutes, and, and the y-axis depicts miles away from home. I motivate students to discuss their ideas with their partners and answer the questions on the paper to the right of the graph.

I scaffold the task just a little, by telling the class that in this activity the speeds are constant, and therefore the segments between points on the graph are straight lines.  I walk around observing students working to make sure that they are reading the time-distance graph correctly.

One helpful hint that I like to give students is to say, "Let each point on the graph be a place in town that a person is visiting." As I observe the class write their stories, I discretely select one to use during the next section of the lesson.

I am also looking out for a pair of common errors that my students make when reading a graph describing motion:

1. Students may think that as you move to the right (on the graph) you are moving away from home. In this case I ask students to focus on the first two consecutive points (0, 1) and (10, 0.75). Then, I ask students to determine the distance from home at these two points. Students quickly realize that the second point is closer to home. It is not always intuitive for students to interpret closer to the x-axis as closer to home.

2. Students may not immediately comprehend that a horizontal line means that there is no motion during the interval shown. I remind students that time doesn't stop, so we need to interpret the change in distance (no change) as "zero" and then calculate the rate of change.

Activity

35 minutes

In this part of the activity, I project two pictures and ask the question below. Picture 1 contains the homeward bound graph and a student sample of their story based on the graph (selected during the Launch) . Picture 2 is a similar graph, but is used only for questions 5 and 6. The goal is to discuss specific parts of a graph in full detail. I want to help my students to appreciate how one focuses in on a particular region of a graph. Iuse the questions below to help focus my students' attention. I ask that they pair up with a partner for the entire activity, so that they can share observations with each other.

Picture 1 Questions

Q1: What does segment BC represent and what does its length mean?

Q2: What does segment EF represent?

Q3: What was this student's speed in mi/min from C to D?

Q4: When did this student travel faster, from the library (A) to the post office (B), or from the library (E) back home (F)? Explain how do you know this from the graph?

Picture 2 Questions

Q5: Is the path from C to D (vertical line) possible? Explain your answer.

Q6: How about the path from E to F, is this traveling possible? Explain.

Next, students are given the Part 2 Activity, a half sheet with a story. They must read the story and create the time-distance graph that represents it. I again walk around the watching the student pairs work together on this. One of the two always takes on the drawing of the graph but I tell the class that I want to see participation on the part of both students in the form of discussion at least. As I walk around I ask the partner not doing the drawing, questions on what is happening with their graph to keep students on their toes. I will use one of the student graphs on the document camera to discuss and close the lesson.

For students struggling or making many mistakes along the way, I help by telling them to first write the coordinates of the points (places) that Nick visits from start to finish. Then all they have to do is plot these on the graph before drawing the lines to connect these. This always works out better for all the students.

Closure

10 minutes

After about 10 minutes, I choose one student's graph to show under the document camera as a correct example. With the work displayed, I tell the class to make any corrections they need to make. Students are also encouraged to ask questions if they have any doubts about the graph. I take advantage of this moment to ask a couple of questions for students to think about:

When did Nick bike faster from A to B, going from home to the first bike shop, or from E to F, from the 2nd shop back home?

How can we tell from the graph at which shop did Nick remain longer?

I question students randomly to make sure they understand what each point and segment of the graph represents, otherwise it is difficult to understand what the graph as whole stands for.

Homework

I expect that all of my students can handle the 3 problems on tonight's Homework. I expect 100% of them to answer the first two.

In the past, it has always been worth discussing this homework at the start of the next class, so we will likely do this during tomorrow's lesson.