Interpreting a Cartesian graph of objects in motion is a process of understanding the representation of the relationship between selected elements. This entry describes how students attempt to make sense of graphs related to problems of relative motion.
|Marna and Peter are two students who are standing one meter apart. They start walking in a straight line one behind the other. Marna is walking behind Peter carrying a calculator and CBR to measure the distance between them. They walk for 7 seconds. Interpret the graph.||Many students will have difficulty interpreting the graph because both students are moving. They may see the horizontal part of the graph as representing that one of the people stopped, when in fact in means that the distance between the two people stayed the same. They may have trouble interpreting what is happening at points A, B, C, and D.|
PROBLEM #1 (Comments in Italics)
|S1:||Peter moves away from Marna for 3 seconds, then he stops for 2 seconds, and then he moves closer to Marna. for 2 seconds.|
|S2:||Well, even though he moves away, he moves back again, I don’t know.|
|S3:||Well, if she walks with him, the graph really doesn’t make sense.
S1’s interpretation is based on Peter moving and Marna remaining in one place.Peter moves away, stops and moves back. S2 makes an unsuccessful attempt to interpret the graph. S3 reminds the group that Marna is also moving.
|S1:||So from A to B Peter is walking faster than Marna. Right?|
|S2:||That could be and then Peter stops from B to C.|
|S3:||No, they are the same distance apart from B to C so they are walking at the same speed.
The interpretation of the graph has changed to Peter walking faster than Marna to Peter stopping to Peter and Marna walking at the same speed. The interpretation that Peter is walking faster than Marna could be coming from the positive slope of AB.
|I:||If Peter is walking faster than Marna, will the distance between them remain the same?|
|S1:||No. What happens between B and C?|
|I:||Is the CBR still moving?|
|S3:||Yes...No I don’t know. Did Marna stop? She has the CBR.
The conversation is changing from relative speed to relative distance.
|I:||What does the point A represent?|
|I:||Is that a distance or a speed?|
|I:||What is the distance measuring?|
|S2:||No, from B to C they are walking at the same speed because the distance between them is staying the same.
A reasonable explanation for BC has been made by S2.
|S3:||Then from C to D the distance between Peter and Marna is decreasing so Marna would be walking faster than Peter. Does that make sense?
S1 and S2 agree Students have come to a reasonable interpretation of the graph that makes sense.
Cartesian graphs are used to describe real world phenomena and convey complex mathematical meanings. The logic of interpreting a Cartesian representation of relative motion can become apparent through discussion as opposed to direct instruction.
Provide time for small group work. Allow students to express and record their interpretation of the graph. Listen to student thoughts. Ask questions that help students clarify their interpretation of the graph as shown below. Break the graph into pieces and establish meaning for each piece.
Questions for Discussion:
1. What does point A represent? (The distance between Marna and Peter at time t=0.)
2. What is happening over segment AB? (The distance between Marna and Peter is increasing.)
3. What is happening over segment BC? (The distance between Marna and Peter is staying the same.)
4. What is happening over the segment CD? (The distance between Marna and Peter is decreasing.)
Radford, L., Miranda, I., & Guzman, J. (2008). Relative motion, graphs, and the heteroglossic transformation of meanings: A semiotic analysis. Paper presented at the Joint 32nd Conference of the International Group for the Psychology of Mathematics Education and the 30th North American Chapter.
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?