This lesson is a continuation of the previous lesson.
The goals for your students are to:
Mathematics is an important tool that will help them meet these challenges. Creating and using mathematical representations is at the heart of a physicist's world view; such representations make it possible to communicate a motion unambiguously to others, compare motions, combine them, identify where acceleration takes place, and create testable hypotheses. It's important to be able to describe motions in words, but you will need mathematical representations to "get under the skin" of the physics.
Richard Feynman expressed the way he thinks about the relationship between mathematics and physics this way:
If you are interested in the ultimate character of the physical world, or the complete world, and at the present time our only way to understand that is through a mathematical type of reasoning, then I don't think a person can fully appreciate, or in fact can appreciate much of, these particular aspects of the world, the great depth of character of the universality of the laws, the relationships of things, without an understanding of mathematics.
(The Pleasure of Finding Things Out, p. 14.)
Science Practices Addressed:
In order to compare and analyze the motions it is helpful to have a graphic representation that shows more directly how speed changes or does not change over time. How might you graphically represent the video strobe data you collected?
Students being by reading About Graphing Speed Over Time, then sketch seven speed over time graphs that correspond with the strobe data gathered in the previous lesson:
Speeding Up (rapidly/gradually)
Slowing Down (rapidly/gradually)
As they draw their graphs, they should note questions they have about particular motions that are challenging to represent. What do they find challenging? In the next part of the lesson, they will work with these graphs in more detail.
In the video below I highlight how this lesson utilizes science practices 4-6. Take a look:
In this part of the investigation, students contrast different variations of each signature motion.
In addition to representing whether the cart is speeding up, slowing down or going at a constant speed, they will represent how quickly each motion is happening. Does the cart speed up gradually? Does it slow down abruptly?
They should use the cart to study the observable differences in each set of motions.
This part of the lesson ends with students combining the individual components of motion to create a complex motion graph. See the images below of student work.
If you have not taught students how to graph motion in this manner before, I'd suggest that you practice these graphs on your own before teaching them. Also, refer to the student samples included here. The abstract nature of making these graphs can be challenging for students at all levels. I have, in the past, taught these skills through direct instruction. I stand in front of the class at the whiteboard and take the students through each graph. They record in their journals the various types of motions – together we create the complex motion graphs and check our answers.
Now that your students have had practice graphing motion in only one direction, it's time to push their thinking and learn to graph motion in the opposite direction. Begin by asking them to create a prediction graph of what they think would represent the following scenario,
“You pull into the garage, stop, then back out.”
Give students time to think and write/draw, then ask for volunteers to come and draw their graphs on the board. If you have access to the individual whiteboards this is a great activity to use those with. Hand each student a whiteboard and marker and asked them to create their prediction graph on the whiteboard. When ready, then have students hold the graph up and show everyone.
Here is the graph:
Most of my students are unable to figure out this graph the first time through. What is surprising to students is that in order to solve the problem, they have to extend the graph into the lower quadrant. It's confusing to them because the values on the y-axis in this quadrant are negative. Physicists use positive and negative numbers to represent many pairs of opposite motions. For this lesson they use positive to mean motion from left to right and negatives describe motion from right to left. In a later lesson, they use positive and negative motion to indicate a different set of opposite notions. But the principle is the same; whichever side the positive motion appears, negative motion is in the opposite direction.
After you have discussed the prediction graph of motion in the opposite direction, have students read the one-page description titled About Graphing Motion in Opposite Directions.
Once they have read this, they proceed with Combining motions-Opposite direction. As in the early part of this lesson, they start by defining the building blocks of graphing motion in the opposite direction focusing on the four signature motions. The students then combine these to create complex motion story involving motions in the opposite direction.
Up to now, students have been representing simple motions. But motions don't exist in isolation. Objects don't just move forward at a constant speed and then stop, or accelerate in the opposite direction and go on forever.
Students create motion stories using the building blocks of motions. They begin by analyzing five Complex Motion Stories in their journals. For each story, they simulate the story using the cart and then draw a velocity over time graph.
They should annotate each segment of the graph, explaining what is happening in their representation. Once students finish with the complex motion stories, they move on to looking at 10 Motion Story Graphs.
For each graph they should label the piece of the graph with the kind of motion it represents. Then for two of the graphs, I ask my students to write a story that corresponds with the graphs. The story should involve forward and backward motion along a straight track, such as a sidewalk, tight rope or a driveway, but they are allowed to be imaginative.
Once created have students share the stories. They come up with some pretty ingenious ideas, that are quite hilarious!
COMPLEX MOTION STORIES
To help your students connect what they are learning in class with day-to-day experiences, ask them to revisit Motions in Your Life, which was completed in lesson one of this unit. They should analyze the motion with an eye towards identifying signature motions and note the parts of the complex motion that don't fit neatly into the four signature motions that have looked at so far. As part of their analysis they should create a motion story graph. This can be a homework assignment to be shared the next day.