standard_description: (+)Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
problem_activity: Activity
This activity prompts students to compare and contrast two representations of equivalent ratios. Students work collaboratively to observe similarities and differences of using a double number line and using a table to express the same situation. Below are some key distinctions:
|                   double number line                   |                                                table                                                 |
|--------------------------------------------------------|------------------------------------------------------------------------------------------------------|
| Distances between numbers and lengths of lines matter. |                   Distances and lengths do not matter because there are no lines.                    |
|       The numbers on each line must be in order.       | Rows of ratios can be out of order; within a column, numbers can go in any order that is convenient. |
|       Each value of a ratio is shown on a line.        |                             Each value of a ratio is shown in a column.                              |
|   Pairs of values of a ratio are aligned vertically.   |                          Pairs of values of a ratio appear in the same row.                          |
You will need The International Space Station blackline master for this activity.
Launch
To help students build some intuition about kilometers, begin by connecting it with contexts that are familiar to them. Tell students that “kilometer” is a unit used in the problem. Then ask a few guiding questions.
“Can you name two things in our town (or city) that are about 1 kilometer apart?” (Consider finding some examples of 1-kilometer distances near your school ahead of time.)
“How long do you think it would take you to walk 1 kilometer?” (Typical human walking speed is about 5 kilometers per hour, so it takes a person about 12 minutes to walk 1 kilometer.)
“What might be a typical speed limit on a highway, in kilometers per hour?” (100 kilometers per hour is a typical highway speed limit. Students might be more familiar with a speed limit such as 65 miles per hour. Since there are about 1.6 kilometers in every mile, the same speed will be expressed as a higher number in kilometers per hour than in miles per hour.)
Arrange students in groups of 2. Give one person a slip with the table and the other a slip with a double number line (shown below). Ask students to first do what they can independently, and then to obtain information from their partners to fill in all the blanks. Explain that when the blanks are filled, the two representations will show the same information.
[IMG]
| distance traveled (km) | elapsed time (sec) |
|------------------------|--------------------|
|           0            |         0          |
|           80           |         10         |
|                        |         1          |
|                        |                    |
|                        |                    |
Student Facing
The International Space Station orbits around the Earth at a constant speed. Your teacher will give you either a double number line or a table that represents this situation. Your partner will get the other representation.
[IMG]
Complete the parts of your representation that you can figure out for sure.
Share information with your partner, and use the information that your partner shares to complete your representation.
What is the speed of the International Space Station?
Place the two completed representations side by side. Discuss with your partner some ways in which they are the same and some ways in which they are different.
Record at least one way that they are the same and one way they are different.
Student Response
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Student Facing
Are you ready for more?
Earth’s circumference is about 40,000 kilometers and the orbit of the International Space Station is just a bit more than this. About how long does it take for the International Space Station to orbit Earth?
Student Response
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Anticipated Misconceptions
Students with the double number line representation may decide to label every tick mark instead of just the ones indicated with dotted rectangles. This is fine. Make sure they understand that the tick marks with dotted rectangles are the ones they are supposed to record in the table.
Activity Synthesis
Display completed versions of both representations for all to see. Invite students to share the ways the representations are alike and different. Consider writing some of these on the board, or this could just be a verbal discussion. Highlight the distinctions in terms of distances between numbers, order of numbers, and the vertical or horizontal orientations of the representations.
Although it is not a structural distinction, students might describe the direction in which multiplying happens as a difference between the two representations. They might say that we “multiply up or down” to find equivalent ratios in a table, and we “multiply across” to do the same on a double number line. You could draw arrows to illustrate this fact:
[IMG]
The vertical orientation of tables and the horizontal orientation of double number lines are conventions we decided to consistently use in these materials. Mathematically, there is nothing wrong with orienting each representation the other way. Students may encounter tables oriented horizontally in a later course. Later in this course, they will encounter number lines oriented vertically.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, highlight the distinctions in terms of distances between numbers, order of numbers, and the vertical or horizontal orientations of the representations.
Supports accessibility for: Conceptual processing; Organization


aligns: no