Discuss the meaning of "invariant" with example of changing area but invariant perimeter.
A loop of string can make different shapes all having the same perimeter but having very different areas. How do you know the area is different? Two loops of equal length can prove it: make a large circle with one string, then make a wiggly shape with the other one inside the circle. Since the second shape is completely contained by the first, it must have a smaller area but an equal perimeter! (I like this example because it exactly reverses the invariance the students will examine in the coming problem.)
Coming Soon: Classroom video of this activity!
A document camera is useful for this quick demonstration.
Once the notion of invariance is understood, hand out the "Tiger, Tiger, Burning Bright..." problem.
Expectation: Students will complete the graph & table on page 1.
Teacher: Some students will need encouragement to see that in this case a 4x9 rectangle may be considered different from a 9x4 rectangle. This is reasonable because we're interested in the full range of values that the length of the rectangle may take on, and both 4 and 9 should be included.
It may be worth asking students to explain why it is reasonable to "connect the dots" on their graph. What does that imply about the number of different rectangles that may be formed?
(By the way, if you or your students are into poetry, you might check out this website for a reading of Blake's Tyger poem. In fact, it might be fun to have this recording playing as students enter the room!)
1) Compare and correct graph & table (parts 1 and 2)
2) Create an equation to model the relationship of length to width.
3) Answer questions 3a and 3b, regarding limitations on the dimensions of the pen and the rate of change of the dimensions with respect to one another.
1) Ensure that students have correct equation.
2) Ask students about mathematical vs. situational limitations relative to question 3a. What does the mathematical model permit that the actual situation does not? Negative lengths? Impossibly small or large lengths? (MP 2 & MP 4)
3) Push students to explain in an intuitive way why the length and width do not change at the same rate or by the same amount? (MP 3) Encourage the best students to explore this question algebraically as you wait for others to complete the three expectations.
Please see this video for reflections: Constant Area Day 1, Video Narrative, Group Time
At this point, different groups or individuals will be asked to come to the front of the room to share and explain their work and then to answer any questions their peers may have. I find it's best to use a document camera to make this go smoothly.
After briefly looking at the graph, the table, and the equation, I'll ask students to volunteer to verbally share their answers to questions 3a and 3b. I'll use these initial explanations as a launchpad for the following discussion. Please see the solutions document with included content standards for details.
Discuss the importance of paying attention to the limitations of mathematical model, as well as its strengths. These three questions will frame the conversation:
1) What does the mathematical model (equation & graph) reveal to us about the situation that we might not have seen otherwise?
2) What is the domain of the function mathematically? What is the domain in the context of the problem?
3) Is the function continuous mathematically? Is it reasonably continuous in context? (Would you ever choose to make one dimension an irrational length?)
Discuss, if time permits, why the area must change if both length and width are changed by the same amount. The sophistication of this conversation will depend on how much time you have and on the interest or ability of the students.