standard_description: Know and apply the Binomial Theorem for the expansion of (x + y)^(n )in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
problem_activity: Warm-up
The purpose of this warm-up is for students to jump back into recognizing functions and determining if two quantities are a function of each other. Students are asked questions similar to these throughout this lesson, and the discussion of this warm-up is meant to get students using the language of functions, which continues throughout the rest of the activities.
Launch
Give students 1–2 minutes of quiet work time, followed by a whole-class discussion.
Student Facing
Here is a graph of the amount of gas burned during a trip by a tractor-trailer truck as it drives at a constant speed down a highway:
[IMG]
At the end of the trip, how far did the truck drive, and how much gas did it use?
If a truck traveled half this distance at the same rate, how much gas would it use?
If a truck traveled double this distance at the same rate, how much gas would it use?
Complete the sentence: ___________ is a function of _____________.
Student Response
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Anticipated Misconceptions
If students struggle with the second and third questions, ask them if this is an example of a proportional relationship. Remind them what they learned previously about proportional relationships and how that can be used to answer these questions.
Activity Synthesis
Invite students to share their answers and their reasoning for why gas burned is a function of distance traveled. Questions to further the discussion around functions:
“Looking at the graph, what information do you need in order to determine how much gas was used?” (We need to know the number of miles traveled.)
“What is the independent variable? Dependent variable? How can you tell from the graph?” (The independent value is the distance traveled. The dependent value is the gas burned. By convention, the independent is on the
\(x\)
-axis and the dependent is on the
\(y\)
-axis.)
“What are some ways that we can tell from the graph that the relationship between gas burned and distance traveled is proportional?” (The graph is a line that goes through the origin, we can see a constant ratio between
\(y\)
and
\(x\)
in some points like
\((80,10), (160, 20)\)
, and
\((240, 30)\)
.)


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