Falling Time Function

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Objective

SWBAT create a function that tells the time it takes an object to hit the ground in terms of the height from which the function was dropped. Students will interpret this function as an inverse function in context.

Big Idea

Drop an object off a cliff. When will it hit the ground? Students generate and analyze a radical function to answer this question.

Lesson Author

Hilary Yamtich

Oakland, CA

Grade Level

Eleventh grade

Twelfth grade

Subjects

Math

inverse functions

quadratic functions

radical algebraic expressions

Standards

HSF-IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

HSF-IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

HSF-IF.C.7b

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

HSF-BF.A.1

Write a function that describes a relationship between two quantities.*

HSF-BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

HSF-BF.B.4

Find inverse functions.

MP1

Make sense of problems and persevere in solving them.

MP2

Reason abstractly and quantitatively.

MP4

Model with mathematics.

MP5

Use appropriate tools strategically.

Warm-Up

30 minutes

While working on this warm-up, students will need to think about some of the following questions:

What is the relationship between a function and its inverse function?

If I have a quadratic function rule, how can I find the rule for its inverse?

How can I verify that I have found the inverse function correctly?

Which inputs can I use in a radical function to help me graph the function without a calculator? How can I choose these inputs systematically?

When students struggle to get started or to think about a new problem, I ask them to find a way to articulate exactly what about that problem confuses them. Often this is quite difficult, but the result of doing this is that it is much easier to make progress. For instance, students are often able to answer their own questions once they have articulated these questions. The questions above are good examples of the types of questions that we want students to be asking. So I ask myself: How can we coach them to ask these questions?

I start by telling them that I will only help them if they can ask me a question that shows you have already thought about the problem. Though this requirement may seem a little wishy-washy, it actually sets a pretty high standard for a question. This is also a great way to create a classroom culture where MP1 happens on a daily basis: if students have to think about a problem in order to even get help, then they have to do a lot of sense-making.

Once students ask such questions, I try to get them to answer their own questions. If they can’t, I ask them to ask their partner. Almost always, my students tell me that they already did ask their partner. Almost always, this is not fully true (perhaps they did, but the partner didn’t listen.) To enforce this expectation, I tell them that I will stand there with them while they ask their partner and listen to their conversation. Though this seems like a lot of time to spend with one pair of students, the more you do this, the more they will actually learn how to talk to each other effectively. Also, if they know that you are going to do this, they will be much more likely to attempt the conversation with their partner before you arrive at their table, which means that they will make much more progress.

If all else fails, I ask the students the questions listed above. Often if you generate the questions, they can more easily think about answers. I think it is better if they come up with the questions and the answers, but as they are still learning how to think this way, I model the process by telling them the questions that I would ask if I were trying to tackle these same warm-up problems.