Fundamental Features of Functions: Activities that Promote Student Problem-Solving for Various Types of Functions
Lee Ann Brown, Oklahoma State University
Katherine Hiebert Brumley, The Jimmy Naifeh Center
Thomas McNamara, Southwestern Oklahoma State University
Melissa Mills, Oklahoma State University
Quan Tran, University of Science and Arts of Oklahoma
This collection of activities investigates fundamental features of mathematical functions from a College Algebra or Pre-Calculus course. Influenced by the work of Dan Meyer, we created interesting problems with varying amounts of information given to the students. Students need to think about what pieces of information they need and use their resources to find reasonable values for the quantities that they deem important. We have written a teacher’s guide for each activity that outlines how we think that the lesson should go and questions that the teacher can pose to emphasize certain aspects of the problem.
Linear Functions
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Pool Party |
Students are asked to write a linear function to describe the volume of water in a pool being filled by a hose and/or a tanker truck. The idea of the rate of change is emphasized in this unit, since it is the flow rate of the water being poured into the pool. Students must decide how long it takes to fill the entire pool, given that it is already partly filled. They must decide between a slow but cheap garden hose or a much more expensive but faster tanker truck. |
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Candle Conundrum |
Students are given two measurements of a candle at different times and are asked to create a model of the candle height over time. They are challenged to find the time in which the candle will burn out, and the time at which the candle was lit. These questions give the student opportunities to work with a linear function that has a negative slope. They also have to grapple with what they want to make t = 0 represent (the time when you first looked at the burning candle, or the time that the candle was lit?) these choices can lead to different mathematical models. |
Exponential Functions
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Grandpa’s Statements |
This activity deals with exponential growth, which is often covered before exponential decay, so instructors may want to take that into account. In this activity, you find three old bank statements and have to answer questions about the initial amount deposited and how much is in the account today. |
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Dollhouse Diner |
Students are given a reduced copy of a menu and asked to determine the percentage by which it was reduced. They will then discuss the number of times that the menu will need to be reduced at the same percentage reduction to be the right size for a Barbie doll menu. This gives students experience with estimating percentage reductions and an example of exponential decay. |
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Decaying Dice |
Students are given several dice and are asked to roll the dice and get rid of all dice displaying a certain number. This process is repeated multiple times until all of the dice have been removed. This gives students a chance to collect real data that can be modeled with an exponentially decaying function. This activity also includes an Excel simulation so that classes can do the experiment with any number of dice. |
Quadratic & Polynomial
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My First Goat Farm |
Students will model a quadratic function based on maximizing the area of a goat pasture using a fixed amount of fencing material. The problem asks students to make one, two, or more separate adjacent enclosures that use fencing on only three sides. |
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Boxing Match |
Students will make open-topped boxes out of sheets of cardstock. They will each make boxes of different volumes. They will collect the height vs. volume data for all of the boxes they create in their groups and as a class and plot them as a scatterplot. Finally, students will create a function for volume as a function of height. This activity gives students hands-on experience with creating boxes, which is a common scenario in Calculus related optimization problems. |
Rational Functions
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Pace Yourself |
This activity centers on a runner who wants to convert their speed in miles per hour to their running pace in minutes per mile. Students are asked to try to construct an initial graph and then refine their graph as they are asked more questions about what happens when the runner runs very slow and very fast. Students will learn to make sense of the asymptotic behavior near zero and as x approaches infinity. |
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Shark Tank |
This activity deals with the salinity of a shark tank as water evaporates. The scenario is based on actual tank dimensions and salinity from the Oklahoma Aquarium Bull Shark exhibit. Students will use information about the shark tank to draw a graph sketch and construct a rational function. The vertical asymptote for this function is not at zero, so it gives students a real scenario for thinking about a vertical asymptote. |
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Painter’s Assistant |
A professional painter needs to choose the best assistant for the job. Assistants have different painting rates and pay rates. The painter can construct a function to find the optimal assistant for the job. This gives a context for a nonzero horizontal asymptote. |
Trigonometric Functions
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Ferris Wheel Frenzy |
This activity involves a fictitious radio contest held at the Ferris wheel located in the Oklahoma City Wheeler district. Students will be asked to determine the winner of the contest based on the heights of riders in each of the 20 gondolas at a given time. There are multiple paths to solutions in this trigonometric problem, where students must use the dimensions of the Ferris wheel and its turning rate to model a trigonometric function. |
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Piering Into Darkness |
A video clip from the television show Bob’s Burgers sets up the problem of saving two men who are tied to a pier as the water level at the beach is rising. Students are tasked with determining how long they have to save the two men. The problem requires students to write a trigonometric function to represent the tide level at a given time, and to solve the equation graphically or algebraically. |
This work is licensed under CC BY-NC-SA 4.0
