# Engaging students: Finding the domain and range of a function

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Sydney Araujo. Her topic, from Precalculus: finding the domain and range of a function.

How can this topic be used in your studentsâ€™ future courses in mathematics or science?

Expanding on finding the domain, this topic is frequently seen in calculus classes. Students need to understand the domain to understand and find limits of functions. Continuity directly expands on domain & range and how it works. We also see domain and range when students are exploring projectile motion. This makes since because when we think about projectile motion, we think about parabolas. With projectile motion there is a definite start, end, and peak height of the projectile. So we can use the domain to show how far the projectile travels and the range to show how high it travels. Looking even further ahead when students start to explore different functions and sets, they start to learn about a codomain and comparing it to the range which is a very valuable concept when you start to learn about injective, surjective, and bijective functions.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Desmos is a great website for students to use when exploring domain and ranges. Desmos has premade inquiry-based lessons for students to explore different topics. Teachers also have the option of creating their own lessons and visuals for their students to interact with. Desmos can also animate functions by showing how they change with a sliding bar or actually animate and show it move. This would be a great tool to use for students to visually understand domain and ranges as well as how they are affected when asymptotes and holes appear. This would also be great for ELLs because instead of focusing on just math vocabulary, they can actually visually see how it connects to the graph and the equation. For example, https://www.desmos.com/calculator/vz4fjtugk9, this ready-made desmos activity actually shows how restricting the domain and range effects the graph and what parts of the graph are actually included with the given domain and range.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Like I discussed earlier, domain and range is directly used in calculus. In the movie Stand and Deliver, they directly discuss the domain and range of functions. The movie Stand and Deliver is about a Los Angeles high school teacher, Jaime Escalante, who takes on a troublesome group of students in a not great neighborhood and teaches them math. He gets to the point where he wants to teach them calculus so they can take the advanced placement test. If they pass the advanced placement test then they get college credit which would motivate them to actually go to college and make a better life for themselves. However through great teaching and intensive studying, the students as a whole ace the exam but because of their backgrounds they are accused of cheating and must retake the exam. There is a few scenes, but one in particular where the students are finally understanding key concepts in calculus and Mr. Escalante is having them all say the domain of the function repeatedly.

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