Engaging students: Defining a function of one variable

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 again comes from my former student Phuong Trinh. Her topic, from Algebra: defining a function of one variable. How have different cultures throughout time used this topic in their society?

The understanding of functions is crucial in the study of both math and science. Not only that, some functions, especially function with one variable, are often used by everyone in their daily life.  For example, a person wants to buy some cookies and a cake. The person will need to figure how much it will cost them to buy a cake and however many cookies they want. If the cost of the cake is $12, and the price for each cookie is$1.50, the person can set up a function of one variable to find the total cost for any number of cookies, expressed as c. The function can be written as f(c) = 1.50c + 12. With this function, the person can substitute any number of cookies and find out how much they would spend for the cookies and cake. Aside from the situation given by this example, function with one variable can also be used in various different scenarios. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Function with one variable can be used in many real life situations. Word problems can be derived from every day scenarios that the students can relate to.

Problem 1: John is transferring his homework files into his flash drive. This is the formula for the size of the files on John’s drive S (measured in megabytes) as a function of time t (measured in seconds): S (t) = 3t + 25

How many megabytes are there in the drive after 10 seconds?

This problem allows the students to get familiar with the function notation as well as letting the students work with a different variable other than x.

Problem 2: (Found at https://www.vitutor.com/calculus/functions/linear_problems.html )

“A car rental charge is $100 per day plus$0.30 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?”

Besides giving the students practice with finding a solution from a function, this problem let the students practice setting up the equation. This also shows the students’ understanding of the subject. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

There are multiple resources that can be used to help the students understand what a function is as well as how they should approach a problem with function. One of the resources can be found at coolmath.com. The layout of the website makes it easy to locate the topic of “Functions” under the “Algebra” tab. By comparing a function with a box, Coolmath defines a function in a way that can be easily understood by students, while also showing how a function can be thought of as visually. The site also provides the explanation for function notation with visuals and examples that are easy to understand. On Coolmath, the students will also have the chance to practice with randomly generated questions. They can also check their answers afterward. On other hands, the site also provides definitions and explanations to other ideas such as domain and range, vertical line tests, etc. Overall, coolmath.com is great to learn for students in and out of the classroom, as well as before and after the lesson.

http://www.coolmath.com/algebra/15-functions

References:

“Linear Function Word Problems.” Inicio, www.vitutor.com/calculus/functions/linear_problems.html.

“Welcome to Coolmath.” Cool Math – Free Online Cool Math Lessons, Cool Math Games & Apps, Fun Math Activities, Pre-Algebra, Algebra, Precalculus, www.coolmath.com/algebra/15-functions.

Vertical line test Engaging students: Computing the composition of two functions

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 again comes from my former student Alexandria Johnson. Her topic, from Algebra II/Precalculus: computing the composition of two functions. The following link is to a worksheet over composition of functions. The worksheet allows students to explore composition of functions without outright telling them what composition of functions is. Instead, the students are working on real world problems about shopping in a store that is having a 20% sale with mystery coupons. In the worksheet, students explore whether or not it matters which discount is applied first and the equations that go along with each scenario. This worksheet is interesting because it approaches composition of functions in an explorative way and it is using a real-world situation students in high school may find relatable, which can help hook students that are math-phobic.

https://betterlesson.com/community/document/1326462/going-shopping-student-materials-docx Computing the composition of two functions requires prior knowledge of basic operations and combining like terms. This topic will expand upon their knowledge of basic operations by applying them to functions. Students will be able to add, subtract, multiply, and divide functions. Students should be able to use the distribution property; this is important when students are writing (fog)(x) and (gof)(x). During this topic, students should be able to expand upon their knowledge of creating functions from real world problems, which can be seen in the worksheet from the link above. Musical composition is a way this topic can appear in high culture. Musical composition is the process of combining notes, chords, and melodies in a particular way. Arranging the notes, chords, or melodies in different ways can change the composition. Function composition is the combining of different functions f(x) and g(x) in different ways like addition, subtraction, multiplication, and division. Order usually matters in function composition just like in musical composition. If you have several band students, or musically inclined students, this would be a good hook to grab students interest.

Heat Index Monotonic Monotony TED-Ed made a very good video describing the Infinite Hotel Paradox, a thought experiment to describe how injective (one-to-one) functions can be used to examine countably infinite sets.

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 Brittany Tripp. Her topic, from Precalculus: finding the domain and range of a function. How could you as a teacher create an activity or project that involves your topic?

One of my favorite games growing up was Memory. For those who haven’t played, the objective of the game is to find matching cards, but the cards are face down so you take turns flipping over two cards and have to remember where the cards are so when you find the match you can flip both of the matching cards. To win the game you have to have the most matches. I think creating an activity like this, that involves finding domain and range, would be a really fun way to get students’ engaged and excited about the topic. You could place the students in pairs or small groups and give each student a worksheet that has a mixture of functions and graphs of functions. Then the cards that are laying face down would contain various different domains and ranges. In order to get a match you have to find the card that has the correct domain and the card that has the correct range for whatever function or graph you are looking at. You could increase the level of difficulty by having functions, graphs, domains, and ranges on both the worksheet and the cards. This would require the students to not only be able to look at a graph of a function or a function and find the domain and range, but also look at a domain and range and be able to identify the function or graph that fits for that domain and range.

These pictures provide an example of something similar that you could do. I would probably adjust this a little bit so that the domain and ranges aren’t always together and provide actual equations of functions that the students’ must work with as well. How can this topic be used in your student’s future courses in mathematics or science?

Finding the domain and range of a function is used and expanded on in a variety of ways after precalculus. For instance, one way the domain and range is used in calculus is when evaluating limits. An example is the limit of x-1 as x goes to 1 is equal to zero, because when looking at the graph when the domain, x, is equal to 1 the range, y, is equal to zero. Finding domain and range is something that is applied to a variety of different type of functions in later courses, like when looking at trigonometric functions and the graphs of trigonometric functions. You look at what happens to the domain of a function when you take the derivative in calculus and later courses. You work with the domain and range of different equations and graphs in Multivariable calculus when you are switching to different types of coordinates such as polar, rectangular, and spherical. There are also multiple different science courses that use this topic in some way, one of those being physics. Physics involves a lot of math topics discussed above. How can technology be used to effectively engage students with this topic?

I found a website called Larson Precalculus that technically is targeted toward specific Precalculus books, but exploring this website a little bit I found that is would be a super beneficial tool to use in a classroom. This website has a variety of different tools and resources that students could use. It has book solutions which if you weren’t actually using that specific textbook could be a really helpful tool for students. This would provide them with problems and solutions that are not exactly the same to what they are doing, but similar enough that they could use them as examples to learn from. This website also includes instructional videos that explain in depth how to tackle different Precalculus topics including finding domain and range. There are interactive exercises which would give the students ample opportunities to practice finding the domain and range of graphs and functions. There are data downloads that give the students to ability to download real data in a spreadsheet that they can use to solve problems. These are only a few of the different resources this website provides to students. There are also chapter projects, pre and post tests, math graphs, and additional lessons. All of these things could be used to engage students and help advance and deepen their understanding of finding domain and range. The only downfall is that it is not a free resource. It is something that would have to be purchased if you chose to use it for your classes.

References:

http://esbailey.cuipblogs.net/files/2015/09/Domain-Range-Matching.pdf

http://17calculus.com/precalculus/domain-range/

http://www.larsonprecalculus.com/pcwl3e/

My Favorite One-Liners: Part 72

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In calculus, the Intermediate Value Theorem states that if $f$ is a continuous function on the closed interval $[a,b]$ and $y_0$ is any number between $f(a)$ and $f(b)$, then there is at least one point $c \in [a,b]$ so that $f(c) =y_0$.

When I first teach this, I’ll draw some kind of crude diagram on the board: In this picture, $f(a)$ is less than $y_0$ while $f(b)$ is greater than $y_0$. Hence the one-liner:

I call the Intermediate Value Theorem the Goldilocks principle. After all, $f(a)$ is too low, and $f(b)$ is too high, but there is some point in between that is just right. Engaging students: Defining a function of one variable

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 again comes from my former student Matthew Garza. His topic, from Algebra: defining a function of one variable. How can this topic be used in your students’ future courses in mathematics and science?

Being able to define a function of one variable is necessary for creating a model that describes the most basic phenomenon in math and science. In math, understanding these parent functions is crucial to understanding more complicated functions and, by considering some variables as temporarily fixed, multivariable equations and systems of equations can be easier to understand. In science, we often observe functions of a single variable.  In fact, even if there are multiple variables coming into play, a good lab will likely control all but one variable, so that we can understand the relationship with respect to that single variable – a function.

Consider in science, for example, the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the quantity in moles of a gas, R is the gas constant, and T is temperature.  This law, taught in high school chemistry, is not taught from scratch.  The proportional, single-variable functions that make up the equation are observed individually before the ideal gas law is introduced. Students will probably be taught Boyle’s, Charles’, Gay-Lussac’s, and Avogadro’s laws first. Boyle’s law states pressure and volume are inversely proportional (for a fixed temperature and quantity of gas).  This law can be demonstrated in one lab by clamping a pipette with some water and air inside, thus fixing all but two variables.  Pressure is applied to the pipette and the volume of air is measured using the length of the air column in the pipette.  Students must then evaluate volume V as a function of the single variable pressure P.  It should be noted that the length of the air column is measured, while the diameter of the pipette is fixed, thus volume must be calculated as a function of the single variable length.  Understanding the single variable, proportional and inversely proportional relationships is crucial to understanding the ideal gas law itself. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Generally speaking, Khan Academy has great videos to help understand math concepts.  Although it’s a little dry, this “Introduction to Functions” video is clear, concise, and touches on several ideas that I was having trouble tying in to every example.  This introductory video begins with the basic concept of a function as a mapping from one value to another single value.  The first examples it uses are a piece-wise function and a less computational function that returns the next highest number beginning with the same letter.  At first I didn’t like that these functions were discontinuous, but this actually gives something else to discuss.  The video links back prior knowledge, explaining that the dependent variable y that students are familiar with is actually a function of x, and represents the two in a table.  The last couple minutes of the video address the fundamental property that a function must produce unique outputs for each x, or it is a relationship. How could you as a teacher create an activity or project that involves your topic?

One idea might be to examine any function in which time is the independent variable.  Basic concepts of motion in physics can supplement an activity – Have groups evaluate position and speed with respect to time of, say, a marble or hot wheels car rolling down a ramp.  Using a stop watch and marking distance on an inclined plane, students could time how long it took to reach certain points and create a graph over time of displacement.  This method might result in some students graphing time as a function of displacement, which could lead to an interesting discussion on independence and dependence, and why it might be useful to view change as a function of time.

Technology could supplement such a lesson as to avoid confusion over whether distance is a function of time or vice versa.  Using motion sensor devices to collect data, such as the CBR2, students can use less time collecting and plotting data and more time examining it.  Different trials resulting in different graphs can lead to discussion on how to model such motion as a function of time – letting an object sit still would result in a constant graph, something rolling down an incline will give a parabolic graph (until the object gets too close to a terminal velocity).

To add variety, students can examine what a graph looks like if they move toward and away from the CBR2 or try to reproduce given position graphs.  This may result in the same position at different times, but since an object can be in only one position at a given time, the utility of using position as a function of time can be represented. Sporadic motion, including changes in speed and direction (like moving back and forth and standing still) also allow discussion of piecewise functions, and that functions don’t necessarily have to have a “rule” as long as only one output is assigned per value in the domain.