# 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.

# Engaging students: Finding x- and y-intercepts

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 Lissette Molina. Her topic, from Algebra: finding $x-$ and $y-$intercepts.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Using this topic, students can now use word problems that involve two variables in our everyday lives. One problem that many scientists often use is population growth. In population growth, we can usually see a trend of a line and determine the slope. We initially begin with a certain population in a certain year, this is considered the y-intercept, since we start at the initial year that we consider to be at x=0. Using the slope of the line when we are speaking in terms of population decay, we may then set our y=0 to find when a population would be equal to zero. We can also consider other examples such as the depreciation of a car, or when a business’s grows out of debt and begins to profit. Word problems include, but are not limited to, problems that involve a trend and wanting to find where that trend will lead to at a certain point, x, when we are given an initially amount or reverse this operation.

How can this topic be used in your students’ future courses in mathematics or science?

This topic crosses multiple courses in mathematics. In general, knowing the x and y-intercepts of equations help students start outlining what the graph of the function might look like. This gives part of the visual representation needed to complete part of the graph. These intercepts usually also give a prediction of what the shape of the graph may look like. A fun assignment would be giving a student two points on the graph and along with the intercepts of that equation that the points belong to. Along with this, these intercepts give us the solutions of the equations. When there are not x or y-intercepts, we would now know that the solutions do not exist or at least are imaginary. Overall, x and y-intercepts help us get a better understanding of what the graphs of almost all equations must look like. This is essentially especially when we are graphing by hand.

How can technology be used to effectively engage students with this topic?

Graphing calculators is one fun essential way of finding intercepts as well as learning functions on a calculator. When a student graphs a function on a graphing calculator, for example, the sine function, we can ask the student where they believe the graph would intercept with the x-axis. We would then ask them to find the intercepts using the calculator by pressing [2nd][trace][4] function and proceed to find the approximated x-intercepts. The student would then find that the intercepts occur at every npi/2. Essentially, using this function is an interesting way of estimating the intercepts along the graph in an interactive way. Other online graphing calculators may do this as well and give students a better understanding of where the intercepts occur.

# Engaging students: Adding and subtracting polynomials

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 Christian Oropeza. His topic, from Algebra: multiplying polynomials.

How could you as a teacher create an activity or project that involves your topic?

The activity would consist of each student being given a bowl with 20 pieces of candy, which has multiple colors (e.g., Skittles or M&M’s) and a worksheet, which by the end will show students how to add and subtract polynomials(Reference 1). The objective for each student is to group all of the pieces of candy by the same color. Once this has been completed, the students will write down on the worksheet for “Part 1”, how many pieces of candy are in each group. Next, the students would be given 10 more pieces of random colored candy. Then, the students will regroup the new pieces of candy and write down the new number of candies in each group for “Part 2”. For “Part 3”, students will eat(or put away) 10 of their candies randomly. Finally, the students will write down the new number of candies in each group. Then the students would be asked, “What did each one of you do to put the candies in groups?”, “what operation was used for Part 2 of the worksheet”, and “what operation was used for Part 3 of the worksheet”. The students’ responses should be somewhere along the lines of “group the candies by the same color”, “addition”, and “subtraction”. Then the students would be told to relabel each group of colored candies into a different variable. For example, green=x, red=x2, yellow=k, blue=y, etc. Knowing the previous information, the students will next repeat the Part 1, 2, and 3, but using the assigned variables instead of the colors. The purpose of this activity is to show students that each variable in a polynomial must be grouped by like terms when performing addition or subtraction.

How does this topic extend what your students should have learned in previous courses?

This topic relates to previous math classes by activating students’ prior knowledge on the concept of adding and subtracting integers. This means knowing the rules of addition and the rules of subtraction. For example, students should know that a 3+2=5=3+2, but 3-2=1$\ne$2-3 (i.e., commutative property). Students should also know that the when subtracting a negative integer, the signs cancel out and all that is left is the addition of a positive integer (e.g., -(-2)=2). Students should also be familiar with grouping anything into specific groups. For example, if students were given colored tiles, then the students should be able to group the tiles into different colored groups. The distributive property is a topic the students should have covered before, which helps out when trying to simplify an expression involving parenthesis (e.g., 2(3+a)=6+2a. The idea of closure for integer properties and operations is the key to adding and subtracting polynomials, so students must have understood this concept prior in order to use the operation of addition and subtraction on like terms.

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.

Technology is always a great way to engage students especially with the newer generation of students where technology is part of their everyday life. The website mathisfun.com (Reference 2) is an excellent piece of technology to introduce this topic to the students because the website breaks down the idea of adding and subtracting polynomials piece by piece in easy manner that will help students see patterns and activate prior knowledge. With the inclusion of examples and non-examples students will learn where to minimize their potential errors. Some of the examples are animated with colors to help the more visual students understand and recognize the pattern for each problem. Another example of effective technology is the website Khan Academy (Reference 3,4,5). Khan Academy has great videos that thoroughly explains this topic. Reference 3 defines the word “polynomial” in math language by breaking the word into two words, which will help students remember and recognize this topic more easily. Also, Reference 2 goes over the vocabulary associated with adding and subtracting polynomials (e.g., coefficients, monomial, binomial, trinomial, and degree). Reference 4 goes over an example of adding a polynomial by going through step by step procedures. Reference 5 does the same thing as Reference 4, but over an example of subtracting polynomials.

References:

# Another poorly written word problem (Part 10)

The current women’s world record for the long jump is 7.52 meters, or 24 feet, 8 inches.

# My Favorite One-Liners: Part 114

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.

I’ll use today’s one-liner whena step that’s usually necessary in a calculation isn’t needed for a particular example. For example, consider the following problem from probability:

Let $X$ be uniformly distributed on $\{-1,0,1\}$. Find $\hbox{Cov}(X,X^2)$.

The first step is to write $\hbox{Cov}(X,X^2) = E(X \cdot X^2) - E(X) E(X^2) = E(X^3) - E(X) E(X^2)$. Then we start computing the expectations. To begin,

$E(X) = (-1) \cdot \displaystyle \frac{1}{3} + 0 \cdot \displaystyle \frac{1}{3} + 1 \cdot \displaystyle \frac{1}{3} = 0$.

Ordinarily, the next step would be computing $E(X^2)$. However, this computation is unnecessary since $E(X^2)$ will be multiplied by $E(X)$, which we just showed was equal to $0$. While I might calculate $E(X^2)$ if I thought my class needed the extra practice with computing expectations, the answer will not ultimately affect the final answer. Hence my one-liner:

To paraphrase the great philosopher The Rock, it doesn’t matter what $E(X^2)$ is.

P.S. This example illustrates that the covariance of two dependent random variables ($X$ and $X^2$) can be zero. If two random variables are independent, then the covariance must be zero. But the reverse implication is false.

# My Favorite One-Liners: Part 113

I tried a new wisecrack when teaching my students about Euler’s formula. It worked gloriously.

# My Favorite One-Liners: Part 112

This was also the story of my childhood.

# My Favorite One-Liners: Part 111

I tried a new wise-crack in class recently, and it was a rousing success. My math majors had trouble recalling basic facts about tests for convergent and divergent series, and so I projected onto the front screen the Official Repository of all Knowledge (www.google.com) and searched for “divergent series” to “help” them recall their prior knowledge.

Worked like a charm.

# My Favorite One-Liners: Part 110

I overheard the following terrific one-liner recently. A teacher was about to begin a lecture on exponential growth. His opening question to engage his students: “What does your bank account have to do with bacteria… other than they both might be really tiny?”

# My Favorite One-Liners: Part 109

I tried a new joke in class recently; it worked gloriously.

I wrote on the board a mathematical conjecture that has yet to be proven or disproven. To emphasize that nobody knows the answer yet despite centuries of effort, I told the class, “If you figure this out, call me and call me collect,” writing my office phone number on the board.

To complete the joke, I said, “Yeah, this is crazy. So here’s my number…”

I thoroughly enjoyed my students’ coruscating groans before I could complete the punch line.