Author: John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

Engaging students: Equations of two variables

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 Zacquiri Rutledge. His topic, from Algebra: equations of two variables.

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Seeing equations with two variables becomes quite common once students have been introduced to independent and dependent variables. However seeing equations in the form x+4y=16 would start as a confusing concept after being taught that equations are written in the format 4x-16=y. However, this concept is highly required when a teacher goes to explain about a system of equations. The reason for this is because a common method that is taught for solving a system of equations is substitution. In order to utilize the substitution method, a student must understand how to solve for a variable by using order of operations to isolate the variable. In fact, a student will use the same skills they did when learning to solve an equation that only has one variable, such as 3x+6=12. However, now the student must apply it to something new.

Another lesson that uses the knowledge from the Equations of Two Variables is interpretation of a graph for an equation with two variables. Before, the students would have learned what independent and dependent variables are, and how they are represented on a graph. Later on the students would further their understanding by finding the graphical representation of equations with two variables. The students would learn that, while the line on the graph during lessons over independent and dependent variables was only to show where the left side of an equation equaled y, the line can also show where x and y combine to equal a certain value. An example of this would be comparing x+4y=16 and (-1/4)x+4=y. They are the same equation, however one equation shows that x and 4y combine to equal 16, so every point on the line are the values of x and y required to equal 16. The second equation says that to find y for a given point x, x must be multiplied by (-1/4) and add 4. Just changing the nature of the equation can change what it is that the equation is saying, as well as give a different perspective one that could be useful when dealing with real life word problems.

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Two variable equations are very subtle, but are all around us. Even when we do not think it is being used, it is. The most common modern example of two variable equations is the American dollar, and how many coins of two different values are needed to make a dollar. Although this is a very easy explanation to use it can be very boring at times. How about classical music or concert music? While it may not seem obvious at first, it is in fact there. The standard set-up for a sheet of music is Four-Four time. What this means is that within every measure there are four beats and a quarter note counts as a whole beat. There are also other kinds of notes which are used in combination with quarter notes to fill a measure, examples being a whole note (four beats), half note (two beats), and eighth notes (half beat). So when a composer sits down to write a piece of music, he/she must keep in mind how many beats are in each measure. This is where the concept of two variable equations comes into play. Suppose the composer wants a measure made up of only half notes and quarter notes in four-four time, then his equation to figure out how many of each note he can have would be 2h+q=4, where h is half notes and q is quarter notes. Then, the next measure is going to be made up of eighth and half notes, therefore 2h+(1/2)e=4 would be the equation, where e is eighth notes. There are many different combinations someone can use when writing music to create a piece that is to be played in front of a live audience. Centuries ago, men like Beethoven and Mozart used this concept every day to create classic pieces such as Beethoven’s Symphony #5 or Mozart’s Moonlight Sonata. This is an excellent example that can be used for classes that include a large number of band students or choir students, to relate the music they are studying in their music classes to their math courses.

 

 

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With the previous response in mind, a teacher could very well use Youtube as an excellent method to engage their students. A lot of children today are not familiar with how classical music is written or how music is written at all. By playing pieces of music for their students that students are likely to have heard befor, via Youtube or even iTunes, such as Ride of the Valkyries or Beethoven’s Symphony #5, can spark an interest not only musically, but mathematically. A teacher could begin by asking students if they had heard the piece before, then go to the next piece and see who has heard it before. Repeat this process for about 2-4 clips of pieces, then ask which of the students know anything about how music is written. This would lead into what was discussed in the previous response. However, by including the technology as a way for the students to hear the music, and not just see it, can have tremendous effects on their attention.

 

What a Quantitative Literacy Course Should Look Like

A recent opinion piece in the New York Times discussed what a good quantitative literacy course should look like. Unlike the algebra-precalculus-calculus sequence or even a class in AP statistics, quantitative literacy is specifically designed for students who are not interested in a STEM major, teaching them how to view numbers to make sense of the world and be an informed citizen.

Engaging students: Negative and zero exponents

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 Jennifer Elliott. Her topic, from Algebra: negative and zero exponents.

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  • Technology Engage
    • I found the website, https://www.mangahigh.com/en-us/math_games/number/exponents/negative_exponents. It is an interactive game that gives a brief explanation of what negative and zero exponents are. Then you can select the difficulty level and the number or questions you wish the children to try. If this a new topic introduced, then the student may miss several. That is ok. As a teacher, you are setting a ground level for the direction of your teach. At the end of the lesson, you can utilize the same game to check the students’ new level of understanding for the topic.

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  • Activity Engage
    • The students will engage in prior knowledge that might be needed to understand the idea behind negative and zero exponents. First I will make different notecards, some with definitions such as negative number, fractions, number line, and reciprocals and others. Then I will have some index cards with different exponents including positive, negative, and zero. The cards will have different values such as one might say 10^-1 and one might say 1/10. Every student will have a note card. I will have different sections set up in the room. Example would be definitions, 1, <1, and >1 and have students find which section they belong in. I could also have them find their card partner (different way of writing the same number) and the word matching the definition. Then maybe from there, that group find their counter-partner (I would maybe not use definitions for this part) such that the group with 10^-2 would find the group with 10^2. This would set up groups for them to explore the idea of negative and zero exponents.
      • This activity came from myself but I had some ideas from different pictures on Pinterest, but nothing in particular to source.)

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  • Curriculum Engage
    • To show how this might be used later in class, I will work on the idea of decay. The idea of decay can be introduced in science and history off the top of my head. Although the students might be years away from the idea of physics and decay value, this will be a fun way to engage students and hopefully recall the information when a lesson on decay comes in the future. The idea is found on several different websites and has to do with the idea of exponential decay using M&M’s. The idea is to create (or use one of the several choices) of a table to record the data from the trials. The group(s) count the total number of M&M’s. The total is the starting number for trial 0. Trial number would be the first column. The second column would be the number of M&M’s. For trial one, you would dump the bag/cup of candy and the student would remove all the M&M’s that do not have the M showing. Shake the candy up again, and dumb out. Continue with trials until you do not have any M&M’s left. Then the third column will be what percentage of the bag they have left (example maybe ½ of the M&M’s remain.) This activity will lead to the discovery of decay and how it uses zero and negative exponents. The starting point of trial 0 has us with “1” bag/cup of candy and then it will decrease from there. Just like x^0=1 which is great than x^-2=1/2 and so on. At the end, of the complete lesson the idea of using negative exponents in sports, sound, radioactive waste, and scientific notation will be a start of what that students will learn in other subjects in the future.

 

 

Another Poorly Written Word Problem: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series poorly written word problem, taken directly from textbooks and other materials from textbook publishers.

Part 1: Addition and estimation.

Part 2: Estimation and rounding.

Part 3: Probability.

Part 4: Subtraction and estimation.

Part 5: Algebra and inequality.

Part 6: Domain and range of a function.

Part 7: Algebra and inequality.

Part 8: Algebra and inequality.

 

 

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. Here’s my series on the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, 2b, and 2c: The 1089 trick.

Part 3a, 3b, and 3c: A geometric magic trick (see also here).

Part 4a, 4b, 4c, and 4d: A trick using binary numbers.

Part 5a, 5b, 5c, 5d: Predicting a digit that’s been erased from a number.

Part 6: Finale.

Part 7: The Fitch-Cheney 5-card trick.

Part 8a, 8b, 8c: A trick using Pascal’s triangle.

 

Stump the Prof: An Activity for Calculus I

After finishing the Product, Quotient, and Chain Rules in my calculus class, I’d tell my class the following: “Next time, we’re going to play Stump the Prof. Anything that you can write on the board in 15 seconds, I will differentiate. Anything. I don’t care how hard it looks, I’ll differentiate it (if it has a derivative). So do your best to stump me.”

At the next lecture, I would devote the last 15-20 minutes of class time to Stump the Prof. Students absolutely loved it… their competitive juices got flowing as they tried to think of the nastiest, hairiest functions that they could write on the board in 15 seconds. And I’d differentiate them all using the rules we’d just covered.. though I never promised that I would simplify the derivatives!

Sometimes the results were quite funny. Every once in a while, a student would write some amazingly awful expression but forgot to include an x anywhere. Since the given function was a constant, the derivative of course was zero.

The worst one I ever got was something like this:

y = \csc^4(\sec^5(\csc^8(\sec^7(\csc^4(\sec^5(x)))))

Differentiating this took a good 3-4 minutes and took maybe 5 lines across the entire length of the chalkboard; I remember that my arm was sore after writing down the derivative. Naturally, some wise guy used his 15 seconds to write y = in front of my answer, asking me to find the second derivative. At that, I waved my white handkerchief and  surrendered.

The point of this exercise is to illustrate to students that differentiation is a science; there are rules to follow, and by carefully following the rules, one can find the derivative of any “standard” function.

Later on, when we hit integration, I’ll draw a contrast: differentiation is a science, but integration is a combination of both science and art.

Math and practice

Sometimes math students view repetition to practice a new skill with the disdain of Allen Iverson, the legendary point guard for the Philadelphia 76ers.

So I found a recently published essay from Math With Bad Drawings to be very, very inspiring. Here are the opening paragraphs:

I’ve always felt conflicted about repetitive practice.

On the one hand, I see how vital practice is. Musicians repeat the same piece again and again. Soccer players run drills. Chefs hone their chopping motion. Shouldn’t math students do the same: rehearse the skills that matter?

But sometimes, I backtrack. “This is just going to bore them,” I fret, scanning a textbook exercise. “I’m emphasizing the rote aspects of math at the expense of the creative ones. They’re going to forget this skill anyway, and be left only with the insidious impression that math is a jackhammer subject of tooth-grinding repetition.”

(Then I assign the exercise anyway, because class starts in five minutes and— despite my repeated petitions—the administration has denied me access to a time turner.)

These two trains of thought suffer daily collisions in my mind: repetition is dull, but repetition is necessary. This inner conflict takes for granted the idea that repetitive practice is a separate endeavor, a distinct stage of the learning process. First, you learn the concept. Second, you practice it. In this view, practice is like cleaning up after a picnic: absolutely essential, but not much fun.

But this summer, a very wise teacher showed me a path forward, a way to reconciliation.

I’m referring, of course, to a two-year-old named Leo.

I highly recommend the entire article.