Engaging students: Solving one-step algebra problems

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 Deetria Bowser. Her topic, from Algebra: solving one-step algebra problems.

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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.)

To create a successful word problem that would both interest, and engage students, the teacher must “know his class.” Knowing one’s class involves knowing the many different students your students have. For example, if one knows that there are a lot of baseball players in the classroom, then creating word problems that involve baseball would be engaging for these students.

Additionally, to benefit all students you could do problems that involve finances. Including more “finance problems” will help students realize the importance of math, and how they can apply it in everyday life. An example of such problem would be “Damon’s earnings for four weeks from a part time are shown in the table. Assume his earnings vary directly with the number of hours worked. Damon has been offered a job that will pay him $7.35 per hour worked. Which job is better pay (Tucker, A.)? Including word problems that students can relate to now or in the future can help students stay engaged while learning, and answer the question that is most commonly asked by students: “When will I ever use this in real life?”

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

As a teacher, creating engaging activities and/or projects can prove to be quite difficult for word problems that are one- or two-step algebra problems, due to the fact that most students completely shut down once a word problem is presented to them. To combat this I have found that making it into a cooperative game can help soothe the anxiety caused by word problems. One game that is great to play with one or two step algebra problems is called rally coach. In this game, students are paired off. Student A is expected to work on solving the problem, while Student B is expected to watch, listen, check, and praise just as a coach would. Once the students think they have the correct answer, they will raise their hand so that the teacher may check it. If they get the answer correct, then the teacher will give them another problem (this time Student A and Student B switch roles). If the answer is incorrect, they must continue working on the problem. The end goal of the game is to answer as many questions as possible before time runs out. By playing this game students are able to help each other solve one or two step word problems.

 

 

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

In future courses many problems will involve one or two step algebra problems. For instance, in science courses like chemistry and physics, one will need to know how to solve for different variables of equations. For example, if one is in a chemistry course and is given a word problem (i.e If a 3.1g ring is heated using 10.0 calories, its temperature rises 17.9°C. Calculate the specific heat capacity of the ring) that provides heat energy (Q) mass of a substance (m) and change in temperature (deltaT), but is asked to solve for the specific heat, students will need to know how to solve for the specific heat either by isolating the variable in the beginning (Cp=Q/mdeltaT) or plugging in the givens and isolating the variable (Daniell, B).

References

Daniell, B. (n.d.). Energy Slides 3 [Powerpoint that contains Specific Heat problem].

Tucker, A. (2016). Direct Variation. Retrieved September 01, 2017, from

http://www.showme.com/sh/?h=PQvPbm4

 

 

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 Austin DeLoach. His topic, from Algebra: negative and zero exponents.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

The topic of negative and zero exponents is very important when or if the students get to calculus. Although that will be several years down the line, having a solid fundamental grasp on the idea of negative and zero exponents will help them understand derivatives a lot better. Because derivatives of “simple” functions just multiply the coefficient by the exponent and then subtract one from the exponent, it is important for the students to have a good understanding of what negative and zero exponents are. If they do not understand already, they will be confused about why, for example, the derivative of 3x is just 3. It also greatly simplifies derivatives of things like 4/x2, as the students will simply be able to recognize that that is the same thing as 4x-2 and follow standard rules instead of needing to think about the quotient rule and waste time with that. It will also help them in the more near future when they work with simplifying expressions with the exponents written in different terms (i.e. with a positive exponent or with a negative exponent in the denominator), as it will help them recognize what simplifications mean the same thing. Explaining that understanding negative exponents will thoroughly help them in the future may be enough for some students to want to solidify their grasp on the topic.

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D2. How was this topic adopted by the mathematical community?

Although this is not about the early adoption of negative and zero exponents in the mathematical community, Geoffrey D. Dietz points out more recent bias for or against the use of negative exponents in textbooks in his Journal of Humanistic Mathematics (linked at the bottom of this answer). Dietz brings up the idea of what is considered “simplified” when it comes to negative exponents vs exponents in denominators. He rated over 20 mathematics textbooks from 1825 to 2012 from “very tolerant” of negative denominators in simplified answers to “very intolerant”. Interestingly, his first encounter with an “intolerant” textbook was not until the 20th century, and textbooks began getting more polarized as very tolerant or very intolerant closer to the end of the 20th century and getting closer to today. This is interesting when it comes to adoption by the mathematical community, as there is a significant inconsistency, even today, about whether negative exponents can be considered “simplified” or not. It will be important to point this out to your students so they can be prepared for their future teachers who may have different preferences on simplification from you, as that will help them understand the polarity in the mathematical community on this topic, as well as hopefully make them want to understand what negative exponents really mean. Dietz recommends giving your students practice with not only converting negative exponents to positive exponents, but also from positive to negative, in order to make sure they are prepared for whatever preferences come up as well as solidifying their understanding of what negative exponents mean.

http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1110&context=jhm

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E1. How can technology be used to effectively engage students with this topic?
This video from Khan Academy does a good job at explaining why negative and zero exponents are what they are. Although Khan Academy videos will likely not be the most engaging for all students, this video is short enough to maintain the attention of the class, and it the logic in it is helpful for the students who don’t understand how the definition of negative and zero exponents was decided on. The presenter does well explaining the idea of “going backwards” and dividing by the number when you decrease the exponent. It’s a good way to explain the “why” for students who ask about it, and it also is a good way to change up the pace for students, as playing videos during class could prevent it from becoming stale for the students, keeping them engaged for longer.

https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals/pre-algebra-negative-exponents/v/negative-exponent-intuition

 

 

My Favorite One-Liners: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on my favorite one-liners.

Mathematical Wisecracks for Almost Any Occasion: Part 2Part 7, Part 8, Part 12, Part 21, Part 28, Part 29, Part 41, Part 46, Part 53, Part 60, Part 63, Part 65, Part 71, Part 79, Part 84, Part 85, Part 100, Part 101Part 108

All-Purpose Anecdotes: Part 38, Part 50, Part 64, Part 70, Part 92, Part 94

Addressing Misconceptions: Part 3Part 4Part 11, Part 14, Part 15, Part 18, Part 30, Part 32, Part 33, Part 37, Part 45, Part 59

Tricky Steps in a Calculation: Part 5, Part 6

Greek alphabet and choice of variables: Part 40, Part 43, Part 56

Homework and exams: Part 39Part 47, Part 55, Part 57, Part 58, Part 66, Part 77, Part 78, Part 91, Part 96, Part 97, Part 107

Inequalities: Part 99

Simplification: Part 10, Part 102, Part 103

Polynomials: Part 19, Part 48, Part 49, Part 81, Part 90

Inverses: Part 16

Exponential and Logarithmic Functions: Part 1, Part 42, Part 68, Part 80

Trigonometry: Part 9, Part 69, Part 76, Part 106

Complex numbers: Part 54, Part 67, Part 86

Sequences and Series: Part 20, Part 35

Combinatorics: Part 27

Statistics: Part 22, Part 23, Part 36, Part 51, Part 52, Part 61, Part 95

Probability: Part 26, Part 31, Part 62, Part 93

Calculus: Part 24, Part 25, Part 72, Part 73, Part 74, Part 75, Part 83, Part 87, Part 88, Part 104

Logic and Proofs: Part 13, Part 17Part 34, Part 44, Part 89, Part 98

Differential Equations: Part 82, Part 105

 

 

 

 

 

 

 

 

 

My Favorite One-Liners: Part 108

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.

Today’s post marks the final entry in this series. When I first came up with the idea of listing some of favorite classroom quips, I thought that this series might last a couple dozen posts. To my surprise, it instead lasted for more than 100 posts. I guess that, in my 21-year teaching career, I’ve slowly developed my own unique lexicon for communicating mathematical ideas, and perhaps this parallels (on a decidedly smaller scale) what a radio talk show host (like local legend Randy Galloway, who was a sports reporter/commentator in the Dallas/Fort Worth area for many years before retiring) does to build rapport with his/her audience.

I’ll use this final one-liner near the end of the semester when it’s time for students complete their evaluations of my teaching. Back in days of yore, professors would take 10-15 minutes to pass out paper copies of these evaluations, students would complete them, and that would be the end of it. In modern times, however, paper evaluations have switched to electronic evaluations, which are perhaps better for the environment but tend to have a decidedly lower response rate than the old paper evaluations. Still, I value my students’ feedback. So I’ll tell them:

Please fill out the student evaluation; the size of my raise depends on this.

After the laughter settles down, I’ll tell them, “Who’s joking?” I can’t say this happens everywhere, but I can honestly say that my department’s executive committee does consider student evaluations of teaching when deciding on the quality of my teaching, and that partially determines the size of my annual merit raise. (The committee also considers other indicators of good teaching other than student evaluations.)

It’s important to note that I don’t tell my students to give me a good evaluation; I just ask them to fill it out and to be honest with their feedback. I also tell them, forgetting my raise, I also want to hear from them about how the semester went. If it went great, I want to know that. If it sucked, I also want to know that. However, if they think the class sucked, just writing “This class sucked” doesn’t give me a lot of information about how to fix things for the next time that I teach the course. So, if they have a criticism, I ask them to give me specific feedback so that I can consider their critiques.

A couple years ago, I served on my university’s committee for reconsidering the way that we conduct student evaluations of teaching. To my surprise, when I interviewed students in focus groups, there was a general consensus that students believed that their evaluations were a waste of time that didn’t actually contribute anything to the university — or if they did contribute something, they had no idea what it was. Ever since then, I’ve made a point of telling my students that their evaluations really do matter and can make a difference in future offerings of my courses.

My Favorite One-Liners: Part 107

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.

When students ask me how long it will take me to grade their exam, I’ll describe my tongue-in-cheek process for grading… I’ll go home, pop on the TV, and watch some movie that gets me in the proper mood for grading exams… perhaps Braveheart… or Gladiator… or The Godfather

My Favorite One-Liners: Part 106

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.

Years ago, when I first taught Precalculus at the college level, I was starting a section on trigonometry by reminding my students of the acronym SOHCAHTOA for keeping the trig functions straight:

\sin \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Hypotenuse}},

\cos \theta = \displaystyle \frac{\hbox{Adjacent}}{\hbox{Hypotenuse}},

\tan \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Adjacent}}.

At this point, one of my students volunteered that a previous math teacher had taught her an acrostic to keep these straight: Some Old Hippie Caught Another Hippie Tripping On Acid.

Needless to say, I’ve been passing this pearl of wisdom on to my students ever since.

My Favorite One-Liners: Part 105

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.

Today’s quip was happily stolen from a former student:

If someone you like is sending you mixed signals, use a Fourier transform.

Not surprisingly, a quick Google search turned up the relevant memes:

My Favorite One-Liners: Part 104

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 use today’s quip when discussing the Taylor series expansions for sine and/or cosine:

\sin x = x - \displaystyle \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \dots

\cos x = 1 - \displaystyle \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \dots

To try to convince students that these intimidating formulas are indeed correct, I’ll ask them to pull out their calculators and compute the first three terms of the above expansion for $x=0.2$, and then compute \sin 0.2. The results:

This generates a pretty predictable reaction, “Whoa; it actually works!” Of course, this shouldn’t be a surprise; calculators actually use the Taylor series expansion (and a few trig identity tricks) when calculating sines and cosines. So, I’ll tell my class,

It’s not like your calculator draws a right triangle, takes out a ruler to measure the lengths of the opposite side and the hypotenuse, and divides to find the sine of an angle.

 

My Favorite One-Liners: Part 103

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 to give students my expectations about simplifying incredibly complicated answers. For example,

Find f'(x) if f(x) = \displaystyle \frac{\sqrt{x} \csc^5 (\sqrt{x} )}{x^2+1}.

Using the rules for differentiation,

f(x) = \displaystyle \frac{[\sqrt{x} \csc^5 (\sqrt{x} )]'(x^2+1) -[\sqrt{x} \csc^5 (\sqrt{x} )](x^2+1)' }{(x^2+1)^2}

= \displaystyle \frac{[(\sqrt{x})' \csc^5 (\sqrt{x} ) + \sqrt{x} (\csc^5(\sqrt{x}))'](x^2+1) - \sqrt{x} \csc^5 (\sqrt{x} )](2x) }{(x^2+1)^2}

= \displaystyle \frac{[\frac{1}{2\sqrt{x}} \csc^5 (\sqrt{x} ) + 5 \sqrt{x}  \csc^4(\sqrt{x}) [-\csc(\sqrt{x})\cot(\sqrt{x})]\frac{1}{2\sqrt{x}}(x^2+1) - \sqrt{x} \csc^5 (\sqrt{x} )](2x) }{(x^2+1)^2}

With some effort, this simplifies somewhat:

f'(x) = -\displaystyle \frac{\left(5 x^{5/2} \cot \left(\sqrt{x}\right)+3 x^2+5 \sqrt{x} \cot \left(\sqrt{x}\right)-1\right) \csc ^5\left(\sqrt{x}\right)}{2 \sqrt{x} \left(x^2+1\right)^2}

Still, the answer is undeniably ugly, and students have been well-trained by their previous mathematical education to think the final answers are never that messy. So, if they want to try to simplify it further, I’ll give them this piece of wisdom:

You can lipstick on a pig, but it remains a pig.

My Favorite One-Liners: Part 102

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 when the final answer is a hideous mess. For example,

Find f'(x) if f(x) = \displaystyle \frac{\sqrt{x} \csc^5 (\sqrt{x} )}{x^2+1}.

The answer isn’t pretty:

f'(x) = -\displaystyle \frac{\left(5 x^{5/2} \cot \left(\sqrt{x}\right)+3 x^2+5 \sqrt{x} \cot \left(\sqrt{x}\right)-1\right) \csc ^5\left(\sqrt{x}\right)}{2 \sqrt{x} \left(x^2+1\right)^2}

This leads to the only possible response:

As all the King’s horses and all the King’s men said when discovering Humpty Dumpty… yuck.