Engaging students: Factoring 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 again comes from my former student Brittnee Lein. Her topic, from Algebra: factoring polynomials.

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

There are many great websites that can help to provide students a conceptual framework for factoring polynomials in lieu of simple lecture. This website lets students explore polynomial equations with online algebra tiles.

https://illuminations.nctm.org/activity.aspx?id=3482

Algebra tiles are effective in teaching factoring because they provide a visual representation of abstract concepts and allow students to understand that the symbol “=” in an equation really means equivalence (i.e. what you do to one side of the equation, you must do to the other side). I also think algebra tiles are very beneficial in teaching students about zero pairs. There are other websites –such as wolfram alpha– that are especially great supplements to go alongside topics such as factoring polynomials because students can see the graphical meaning of the roots of a quadratic equation. When combined, these websites can aid students in gaining a both conceptual and procedural understanding of the topic.

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

There is an activity called “Factor Draft” where students set up a ‘playing field’ of cards. In this field, there are factor cards such as (x+2), (x-12), etc. sum (5x), (12x), etc., and product cards (1), (42), and so on. The goal of the game is to draw a winning hand of two factor cards and a corresponding sum and product card. Each card is color coded to their type. Each turn a player draws one card from the field of face up cards. The player must pay mind to not only his/her own cards but also those of their opponent’s –as the first person to get two factor cards and their corresponding sum and product card wins. This activity is beneficial in furthering student understanding between the relationships between each term in a quadratic polynomial. For example $(x+4)(x-3) = x^2 + 1x - 12$ and the corresponding factor cards would be (x+4) and (x-3) the sum card would be (1x) and the product card would be (-12). This activity allows students to intuitively get a sense of the process of factoring and gives them practice multiplying out polynomials.

2. How can this topic be used in your students’ future courses in mathematics or science?
• Factoring polynomials is used in many important future science and mathematics concepts. When a quadratic equation cannot be factored simply, teachers must introduce the quadratic formula. This slides into the introduction of complex roots of an equation and complex numbers. When factoring polynomials of higher degree than 2, synthetic division (another topic in high school mathematics) is useful in finding the roots of the equation. If a student is able to understand the meaning of the roots of an equation, that will aid in solving many interesting physics and mathematics problems. Factoring is used quite often to find the domain of a rational equation such as $f(x) = (x+2)/ (x^2+ 4x+3)$. A student must also have a strong basis in factoring polynomials to learn concepts such as completing the square.

References

• National Library of Virtual Manipulatives, nlvm.usu.edu/en/nav/vlibrary.html.

• Cleveland, James. “The Factor Draft.” The Roots of the Equation, 23 May 2014, rootsoftheequation.wordpress.com/2014/05/22/the-factor-draft/.

Engaging students: Graphing Square Root 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 student Alexandria Johnson. Her topic, from Algebra II: graphing square root functions.

An interesting word problem that students should be able to solve after completing a lesson on graphing square root functions would be: “The Chandra satellite detected x-rays coming from the region of the sky containing the galaxy cluster JKS041. The electrons in the gas are emitting the X-rays, and colliding at high speed with the protons in the gas. The energy of the x-rays at the time they were emitted by the hot gas was 21,400 electron Volts (eV). This energy is shared equally between the electrons and protons. The speed of a proton is related to its kinetic energy by E = 1/2mV^2 where E is the energy in Joules, V is the proton speed in meters/sec, and m is the mass of a proton (m = 1.7 x 10-27 kg). About how fast are the protons moving? (Note: 1 eV = 1.6 x 10^-19 Joules)”. Students can arrange the problem into a square root function to solve for velocity: V=sqrt(2E/m). Using the information provided students can convert eV to E and solve for m. Once this information is found, students can plug in the numbers to solve for V. Note: this question is difficult and some students may struggle with the calculations. A simpler question about the relationship between kinetic energy and velocity could be used in place of this one. Question provided by https://spacemath.gsfc.nasa.gov/weekly/6Page70.pdf.

In Physics, students will be able to use square root functions to describe the relationship between different variables. Having the knowledge of graphing square root functions will allow students to represent these relationships graphically. For example, to find kinetic energy, students use the formula E=(1/2)*m*v^2, where m=mass and v=velocity. Students can manipulate the equation to find v which would be v=sqrt(2E/m). Given m, students should be able to graph the relationship between v and E. When solving for volume, students can rearrange the equation into the form y=a*sqrt(x-h)+k, where h=0, k=0 y=v, x=E, and a=sqrt(2/m). knowing how to graph a square root function, students can graph this equation.

A useful resource when creating a lesson about graphing square root functions is https://teacher.desmos.com/. This website provides teachers with existing activities that the students can complete. Also, it allows the teacher to create activities for the student. An activity that is already created for teacher use is called Polygraph: Square root functions. In this activity, students play a game similar to the board game Guess Who. Students pair up and are given a set of graphs of square root functions. Partner 1 chooses a graph. Then, Partner 2 asks questions about the graphs to try to find the graph that Partner 1 chose. Students compare various graphs and communicate these differences. Though the website doesn’t offer any other premade activities at this time, teachers can use the activity type “marble slides” to create an activity that shows how a, h and k affect the parent function of square roots.
Work cited

“Chandra Spies the Most Distant Cluster in the Universe.” Space Math, NASA, Chandra Spies the Most Distant Cluster in the Universe. Accessed 15 Sept. 2017.
“Square Root Functions.” Desmos Classroom Activities, teacher.desmos.com/polygraph/custom/560ad29158fd074d156300b6. Accessed 15 Sept. 2017

Happy E Day!

In the United States, today is 2/7/18, matching the first four significant digits of $e$.

The next time that this date can be celebrated is July 2, 2018 (using the day/month/year format of abbreviated dates.) After that, we’ll have to wait until 27/1/82, or January 27, 2082. (Sadly, I knew about the number $e$ back in 1982 but was then unaware of the day/month/year method of abbreviating dates, and so this day went unrecognized by me on January 27, 1982.)

Borwein integrals

When teaching proofs, I always stress to my students that it’s not enough to do a few examples and then extrapolate, because it’s possible that the pattern might break down with a sufficiently large example. Here’s an example of this theme that I recently learned:

Babylonian trigonometry

An interesting article that I read on Babylonian mathematics.

Euler’s Equation

This was hands-down my favorite variant of the “distracted boyfriend” meme that went around the internet last year.

Cheat Sheet

The news clip below shows why, when I allow my students to use a 3×5 card on an exam, I specify — “that’s in inches. It must be handwritten. And no magnifying glasses.”

Student Outwits Prof By Bringing 3×5 FOOT Cheat Sheet To Exam