Engaging students: Parallel and perpendicular lines

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 Cody Jacobs. His topic, from Algebra: parallel and perpendicular lines.

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid is one of the most famous mathematicians of all time. His fame rests mostly on his 13 books commonly referred to as Euclid’s Elements. Euclid’s Elements are said to have a greater impact on the human mind that any other book except for the bible. Euclid contributed to the development of this topic based off the fact that his Elements have been used for centuries for teaching foundational geometry. The importance of Euclid’s books come from the minimal assumptions made, and the natural progression from simple results to more complex results. Euclid starts of listing 23 definitions and 5 postulates in which uses to prove theorems. His books contain over 400 theorems and proofs which layout the guidelines for how we use geometry today.

E1. How can technology be used to effectively engage students with this topic?

Desmos.com is a great website website that allows you to pick out activities your students can do. They have some activities regarding parallel and perpendicular lines where students shift the lines to make them parallel or perpendicular. I have used this website before regarding parabolas and students are fully engaged. Desmos has plenty of activities to choose from to find the right fit for your class, so do not be afraid to look around for a while. You can sign in as a teacher and make a code for your students to get into the activity. There are even some word problems so you can get a better understanding of what your students are thinking. I think Desmos is best used at the end of a topic, more as a general review over everything because the activities go through topics pretty fast.

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

Students will continue to use parallel and perpendicular equations throughout their mathematical career. I am now in vector calculus and I am still using parallel and perpendicular lines in 3-dimensional planes. With that being said parallel and perpendicular lines are not going to disappear as you go further into math, in fact you have to start using different methods to find the parallel and perpendicular lines the farther you go. Soon it will no longer be as simple as duplicating the slope or finding the reciprocal. Parallel and perpendicular lines also play a key part in physics regarding vectors just as they do in vector calculus, when you try to find equilibrium forces.

Engaging students: Approximating data by a straight line

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 Caroline Wick. Her topic, from Algebra: approximating data to a straight line.

B1. Curriculum

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

Though approximating data by a straight line is a subject that is brought up in Algebra 2, it is something that students will need to use in a number of subjects down the line. Probably the most obvious subject would be statistics. Finding an approximate trend line is extremely important for a statistician so that they can predict future, unobserved data. Another example that might not be as readily noticeable would be anthropology. Anthropology is the study of humans in various parts of life. In this case, according to Brian Hopkins, anthropology can be used by stores to figure out what types of products they should stock on their shelves during different types of the year. They do this by collecting the data, then approximating the trend lines to predict how the product will sell during the same season of the next year. For example, Orange Juice and tissues are known to be sold more often during the winter seasons, so stores know that they want to stock up on orange juice and tissue during the colder season each year.

A1: Applications

What interesting (i.e., uncontrived) word problems using this topic can your students do now?
Using the data given below:
(a) plot the points on a graph
(b) Then, using a ruler, do your best to approximate a trend line that fits the points
(c) Write an equation (y=mx+b) that best fits the trend line
(d) Approximate the next four numbers on the line using the equation you created.

Population growth in squirrels in TX from 1950-1980 (in millions)*
Year (x) 1950 1955 1960 1965 1970 1975 1980
Pop. (y) 12 12.7 13.1 13 13.6 13.7 14

From here the student would create his/her graph with the plotted points, find a line that best fits the points with equal numbers over and under the line. They would then use the data and the line to find an equation that best fits the scatter plot data that they graphed. They would then find the approximate squirrel population for 1985, 1990, 1995, and 2000.

This could be either an assignment or it could turn into a project for students with different sets of data. Students could even collect their own data to formulate the graph and equation.

*not real data, fabricated for this problem specifically.

Culture
How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The approximation of data through trend lines has been used in pop culture since the birth of popular culture in the mid twentieth century. More relevantly, it is used to map certain cultural trends. When a new movie is coming out, statisticians use previous data from people who watched/reviewed the movie before its release to map out how they believe it will be appreciated by the public. A movie that did will before its release will likely have a positive trend line that continues upward at a somewhat steady rate. It will get more tickets at the box office than a movie that was not as well liked that might have a less-steep slope. Statisticians use this same trend approximation with TV shows and whether they should run another season, or in music when it hits the top of the charts. The more people listen to a song, the more likelihood it has to be listened to other people, thus the trend continues upward until is slowly dies off.

Take for instance, Taylor Swift’s “Look What You Made Me Do” that was released August 25th of this year. From its release and popularity, statisticians were able to track the data and predict that the song would be number 1 on the top 100 just a few weeks after its release.

References:

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: