Engaging students: Deriving the proportions of a 45-45-90 right triangle

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 Amber Northcott. Her topic, from Geometry: deriving the proportions of a 45-45-90 right triangle.

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

There are ways to make the 45-45-90 right triangle not only interesting, but make it fun. A project or activity that I made up involves architecture using the special right triangle 45-45-90. In the project the students become architects. Their job is to create their own architecture, whether it is a bridge or house, etc. by using 45-45-90 right triangles. They must use a three to ten 45-45-90 right triangles. Once the students figured out how many they will use, they are going to draw their architecture. Then the students will label the sides and angles of what they drew. At the end of the activity or project they will solve the 45-45-90 triangles they used. An option for a long project is to actually build the architecture using measurable materials. The project will allow them to be creative and connect real life to the 45-45-90 right triangle. The students will also present their projects.

Another way to do the activity or project is make it a group activity and give the students some word problems dealing with architecture and have them choose one of those word problems. The students will then take the word problem and create the architecture in the word problem. They can draw it or create it, but it has to be measured and labeled along with finding the missing piece. Then they can present their findings, which includes how they came up with their measurements of sides and angles.

All the ways to do the activity or project will still need the student to be able to answer any questions that their peers or myself may ask. Also, at the end their will be a reflection on the project and their interpretation of how to solve the 45-45-90 right triangle.

 

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How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Triangles can be seen everywhere. For example, they can be seen on bridges and buildings. The website geometrinarchitecture.weebly.com has a section talking about the special right triangles, which includes the 45-45-90 right triangle. On the bottom of the page the website shares pictures of windows, roofs, and even a front door is seen within a triangle. The webpage also gives examples of how the special triangles can be used in architecture.

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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.

The dynamicgeometry.com website talks about the Geometers Sketchpad. After checking it out, I find that the program can be useful. The students can create their own 45-45-90 right triangles and explore the idea of 45-45-90 right triangles on their own after instructions on how to use the program. This engages them because the student will be able to think, how can I create a 45-45-90 right triangle? What is a 45-45-90 right triangle?  The students will have these questions and more, but those questions will soon be answered throughout the lesson itself.

References

http://geometrinarchitecture.weebly.com/special-triangles.html

http://www.dynamicgeometry.com/index.html

 

 

 

 

 

 

Finding the Regression Line without Calculus

Last month, my latest professional article, Deriving the Regression Line with Algebra, was published in the April 2017 issue of Mathematics Teacher (Vol. 110, Issue 8, pages 594-598). Although linear regression is commonly taught in high school algebra, the usual derivation of the regression line requires multidimensional calculus. Accordingly, algebra students are typically taught the keystrokes for finding the line of best fit on a graphing calculator with little conceptual understanding of how the line can be found.

In my article, I present an alternative way that talented Algebra II students (or, in principle, Algebra I students) can derive the line of best fit for themselves using only techniques that they already know (in particular, without calculus).

For copyright reasons, I’m not allowed to provide the full text of my article here, though subscribers to Mathematics Teacher should be able to read the article by clicking the above link. (I imagine that my article can also be obtained via inter-library loan from a local library.) That said, I am allowed to share a macro-enabled Microsoft Excel spreadsheet that I wrote that allows students to experimentally discover the line of best fit:

http://www.math.unt.edu/~johnq/ExploringTheLineofBestFit.xlsm

I created this spreadsheet so that students can explore (which is, after all, the first E of the 5-E model) the properties of the line of best fit. In this spreadsheet, students can enter a data set with up to 10 points and then experiment with different slopes and y-intercepts. As they experiment, the spreadsheet keeps track of the current sum of the squares of the residuals as well as the best guess attempted so far. After some experimentation, the spreadsheet can also provide the correct answer so that students can see how close they got to the right answer.

Dabbing and the Pythagorean Theorem

I enjoyed this article from Fox Sports. Apparently, a French Precalculus textbook created a homework problem asking if football (soccer) superstar Paul Pogba is doing the perfect dab by creating two right triangles.

 

Units

Photo courtesy of Dr. Fredrick Olness, a professor of physics at SMU:

Engaging students: Solving two-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 Jessica Bonney. Her topic, from Pre-Algebra: solving two-step algebra problems.

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

A great activity to use in the classroom with students for this topic would have to be algebra tiles. The tiles are a good manipulative that can be used to improve the students’ understanding and offer contact to representative manipulation for students that are more kinesthetic learners. The algebra tiles can be used to help justify and explain the process of solving two-step equations. They were developed on the basis of two ideas: (1) we can isolate variables by using “zero pairs” and (2) equations don’t change when equal amounts of tiles are used on both sides of the equation. Algebra tiles come in different colors and sizes, which can be used to represent different parts of an equation that can help students solve two-step algebra problems.  I think this would be a fun and interactive activity to help students learn and understand how to go about solving these types of problems.

 

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

Once a student gets to a certain grade level, they constantly start building upon what they learn. This material can be carried into high school and even college level courses.  Before a student learns two-step equations, they must master one-step equations, and even before that they need to understand basic arithmetical operations. Once mastery has been achieved, students will move onto solving larger polynomials, which can later be used in future algebra, geometry, and calculus courses. Another interesting use for two-step algebra problems is for future science and even computer science courses. In science, let’s say physics or chemistry, the students can use the two-step method for solving how fast a ball fell from a rooftop or for solving how fast a chemical evaporated at a certain temperature. Now in computer science students can learn how to develop algebraic functions in a computerized setting.

 

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

Rene’ Descartes, born in March of 1596, was a French mathematician, philosopher, and scientist. He is widely known for the statement, “I think, therefore I am,” deriving it from the foundation of intuition that, when he thinks, he exists. After obtaining a degree in law, his father wanted him to join Parliament, but sadly he was only 20 and the minimum age to join was 27. In turn, he moved to the Netherlands where he was influenced to study science and mathematics. During this time he formulated a common method of logical reasoning, centered on mathematics, which can be related to all sciences. This method is discussed in Discourse on Method, and is comprised of four rules: “(1) accept nothing as true that is not self-evident, (2) divide problems into their simplest parts, (3) solve problems by proceeding from simple to complex, and (4) recheck the reasoning.” We use these rules everyday when directly apply them to mathematical procedures.

 

References:

“Rene Descartes”. Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia

Britannica Inc., 2016. Web. 07 Sep. 2016 <https://www.britannica.com/biography/Rene-

Descartes>.

 

 

 

 

 

Math Graduation Message

Source: https://www.instagram.com/p/BFMe-koR6ZC/

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: