# Engaging students: Finding the area of a 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 again comes from my former student Ashlyn Farley. Her topic, from Geometry: finding the area of a right triangle.

Music is a large part of entertainment in today’s society, thus bringing in music to the classroom can help students relate to the material more. There have been studies that shows music activates both the left and right brain, which can maximize learning and improve memory. Along with the fact that it’s easier to memorize lyrics to a song than a fact, music-based learning can be engaging and impactful. It’s the same reason why musicians put a hook in their songs; brains look for patterns to better understand and process information. For the area of a triangle there are two examples, one is a rap by PBS, “Area of a Triangle Musically Interpreted,” the other is a pop parody, “Half It Baby.” By having multiple types of songs, students who have a variety of musical interest can each make a personally connection, and having a parody makes memorizing the lyrics even easier since the students will already have a reference of the melody in their brains. These songs, and other types, can be found on YouTube.

Origami is heavily based in geometry, so many lessons, such as finding area, can be created. One activity that could be engaging for the students, and have the students find the area of a triangle themselves, is with origami. The idea is that the students will create their own origami figures, after taking the area of the paper they are working with. After folding the shape, the students are to find the area of each shape, which should add up to be to total of the paper. Therefore, this project, applies the ideas of finding the area of a triangle, and finding the area of composite figures. Since origami is mainly quadrilaterals and triangles, the students are using what they know and see to figure out what the triangles’ areas equal. Because the students get to choose the origami figures, the material becomes personalized by their choices. However, this can be a difficult task if not scaffolded correctly, thus the teacher should take precautions. Done correctly, this project can be done as PBL if desired, not just group work.

Finding the area of a triangle, as well as many other shapes, is very important in architecture. However, architecture, and its designers, have very different understandings of the triangle’s meaning. A basis for all architecture, is the fact that triangles are common because the design and symmetry aid in distributing weight. Some examples of famous long-standing triangles in architecture are the Egyptian pyramids, The east Building in the National Gallery of Art in Washington, the Hearst Tower in Manhattan, the Louve in France, and the Flatiron Building in New York City. Some of these designs are using triangles as support, while others are used for decoration. However, according to Feng Shui, the triangle should be avoided, both in terms of architecture and interior design. The triangle is associated to fire energy which is chaotic energy. When triangles are used, they should point upward, implying the upward movement of energy. As seen, there are many times that the area of triangle is needed in architecture.

Resources:

# Engaging students: Writing if-then statements in conditional form

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 Bri Del Pozzo. Her topic, from Geometry: writing if-then statements in conditional form.

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

There are numerous examples of conditional statements in pop culture including movies, tv shows, and video games. I think that a fun activity to introduce students to conditional statements is to have students play a matching card game where they match the “if” strand of a famous quote to the “then” strand. For example, students would match the phrase: “If you’re happy and you know it” to “then clap your hands!” This would allow the opportunity for students to discover if-then statements in a fun and interactive way! A couple more examples that I would consider including would be from Justin Bieber’s “Boyfriend”: “If I was your boyfriend, (then) I’d never let you go.” I would also include a line from the famous children’s book, “If You Give a Mouse a Cookie.” I want to include relatable and fun examples that also help students get a clear idea of what a conditional statement is. After the matching activity, I would have students pair up and determine the definition of a conditional statement and what their general structure looks like. Including pop culture references is a fantastic way to keep the lesson fun while engaging students in the lesson material.

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

As an introduction to writing inverses, converses, and contrapositives, I could help students create graphic-organizer. Conditional statements can start to get confusing when introducing inverses, converses, and contrapositives, so a graphic organizer would be a fantastic way for students to differentiate the vocabulary and the structures of each type of statement. I would encourage students to include examples (possibly from the card sort activity), drawings, and the mathematical representation of each type of statement. The graphic organizer can also serve as a guide for students as they work through practice problems and start to develop their skills in writing conditional statements in a geometric context. As students progress through the content, I would allow students the time to go back to their organizer and include geometric examples and pictures. The organization of concepts serves as an excellent scaffold for more difficult concepts and serves as a fun way for students to practice their statement writing.

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.

This Desmos Activity can be an effective resource for students to gain some practice with conditional statements. What I like most about this website is that the questions come in different formats and ask students to utilize different skills. It is beneficial to students’ development in the subject matter that some questions ask them to write conditional statements and their converse, inverse, or contrapositive, and other questions that ask students to underline keywords. This activity would fit into this lesson topic after students have learned conditional statements, inverses, converses, and contrapositives. The interactive Desmos Activity would go well with the foldable and students can complete both lesson components simultaneously. Additionally, the interactive Desmos Activity includes examples of the different types of statements with symbols included. The combination of visuals and words is very beneficial to students who may have trouble understanding the difference between the different types of statements. Finally, the card sort activity can encourage students to work in pairs and complete an activity similar to their entry activity.

(Here is the link to the Desmos Activity https://teacher.desmos.com/activitybuilder/custom/5b909548262be93b79d1e056)

# Engaging students: Defining the terms corresponding angles, alternate interior angles, and alternate exterior angles

A quick programming note: I am transitioning to another administrative role at my university, and I expect that I’ll have much less time to post original content to this blog in the future. For this reason, I’ll only be posting on Fridays for the foreseeable future.

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 Sydney Araujo. Her topic, from Geometry: defining the terms corresponding anglesalternate interior angles, and alternate exterior angles

.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Geogebra is a great source for this topic. It’s an interactive program where students can make their own geometric shapes. Geogebra even has ready-made animations and programs that correspond with different geometry concepts. I found several ready-made explorations and animations that explain and visually show corresponding, alternate interior, and alternate exterior angles. Some of them come with questions for students to answer which would be a great activity for students to do. They have the ability with the program to adjust angles, shapes, and see how much of a difference a small change makes. It’s great for students for them to make their own discoveries and they have the ability to with this program and the different activities available. Instead of students simply being told about these angles and doing a simple worksheet, they can explore on their own which is more organic and engaging for them.

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

Using the program Geogebra that I describe, there’s several different activities already prepared on the website that can used to define corresponding, alternate interior, and alternate exterior angles. Because of the technology resources available, I could either do a jigsaw activity or a stations activity. Using a jigsaw activity I could have students form groups of 3 and each student would be in charge of learning one of the three angles. They would each complete a Geogebra activity that corresponds to their topic they are responsible for. Then after they have mastered their topic they will come back to their original groups and teach the other group members what they have learned. They could also do a stations activity where they rotate around during the class time doing a Geogebra activity for corresponding, alternate interior, and alternate exterior angles.

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

Euclid is known as the father of geometry and wrote The Elements. He was a Greek mathematician who lived from 325 BC to 265 BC. The Elements is divided into 13 books is widely famous and used among mathematicians, even in current times. It is quite amazing the discoveries Euclid made and proved during that time. In total, The Elements contains 465 theorems and proofs in which Euclid only used a compass and a straight edge. He reworked the math concepts of his predecessors, like Plato and Hippocrates, into a whole which would later become known as Euclidean geometry. Which still holds today, 2,300 years later. We actually see his proof of alternate angles in Book 1 of The Elements, it is proposition 29. It is actually the first proposition in The Elements that depends on the parallel postulate.

References

# Predicate Logic and Popular Culture (Part 236): Dirty Dancing

Let $P$ be the set of all people, and let $C(x)$ be the statement “$x$ puts Baby in a corner.” Translate the logical statement

$\forall x \in P (\sim C(x))$.

This matches a line from the movie Dirty Dancing.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Engaging students: Multiplying binomials

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 Emma White. Her topic, from Algebra: multiplying binomials.

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

When it comes to multiplying binomials, there are various activities that can make this topic interesting and fun. Furthermore, I believe this activity will make the topic of multiplying binomials stick in the students’ heads. For those reading, note that when I refer to FOIL, this is a method that lets you multiply two binomials in a particular order. It stands for: First, Outer, Inner, and Last (for more information on this concept, “Multiplying Binomials by the FOIL Method” by Professor Dave Explains on YouTube does a wonderful job of explaining the concept. The link is down below and skip to time stamp 1:00 for binomials). One resource that makes multiplying binomials more tangible is “FOIL Bingo”. In the resource provided below, a teacher took the time to create various bingo cards with two binomials in each square. The students would have to solve the binomials and when the teacher calls out the product of the binomials, the students would cover that spot and so forth. It is like regular bingo where you want to get a certain amount in a row, blackout the card, get a certain design, etc. The choice is up to the teacher. Another way to do this game (if you’re wanting to conserve time), if give the students a bingo sheet for homework the day before or even as an entry ticket the day of. Then the student could solve the binomials prior to playing the game and will have the answers in front of them instead of having to wait for each student to solve the problem during the game. This could eliminate the risk of going too slow and having students get bored or going too fast and having those who need more time to solve left behind. Lastly, the layout provided in Excel can be altered. Therefore, you could change the values if you wanted to do this activity with your class more than once.

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

Around 600-700 AD, the Hindu mathematicians had taken the Babylonia methods of approaching equations a step further when it came to introducing unknowns, sometimes more than one unknown in a single problem. It wasn’t until the Medieval times did the Islamic mathematicians discuss the variable x and how important it was. This is when the binomials theorems evolved. Furthermore, the Islamic mathematicians were able to use many operations on polynomials and soon binomials, such as multiplication, division, finding roots, and more! One thing I find highly fascinating is the Islamic mathematicians advanced the study of algebra, which “flourished during the golden age”. Evermore so, private collections were found in a lost Islamic library, which was destroyed in the 13th Century. These private collections “altered the course of mathematics.” An example of a concept that was furthered studied was the Fibonacci sequence (which is, in my opinion, one of the most fascinating things in math history and how it relates to the world and finding mathematics around us, but that is for another time…). All I can say is the Babylonians, the Hindu and Islamic mathematicians were a driven and mathematically inclined people and it blows my mind how far these people brought the world of mathematics.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

When it comes to finding ways to use technology for multiplying binomials, I truly believe visuals are essential. I’m a little biased since I was introduced to a way of multiply binomials just last semester in one of my teaching classes and it BLEW MY MIND. I wish I knew how to do this earlier in high school!  Essentially, this online source allows the student to use algebra tiles without having them physically in front of them. Therefore, they can use this source if they have technology capable of doing so (such as a phone, computer, tablet, etc.). This source is visual and easy for students to understand and manipulate. The student starts by placing the corresponding tiles for one binomial across the top like a table (would be 4 x-tiles and 2 1-tiles). Along the left side, the other binomial is represented (long ways/up-and-down). You then multiply corresponding values and where they meet in the open area (example: where an x-tile and another x-tile meet, it would become since x times x is ). Algebra tiles can also be used for upcoming topics the students would learn, such as completing the square. For a student who may have trouble grasping the idea of multiplying binomials and struggling to understand the concept of abstracts, using algebra tiles will hopefully help with the misunderstandings and confusion. All I’m saying is if this concept of online algebra tiles assisted a college student and made the topic MUCH easier to visualize and explain, I’m sure most high school students will find the use of technology in their math class interesting. Who knows, some students may come to love math more because of it!

Reference(s):

“Multiplying Binomials by the FOIL Method” by Professor Dave Explains:

“History of Polynomials”: https://polynomialshistory.weebly.com/history.html

“How modern mathematics emerged from a lost Islamic libray”: https://www.bbc.com/future/article/20201204-lost-islamic-library-maths

Algebra Tiles: https://technology.cpm.org/general/tiles/

# Predicate Logic and Popular Culture (Part 235): Suits

Let $p$ be the statement “Winners make excuses,” and let $q$ be the statement “The other side plays the game.” Translate the logical statement

$q Rightarrow \sim p$.

This matches a line from the TV show Suits.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Engaging students: Defining a function of one variable

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 Lydia Rios. Her topic, from Algebra: defining a function of one variable.

How does this topic extend what your students should have learned in previous courses?

From Prekindergarten and up, students have been practicing skills that prepared them from the concepts of a function. By counting they knew that they were adding that same number to every other number in the same sequence. By doing 1,2,3,4,5,… counting by ones they realized that every left number was being added by one to get the right number. They were taking the input 2 and doing the operation of addition by 1 to get the output of 3. The same thing was happening for other counting sequences, or even general operation statements such as 1+7=8. They have been building up to the idea of functions without recognizing that they were. You can use this no simple idea that’s been installed in them to understand what functions are. You can build them up from here and then start giving them statements with a missing component so they can find a missing variable. Then finally, building them towards defining a function where you give them similar statements with a missing component so that they can start writing out their own equations. *Don’t forget to introduce input and output and that are function represent the relationship between out input (x) is having this operation done to it to get our output (y).

Mathematics Vertical Alignment, Prekindergarten-Grade 2 (texas.gov)

Introduction to Functions | Boundless Algebra (lumenlearning.com)

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

You could use different appearances in pop culture to get students to understand input and output, such as when you are playing video games you are putting your input on the controller to get the output on the screen. However, this may not have an association with function unless you want to start getting into detail about programming. Therefore, to bring about the topic of functions I would just use a word problem that associates with pop culture. You could also bring the business side of pop culture into the class, such as setting up an equation that shows how the more tickets bought makes and increased revenue for the production of a movie. For example, lets say a ticket cost $8.50 and the production get’s 40% of the profit. Then you could set up the equal as 0.40(8.5X)=Y with 0.40 representing 40% of the profit that the production team will receive of the$8.50 tickets.

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

The topic of inputs and outputs can be touched on in reference to theatre. Both in lighting and sound, inputs and outputs are used. Therefore, the concept of this can be taught to the students. For lighting, you can talk about DMX which is what LED lights use so that the technology in the lights can pick up the functions that the computer is telling it to do. You connect the DMX in cord to the DMX in into the lighting board and then the DMX out of the lighting board to the DMX out on the lights. The same idea works with audio. However, the inputs are the microphones and the outputs are the speakers. You would take the microphone aux cord and plug that into the inputs on the Sound Board and then you would take the speaker cord and plug that into the outputs on the Sound Board. Therefore, that particular microphone is connected to that speaker and will only come out of that speaker.

“Welcome to Coolmath.” Cool Math – Free Online Cool Math Lessons, Cool Math Games & Apps, Fun Math Activities, Pre-Algebra, Algebra, Precalculus, www.coolmath.com/algebra/15-functions.

# Engaging students: Graphs of linear equations

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 Morgan Mayfield. His topic, from Algebra: graphs of linear equations.

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.)

Given a rather vague statement such as ”Graphs of Linear Equations”, I was unsure if it meant only the technique of analyzing graphs or being able to have the ability to build a graph of a linear equation. In A1, I attempt to rely on analysis. Here are the problems I encountered on Space Math @ NASA:

• Problem 1 – Calculate the Rate corresponding to the speed of the galaxies in the Hubble Diagram. (Called the Hubble Constant, it is a measure of how fast the universe is expanding).
• Problem 2 – Calculate the rate of sunspot number change between the indicated years.

Space Math has these problems listed as “Finding the slope of a linear graph”, the two key phrases being “Finding the slope” and “linear graph”. The students must be able to do both. Students are given three sets of graphical data to analyze (shown below). I am not an expert in any of these fields, but I suspect these graphs were made using real data scientists collected. The Space Math team gave students two points on the data to aid in calculations. What makes these graphs interesting is the fact that they are messy, but real compared to a graph of a linear equation in a classroom. These graphs can be analyzed further than the problems Space Math offered. Students could see how that data can be collected and put into a scatterplot, like in the case of graph 2, and have an approximately linear correlation. Sadly, most things don’t follow a neat model of what we see in our math class, yet we can still derive meaning from real-world phenomena because of what we learn in math class. Scientists use their understanding of graphs of linear equations and linear models to analyze data and come to conclusions about our real-world environment. In graph 2, a scientist would clearly see that there is a linear proportional relationship between the speed and distance from the Hubble space telescope of a galaxy, or more meaningfully understood as a rate, 76 km/sec/mpc. Now, if a scientist encountered a new galaxy, they could determine an approximate speed or distance of the galaxy given the other variable.

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

Students will formalize learning about graphing linear function in Algebra I. Graphs of linear equations are important in solving linear inequalities in two variables, solving systems of linear inequalities, solving systems of linear equations, and solving systems of equations involving linear and nonlinear equations which are all topics in Algebra I and II. Solving systems can be done algebraically, but graphing systems give students a more concrete way in finding a solution and is an excellent way of conveying information to others. If a student ever found themself in a business class, they may be asked to make “business decisions” on a product to buy. If I were the student explaining my decision to my teacher and potential “investors”, I would be making a graph of linear systems to help explain my “business decisions”. Generally, a business class would also introduce “Supply and Demand” graphs, where the solution is called the “equilibrium”. Many graphs in an intro class depict supply and demand as a system of linear equations.

In the high school sciences, a student will come across many linear equations. Students in a regular physics course and an AP physics course will come across simplified distance vs. time graphs to represent velocity, velocity vs. time graphs to represent acceleration, and force vs. distance graphs to represent work and energy (khan academy link included below). Note, just because many of the examples used in a physics class are graphs of linear equations, real life rarely works out like this.

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

We are shown data daily that our climate is changing fast through infographics on social media, posters set up by environmentalists, and news broadcasting. Climate change is one of the most important issues that society faces today and is on the collective consciousness of my generation as we have already seen the early consequences of climate change. Climate change, like most real-world data collecting does not always follow a good linear fit or any other specific fit with 100% accurately. However, a way scientists and media want to convey a message to us is to overlay a “trend line” or a “line of best fit” over the graphed data. Looking at the examples below, we can clearly understand that average global temperatures have been on the rise since 1880 despite fluctuations year-to-year and comparisons to the expected average global temperature. The same graph also gives insight on how the same data can also be cherrypicked to fit another person’s agenda. From 1998 – 2012, the rate of change, represented by a line, is lower than both 1970 – 1984 and 1984 – 1998. In fact, the rate is dramatically lower, thus climate change is no more! Not so, this period of slowing down doesn’t immediately refute the notion of climate change but could be construed as so. Actually, in the NOAA article linked below and its corresponding graph actually finds that we were using dated techniques that led to underestimates and concluded that the IPCC was wrong in it’s original analysis of 1998-2012 and that the trend was actually getting worse, indicated by the trend line on the second graph, as the global temperature departed from the long-term average.

Look at how much information could be construed by a few linear functions represented on a graph and some given rate of changes.

References:

(or Problem 226 from https://spacemath.gsfc.nasa.gov/algebra1.html)

https://d1yqpar94jqbqm.cloudfront.net/documents/Gateway5A1VAChart.pdf (or grade 5 – Algebra II Vertical Alignment https://www.texasgateway.org/resource/vertical-alignment-charts-revised-mathematics-teks)

https://bim.easyaccessmaterials.com/index.php?location_user=cchs

https://www.ncdc.noaa.gov/news/recent-global-surface-warming-hiatus

https://www.climate.gov/news-features/climate-qa/did-global-warming-stop-1998

# Thoughts on Numerical Integration: 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 numerical integration.

Part 1 and Part 2: Introduction

Part 3: Derivation of left, right, and midpoint rules

Part 4: Derivation of Trapezoid Rule

Part 5: Derivation of Simpson’s Rule

Part 6: Connection between the Midpoint Rule, the Trapezoid Rule, and Simpson’s Rule

Part 7: Implementation of numerical integration using Microsoft Excel

Part 8, Part 9, Part 10, Part 11: Numerical exploration of error analysis

Part 12 and Part 13: Left endpoint rule and rate of convergence

Part 14 and Part 15: Right endpoint rule and rate of convergence

Part 16 and Part 17: Midpoint Rule and rate of convergence

Part 18 and Part 19: Trapezoid Rule and rate of convergence

Part 20 and Part 21: Simpson’s Rule and rate of convergence

Part 22: Comparison of these results to theorems found in textbooks

Part 23: Return to Part 2 and accuracy of normalcdf function on TI calculators