Engaging students: Finding the area of a square or rectangle

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 Christian Oropeza. His topic, from Geometry: finding the area of a square or rectangle.

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

Finding the area of a square or rectangle comes up again later in Geometry when solving for the surface area of a prism. The reason behind this is because any n-sided prism, where n is the number of sides the base of the prism has, will have n many squares or rectangles. Therefore, in order to calculate the surface area of any prism, the use of finding the area of a square or rectangle is required (Reference 1). Another course that involves this topic is Calculus. When approximating the area under a curve, one strategy is to use left or right endpoint approximation which is just the sum of the areas of the rectangles under or over the curve (Reference 2). This topic is also used in physics when covering measurements. The idea of finding the area of a square or rectangle in the measurements section is to precisely and accurately find the area.

How has this topic appeared in the news?

Steiner Ranch is a hair studio that just recently added 1600 square feet, thus bringing their total to 3468 square feet. With the addition of more space the studio now holds: 19 stylist chairs, 8 shampoo bowls, 3 restrooms, and a color mixing room. All in all, this could not have been done without the use of finding the area of a square or rectangle because then the owner, Brian Charles, would not know how much of each studio equipment would be able to fit in a way that was fitting for him (Reference 3). In other news, state deputies of the Legislative Assembly of Rondonia decided to try creating 11 new protected area in the Brazilian Amazon, which amounted to a total of 2,316 square miles. Therefore, the use of the area of a square was used to determine how much area would go to the new protected areas. However, the bacanda ruralista agribusiness lobby opposed this decision and passed a bill that did not allow the process of making the protected areas (Reference 4).

How have different cultures throughout time used this topic in their society?

During 570-495 BC, the use of finding the area of a square impacted math in Greek culture. More specifically, a man by the name of Pythagoras created what is known now to be the Pythagorean Theorem. He discovered this theorem by noticing that the area of the square created by the hypotenuse of a right triangle is equal to the sum of the area of the squares created by the other two sides of the same right triangle (Reference 5). Also, there were different cultures who had discovered the same formula as the Pythagorean Theorem, but were not the first to publish their findings. These different cultures include: Mesopotamian, Indian, and Chinese (Reference 6). Finding the area of a square or a rectangle comes up immensely in computing the cost for installation of hardwood floors. The cost is computed by charging the customer for the price of each square foot of wood used and the labor for each square foot of wood installed (Reference 7).

References:

Engaging students: Using a truth table

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 Chris Brown. His topic, from Geometry: using a truth table.

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

Truth tables apply directly to the field in Computer Science, as in its essence, it runs on Boolean logic. Boolean logic simply means that everything has a result of True or False. This can be seen explicitly when dealing with logic gates, which are different paths that a computer program follows as it tests whether inputs are true or false based on given conditions. Based on the results, the program will continue to run, testing different cases, based on each result in a complex chain of tests. For example, for a simple program, let’s say you may input any integer, n, between 10 and 20 inclusive. If the number is divisible by 2, then it will compute n divided by 2. If the number is not divisible by 2, then it will return the original number. Then, if the resulting number is divisible by 2 as well, it will once again compute n divided by 2. If the resulting number is not divisible by 2, then it will return the resulting number. This sequence of tests follows the conditional statement, “If an integer between 10 and 20 inclusive is divisible by 2, and it’s resulting value is also divisible by 2, then the chosen integer has 2² within its prime factorization.” For the “and” truth table: if the integer chosen was 10, we see the True & False = False case; if the integer 16 was chosen, we see the True & True = True case; if the integer 19 was chosen, we see the False & False = False case. With variations and chains of logic gates, Computer Science has every single type of truth table embedded within the Boolean logic it uses.

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

Logical humor has often been used in the more intelligent based humor of popular culture, and truth tables and arguments are even more so apart of this. In the movie “Get Smart,” released in the year 2008, features a quirky, and humorous data analyst named Maxwell Smart who by an odd turn of events was promoted to field agent. On one of Smart’s missions to infiltrate the enemy base, he, Siegfried, and Shtarker wittingly enters into a logical argument that is a beautifully crafted logical argument. I have written the lines below.

Smart: I understand that you are the man to see if someone is interested in acquiring items of a nuclear nature

Siegfried: How do I know you are not Control

Smart: If I were Control, you would already be dead

Siegfried: If you were Control, you would already be dead

Smart: Since Neither of us are dead, so I guess I am not Control

Shtarker: That actually makes sense!

While this is not an example of a truth table per say, truth tables and propositional logic was the foundation of how this argument was created. What we see in lines 3-5 is the following propositional formula:

((p → q) ∧ (p → s))

Such that:

p = Smart being Control

q = Siegfried Being Dead

s = Smart Being Dead

By viewing the truth table, we see that when q and s are false, then p must be false; as stated in Line 5 of the movie.

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

The technology tool that I found was listed on the Stanford University website and is one that the students can easily use to check over their work. The website, attached below, allows students to enter in their propositional logic formulas for any complex length and has functionality for all necessary, binary logical operators. The site also allows for the usage of many logical expressions, not just 2. Inputting the formulas is very user friendly and allows for multiple representations of each logical operator. For instance, “or” can be represented by “\/” and also “or,” and can even both be used within the same formula chain. If a character or statement is used that the system does not recognize, the system will highlight the symbol in red and say, “illegal character,” which I personally find easily understandable for all ages. What I love most about this website is that as the formula is being entered, the student is able to see the table being created as it is being entered.

http://web.stanford.edu/class/cs103/tools/truth-table-tool/

Engaging students: Finding the volume and surface area of spheres

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 Cameron Story. His topic, from Geometry: finding the volume and surface area of spheres..

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

As a geometry teacher, manipulatives and visuals are important for conceptual understanding. Rather than handing out a formula sheet, it is far more rewarding to have your students derive volume and surface area formulas for themselves using some kind of physical representation. Not only is this more engaging for students, but the concepts behind the formula are emphasized. Yes, the volume of a sphere is $V = \frac{4}{3} \pi r^3$ , but why? Where does the fraction come from? These are important questions.

An example of an activity that could be useful when teaching the volume of a sphere is best shown by Megan Millan in the following YouTube Video:

Here, students fill up hollow solids with water and find ratios between the volumes of several different shapes.

Assuming students already know the formulas for cones and cylinders, it would make it much easier to visualize those volumes with water. Through pure experimentation, students conclude that the volume inside of a cone (whose height is twice the radius) plus the volume of a sphere is equal to the total volume of a cylinder equal height and radius.

From the student’s own experimentation (and some specifically sized manipulatives), the formula is found instead of given.

How has this topic appeared in the news?

An interesting news story by the Daily Galaxy reports that Saturn’s moon Titan has a methane cycle analogous to the water cycle on Earth; Titan has methane rain, methane clouds, and methane lakes. Ligeia Mare, Titan’s second largest methane lake, “occupies roughly the same surface area as Earth’s Lake Huron and Lake Michigan together,” (The Daily Galaxy, 2018). This news story is exciting as it hits on possible life outside earth, one that may even live in these liquid-methane lakes. As a math teacher, we can follow up this story with the following visual, illustrating the size of Earth compared the size of Titan. If these lakes are the same size, what fraction of the total surface area is the lake on Earth compared to the lake on Titan?

This can lead into how surface area changes as spheres grow or shrink. It also leads to some curiosity in the student. For example, what would Texas look like on Titan?

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

The Greek mathematician Archimedes discovered many things about solids and their properties long before calculus, and this is perfect for students in geometry; they can’t use calculus yet either. Archimedes is known for many mathematical discoveries, but in particular he is famous for finding that “…the volume of a sphere with radius r is two-thirds that of the cylinder in which it is inscribed,” (Toomer, 2018). This fact leads directly to the standard formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$ . Supposedly, Archimedes was proud enough of this discovery to “leave instructions for his tomb to be marked with a sphere inscribed in a cylinder,” (Toomer, 2018).

What I like about this bit of history is that your students can discover this formula on their own with some support from the teacher. The great mathematician Archimedes found the same formula and found it so important that he had it be inscribed in his final resting place, so your students will have a sense of pride knowing that they overcame the same challenge that only the best mathematicians from 2,000 years ago could tackle.

References:

YouTube video by Megan Millan – “Cylinder, Cone, and Sphere Volume” https://www.youtube.com/watch?v=RZkhnIzBC_k

Toomer, Gerald J. “Archimedes.” Encyclopedia Britannica, Encyclopedia Britannica, Inc., 28 Mar. 2018, www.britannica.com/biography/Archimedes#ref=ref383380&tocpanel=sectionId~toc214869,tocId~toc214869.

“Cassini’s Final Encounter with Saturn’s Giant Moon Titan –‘Like the Early Earth.’” The Daily Galaxy, The Daily Galaxy, 14 Sept. 2018, dailygalaxy.com/2018/09/cassinis-final-encounter-with-saturns-giant-moon-titan-like-the-early-earth/.

Engaging students: Defining the terms prism, cylinder, cone, pyramid, and sphere

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 Alejandro Rivas. His topic, from Geometry: defining the terms prism, cylinder, cone, pyramid, and sphere.

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

I as a teacher can create a research activity or project with prisms, cylinders, cones, pyramids, and spheres. The activity would entail having the students do some research over a particular building or structure of their choice. Once the students have decided on which building or structure I will ask them to identify all of the prisms, cylinders, cones, pyramids, and spheres the building or structure contain. The students will have to count the quantity of each, figure out a way that all of the 3-dimensional figures hold the building or structure together, have a picture, and to present to the class. After the students have presented their projects, I will then explain how prisms, cylinders, cones, pyramids, and spheres are involved in our everyday lives. I will tie it in and explain that certain professions use these 3-dimensional figures such as Engineering, Architecture, Art, Graphic Design, etc.

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

This topic extends what my students should have learned in previous courses by them being able to identify simple shapes that form prisms, cylinders, cones, pyramids, and spheres. For examples, the most common and referred prism is the rectangular prism. The prior knowledge of the shapes the students need to have are rectangles and squares. To expand my student’s knowledge from previous courses I will have them build prisms, cylinders, cones, pyramids, and spheres out construction paper. Before they cut out and form the 3-dimensional figures the students will have to identify each shape. I will split the students up into different groups. Once the groups have been formed I will let the students choose between a prism, cylinder, cone, pyramid, and sphere. Once they choose the 3-dimensional figure they will create a poster that must contain the shapes that are being used in order to form the 3-dimensional shape, and the steps the students took to get the end result.

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How can technology be used to effectively engage students with the topic?

A way that technology can be used to effectively engage students with defining the terms prism, cylinder, cone, pyramid, and sphere is by playing a game of Kahoot! I would begin the class with giving the students the definitions of the different 3-dimensional figures. Once they know the definitions I will break the students off into groups of 2 or 3 depending on the class size and have them come up with a team name. The Kahoot! will have different questions pertaining to the definition of prism, cylinder, cone, pyramid, and sphere. This should be able help me, the instructor, gauge how much the students know about prisms, cylinders, cones, pyramids, and spheres. This will also give me an opportunity to help the students understand major differences between the 3-dimensional figures. This will allow me to go into detail about the bases of certain 3-dimensional figures and how that ties into the reasoning behind their specific name.

Engaging students: Completing the square

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 Victor Acevedo. His topic, from Algebra: completing the square.

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

Completing the square is an Algebra II topic that builds on students’ prior knowledge of areas and shapes. With a given quadratic equation, students can make a visual representation of what it looks like by using Alge-blocks or Algebra tiles. The x-squared term becomes the starting point for the model. The x term gets split in half and placed on 2 adjacent sides of the x-squared term. The next step in the process requires the student fill in what is missing of the square. Students use their knowledge of squares and packing to complete the square and make the quadratic equation easily factorable.

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

Eddie Woo is an Australian High School Math teacher that also uploads videos to YouTube. He uploads his class lectures that he thinks will help others appreciate and understand math concepts better. He made this video where he makes a visual representation and informal proof for why the “Completing the Square” method works. By using the student’s knowledge of equations and shapes he can construct the square that appears when completing the square for a quadratic equation. The moment that he puts the blocks together you can hear the amazement by his students. Many of his videos have this some feeling to them in which he explores the beauty of math and makes logical connections between what students already know and what they need to know.

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

Completing the square was a method that was discovered in order to solve quadratic equations. This method was discovered by Muhammad ibn Musa al-Khwarizmi, a Persian mathematician, astronomer, and geographer. Al-Khwarizmi, also known as the father of Algebra, wrote “The Compendious Book on Calculation by Completion and Balancing” in which he presented systematic solutions to solving linear and quadratic equations. At the time Al-Khwarizmi’s goal was to simplify any quadratic equation to be expressed with squares, roots, and numbers (ax², bx, and c constants respectively) to one of six standard forms. The method of completing the square is a simple one to follow, but it had not been put into words formally until Al-Khwarizmi laid out the steps. In his book he progressed through solving simple linear equations and then simple quadratic equations that only required roots. This method only came up once he got to quadratic equations of the form ax²+bx+c=0 that could not be solved simply with roots. The discovery of this method leads to a simpler way of visually representing quadratic equations and applying it to parabolic functions.

References

Hughes, Barnabas. “Completing the Square – Quadratics Using Addition.” MAA Press | Periodical | Convergence, Mathematical Association of America, Aug. 2011, www.maa.org/press/periodicals/convergence/completing-the-square-quadratics-using-addition.

Mastin, Luke. “Al-Khwarizmi – Islamic Mathematics – The Story of Mathematics.” Egyptian Mathematics – The Story of Mathematics, 2010, www.storyofmathematics.com/islamic_alkhwarizmi.html.

Engaging students: Midpoint

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 Tinashe Meki. His topic, from Geometry: deriving the term midpoint.

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

During political elections, we usually hear how candidates are projected to do as the election moves forward. An important marker that usually separates likely candidates to win is the midpoint. Different new channels and news castor tend to use the phrase “midpoint of the election…”, or “midway through the election…” as ways to signify a halfway marker in time or events. The use of midpoint in news is used to describe halfway mark of time, events, distance etc. It’s a flexible word which gives its viewers a marker of how they can predict future events, time or distance. The uses of midpoint is inherently powerful because it simplifies and organizes ideas for views. For example, during time election there are so many stories being reported, different polls and various interpretation of how candidates are doing. Once the midpoint of the elections is reached, news anchors and new outlets provide the viewers with a consensus on how the election is going. That information is better received by the viewers because they can organize all the information they have received and create the own opinions for the second half of the election.

How could you as a teacher create an activity or project that involves your topic? What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

https://mathcs.clarku.edu/~djoyce/elements/bookIII/bookIII.html

This topic allows the teacher to simultaneously teach students about mathematical history and provide an engaging activity. I think the best way introduce students to the definition of a midpoint would be to have the students find the midpoint themselves, describe what they have found in their own words then provide them with a formal definition. A way to do that would be to show students how to bisect a line using Euclidian tool (ruler and compass) as the ruler, then have the students name the point where the line is bisected. Ask students to describe that point in their own words about the line. This activity would allow the instructor to introduce students to Euclidean geometry. The cool thing about using Euclidean geometry is that it allows students to visualize geometric concepts. It would provide them concrete understating of geometric topics.

How have different cultures throughout time used this topic in their society?

https://www.learner.org/courses/learningmath/geometry/session1/part_c/index.html

https://www.ics.uci.edu/~eppstein/junkyard/origami.html

https://plus.maths.org/content/power-origami

An interesting approach to define midpoint would be to use origami geometry. Much like Euclid constructions, Origami offers similar constructions and definitions for geometry terms. Origami is Japanese art form that has been around since 200.AD. “Modern mathematicians Humiaki Huzita and Koshiro Hatori devised a complete set of axioms to describe origami geometry — the Huzita–Hatori axioms.” Among these axioms, one of them defines and constructs a proof for the midpoint. Having students construct the midpoint using Huzita and Hatori would be an interesting way to not only introduce the definition of midpoint, but also provide a different approach of explaining geometric concepts.

Engaging students: Defining a function of one variable

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 Phuong Trinh. Her topic, from Algebra: defining a function of one variable.

How have different cultures throughout time used this topic in their society?

The understanding of functions is crucial in the study of both math and science. Not only that, some functions, especially function with one variable, are often used by everyone in their daily life. For example, a person wants to buy some cookies and a cake. The person will need to figure how much it will cost them to buy a cake and however many cookies they want. If the cost of the cake is $12, and the price for each cookie is $1.50, the person can set up a function of one variable to find the total cost for any number of cookies, expressed as c. The function can be written as f(c) = 1.50c + 12. With this function, the person can substitute any number of cookies and find out how much they would spend for the cookies and cake. Aside from the situation given by this example, function with one variable can also be used in various different scenarios.

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.)

Function with one variable can be used in many real life situations. Word problems can be derived from every day scenarios that the students can relate to.

Problem 1: John is transferring his homework files into his flash drive. This is the formula for the size of the files on John’s drive S (measured in megabytes) as a function of time t (measured in seconds): S (t) = 3t + 25

How many megabytes are there in the drive after 10 seconds?

This problem allows the students to get familiar with the function notation as well as letting the students work with a different variable other than x.

Problem 2: (Found at https://www.vitutor.com/calculus/functions/linear_problems.html )

“A car rental charge is $100 per day plus $0.30 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?”

Besides giving the students practice with finding a solution from a function, this problem let the students practice setting up the equation. This also shows the students’ understanding of the subject.

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

There are multiple resources that can be used to help the students understand what a function is as well as how they should approach a problem with function. One of the resources can be found at coolmath.com. The layout of the website makes it easy to locate the topic of “Functions” under the “Algebra” tab. By comparing a function with a box, Coolmath defines a function in a way that can be easily understood by students, while also showing how a function can be thought of as visually. The site also provides the explanation for function notation with visuals and examples that are easy to understand. On Coolmath, the students will also have the chance to practice with randomly generated questions. They can also check their answers afterward. On other hands, the site also provides definitions and explanations to other ideas such as domain and range, vertical line tests, etc. Overall, coolmath.com is great to learn for students in and out of the classroom, as well as before and after the lesson.

http://www.coolmath.com/algebra/15-functions

References:

“Linear Function Word Problems.” Inicio, www.vitutor.com/calculus/functions/linear_problems.html.

“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: Slope-intercept form of a line

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 Michael Garcia. His topic, from Algebra I: the point-slope intercept form of a line.

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

Writing linear function in slope-intercept form is an important topic in Algebra I. The slope-intercept form of a line consists of an independent and dependent variable, a slope, and a y-intercept. In previous courses, the students should have learned slope. They may not have learned specifically about the word “slope,” but they should have been introduced to the topic of rate of change. The students also should have been introduced to the topic of graphing, specifically graphing a point on a Cartesian coordinate plane. Finally, the students should have learned about independent and dependent variables.

The slope-intercept form of a line extends the concept of rate of change, graphing, and independent/dependent variables by “putting it all together.” Students now have their first glimpse into the world of graphing equations. They can now visually see the representation of the rate of change (or slope) between the different points of a line. The students can see how the slope is a constant rate that goes through our points of data.

How has this topic appeared in the news?

On September 12, 2018, Apple held its annual news conference. They announced plenty of new gear and updates to their IOS, but everyone tuned in to hear about the new iPhone. The world freaked out when the new iPhone XS, iPhone XS Max, and iPhone XR were announced. So, how does this announcement relate to the slope-intercept form of a line? If we wanted to purchase a new iPhone and have a cell service plan with it, we can write a linear equation.

According to the Apple website, you can purchase the iPhone XS for $999, while you can purchase the iPhone XS Max for as little as $1,099. However, those price points are for the 64 GB model. If we are going to purchase an iPhone, we are going to buy the biggest and flashiest model. Since the iPhone XR is not currently taking pre-orders, we are going to purchase an iPhone XS Max with 512 GB of storage for $1,449. Since most people cannot afford to spend $1,449 on a single item, we are going to have monthly payments of $68.66.

According to an Apple sales representative I spoke on the phone with, there would not be a down payment on this iPhone model.* Also, according to my mom’s phone bill, it would cost $40 a month for one cell phone line that has unlimited talking, but 0.05 cents per text (my mom doesn’t text) . Our linear equation would be y= 0.05x +68.66 + 40, which is the same as y= 0.05x + 108.66.

This is a great way of viewing linear functions in slope-intercept form because it makes the problem more personable to the student.

*given that the customer signs up for the Apple iPhone Upgrade program

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

With new technology coming out every day, there are plenty of resources available for teachers. One tool that can be used to engage students with the slope-intercept form of a line is an application called GeoGebra. GeoGebra is “dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package.” There are online resources already created (very similar to Desmos), but I would mainly use their external app. I would use GeoGebra because of the many possibilities that are provided within the app.

The beauty of GeoGebra is student can utilize this app in their studying time as well. When you plot two points, the application automatically writes the equation associated with the line. This is a great way for the student to check their work when graphing/writing linear equation in slope-intercept form.

References:

https://www.apple.com/shop/buy-iphone/iphone-xs/6.5-inch-display-512gb-space-gray-sprint#01,11,31,42

https://www.geogebra.org/about