Engaging students: Perimeters of polygons

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 Brittnee Lein. Her topic, from Geometry: perimeters of polygons.

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

Finding the perimeter of a polygon has been a necessity since the implementation of architecture and engineering in society. Every advanced society has used this topic to their benefit. For example, if a person wanted to build a fence around their rectangular garden, but wanted to use the least amount of building materials possible, they could find the perimeter around the garden and then calculate how many planks of wood they would need to use before buying that wood. This is a much more effective method than buying the wood and then finding out how much one would need to use.

 

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2. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

A few interesting word problems I found online involve real world applications of decorating. Both of the problems I found are off of this website: http://www.bbc.co.uk/skillswise/worksheet/ma31peri-e3-w-perimeter-problems

Problem 1 is good for enforcing student understanding because it challenges them to think more abstractly than if the problem were merely stated as “what is the perimeter of the cake?”. This problem tests student understanding of both the meaning of a square and the meaning of perimeter.

Problem 5 is beneficial to students because it includes both a closed and an open question. The closed question allows students to practice what they have learned about finding the perimeter of a polygon and the open-ended question is worded in a way that challenges the student’s conceptual understanding. The student must not only compute the perimeter but also must explain his/her thinking. This problem also forces students to visualize the problem in their head.

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

An engaging activity that a teacher could create to reinforce the topic of finding the perimeter of a polygon is a game where students set out to stop a criminal from entering an a given area/robbing a bank. The students would have to find the perimeter of a building from a simple blueprint mapping the building’s structure (in this case it would just be the outline of the shape of the building with given dimensions). The student would be informed of how much area each officer can cover and they would then be expected to “secure the perimeter” by allotting a certain amount of police officers to the building and placing them along the perimeter (denoted by a given symbol). To ramp up the difficulty of the game, you could set a time limit for each building and have students compete against the clock to stop the robber and you could also increase the variety of officers in the game where each type has a specialty and can cover a different amount of area.

The idea for this activity is found on the website: https://www.teacherspayteachers.com/Product/Secure-the-Perimeter-Cover-the-Area-Hands-on-police-trainee-activity-2770696

 

References

“Secure the Perimeter! Cover the Area! Hands-on ‘Police Trainee’ Activity.” Teachers Pay Teachers, http://www.teacherspayteachers.com/Product/Secure-the-Perimeter-Cover-the-Area-Hands-on-police-trainee-activity-2770696.

Perimeter Problems.” BBC News, BBC, http://www.bbc.co.uk/skillswise/worksheet/ma31peri-e3-w-perimeter-problems.

 

Engaging students: Finding the volume and surface area of spheres

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 Austin DeLoach. His topic, from Geometry: finding the volume and surface area of spheres..

green lineWhat interesting word problems using this topic can your students do now?

I found an interesting word problem that has to do with finding the size and density of Pluto using satellite images and data at https://spacemath.gsfc.nasa.gov/Geometry/6Page143.pdf that would be a good way for students to practice finding the volume of a sphere among other things. This problem could not be used at the very beginning of the section, but it is definitely interesting and could be very engaging for some students. There are multiple parts to the problem, but the third part has students calculate the volume of Pluto using the scale of measurement that they discovered in an earlier part. Students would then use their calculated volume to determine the density of the planet and compare it to other common things by using the given mass of the planet. Not only is this practice for the students to be able to calculate volume of spheres, but it helps them by showing further applications and how their calculated volume can be used to make more scientific discoveries. Problems like this are very good for students to see so that they can recognize real-world application for what they are learning in school, even if it is simplified for the sake of the class.

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

One idea that I think is interesting and engaging for students is taking an orange, measuring the diameter, then seeing how many circles of the same diameter the removed orange peel can fit in. There is a short demonstration video at https://www.youtube.com/watch?v=FB-acn7d0zU to see what I mean. This is a good activity because it is very hands-on for students to be actively engaged, and it also helps students recognize that mathematical formulas are not just thrown together, but there is reasoning behind all of them. This will also help the students remember the formula for the surface area of a sphere, as they will be able to think back to this activity and remember the time that they discovered the formula on their own. There is potential to be messy with this activity, but because it is such a memorable activity and will genuinely engage the students and let their curiosity about mathematics come to life, it is worth it if you can set aside the time for clean-up afterwards.

green lineHow has this topic appeared in pop culture?

A big place for volume and surface area of spheres to come up in pop culture is in sports. One recent situation can be seen at http://www.espn.com/espn/wire/_/section/ncw/id/18605942 where the Charleston women’s basketball team had to forfeit two victories because their basketballs for those games were not regulation size. The team accidentally used NCAA men’s basketballs (which have a circumference of 29.5-30 inches) instead of the standard women’s basketballs (circumference of 28.5-29 inches). Because the balls were not regulation size, the victories did not count. Students could use the given circumferences to find the surface area and volumes of each ball and see how significant the difference is, then discuss with their peers what the significance of different sized basketballs is. Although this is not an advanced practice idea, it is still a way for students to compute volume and surface area, as well as discover the significance of each of those properties in a way that could interest them, as many students are interested in sports and do not often think of math as playing a significant role in them. Computing the volume and surface area of the basketballs would also help them recognize the relationship between those and circumference.

 

 

Engaging students: Equations of two variables

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 Trent Pope. His topic, from Algebra: equations of two variables.

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A1. 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.)

I found a website that has many word problems where students can solve for two variables. An example of one of these problems is “If a student were to buy a certain number of $5 scarfs and $2 hats for a total amount of $100, how many scarfs and hats did they buy?”. This example would give students a real world application of how we use two variable equations. It would show students that there are multi variable problems when we go to the store to shop for things, like food or clothing. An instance for food would be when a concession stand sells small and large drinks at a sporting event and want to know how many drinks they have sold at the end of the night. After using a two variable linear equation and knowing the price of the cups, total amount earned, and total cups sold, students would be able to solve for the number of small cups as well as large cups sold.
https://sites.google.com/site/harlandclub/Home/math/algebra/word2var

 

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B2. How does this topic extend what your students should have learned in previous courses?

This topic extends on the students’ ability to graph and solve a linear equation, which should have been taught in their previous classes. The only difference is that the variable, y, that you solved for in Pre-Algebra is now on the same side as the other variable. For instance, the equation y =(-1/4) x + 4 is the same as x + 4y = 16. We see that we solve for the same variables, but they are both on the same side. This is because you are solving the same linear equation. A linear equation can be written in multiple forms, as long as the forms have matching solutions. This is something that students could prove to you by graphing and solving the equations. They would solve the equations to see that they have the same variables. This makes students more aware that they need to be able to compute for other variables besides x if the question asks for it.

 

 

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E1. 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 most effective way to engage a student about this topic is by using a graphing calculator. This is to help students make the visual connection with the topic and check to see if they have graphed the equations the correct way. Students learn more effectively through visual demonstration. Because students are the ones to solve for the equation and plug it into the calculator to check their work, they are going to be able to make that connection, and we will be able to verify that they understand the material. As teachers, we need to incorporate more technology into the ways of learning because we are surrounded by it daily. Using graphing calculators would be a great way to show and check the work of a two variable equation. This gives students a chance to see what mistakes they have made and what lose ends need to be tied up.

References

Solving Word Problems using a system with 2 variables. n.d. <https://sites.google.com/site/harlandclub/Home/math/algebra/word2var&gt;.

Solutions to Exercises in Math Textbooks

I read a very thought-provoking blog post on the pros and cons of having answers in the back of math textbooks. The article and comments on the article are worth reading.

https://blogs.ams.org/bookends/2017/10/11/solutions-to-exercises-in-math-textbooks/

5 Ways to go Beyond Recitation

Most students will encounter recitation in a math class during their academic career. How can math professors make the experience more meaningful? MAA Teaching Tidbits blog has 5 ways educators can enhance the student experience during recitation.

  1. Focus on getting students to do the work instead of doing it for them.
  2. Incorporate group work into your sessions.
  3. Get students to communicate what they understand to each other and to the class.
  4. Have students relate mathematics to their own experiences.
  5. Cultivate an environment where failure is ok and experimentation is encouraged.

Full article: http://maateachingtidbits.blogspot.com/2017/09/5-ways-to-go-beyond-recitation.html

Engaging students: The quadratic formula

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 Megan Termini. Her topic, from Algebra: the quadratic formula.

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

The Quadratic Formula came about when the Egyptians, Chinese, and Babylonian engineers came across a problem. The engineers knew how to calculate the area of squares, and eventually knew how to calculate the area of other shapes like rectangles and T-shapes. The problem was that customers would provide them an area for them to design a floor plan. They were unable to calculate the length of the sides of certain shapes, and therefore were not able to design these floor plans. So, the Egyptians, instead of learning operations and formulas, they created a table with area for all possible sides and shapes of squares and rectangles. Then the Babylonians came in and found a better way to solve the area problem, known as “completing the square”. The Babylonians had the base 60 system while the Chinese used an abacus for them to double check their results. The Pythagoras’, Euclid, Brahmagupta, and Al-Khwarizmi came later and all contributed to what we know as the Quadratic Formula now. (Reference A)

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A2. How could you as a teacher create an activity or project that involves your topic?

A great activity that involves the Quadratic Formula is having the students work in groups and come up with a way to remember the formula. It could be a song, a rhyme, a story, anything! I have found a few examples of students and teachers who have created some cool and fun ways of remembering the Quadratic Formula. One that is commonly known is the Quadratic Formula sung to the tune of “Pop Goes the Weasel” (Reference B). It is a very catchy song and it would be able to help students in remembering the formula, not just for this class but also in other classes as they further their education. Now, having the students create their own way of remembering it will benefit them even more because it is coming from them. An example is from a high school class in Georgia. They created a parody of Adele’s “Rolling in the Deep” to help remember the Quadratic Formula (Reference C). It’s fun, it gets everyone involved, it engaging, and it helps student remember the Quadratic Formula.

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E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Technology is a great way of engaging students in today’s world. Many students now have cell phones or the school provides laptops to be used during class. Coolmath.com is a great website for students to use to learn about the quadratic formula and great way to practice using it. They show you why the formula works and why it is important to know it because not all quadratic equations are easy to factor. There are a few examples on there and then they give the students a chance to practice some random problems and check to see if they got the right answer. This website would be good for student in and out of the classroom (Reference D). Khan Academy is another great way for students to learn how to use the quadratic formula. They have many videos on how to use the formula, proof of the formula, and different examples and practices of applying the quadratic formula (Reference E). Students today love when they get to use their phones in class or computers, so technology is a great way to engage students in learning and applying the quadratic formula.

 

References:

A. Ltd, N. P. (n.d.). H2g2 The Hitchhiker’s Guide to the Galaxy: Earth Edition. Retrieved September 14, 2017, from https://h2g2.com/approved_entry/A2982567
B. H. (2011, April 04). Retrieved September 14, 2017, from https://www.youtube.com/watch?feature=youtu.be&v=mcIX_4w-nR0&app=desktop
C. E. (2013, January 13). Retrieved September 14, 2017, from https://www.youtube.com/watch/?v=1oSc-TpQqQI
D. The Quadratic Formula. (n.d.). Retrieved September 14, 2017, from http://www.coolmath.com/algebra/09-solving-quadratics/05-solving-quadratic-equations-formula-01
E. Worked example: quadratic formula (negative coefficients). (n.d.). Retrieved September 14, 2017, from https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-using-the-quadratic-formula/v/applying-the-quadratic-formula

 

 

 

 

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 Saundra Francis. Her topic, from Algebra: graphs of linear equations.

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

Learning how to graph linear equations is the basis for many topics that students will learn later in Algebra and future mathematics and science courses. Students will now be able to solve word problems using graphs to model the situation describe in the problem. Being able to graph linear equations will help students graph non-linear equations since they will be able to apply the steps they learn on how to graph to different types of equations, Students will also be able to graph inequalities to find solutions for an equation since graphing equations is the first step in graphing inequalities. Another application of graphing linear equations is when students need to make graphs when completing science labs, many times students need to graph their data collected and find an equation that represents the data.

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C3. How has this topic appeared in the news?

Graphs of linear equations are displayed in the markets sections on The New York Times. Segments of different linear equations can be put together match the graphs that display the rise and fall of different markets and stocks. Time is displayed on the x-axis while the y-axis list the price of the stock. The slope of the line is the percent change in the price of the stock and can be positive or negative depending if the price rose or fell. The y-intercept would be the price that the stock or market was at before the percent change. This will engage students because it is an example of how graphs of linear equations is displayed in the real world and they get a chance to see how they can use this concept in the future. This could also be made into an activity where students discover the linear equations that are combined to make a certain market or stock graph.

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

René Descartes was born in 1596 and was a French scientist, philosopher, and mathematician. He is thought to be the father of modern philosophy. Descartes started his education at age nine and by the time he was twenty-two he had earned a degree in law. Then Descartes tried to understand the natural world using mathematics and logic, which is when he discovered how to visually represent algebraic equations. Descartes was the first to use a coordinate system to display algebraic equations. In 1637 Descartes published La Géométrie, which was where he first showed how to graph equations. He linked geometry and algebra in order to represent equations visually. While thinking about the nature of knowledge and existence Descartes stated, “I think; therefore I am”, which is one of his most famous thoughts. Students will gain interest in graphing equations when they are told about Descartes since he was an interesting person and he discovered things not only in the field of mathematics but philosophy too.

References
https://www.biography.com/people/ren-descartes-37613
http://www.classzone.com/books/algebra_1/page_build.cfm?content=links_app4_ch4&ch=4
https://markets.on.nytimes.com/research/markets/overview/overview.asp

 

 

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 Sarah McCall. Her topic, from Algebra: multiplying binomials.

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B2. How does this topic extend what your students should have learned in previous courses?

My hope is that this topic may be easier to understand if student’s can first recall an easier concept that they have already mastered, and then build upon that foundation to learn new skills. For example, at this point students should have already learned the distributive property. To introduce this new concept, I would begin by writing 4(x-5)=4 on the board and asking students what the very first step would be to solve for x. They should know to start by distributing the four to both x and -5, to get 4x-20=4. After completing a few similar examples as a class and/or in groups, then the idea of multiplying binomials would be introduced. This way, students are less intimidated when presented with new material, and they will have a good understanding of how to distribute to each term.

 

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

Teaching students some of the history behind what they are learning can be a great engaging tool. In this case it is helpful to know where the foil method first originated. I would incorporate this by discussing how it first was used in 1929; in William Bentz’ Algebra for Today. In Algebra for Today, Bentz was the first person to mention the “first terms, outer terms, inner terms, last terms” rule. Students should be knowledgeable about the history behind the math they are using, so that they realize the importance of this method. I also believe that it will be cool for students to see how a method developed is still relevant 88 years later. This technique was created in order to provide a memory aid, or “mnemonic device” to help students learn how to multiply binomials. The fact that it is still being used even today proves what an influential concept it was at its time, and throughout the years.

 

green lineE1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I am a huge fan of incorporating technology in the classroom, and YouTube is especially great because most students already use YouTube outside of school. The following clip (stopped at 1:48) provides a clear, concise explanation and demonstration of the FOIL method for multiplying binomials. It explains how factoring and foiling are related, and shows students which order to distribute in (first, outer, inner, last). The acronym FOIL is easy for students to remember, and gives them something that they can write down each time they complete a problem to help them distribute properly. Additionally, the clip is just under two minutes, which is the perfect time to ensure that students don’t zone out or lose interest before the end of the video. I would choose to follow up this video by completing a few examples as a class, emphasizing the four steps of foiling as mentioned in the video and how to use them.

References

http://pballew.blogspot.com/2011/02/origin-of-foil-for-binomial.html

Engaging students: Graphing inequalities

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 Lyndi Mays. Her topic, from Algebra: graphing inequalities.

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A1. Once students get to the point where they’re graphing inequalities, they should have a pretty good understanding of how to graph regular functions. I’ve noticed that where students have issues graphing inequalities is knowing which side of the graph should be shaded. Students get confused thinking that the graph should be shaded depending on the direction of the line instead of checking specific points. One activity that I would like to try in the classroom is giving them a worksheet where they graph inequalities on a small graph and when all the little graphs are graphed and shaded it creates a mosaic picture. I feel like there needs to be some sort of pattern or picture so that the students are sure that they’re doing the questions correctly. Another reason I like this activity is because it reaches the intelligence of artistic students. It’s not often that a math lesson can reach artistic intelligences.

 

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C1. One thing the students might find interesting about linear inequalities is that they appeared in the popular TV series, Numbers. In this particular episode, there is a blackout from attacks on an electrical substation. In order to figure out where the attack was located they mapped out where the blackouts were happening. Once they filled in all the different places that were blacking out, they realized it was one big section. Then they drew lines as if the map was on the coordinate plane. From there they are able to target the location where the attack happened.
Students also might be interested in knowing that this is also the way that policeman use to locate a cell phone. They mark the three closest cell towers that the cell phone pinged off of and are then able to draw a section and use linear functions to find the cell phone.

 

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E1. https://us.sofatutor.com/mathematics/videos/graphing-linear-inequalities

This video shows students how to solve for a variable and graph with inequalities. I liked the way it was set up because it was a word problem set up like a story and then solved. I know that students can become intimidated by having to learn new material and then having to apply it to a word problem. But this video kind of walks them through it which I believe could be helpful. Another thing was that the thing we were solving for was very realistic and might help students see why they would need to know how to graph linear equations in the future. The video also showed what x represented (cookies) and what y represented (lemonade). This lets the students know that x and y actually mean something instead of just being an arbitrary variable. I also liked that the video checked for specific points for the shading portion since many students forget that that’s a possibility and end up guessing where to shade.

 

References:
Sayfan, Sayfan. Graphing Linear Inequalities. https://us.sofatutor.com/mathematics/videos/graphing-linear-inequalities.

 

 

 

Engaging students: Simplifying rational expressions

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 Peter Buhler. His topic, from Algebra II/Precalculus: simplifying rational expressions.

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A2. How could you as a teacher create an activity or project that involves your topic?

One activity that could be performed when introducing rational expressions is to demonstrate the reason for simplifying. Before teaching students to simplify, instead ask them to evaluate the expressions given various x values. As they struggle through the painstaking process of taking squares, distributing, multiplying, adding and subtracting as they attempt to evaluate the rational expression, take note of how long it may take the students. Then have several students share their method. Following the student sharing, show your efficient method that allows you to simplify the expression before beginning to evaluate.
This not only shows the students that it is quicker, but it often provides more accurate answers to the process that must be taken to “cancel” the terms and then evaluate. Students should be more willing to participate in the following lesson on simplification due to the desire to do less work. This could also be an opportunity to discuss why it is often helpful to look for “shortcuts” or tools that can be used to simplify long or tricky problems into something manageable, even by high school students.

 

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B2. How does this topic extend what your students should have learned in previous courses?

This topic actually extends several previous topics seen in middle school mathematics. One of these topics is reducing fractions. This actually builds on the topic of finding the greatest common factor (GCF), which students learn in elementary school. To reduce a fraction, students find a GCF from both the top and bottom of the fraction, and then simply eliminate that factor leaving the expression in a simplified form. This could be utilized to introduce the idea of simplifying rational expressions, as students will likely be familiar with reducing fractions to their most simplified form.
This can also be applied to multiplying by fractions, as the GCF can be pulled out of the top and bottom of the fractions and simplified, making the multiplication of the fraction simpler. One last possible application could be in solving proportions, as students are typically taught to simplify the proportions before attempting to solve. The common theme in all of these is simplifying in order to make a problem easier and is a more efficient process for most students.

 

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D2. How was this topic adopted by the mathematical community?

There are many advanced applications of simplifying rational expressions. One such function is the Pade approximant, which is an approximation of a rational function of a given order. It was created by Henri Pade in 1890 and has been used to model certain rational functions. While this is certainly an advanced rational expression, it still holds true as there is a polynomial on the top and the bottom, which can be factored and simplified.
Rational functions have also been commonly used to model certain equations in STEM field such as functions of wave patterns for molecular particles, various forces in physics, and other fields that take mathematical ideas and apply them to a science. As a teacher introducing the topic of simplifying these expressions, one could display various applications of these functions and how they are used in a day-to-day setting. Students should be able to see beyond the cut-and-dry steps of simplifying the expressions and understand the implications beyond what they are doing.

References:

http://blog.mrmeyer.com/2015/if-simplifying-rational-expressions-is-aspirin-then-how-do-you-create-the-headache/
https://en.wikipedia.org/wiki/Rational_function