I tried a new wisecrack when teaching my students about Euler’s formula. It worked gloriously.

## All posts in category **Algebra II**

# My Favorite One-Liners: Part 113

*Posted by John Quintanilla on May 4, 2018*

https://meangreenmath.com/2018/05/04/my-favorite-one-liners-part-113/

# My Favorite One-Liners: Part 112

This was also the story of my childhood.

*Posted by John Quintanilla on April 30, 2018*

https://meangreenmath.com/2018/04/30/my-favorite-one-liners-part-112/

# My Favorite One-Liners: Part 110

I overheard the following terrific one-liner recently. A teacher was about to begin a lecture on exponential growth. His opening question to engage his students: “What does your bank account have to do with bacteria… other than they both might be really tiny?”

*Posted by John Quintanilla on April 23, 2018*

https://meangreenmath.com/2018/04/23/my-favorite-one-liners-part-110/

# 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.

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)

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.

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

*Posted by John Quintanilla on April 13, 2018*

https://meangreenmath.com/2018/04/13/engaging-students-the-quadratic-formula-4/

# 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.

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.

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.

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

*Posted by John Quintanilla on April 9, 2018*

https://meangreenmath.com/2018/04/09/engaging-students-graphs-of-linear-equations-3/

# Pascal’s Triangle and a British game show

So this happened on the popular British game show “University Challenge” on Monday, April 2. This game show pits teams of four from various British universities and is a severe test of the breadth and depth of their knowledge of many fields, including mathematics. A contestant’s response to one math question, asking for the seventh row of Pascal’s triangle, took the UK by storm this week (start at the 26:42 mark of the video below).

Twitter immediately went ablaze. Amazingly, a write-up of this encounter made it into the Times of London, one of the world’s most venerated newspapers (as opposed to the tawdry English tabloids). The above link requires a subscription; here’s a photo of page 13 from the April 4 edition:

I must admit that I’m a little amused by the amount of press that this little encounter received. When I was a kid, I memorized the first few rows of Pascal’s triangle simply from working with it so often, so when a family member told me about this story earlier this week, I knew the answer to the question instantly. I suspect that’s exactly what the contestant did here. (Whether I could have gotten the answer right under the pressure of a quiz show and a national TV audience, on the other hand, is another matter entirely.)

I have a theory as to why this appeared to be a mighty feat of mental arithmetic. The audience may have thought that he was adding the numbers quickly, but I’m guessing that the real purpose of the introductory clause “If 1,1 is the second row of Pascal’s triangle…” is to label that row as the second row instead of the first row (following the usual convention of starting the row and column counts with 0.)

*Posted by John Quintanilla on April 6, 2018*

https://meangreenmath.com/2018/04/06/pascals-triangle-and-a-british-game-show/

# 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.

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.

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.

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?

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

*Posted by John Quintanilla on April 6, 2018*

https://meangreenmath.com/2018/04/06/engaging-students-multiplying-binomials-4/

# Engaging students: Graphing inequalities

*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.

This student submission again comes from my former student Lyndi Mays. Her topic, from Algebra: graphing inequalities.

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.

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.

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.

*Posted by John Quintanilla on April 2, 2018*

https://meangreenmath.com/2018/04/02/engaging-students-graphing-inequalities/

# Engaging students: Simplifying rational expressions

*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.

This student submission comes from my former student Peter Buhler. His topic, from Algebra II/Precalculus: simplifying rational expressions.

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.

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.

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

*Posted by John Quintanilla on March 30, 2018*

https://meangreenmath.com/2018/03/30/engaging-students-simplifying-rational-expressions/

# Engaging students: Solving one- or two-step inequalities

*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.

This student submission again comes from my former student Michelle Contreras. Her topic, from Algebra: solving one- or two-step inequalities.

A.2 How could you as a teacher create an activity or project that involves your topic?

One way that I could create an activity for my students is by modifying a worksheet that I’ve seen around the internet and make it something enjoyable and engaging for students. The original worksheet is called “Who broke the Vase?” from teacherspayteachers.com and the students are supposed to solve the one and two step inequalities and match the answer with the letter of the question to figure out who broke the vase. The students are to cut the lettered puzzle and match them to the answer. I believe tweaking this worksheet and make it a group scavenger hunt activity will be a good idea because the students will be split into a group of 3 or 4 and each group will have particular letters assigned to solve.

The scavenger hunt will be around the class so the students have an opportunity to work with other students but also to walk around and be active. I will have the lettered puzzle cut into pieces so each group can match their answerers to a letter and put it up in the overhead so everyone can see everyone else’s answers and progress. I believe this 20 minute activity will be best used after a lesson in one or two step inequalities giving the students an opportunity to work with their peers, to ask questions, and to address any misconceptions. This gives the teacher an opportunity to clarify ideas and to see how well students are understanding inequalities.

B.1 How can this topic be used in your students’ future courses in mathematics or science?

Having a good understanding on how to solve for one or two step inequalities is an important skill to acquire. There are many classes that use this concept of inequalities over and over again, so if the foundation of this topic is not set right other math topics and concepts may not make sense. Personally I have made use of my knowledge of inequalities in calculus 1, solving for inequalities trying to prove limits and the squeeze theorem. Last semester in real analysis class there was a theorem called the triangle inequality which just by the name you have an idea of what it’s about. The theorem compares the sum of two lengths of a triangle to the length of the third side. Talking with your students about different instances that you will come across a certain topic may help them want to learn and gain a better understanding.

Comparing inequalities and equations is important and helps the students draw connections and remember better what to do since the properties of inequalities are very similar to equations. Stressing to your students that when you divide by a negative number on both sides that you should always flip the sign is essential. Students struggle to remember this properties since with equations you normally don’t do anything when you divide by a negative number. Having all the properties imbedded into your student’s memory will benefit them and prepare them for the future.

E.1 How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

You Tube in my opinion has great learning videos, which are a great tool for the classroom. Most often than not I refer to You Tube to get a deeper understanding about a topic, even more now that I’m in college. Searching for inequality videos that would be engaging was a pretty tough because I was looking for a precise video that was not only educational but I guess “fun” to watch. The video that I believe is a great tool for the students who are trying to remember all the rules for solving inequalities is called “Inequalities Rap”. This video makes reference to a show that I used to watch when I was little “Power Puff Girls” so automatically it grabbed my attention.

The video which was made by a group of students for a math project contains the voices of the actual students rapping about the properties of inequalities and going over the steps to solve one or two step inequalities. The video is just short of 2 minutes and is very enjoyable to watch which I believe will grab the students attention since there is some rapping/singing involved. I could also ask my students to memorize all the lyrics to the rap song and rap it to the entire class if their up for it. Giving those particular students 5-6 free homework passes that could be used throughout the year.

References

“Who broke the vase?” https://www.teacherspayteachers.com/Product/Solving-One-Step-Equations-Fun-Engaging-Worksheet-Activity-124604

“Inequalities Rap” https://www.youtube.com/watch?v=FpWm_wL73LY

*Posted by John Quintanilla on March 23, 2018*

https://meangreenmath.com/2018/03/23/engaging-students-solving-one-or-two-step-inequalities/