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.

green line

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)

green line

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.

green line

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.

green line

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.

green line

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.

green line

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.

green line

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.

 

green line

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.

green line

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.

 

green line

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.

 

green line

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: Slope-intercept form of a line

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 Jessica Williams. Her topic, from Algebra I: the point-slope intercept form of a line.

green line

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

In order to teach a lesson regarding slope intercept form of a line, I believe it is crucial to use visual learning to really open the student’s minds to the concept. Prior to this lesson, students should know how to find the slope of a line. I would provide each student with a piece of graph paper and small square deli sheet paper. I would have them fold their deli sheet paper into half corner to corner/triangle way). I would ask each student to put the triangle anywhere on the graph so that it passes through the x and the y-axis. Then I will ask the students to trace the side of the triangle and to find two points that are on that line. For the next step, each student will find the slope of the line they created. Once the students have discovered their slope, I will ask each of them to continue their line further using the slope they found. I will ask a few students to show theirs as an example (picking the one who went through the origin and one who did not). I will scaffold the students into asking what the difference would look like in a formula if you go through the origin or if you go through (0,4) or (0,-3) and so on. Eventually the students will come to the conclusion how the place where their line crosses the y-axis is their y intercept. Lastly, each student will be able to write their equation of the line they specifically created. I will then introduce the y=mx+b formula to them and show how the discovery they found is that exact formula. This is a great way to allow the students to work hands on with the material and have their own individual accountability for the concept. They will have the pride of knowing that they learned the slope intercept formula of a line on their own.

 

green line

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?

Graphing calculators are a very important aspect of teaching slope-intercept form of a line. It allows the students to visually see where the y-intercept is and what the slope is. Also, another good program to use is desmos. It allows the students to see the graph on the big screen and you can put multiple graphs on the screen at one time to see the affects that the different slopes and y intercept have on the graph. This leads students into learning about transformations of linear functions. Also, the teacher can provide the students with a graph, with no points labeled, and ask them to find the equation of the line on the screen. This could lead into a fun group activity/relay race of who can write the formula of the graph in the quickest time. Also, khan academy has a graphing program where the students are asked to create the graph for a specific equation. This allows the students to practice their graphing abilities and truly master the concept at home. To engage the students, you could also use Kahoot to practice vocabulary. For Kahoot quizzes, you can set the time for any amount up to 2 minutes, so you could throw a few formula questions in their as well. It is an engaging way to have each student actively involved and practicing his or her vocabulary.

green line

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

Learning slope intercept form is very important for the success of their future courses and real world problems. Linear equations are found all over the world in different jobs, art, etc. By mastering this concept, it is easier for students to visualize what the graph of a specific equation will look like, without actually having to graph it. The students will understand that the b in y=mx+b is the y-intercept and they will know how steep the graph will be depending on the value of m. Mastering this concept will better prepare them to lead into quadratic equations and eventually cubic. Slope intercept form is the beginning of what is to come in the graphing world. Once you grasp the concept of how to identify what the graph will look like, it is easier to introduce the students to a graph with a higher degree. It will be easier to explain how y=mx+b is for linear graphs because it is increases or decreases at a constant rate. You could start by asking,
1.What about if we raise the degree of the graph to x^2?
2.What will happen to the graph?
3.Why do you think this will happen, can you explain?
4.What does squaring the x value mean?
It really just prepares the students for real world applications as well. When they are presented a problem in real life, for example, the student is throwing a bday party and has $100 dollars to go to the skating rink. If they have to spend $20 on pizza and each friend costs $10 to take, how many friends can you take? Linear equations are used every day, and it truly helps each one of the students.

References:
https://www.khanacademy.org/math/algebra/two-var-linear-equations/graphing-slope-intercept-equations/e/graph-from-slope-intercept-equation

 

 

Engaging students: Solving one- or two-step 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 Michelle Contreras. Her topic, from Algebra: solving one- or two-step inequalities.

green line

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.

 

green lineB.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.

green lineE.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

 

 

 

 

 

Engaging students: Completing the square

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 Kelsi Kolbe. Her topic, from Algebra: completing the square.

green line

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

When students are learning how to complete the square they are usually told the algorithm take b divide it by two and square it, add that number to both sides. To the students this concept seems like a ‘random trick’ that works. This can lead to students forgetting the formula with no way to get it back. However, if we show students how to complete the square using algebra tiles they will be able to understand how the formula came to be (pictured to the left). This will allow the students to be able to have actual concrete knowledge to lean on if they forget the algorithm.

For an engage I would introduce them how to use the algebra tiles by representing different equations on the tiles. I would mix perfect squares and non-perfect squares. I would wait to do the actual completing the square as the explore activity. This way it’s something they can experiment with and really learn the material themselves.

 green line

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

Muhammad Al-Khwarizmi was a Persian mathematician in the early 9th century. He oversaw the translation of many mathematical works into Arabic. He even produced his own work which would influence future mathematics. In 830 he published a book called: “Al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala” Which translates to “The Compendious Book on Calculation by Completion and Balancing” This book is still considered a fundamental book of modern algebra. The word algebra actually came from the Latinization of the word “al-jabr” which was in the title of his book. The term ‘algorithm’ also came from the Latinization of Al-Kwarizmi. In his book he solved second degree polynomials. He used new methods of reduction, cancellation, and balancing. He developed a formula to solving quadratic equations. As you can see to the right this is how Al-Khwarizmi used the method of ‘completing the square’ in his book. It is very similar to how we use algebra tiles in modern day. You can really see the effect he had on modern algebra, especially in solving quadratic equations.

green line

E1) How can technology be used to effectively engage students with this topic?

I found a fun YouTube video of the Fort Collins High School Math Department singing a parody of Taylor Swift’s song “blank space”. In the video they are teaching the steps for completing the square. It also addresses imaginary numbers for more complex problems. I think this could be a fun engage to get the students attention. The video incorporates pop culture into something educational. I have always liked watching mathematical parodies videos on YouTube. It not only engages the students, but if they already know the words to the song, they could also get the song stuck in their head, which will help them solve the problems in the future.

References:
Completing the Square. (n.d.). Retrieved September 14, 2017, from http://www.mathisradical.com/completing-the-square.html
Mastin, L. (2010). Islamic Mathmatics – Al-Khwarizmi. Retrived September 14, 2017, from http://www.storyofmathematics.com/islamic_alkhwarizmi.html

Engaging students: Adding and subtracting polynomials

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 Kelly Bui. Her topic, from Algebra: multiplying polynomials.

green line

A2. How could you as a teacher create an activity or project that involves your topic?
The main idea of the activity will be finding an expression to represent the area of the border given the dimensions of the outer rectangle and the inner rectangle. The students will need to know how to multiply binomials and add or subtract polynomials. Therefore, this activity would be towards the end of the unit. Students will be asked to roam around the classroom or the hallway in search of items that already have dimensions labeled. For example, in the hallway there may be a bulletin board with the dimensions (2x² – 7) for the length and (3x – 4) for the width. Inside of the bulletin board, there will be the dimensions of a smaller rectangle. The question will be asked: What expression will represent the area I want to cover if I want to cover the only the border with paper?
Students may work in partners or groups to put minds together to solve this problem. Every object labeled with dimensions will be in the shape of a rectangle and the math involved will require students to multiply binomials and subtract polynomials.

 

green lineB2. How does this topic extend what your students should have learned in previous courses?
Before students learn to add and subtract polynomials, they learn how to combine like terms such as 3x and 5x. When we add and subtract polynomials, it is very similar to combining like terms in algebraic expressions. Students will need be familiar with the concept of combing like terms before they add or subtract polynomials. To introduce the topic of combining polynomials, it can be set up horizontally.

Such as: (3x² – 5x + 6) – (6x² – 4x + 9)

By setting it up this way, students can determine which terms can be combined and which terms need to be left alone. Additionally, students will build on the concept of combining like terms as it applies to this process as well. Setting it up horizontally will also increase the chance of preventing the mistake of forgetting to distribute the negative sign throughout the second polynomial. Once students are comfortable doing it this way, the addition and subtraction can be set up as a vertical problem where students must now take the step to align the like terms together in order to add or subtract. By taking the step to set up the polynomials horizontally before vertically, it will give the students a deeper understanding of what concept is actually behind adding and subtracting polynomials.

 

green line

E1. How can technology be used to effectively engage students with this topic?

The website http://www.quia.com provides a multitude of activities relating to different subjects. The game I chose to correlate adding and subtracting polynomials is identical to the actual game Battleship. The game can be played by anyone with access to the internet and Adobe. This game is interactive because you won’t have to perform math on every single shot fired at the enemy. If the student does hit one of the vessels, in order to actually “hit” the enemy’s ship, the student must successfully add or subtract two polynomials. If a student hits a vessel but is unable to solve the polynomial correctly, the game will highlight the hit area so that the student can try again. This game can either engage the students to see who can sink all of the enemy’s ships first, or it can be assigned as a homework assignment that requires showing work and screenshotting the end result of the game. Lastly, you can choose the level of difficulty of the game. For example, on the hard level, you must determine the missing addend or minuend to the expression, or add or subtract polynomials of different degrees.
The website also offers an option to create your own activities, so if Battleship isn’t panning out as desired, it is possible to create your own game for your students.
Game: https://www.quia.com/ba/28820.html

References:

Area of the Border: https://www.sophia.org/concepts/adding-and-subtracting-polynomials-in-the-real-world

Combining Like Terms: https://courses.lumenlearning.com/boundless-algebra/chapter/introduction-to-polynomials/

Battleship: https://www.quia.com/ba/28820.html

Engaging students: Using the point-slope equation of a line

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 Rachel Delflache. Her topic, from Algebra: using the point-slope equation of a line.

green line

 

A2: How could you as a teacher create an activity that involves the topic?

An adaptation of the stained-glass window project could be used to practice the point-slope formula (picture beside). Start by giving the students a piece of graph paper that is shaped like a traditional stained-glass window and then let they students create a window of their choosing using straight lines only. Once they are done creating their window, ask them to solve for and label the equations of the lines used in their design. While this project involves the point slope formula in a rather obvious way, giving the students the freedom to create a stained-glass window that they like helps to engage the students more than a normal worksheet. Also, by having them solve for the equations of the lines they created it is very probable that the numbers they must use for the equation will not be “pretty numbers” which would add an addition level of difficulty to the assignment.

green line

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

The point-slope formula extends from the students’ knowledge of the slope formula

m = (y2-y1)/(x2-x1)
(x2-x1)m = y2-y1
y-y1 = m(x-x1).

This means that the students could solve for the point-slope formula given the proper information and prompts. By allowing students to solve for the point-slope formula given the previous knowledge of the formula for slope, it gives the students a deeper understanding of how and why the point-slope formula works the way it does. Allowing the students to solve for the point-slope formula also increases the retention rate among the students.

 

green line

C1&3: How has this topic appeared in pop culture and the news?

Graphs are everywhere in the news, like the first graph below. While they are often time line charts, each section of the line has its own equation that could be solved for given the information found on the graph. One of the simplest way to solve for each section of the line graph would be to use point slope formula. The benefit of using point slope formula to solve for the equations of these graphs is that there is very minimal information needed—assuming that two coordinates can be located on the graph, the linear equation can be solved for. Another place where graphs appear is in pop culture. It is becoming more common to find graphs like the second one below. These graphs are often time linear equation for which the formula could be solved for using the point slope formula. These kinds of graphs could be used to create an activity where the students use the point slope formula to solve to the equations shown in either the real world or comical graph.

 

 

References:

Stained glass window-
http://digitallesson.com/stained-glass-window-graphing-project/

iPhone sales-
https://www.usatoday.com/story/tech/news/2017/06/28/iphones-smartphone-revolution-4-graphs/103216746/

Halloween graph-
https://www.buzzfeed.com/agh/halloween-charts-and-graphs?utm_term=.hpXrNWPm9#.qpvwGmxp0

 

 

Engaging students: Finding x- and y-intercepts

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 Deetria Bowser. Her topic, from Algebra: finding x- and y-intercepts. Unlike most student submissions, Maranda’s idea answers three different questions at once.

green line

 

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.

One example of an engaging form of technology that involves finding x- and y-intercepts of lines is mangahigh.com. Under the algebra section, there is a tab for finding x and y intercepts which once clicked provides an option to start a game (“Algebra.”). In this game, the student is expected to look at lines and quickly decipher what is known about the x and y intercepts of the line in question. Before the game begins, the student is able to choose the difficulty of the game as well as the number of questions. After the game is completed students are able to review their answers. Implementing this website into the classroom will help students gain quickness in identifying x and y intercepts. Additionally, this game is also a quick and fun way to evaluate students understanding of x and y intercepts, without forcing them to take a quiz.

green lineD1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

The topic of x and y intercepts falls under a much broader topic called analytical geometry.The article “Analytic geometry” defines analytical geometry as “[a] mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry” (D’Souza). One of the people who discovered this topic was René Descartes. René Descartes was actually a french modern philosopher who also made discoveries in the realms of science as well as mathematics. Descartes “dismissed apparent knowledge derived from authority,” meaning that he made his discoveries based on what he thought rather than taking ideas from scientists, philosophers and mathematicians (Watson). He discovered analytical mathematics (along with Fermat) in the 1630s (D’Souza). He also “he stressed the need to consider general algebraic curves—graphs of polynomial equations in x and y of all degrees” (D’Souza). Mentioning Descartes in class, and explaining his accomplishments in Mathematics as well as modern philosophy and science, will encourage students to realize that they can succeed in more than one subject . Also, Descartes can be used as an influence in the building of ideas in the classroom, since he did not just accept ideas already created.

green line

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The topic of x and y intercepts appeared on a “pop culture blog” called the comeback.com. In an article posted in November 2016, a former UCLA and current Cleveland Indians baseball player named Trevor Bauer helped one of his fans with her math homework (Blazer). This article describes a girl asking Bauer for help determining the slope of a line and the y – intercepts via Twitter. Her specific question involves the equation 2y=x (Blazer). He then explains that “for every 1 unit on the x axis go 2 units on the y axis. y intercept is where it crosses the y axis. Make y 0 and figure x” (Blazer). Since Bauer is a professional baseball player, he already has a great influence over people. Showing students this article about Bauer will show students that even people who play baseball for a living still have the knowledge of Algebra.

 

References
“Algebra.” Mangahigh.com – Algebra,
http://www.mangahigh.com/en-us/math_games/algebra/straight_line_graphs/find_the_x_and_y_intercepts_of_lines. Accessed 15 Sept. 2017.

Blazer, Sam, et al. “Trevor Bauer helped a fan do their math homework on Twitter.” The
Comeback, 13 Nov. 2016,
thecomeback.com/mlb/trevor-bauer-twitter-math-homework.html. Accessed 15 Sept.
2017.

D’Souza, Harry Joseph, and Robert Alan Bix. “Analytic geometry.” Encyclopædia Britannica,
Encyclopædia Britannica, inc., 6 June 2016,
http://www.britannica.com/topic/analytic-geometry. Accessed 15 Sept. 2017.

Watson, Richard A. “René Descartes.” Encyclopædia Britannica, Encyclopædia Britannica, inc.,
27 Jan. 2017, http://www.britannica.com/biography/Rene-Descartes. Accessed 15 Sept. 2017.