Engaging students: Solving quadratic 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 comes from my former student Elizabeth (Markham) Atkins. Her topic, from Algebra II: solving quadratic equations.

D. History: Who were some of the people who contributed to the discovery of this topic?

Factoring quadratic polynomials is a useful trick in mathematics. Mathematics started long ago. http://www.ucs.louisiana.edu/~sxw8045/history.htm stated that the Babylonians “had a general procedure equivalent to solving quadratic equations”. They taught only through examples and did not explain the process or steps to the students. http://www.mytutoronline.com/history-of-quadratic-equation states that the Babylonians solved the quadratic equations on clay tablets. Baudhayana, an Indian mathematician, began by using the equation $ax^2+bx=c$ . He provided ways to solve the equations. Both the Babylonians and Chinese were the first to use completing the square method which states you take the equation $ax^2+bx+c$ . You take $b$ and divide it by two. After you divide by two you square that number and add it to $ax^2+bx$ and subtract it from $c$ . Even doing it this way the Babylonians and Chinese only found positive roots. Brahmadupta, another Indian mathematician, was the first to find negative solutions. Finally after all these mathematicians found ways of solving quadratic equations Shridhara, an Indian mathematician, wrote a general rule for solving a quadratic equation.

C. Culture: How has this topic appeared in the news?

USA today (http://www.usatoday.com/news/education/2007-03-04-teacher-parabola-side_N.htm) had a news article that talks about students who used quadratic equations to cook marshmallows. A teacher had students in teams choose a quadratic equation. The teams then used the quadratic equation choosen to build a device to “harness solar heat and cook marshmallows”. http://www.kveo.com/news/quadratic-equations-no-problem talks about a 6 year old who learned to solve quadratic equations. Borland Educational News (http://benewsviews.blogspot.com/2007/03/memorize-quadratic-formula-in-seconds_3620.html) talks about someone who came up with a song for the quadratic formula, which is a way to solve a quadratic equation. They sing the following words to the tune of Pop Goes the Weasel: “X is equal to negative B plus or minus the square root of B squared minus 4AC All over 2A.” It may be an elementary way to solve the equation, but it sure does work. Mathematics is all around us. It is in our everyday lives. We use it without even knowing it sometimes!

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

Lesson Corner (http://www.lessoncorner.com/Math/Algebra/Quadratic_Equations) is an excellent resource for finding lesson plans and activities for quadratic equations. One lesson (http://distance-ed.math.tamu.edu/peic/lesson_plans/factoring_quadratics.pdf) talking about engaging the students with a game called “Guess the Numbers”. The students are given two columns, a sum column and a product column. They are then to guess the two numbers that will add to get the sum and multiply to get the product. This is an excellent game because it gets the students going and it is like a puzzle to solve. Learn (http://www.learnnc.org/lp/pages/2981) has a lesson plan for a review of quadratic equations. The students are engaged by playing “Chutes and Ladders”. The teacher transformed it. The procedures are as follows:

Draw a card.
Roll the dice.
If you roll a 1 or a 6, then solve your quadratic equation by completing the square.
If you roll a 2 or 5, then solve your quadratic equation by using the quadratic formula.
If you roll a 3, then solve your quadratic equation by graphing.
If you roll a 4, then solve your quadratic equation by factoring if possible. If not, then solve it another way.
If you solve your equation correctly, then you may move on the board the number of spaces that corresponds to your roll of the die.
If you answer the question incorrectly, then the person to your left has the opportunity to answer your question and move your roll of the die.
The first person to reach the end of the board first wins the game!
Good luck!!

I think this is an excellent idea because it brings back a little of the students’ childhood!

Engaging students: Truth tables

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 Elizabeth (Markham) Atkins. Her topic, from Geometry: truth tables.

D. History: Who were some of the people who contributed to the development of this topic?

In “Peirce’s Truth-Functional Analysis and the Origin of Truth Tables” it is said that Charles Peirce was the first to start studying truth tables or rather developing the idea. He created the truth table in 1893. Peirce stated “the purpose of reasoning is to establish the truth or falsity of our beliefs, and the relationship between truth and falsity”. Nineteen years later, two mathematicians developed the truth table as we know it today. Ludwig Wittgenstein and Bertrand Russell both knew of truth tables but formalized them into the form we know today. In “The Genesis of the Truth-Table Device” it is said that George Berry stated “Peirce developed the technique, but not the device”. Wittgenstein developed the terminology that we today associate with truth tables. All in all it is the work of many people that finally developed the truth tables that we know today.

APPLICATIONS: What interesting word problems using this topic can your students do now?

Truth tables state that if P is true and Q is true then both P and Q are true. If either P or Q or both are false then P and Q are false. So I could have the students construct many truth tables to demonstrate their knowledge of the subject or I could come up with some interesting word problems. Word problems such as “True or false: If Billy Joe graduated and Shawn graduated then both Billy Joe and Shawn graduated.” There are not many word problems you could create that would deal with truth tables. You can have the students begin to think logically. You could give them a statement to complete such as, “Good apples are red. Granny Smith apples are green. Thus ____” This enables the teacher to get the students in the logical process of thinking in order for them to correctly understand truth tables.

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

By teaching my students truth tables and how to use them correctly it prepares them for future classes and for everyday life. In high schools now the students are learning twenty first century skills. To learn truth tables it will help with the twenty first century skills. When you learn truth tables you learn to think logically. The students need to learn logical thinking for science and economics. In Science, they need to learn logical thinking for when they do experiments. It will allow them to process, “well if I do this then this might happen.” In economics students need logical thinking so that when they learn to invest money they can weigh their options. In everyday life students make decisions that they need to think about. Teenagers in the modern day are moving so fast that they often do and say things without thinking. If they learn to think logically then they might be able to think, “If I say or do this then this might happen.”

Irving H. Anelli’s

PEIRCE’S TRUTH-FUNCTIONAL ANALYSIS AND THE ORIGIN OF TRUTH TABLES http://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

THE GENESIS OF THE TRUTH-TABLE DEVICE http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal

Engaging students: Solving exponential equations

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 Elizabeth (Markham) Atkins. Her topic, from Precalculus: solving exponential equations.

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

Exponential equations can be different topics. You can use exponential equations for bacterial growth or decay, population growth or decay, or even a child eating their Halloween candy. Another example would be minimum wage. A good word problem would be at one point minimum wage was $1.50 an hour. Use A= $1.6 e^{rt}$ to figure out when minimum wage will reach $10.25 an hour. Another good word problem would be Billy Joe gets a dollar on his first day of work. Every day he works his salary for that day doubles. How much money does he have at the end of 30 days? A good money example would also be banking. “Use the equation $A=Pe^{rt}$ . Shawn put $100 in a savings account, which has a rate of 5% per year. How long will it take for his savings to grow to $1000? There are many ways to show exponential growth and decay.

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

Exponential equations can be used in science and life for many years from now. Students will see exponential equations when they begin to study bacteria. They will have to find the decay of growth. Students will also have to see population growth and decay throughout history. They may be asked to find out what the population will be in twenty years. When students take economics, or do their own banking, they will need to calculate interest and principal. Students will also need to do the stock market which uses exponential equations. If students go into field where they are concerned with the population of species that may be becoming extinct then the student would predict when the species would become distinct by using an exponential formula. They could also calculate how long until a certain species may take over the world, such as tree frogs or rabbits. Exponential equations are everywhere in the world and in other subjects, besides mathematics.

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

Exponential equations are used with technology everyday and every which way. Khan Academy has a few examples of exponential growth and exponential decay. Youtube has many great examples of exponential equations. Crewcalc’s exponential rap is an excellent example. They are very creative high school who found a way to express a mathematical concept through music.

Zombie Growth shows another interesting way to portray the mathematical concept of exponential equations. They use the phenomenon of zombies to demonstrate how exponential equations work.

Math project on Youtube showed another way to demonstrate how exponential equations work. They posed a problem and then stated the steps to solve the problem. Students need to use graphing calculators to check whether or not they have the right graph based on information given. They also need calculators to calculate equations and check their equations.

Football and the hypotenuse

An amusing video from Training Camp 2013 for illustrating that each side of a triangle (including right triangles) is shorter than the sum of the other two sides. The speaker is Jason Garrett, head coach of the Dallas Cowboys.

Engaging students: Finding points in the coordinate plane

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 Derek Skipworth. His topic, from Pre-Algebra: finding points in the coordinate plane.

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

When I think of the coordinate plane, one of the first things that come to mind is mapping. When I think of my teenage years, I think of how I always wanted more money. By using these two ideas together, an activity could easily be created to get the students involved in the lesson: a treasure map!

The first part of the activity would be providing the students with a larger grid. Then provide them with a list of landmarks/items at different locations (i.e. skull cave at $(3,2)$ ) that would then be mapped onto the grid. By starting out with one landmark, you could also build off previously identified landmarks, such as “move 3 units East and 4 units North to find the shipwreck. The shipwreck is located at what coordinates?” These steps could also be based off generic formulas with solutions for $x$ and $y$ . After all landmarks were identified, there would be a guide below that would trace out a path to find the treasure, which is only discovered after the full path is completed.

Courtesy of paleochick.blogspot.com

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

One of the good things about the exercise above is that it integrates several different ideas into one. A big one that stands out to me is following procedures. This is vital once you get into high school sciences. By building the map step-by-step, which each one building off the previous step, you cannot find the treasure without replicating the map exactly if you miss/misinterpret a step along the way.

As far as the coordinate plane, finding locations on the plane is important when graphing functions. Being able to find the intercepts and any asymptotes gives you starting points to work with. From there you generally only need a few more points to create a line of the function based off plotted points. This also has applications in science/math when creating bar graphs/line graphs and similar graphs.

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

As discussed in my Geometry class this semester (Krueger), the Cartesian plane opened up a lot of doors in the world of Geometry. Euclid had already established a great working knowledge of a vast amount of Geometric ideas and figures. One thing he did not establish was length. In his teachings, there were relative terms such as “smaller than” or “larger than”. No values were ever assigned to his figures though. By introducing the Cartesian plane (and in effect, being able to plot points on said plane), we were able to actually assign values to these figures and advance our mathematical knowledge. The Cartesian plane acts as a bridge between Algebra and Geometry that did not exist before. Because of this, we can know solve problems based in Geometry without ever even needing to draw the figure in the first place (example: Pythagorean Theorem).

Engaging students: Solving linear systems of equations with matrices

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 Alyssa Dalling. Her topic, from Algebra II: finding the area of a square or rectangle.

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

A fun way to engage students on the topic of solving systems of equations using matrices is by using real world problems they can actually understand. Below are some such problems that students can relate to and understand a purpose in finding the result.

The owner of Campbell Florist is assembling flower arrangements for Valentine’s Day. This morning, she assembled one large flower arrangement and found it took her 8 minutes. After lunch, she arranged 2 small arrangements and 15 large arrangements which took 130 minutes. She wants to know how long it takes her to complete each type of arrangement.

(Idea and solution on http://www.ixl.com/math/algebra-1/solve-a-system-of-equations-using-augmented-matrices-word-problems )

The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make?

(Idea and solution on http://www.algebra-class.com/system-of-equations-word-problems.html )

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

For this topic, creating a fun activity would be one of the best ways to help students learn and explore solving systems of equations using matrices. One way in which this could be done is by creating a fun engaging activity that allows the students to use matrices while completing a fun task. The type of activity I would create would be a sort of “treasure hunt.” Students would have a question they are trying to find the solution for using matrices. They would solve the system of equations and use that solution to count to the letter in the alphabet that corresponds to the number they found. In the end, the solution would create different blocks of letters that the student would have to unscramble.

For Example: The top of the page would start a joke such as “What did the Zero say to the Eight?…

Solve $x+y=26$ and $4x+12y=90$ using matrices.

To solve this, the student would put this information into a matrix and find the solution came out to be $x=12$ and $y=14$ . They would count in the alphabet and see that the 12^th letter was L and the 14^th letter was N. Then at the bottom of their page, they would find where it said to write the letters for x and y such as below-

N __ __ __ __ __ L __! (Nice Belt!)

x a c z d z y w

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

This activity would be used after students have learned the basics of putting a matrix into their calculator to solve. The class would be separated into small groups (>5 or more if possible with 2-3 kids per group) The rules are as follows: a group can work together to set up the equation, but each individual in the group had to come up to the board and write out their groups matrices and solution. The teacher would hand out a paper of 8-12 problems and tell the students they can begin. The first group to finish all the problems correctly on the board wins. There would be problems ranging from 2 variables to 4.

Ex: One of the problems could be and . The groups would have to first solve this on their paper using their calculator then the first person would come up to the board to write how they solved it-

Written on the board:

The technology of calculators allows this to be a fun and fast paced game. It will allow students to understand how to use their calculator better while allowing them to have fun while learning.

Engaging students: Mathematical induction

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 Dale Montgomery. His topic, from Precalculus: mathematical induction.

Technology

https://www.khanacademy.org/math/trigonometry/seq_induction/proof_by_induction/v/proof-by-induction

Looking at Khanacademy’s video on mathematical induction, I feel like he has one of the better explanations of mathematical induction that I have heard. This lends itself well to starting class off with a video to engage, and then moving on to an explore where the students test what can or can’t be proved by induction. This quick explanation by Khan gives a good starting point, and the fact that his videos are interesting should be sufficient enough to engage the students. Another possibility is to have the students watch this at home, that way you have more time during class do work on learning how to use the principle of induction.

Application

This problem, and proof (taken from Wikipedia) has flawed logic. In it, it uses the principle of mathematical induction. This would be a good engage because it has supposedly sound logic but it says something that is obviously not true. This will engage the students by showing them something that doesn’t make sense. This will cause a imbalance in their thinking, and make them want to make sense of the situation. I would probably present it as a bell ringer or similar problem, after induction has been introduced.

All horses are the same color

The argument is proof by induction. First we establish a base case for one horse ( $n = 1$ ). We then prove that if $n$ horses have the same color, then $n+1$ horses must also have the same color.

Base case: One horse

The case with just one horse is trivial. If there is only one horse in the “group”, then clearly all horses in that group have the same color.

Inductive step

Assume that $n$ horses always are the same color. Let us consider a group consisting of $n+1$ horses.

First, exclude the last horse and look only at the first horses; all these are the same color since horses always are the same color. Likewise, exclude the first horse and look only at the last horses. These too, must also be of the same color. Therefore, the first horse in the group is of the same color as the horses in the middle, who in turn are of the same color as the last horse. Hence the first horse, middle horses, and last horse are all of the same color, and we have proven that:

If $n$ horses have the same color, then $n+1$ horses will also have the same color.

We already saw in the base case that the rule (“all horses have the same color”) was valid for $n=1$ . The inductive step showed that since the rule is valid for $n=1$ , it must also be valid for $n=2$ , which in turn implies that the rule is valid for $n=3$ and so on.

Thus in any group of horses, all horses must be the same color.

(taken from http://en.wikipedia.org/wiki/All_horses_are_the_same_color )

The explanation relies on the fact that a set of a single element cannot have 2 different sets with the same element. Because this assumption cannot be made, the case of $n=2$ falls apart and tears the argument apart.

Application

Dominoes have been talked about as a way to explain mathematical induction. The idea that if you can prove that the first one falls, and you can prove that in general if a domino falls, the one after it will fall, you can prove that the entire row of dominoes would fall. I think it would be fun to students to actually demonstrate this idea. It would even be fun to illustrate what would happen if you cannot prove that the first one falls by gluing the dominoes to whatever surface that you are using (not the table).

The idea would be to have it set up as the students walked in and ask them what would happen if you pushed over the first domino. After that test the hypothesis with one row (you should probably have multiple rows set up for this). Then introduce the concepts of base case and induction step using the dominos. Then you can ask well what if we cannot push the first domino over, does that mean we cannot show that all of the dominos will fall? After this you can start taking the concept of dominos and applying it to Mathematical induction.

Engaging students: Computing inverse functions

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 Derek Skipworth. His topic, from Algebra II: computing inverse functions.

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

In essence, an inverse function is supposed to “undo” what the original function did to the original input. Knowing how to properly create inverse functions gives you the ultimate tool for checking your work, something valuable for any math course. Another example is Integrals in Calculus. This is an example of an inverse operation on an existing derivative. A stronger example of using actual inverse functions is directly applied to Abstract Algebra when inverse matrices are needed to be found.

C. How has this topic appeared in high culture?

The idea of inverse functions can be found in many electronics. My hobby is 2-channel stereo. Everyone has stereos, but it is viewed as a “higher culture” hobby when you get into the depths that I have reached at this point. One thing commonly found is Chinese electronics. How does this correlate to my topic? Well, the strength of the Chinese is that they are able to offer very similar products comparable to high-end, high-dollar products at a fraction of the costs. While it is true that they do skimp on some parts, the biggest reason they are able to do this is because of their reverse engineering. Through reverse engineering, they do not suffer the massive overhead of R&D that the “respectable” companies have. Lower overhead means lower cost to the consumer. Because of the idea of working in reverse, “better” products are available to the masses at cheaper prices, thus improving the opportunity for upgrades in 2-channel.

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

A few years ago, there was a game released on Xbox 360 arcade called Braid. It was a commercial and critical success. The gameplay was designed around a character who could reverse time. The trick was that there were certain obstacles in each level that prevented the character from reversing certain actions. To tie technology into a lesson plan, I would choose a slightly challenging level and have the class direct me through the level. This would tie into a group activity where the students are required to calculate inverse functions to reverse their steps (like Braid) and eventually solve a “master” problem that would complete the activity. This activity could be loosely based off a second level that could wrap up the class based off the results that each group produced from the activity.

http://braid-game.com/

Mathematics and The Price Is Right

I just read a very entertaining article on the use of game theory for improving contestants’ odds of winning the various games on the long-running television game show “The Price Is Right.” Quoting from the article:

On a crisp November day eight years ago, I took the only sick day of my four years of high school. I was laid up with an awful fever, and annoyed that I was missing geometry class, which at the time was the highlight of my day. I flipped on the television in the hope of finding some distraction from my woes, but what I found only made me more upset: A contestant named Margie who was in the process of completely bungling her six chances of making it out of Contestants’ Row on The Price is Right.

Many contestants fail to win anything on The Price is Right, of course. But as I watched the venerable game show that morning, it quickly became clear to me that most contestants haven’t thought through the structure of the game they’re so excited to be playing. It didn’t bother me that Margie didn’t know how much a stainless steel oven range costs; that’s a relatively obscure fact. It bothered me, as a budding mathematician, that she failed to use basic game theory to help her advance. If she’d applied a few principles of game theory—the science of decision-making used by economists and generals—she could have planted a big kiss on Bob Barker’s cheek, and maybe have gone home with … a new car! Instead, she went home empty-handed…

To help future contestants avoid Margie’s fate, I decided to make a handy cheat sheet explaining how to win The Price Is Right—not just the Contestants’ Row segment, but all of its many pricing games. This guide, which conveniently fits on the front and back of an 8.5-by-11-inch piece of paper, does not rely on the prices of items.

The full article can be found at http://www.slate.com/articles/arts/culturebox/2013/11/winning_the_price_is_right_strategies_for_contestants_row_plinko_and_the.html.

Engaging students: Factoring quadratic polynomials

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 Kelsie Teague. Her topic, from Algebra I and II: factoring quadratic polynomials.

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

In Renaissance times, polynomial factoring was a royal sport. Kings sponsored contests and the best mathematicians in Europe traveled from court to court to demonstrate their skills. Polynomial factoring techniques were closely guarded secrets.

http://www.ehow.com/info_8651462_history-polynomial-factoring.html

When reading this article, I found the fact that this topic was considered a royal sport very interesting. Students would also find that interesting because it would get their attention with the fact that kings thought this was very important. We could even have our own royal game for it. I think we could start off with a scavenger hunt to work on factoring just basic integers. Also, I think we could use the same idea to start the explore except to do it backwards and give them the polynomial already factored and have them FOIL it and get their polynomial. I want to see if they can see how to do it the other way around without being taught how. This game could show them that factoring is just the reverse of foiling.

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 looked up factoring quadratic polynomials on Khan Academy and I found some really great videos. They have videos that show detail steps and also after a few videos they have parts where you can practice what you just watched and see if you understand it. This website is great for at home practice or in class practice because with the practice sections it tells you if you are correct or not and will also give you hints if you don’t know where to start. Also, if you don’t have a clue how to do the problem given, you can hit “show me solution” and it will redirect you to a similar problem in a video to help out. I think this website is a great tool to let students know about to learn and practice.

Also I found a great video on YouTube it’s a rap about factoring that would certainly get gets engaged.

Curriculum

Students first learn about the basic idea of factoring in elementary school and continue to learn and use this topic all the way through college. You need to factor polynomials in many different contexts in mathematics. It’s a fundamental skill for math in general and can make other calculations much easier. You use factoring for finding solutions of various equations, and such equations can come up in calculus when find maxima, minima, inflection points, solving improper integrals, limits, and partial fractions. Students will need to know factoring all the way up in to their higher-level math classes in college, and also be able to use it in a career that is related to engineering, physics, chemistry and computer science.