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 Chais Price. His topic, from Algebra: the quadratic formula.

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

What is the quadratic formula you ask? It is basically a tool used to find roots of the quadratic equation. It all started around 3000 BC,when the Ancient Babylonians needed a method for calculating how much to pay the tax collector. Imagine that you are an Ancient Babylonian farmer with a square field that is placed in the middle of your farm. On this field you plan to plant some crops. After some experimenting you discover that if you double the length of each side of the square field, you end up with 4 times as many crops as before. This observation demonstrated the area of the field and the amount of crops that you can grow and the square of one of the side lengths are all proportional. From here we come up with the first quadratic equation. Let “x” represent the length of a side and “m” be the amount of crops you can grow on a square field of side length 1. Then we have “c” which is the entire area of crop that can be grown. Thus c= mx^2. It is now April 15 in Babylonia and the tax collector comes and says to give him “c” crops to pay your taxes. Now you ask yourself, how big does your square field need to be to grow that amount of crops? Well as it turns out, we just take x= (c/m)^1/2. The Babylonians however, did not have a way to solve square roots accurately. They were just estimations. This square field representation is just a basic representation. Let’s say that your field was not a square but a triangle divided into 2 right triangles where a and b are the amount of crops that you can grow in each field.

To calculate the amount of crops, you being a very intelligent Babylonian farmer, you come up with the equation

$c= ax^2+bx$ .

The next step is to divide by “a” and then complete the square yielding

$a \displaystyle \left(x + \frac{b}{2a} \right)^2 = x^2 + \displaystyle \frac{b}{a}x + \displaystyle \frac{b^2}{4a^2}$

Now we substitute into the previous equation. We now have an equation of the form

$a \displaystyle\left( x + \frac{b}{2a} \right)^2 = c + \displaystyle \frac{b^2}{4a^2}$

Solving again for the tax collector, we need to solve for “x.” This gives us what we know as the quadratic formula:

$x = \displaystyle \frac{-b \pm \sqrt{b^2 + 4ac}}{2a}$

Something worth noting since you are an Ancient Babylonian farmer, is that all the roots you find are positive since negative numbers have not been discovered yet. In addition, the quadratic formula shown above is just an illustration broken down step by step. The Babylonians had no general formula for the quadratic formula, but there method for the quadratic formula can be closely associated with the method of completing the square.

How has this topic appeared in the news?

There is a pretty interesting article I read which had to do with the quadratic equation. A teacher provided a list of quadratic equations that the students would pick from. This equation was to be to model for a parabolic device that the student was responsible for building. The student was instructed to spend no more than 12 dollars on this project. They would use the equation to calculate a positional focus which suspended a marshmallow which would cook slowly due to the rays of the sun reflected upon it. This lesson stretched using a quadratic equation to form a focus. Then the student would graph the data and calculate the rate of temperature increase. Student were also asked to make predictions on what temperature the marshmallow would get to. Once the experiment is complete, the student is anxious to see how accurate their model is to the actual equation they chose. This is a very good lesson that covers a pretty broad range of topics.

So the concept of the quadratic formula would develop beyond what the Ancient Babylonians discovered. 1500 years after the Babylonians, the Egyptians came along with a different approach for similar type problems where the quadratic formula was used. The mathematics behind the calculations were known to be very complicated. However, they recorded these calculation on a table ( much like our multiplication tables) so that when a farmer or an engineer required some kind of proportion or dimensional measurement using the quadratic formula, rather than solving the problem again, they would just look it up on the table that had already been calculated. They reproduced this table and made it a master look up guide to follow for such solutions.

Then the Babylonians would reappear in time and discover the method of completing the square. This method still required somewhat of an educated guess. However, since the Babylonians had a 60 base numerical system ( which the Egyptians did not) addition and multiplication were easier to compute and check calculations. Even still there was not a general formula or equation established. Then Pythagoras and Euclid came along and developed a general formula to solve the quadratic equation. Pythagoras argued that the ratio of the area of a square and the length of a side does not always produce an integer and indeed can have a rational solution. Euclid advanced beyond Pythagoras and claimed you could have irrational results. These were logical claims made from a theoretical point of view because at that time there was no way to calculate the square root of a number by hand. Finally in 700 AD, a mathematician named Brahmagupta from India came up with a general solution to the quadratic equation using numbers. He also was on board with Euclid and his irrationals as well as an equation producing 2 roots. His work more than likely inspired another Hindu Mathematician named Baskhara who around 1100 AD formulated the complete solution we are familiar with today. He was also the first to acknowledge that any number greater than 0 has 2 square roots. It was around this same time in history that a Jewish mathematician Abraham bar Hiyya derived the quadratic formula and brought it to Europe. It wouldn’t be until another 500 years that the quadratic formula would be adopted into the formula we know today.

Works Cited

Budd, Chris, and Chris Sangwin. “101 Uses of the Quadratic Equation.” Plus Math. Plus Magazine… Living Mathematics , 01 Mar 2004. Web. 10 Sep 2014. http://plus.maths.org/content/101-uses-quadratic-equation

Tracey, Wong Briggs. “Students use quadratic equations to cook marshmallows.” USA Today. USA Today, 05 Mar 2007. Web. 10 Sep 2014. http://usatoday30.usatoday.com/news/education/2007-03-04-teacher-parabola-side_N.htm?csp=34

Hell, Dr. . “The History Behind the Quadratic Formula .” . BBC H2G2, 13 October 2004. Web. 10 Sep 2014. http://news.bbc.co.uk/dna/place-lancashire/plain/A2982567

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 again comes from my former student Carissa Birdsong. Her topic, from Algebra: computing inverse functions.

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

When students are learning any algorithm in math, it helps keep their interest if they know what this can be possibly used for in the future. In pre-calculus, students need to find the inverse of cosine, sine, tangent, etc. to find certain angles. In order to grasp the students’ attention, the teacher can show videos of bottle rockets being shot off at different angles. Then the teacher will explain that in order to find most of these angles, one must use the inverse property. Then the teacher can go into depth of how to find the inverse of a function. But, the students must understand that using inverse to find angle measurements will not happen in this curriculum, but in future classes such as pre-calculus, trigonometry and physics.

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

Human Representation of Inverse Function

Move the desks to the sides of the room, making a big open space in the middle.
Assign each student a partner.
Have a strip of tape down the middle of the room prior to class. Have the students line up facing their partner with the strip of tape in between them.
Have the side on the “right” be side A and the side on the “left” be side B. (The teacher will choose which side is the right or left, depending on where the front of the classroom is)
Side A will pick a position to stand in (the teacher must monitor to make sure the students are being appropriate). The students are encouraged to change their face, arms, head, etc. to pick the most creative position possible.
Now side B will mimic their specific partner on side A.
Once the students have locked in their position, the teacher will point out that side B is reflective of side A. Therefore, side B is the inverse function of A.

*Make sure that the students understand that side B is not doing the exact same thing that side A is doing, but the opposite, the reflection. The inverse of a function “undoes” the function itself. If someone were to take away side A, and bring in a new crop of people to reflect side B, it should be EXACTLY what side A had done. The inverse of the inverse of a function must take you back to the original function.

*After the teacher teaches how to find the inverse of a function, and can elaborate on the graphing of each function, he or she can refer back to this activity and show that there is an invisible line between the function and the inverse function, making clear that they reflect each other, just as the students did.

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

Even though most students probably haven’t seen Top Secret, they will probably appreciate watching any sort of movie or television during class. In the making of Top Secret, the actors film a scene walking backwards and saying lines in reverse order. In the movie, this scene is played in reverse, so they look like they were just speaking gibberish and walking forward. They did this so Val Kilmer can do cool tricks like throw a book on the top shelve and slide up a pole.

The teacher could show his or her class the original scene, straight from the movie.

Then ask, “How do you think the actors did this?” “What language are they speaking?” Hopefully a student will catch on fast and say that they just filmed it backwards. Then the teacher can show the scene played forwards.

These two scenes are inverse each other. Going from the beginning to the end of one takes you to the beginning of the other. And going from the beginning to the end of the other, takes you to the beginning of one. Most functions have an inverse function. This means there is a function that is reverse of its inverse. This does NOT mean that the inverse of a function is just the original backwards (i.e. y=3+x and x+3=y). The function of f has the input x and the output y, whereas the inverse of the function f has the input y and the output x.

Resources:

https://www.youtube.com/watch?v=jXlucE4iUDE

https://www.youtube.com/watch?v=2Mr_XAM8CMw

Engaging students: Finding the inverse of a matrix

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 Donna House. Her topic, from Algebra: finding the inverse of a matrix.

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

Engage the students by asking them how they think our military (or a secret agent) sends and receives messages without the enemy knowing what message is being sent. Then the discussion can be guided by asking how math is used in encoding and de-coding secret messages. Since they already will have learned about matrices, tell them they are going to learn how to use matrices to create a secret message and de-code a secret message from a classmate.

First they need to learn to compute the inverse of a simple matrix A (provide this matrix to be certain it has an integer inverse.) I prefer a three-by-three, but this can also be done with any size matrix – even a two-by-two. Next, they create their own short message and code it using numbers to represent the letters of the alphabet (A=1, B=2, etc., with 0=space). This coded message should be written into a matrix form, filling in one row at a time (the number of columns MUST match the number of rows in matrix A.) If the secret message does not fill the last row add zeros for spaces. Now, multiply the message matrix by matrix A (with matrix A on the right.)

Message: 7 15 0 21 14 20 0 5 1 7 12 5 19

$\displaystyle \left[ \begin{array}{ccc} 7 & 15 & 0 \\ 21 & 14 & 20 \\ 0 & 5 & 1 \\ 7 & 12 & 5 \\ 19 & 0 & 0 \end{array} \right] \left[ \begin{array}{ccc}3 & 1 & 3 \\ 7 & 10 & -3 \\ 8 & 5 & 5 \end{array} \right]$

This will result in your encoded message:

$\displaystyle \left[ \begin{array}{ccc} 126 & 157 & -24 \\ 321 & 261 & 121 \\ 43 & 55 & -10 \\ 145 & 152 & 10 \\ 57 & 19 & 57 \end{array} \right]$

Now have each student pass this encoded message to another student. Each student must use the inverse of matrix $A$ to de-code the message!

Have them multiply this message matrix by $B A^{-1}$ with the inverse on the right. They will get the de-coded Message matrix. From this they can discover the message!

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

Written as an engage:

We are going to begin with a short video today!

(Published on Feb 21, 2013)

This video introduces the Computer Graphics chapter of the “Computer Science Field Guide”, an online interactive “textbook” about computer science, written for high school students. The guide is free, and is available from cosc.canterbury.ac.nz/csfieldguide/ . This video may be downloaded if you need to play it offline.)

What did you notice about the movement of the objects in the video? Does this movement – rotation, position, size – remind you of anything you have done in math class before? What happened to the graph of a function when we multiplied the x value? What about when we multiplied the y value? What happened when we added or subtracted a number to x or y? Do these transformations of functions move in a similar manner as the computer graphics in the video? (Of course, the video shows three-dimensional movement while our graphs only showed two-dimensional movement.)

So what kind of transformations do you think are used to create computer graphics? The graphics you see in your video games, in the movies, on TV, in flight simulators for training pilots, and in many other applications are all created with the transformations of matrices. Matrix multiplication is used in computer graphics to size and scale objects as well as rotate and translate them. Today we are going to learn to compute a special matrix transformation – the inverse of a matrix!

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

After the students have learned how to calculate the inverse of a 3 x 3 matrix by hand, you could tell them they are now going to calculate the inverse of a 4 x 4 matrix. After they all roll their eyes and groan, you can ask if they would rather do the calculations by hand or on their graphing calculators.

Now you can introduce a method for entering the data into a calculator (such as the TI83 or TI84.) Since many graphing calculators can handle large matrices, the matrix and the identity matrix can be entered together as a 4 x 8 matrix. By using the “rref(” application, the inverse matrix will automatically be calculated. Another way to calculate the inverse is to enter the matrix then press the x^-1 key.

However, you may want to wait before teaching this “short-cut” method. You may choose to have the students enter the 4 x 8 (matrix and identity matrix) and show them how to do the row operations on the calculator. This is useful in helping them see the steps involved in the calculation (and tortures them just a little.)

Engaging students: Finding the asymptotes of a rational function

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 Belle Duran. Her topic, from Algebra: finding the asymptotes of a rational function.

How has this topic appeared in high culture?

Although the topic itself has not appeared in high culture, idea of asymptotes brings me the idea of the myth of Tantalus. In a nutshell, Tantalus was always committing crimes against the Gods of Olympus but always going unpunished. One day, he invites the Gods to his home for a feast in which he serves the Gods a rather vile dish. This ultimately angered the Gods to the point of punishing Tantalus by hanging him from a fruit tree amidst a lake, sentencing him to suffer eternal hunger and thirst. Tantalus was always so close to the water and fruits, yet they stayed beyond his reach. In the same way, when a graph has an asymptote then a part of the graph will approach that asymptote without ever touching it or being equal to it.

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

The word, “asymptote” derives from the Greek word, “asumptotos” which translates to “not falling together.” The term was first introduced by Apollonius of Perga in his work on conic sections, but used the term to represent a line that will not meet the curve in any finite point. Other achievements by Apollonius includes the introduction of eccentric and epicyclic motion to explain the motion of the planets as well as the hemicyclium which is a sundial with hour lines drawn on the surface of a conic section to give greater accuracy.

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

One way finding asymptotes can be used in students’ future courses are to understand finding the limits of a function. When it comes to limits, it can be shown that vertical asymptotes are concerned with objectives in which the function is not usually defined and near which the function becomes large positively or negatively, or if a line x=a is called a vertical asymptote for the graph of a function of either the limit to positive infinity as x approaches positive a or negative a. Likewise, horizontal asymptotes are concerned with finite values approached by the function as the independent variable grows large positively or negatively. In other words, a line y=b is a horizontal asymptote for the graph is either the limit of the function is b as x approaches positive infinity or negative infinity.

References

The myth of Tantalus

http://www-history.mcs.st-and.ac.uk/Biographies/Apollonius.html

http://jwilson.coe.uga.edu/emat6680/greene/emat6000/greek%20geom/Apollonius/apollonius.html

http://www.education.com/study-help/article/horizontal-vertical-asymptotes/

http://oregonstate.edu/instruct/mth251/cq/Stage3/Lesson/asymptotes.html

Design Zone at Fort Worth Museum of Science and History

If you live in the Dallas-Fort Worth metroplex or are visiting this summer, I highly recommend Design Zone, which is on exhibit at the Fort Worth Museum of Science and History until September 7. I’ve been to a lot of science museums, so I don’t make the following statement lightly: this may well be the most fun and most engaging physics exhibit for children to enjoy that I’ve ever seen. There’s all kinds of things to lift, throw, balance, and blast off so that children have so much fun that they don’t even realize that they’ve learned something.

Here’s the information posted by the museum: http://fwmuseum.org/design-zone

Here’s a short promotional video:

And here’s a longer video describing the exhibit:

Engaging students: Multiplying binomials

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 Andy Nabors. His topic, from Algebra: multiplying binomials.

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

Multiplying binomials is an interesting concept because there are so many ways in which this can be done. I can think of five ways that binomials can be multiplied: FOIL, the box method, distribution, vertical multiplication, and with algebra tiles. I would incorporate these methods into one of two different ways. In either case, I would split the class into five groups.

In the first way, I would assign each group a different method of multiplication. The groups would each be responsible for exploring their method, working together to master it. Then each group would be responsible for making a poster describing their method in detail. Then would then present their poster to the class, and the students not presenting would be taking notes. Already having one concept of binomial multiplication, the students would be seeing other methods and deciding which makes most sense to them.
In my second idea, I would have five stations in the classroom each with their own method. The groups would rotate station to station figuring out the different methods collaboratively. The groups would rotate every 7-10 minutes until they had been to every station. Then the class would discuss the strengths/weaknesses of each method compared to the others in a class discussion moderated by the teacher.

These activities rely on the students being able to work and learn in groups effectively, which would present difficulty if the class was not used to group work.

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

I had the privilege of teaching a multiplying binomial lesson to a freshmen algebra one class in CI last spring. My partner and I focused on the box method first, and then used that to introduce FOIL. The box method was easier to grasp because of the visual nature of it. In fact, it looks a lot like something that the students will definitely see in their biology classes. The box method looks almost identical to gene Punnet Squares in biology. In fact, my partner and I used Punnet Squares in our Engage of that lesson. We reminded the students of what a Punnet Square was, and then showed them a filled out square. We went over how the boxes were filled: the letter on top of each column goes into the boxes below and the letters to the left of the box go in each box to the right. Then we showed them an empty Punnet Square with the same letters before. We inquired about what happens when two variables are multiplied together, then filled out the boxes with multiplication signs in between the letters. The students responded well and were able to grasp the concept fairly well from the onset.

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

The internet is fast becoming the only place students will go for helpful solutions to school problems. This activity is designed to be a review of multiplying binomials that would allow students to use some internet resources, but make them report as to why the resource is helpful. The class will go to the computer lab or have laptops wheeled in and they will be given a list of sites that cover binomial multiplication. They will pick a site and write about the following qualities of their chosen site: what kind of site? (calculator, tutorial, manipulative, etc.), how is it presented? (organized/easy to use), was it helpful? (just give an answer opposed to listing the steps), did it describe the method it used?, can you use it to do classwork?, etc.

This is a sample list, I would want more sites, but it gives the general idea I’m going for. (general descriptions in parentheses for this project’s sake)

http://www.mathcelebrity.com/binomult.php (calculator, shows basic steps of FOIL of inputted problem)

http://www.webmath.com/polymult.html (calculator, shows very detailed and specific steps of FOIL of inputted problem)

http://calculator.tutorvista.com/foil-calculator.html (calculator, shows general steps of FOIL, not the inputted problem)

http://www.coolmath.com/crunchers/algebra-problems-multiplying-polynomials-FOIL-1.html (calculator but only problems it gives itself, more of a practice site)

http://www.mathwarehouse.com/algebra/polynomial/foil-method-binomials.php (FOIL tutorial site with practice problems with hidden steps)

http://www.themathpage.com/alg/quadratic-trinomial.htm (wordy explanation, lots of practice problems with hidden answers)

https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/multiplying-binomials/v/multiplying-polynomials-2 (many tutoring videos, just the writing no person)

http://www.zooktutoring.com/now-available-my-very-first-instructional-math-video/ (many tutoring videos, tutor is seen with the work)

http://illuminations.nctm.org/Activity.aspx?id=3482 (algebra tile manipulator)

I will assume as a teacher that my students already look for easy solutions online, so I want to make sure they look in places that will help them gain understanding. I would stress that calculator sites are dangerous because if you just use them then you will not be able to perform on your own, but could be helpful to check your answer if you were worried. At the end of the lesson they would have a greater understanding of how to use internet sources effectively and have reviewed multiplying binomials.

Resources:

http://www.mathcelebrity.com/binomult.php

http://www.webmath.com/polymult.html

http://calculator.tutorvista.com/foil-calculator.html

http://www.coolmath.com/crunchers/algebra-problems-multiplying-polynomials-FOIL-1.html

http://www.mathwarehouse.com/algebra/polynomial/foil-method-binomials.php

http://www.themathpage.com/alg/quadratic-trinomial.htm

https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/multiplying-binomials/v/multiplying-polynomials-2

http://www.zooktutoring.com/now-available-my-very-first-instructional-math-video/

http://illuminations.nctm.org/Activity.aspx?id=3482

John Nash: A Tragic but Meaningful Life

Princeton University’s Office of Communications put together a well-written news release concerning the life and tragic death of John Nash: http://www.princeton.edu/main/news/archive/S43/27/52G52/index.xml?section=featured. This very readable article discusses not only his mathematical contributions that ultimately led to his winning the 1994 Nobel Prize in Economics but also the very human side of his mental illness.

How to Avoid Thinking in Math Class (Part 6)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

I had a surreal moment this year. I’d almost finished a lesson when one boy, usually a hyperkinetic little bundle of enthusiasm, raised his hand.

“So, like, I don’t really understand anything you’re saying,” he informed me, “But I can still get the right answer.”

He smiled, waiting.

“Which part is giving you trouble?” I asked.

“Oh, you were talking about this extra stuff,” he said, “like the ideas behind it and everything. I don’t… you know… do that.”

I blinked. He blinked. We stood in silence.

“So is that okay?” he concluded. “I mean, as long as I can get the right answer?”

Here is Part 6: http://mathwithbaddrawings.com/2015/02/11/the-church-of-the-right-answer/

How to Avoid Thinking in Math Class (Part 5)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

As for students, it can be frightening to start a math problem. You don’t know quite where it will lead. Will my approach be fruitful? Will it falter? Where do I even begin?

But unlike my desk-perching student, most kids don’t recognize that one rope holding them back is fear of the unknown. They just hesitate: too afraid to leap without a net, but never bothering to go in search of a net for themselves…

In all these cases, students are refusing to engage with their uncertainty. But if you’re uncomfortable with doubt, you’ll never break through to the other side. You’ll never have a “Eureka!” moment or an intellectual “Aha!” You’ll never… well… learn. After all, if you can’t bear to face the unknown, how will you ever come to know it?

I find that my desk-percher has it right. At times like these, the mere presence of an expert can supply the confidence you’re lacking.

Here is Part 5, introducing what happens when students get stuck getting started on a problem: http://mathwithbaddrawings.com/2015/02/04/fearing-the-unknown/

How to Avoid Thinking in Math Class (Part 4)

Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

This speaks more to my naiveté as a first-year teacher than anything else, but I was shocked to find how fervently my students despised the things they called “word problems.”

“I hate these! What is this, an English lesson?”

“Can’t we do regular math?”

“Why are there words in math class?”

Their chorus: I’m okay with math, except word problems.

They treated “word problems” as some exotic and poisonous breed. These had nothing to do with the main thrust of mathematics, which was apparently to chug through computations and arrive at clean numerical solutions.

I was mystified—which is to say, clueless. Why all this word-problem hatred?

Here is Part 4, addressing students’ fears of word problems: http://mathwithbaddrawings.com/2015/01/28/the-word-problem-problem/