# Difference of Two Squares (Part 1)

In Algebra I, we drill into student’s heads the formula for the difference of two squares:

$x^2 - y^2 = (x-y)(x+y)$

While this formula can be confirmed by just multiplying out the right-hand side, innovative teachers can try to get students to do some exploration to guess the formula for themselves. For example, teachers can use some cleverly chosen multiplication problems:

$9 \times 11 = 99$

$19 \times 21 = 399$

$29 \times 31 = 899$

$39 \times 41 = 1599$

Students should be able to recognize the pattern (perhaps with a little prompting):

$9 \times 11 = 99 = 100 - 1$

$19 \times 21 = 399 = 400 - 1$

$29 \times 31 = 899 = 900 - 1$

$39 \times 41 = 1599 = 1600 - 1$

Students should hopefully recognize the perfect squares:

$9 \times 11 = 99 = 10^2 - 1$

$19 \times 21 = 399 = 20^2 - 1$

$29 \times 31 = 899 = 30^2 - 1$

$39 \times 41 = 1599 = 40^2 - 1$,

so that they can guess the answer to something like $59 \times 61$ without pulling out their calculators.

Continuing the exploration, students can use a calculator to find

$8 \times 12 = 96$

$18 \times 22 = 396$

$28 \times 32 = 896$

$38 \times 42 = 1596$

Students should be able to recognize the pattern:

$8 \times 12 = 10^2 - 4$

$18 \times 22 = 20^2 - 4$

$28 \times 32 = 30^2 - 4$

$38 \times 42 = 40^2 -4$,

and perhaps they can even see the next step:

$8 \times 12 = 10^2 - 2^2$

$18 \times 22 = 20^2 - 2^2$

$28 \times 32 = 30^2 - 2^2$

$38 \times 42 = 40^2 -2^2$.

From this point, it’s a straightforward jump to

$(10-2) \times (10+2) = 10^2 - 2^2$

$(20-2) \times (20+2) = 20^2 - 2^2$

$(30-2) \times (30+2) = 30^2 - 2^2$

$(40-2) \times (40+2) = 40^2 -2^2$,

leading students to guess that $(x-y)(x+y) = x^2 -y^2$.

# Engaging students: Equations of two variables

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Zacquiri Rutledge. His topic, from Algebra: equations of two variables.

Seeing equations with two variables becomes quite common once students have been introduced to independent and dependent variables. However seeing equations in the form x+4y=16 would start as a confusing concept after being taught that equations are written in the format 4x-16=y. However, this concept is highly required when a teacher goes to explain about a system of equations. The reason for this is because a common method that is taught for solving a system of equations is substitution. In order to utilize the substitution method, a student must understand how to solve for a variable by using order of operations to isolate the variable. In fact, a student will use the same skills they did when learning to solve an equation that only has one variable, such as 3x+6=12. However, now the student must apply it to something new.

Another lesson that uses the knowledge from the Equations of Two Variables is interpretation of a graph for an equation with two variables. Before, the students would have learned what independent and dependent variables are, and how they are represented on a graph. Later on the students would further their understanding by finding the graphical representation of equations with two variables. The students would learn that, while the line on the graph during lessons over independent and dependent variables was only to show where the left side of an equation equaled y, the line can also show where x and y combine to equal a certain value. An example of this would be comparing x+4y=16 and (-1/4)x+4=y. They are the same equation, however one equation shows that x and 4y combine to equal 16, so every point on the line are the values of x and y required to equal 16. The second equation says that to find y for a given point x, x must be multiplied by (-1/4) and add 4. Just changing the nature of the equation can change what it is that the equation is saying, as well as give a different perspective one that could be useful when dealing with real life word problems.

Two variable equations are very subtle, but are all around us. Even when we do not think it is being used, it is. The most common modern example of two variable equations is the American dollar, and how many coins of two different values are needed to make a dollar. Although this is a very easy explanation to use it can be very boring at times. How about classical music or concert music? While it may not seem obvious at first, it is in fact there. The standard set-up for a sheet of music is Four-Four time. What this means is that within every measure there are four beats and a quarter note counts as a whole beat. There are also other kinds of notes which are used in combination with quarter notes to fill a measure, examples being a whole note (four beats), half note (two beats), and eighth notes (half beat). So when a composer sits down to write a piece of music, he/she must keep in mind how many beats are in each measure. This is where the concept of two variable equations comes into play. Suppose the composer wants a measure made up of only half notes and quarter notes in four-four time, then his equation to figure out how many of each note he can have would be 2h+q=4, where h is half notes and q is quarter notes. Then, the next measure is going to be made up of eighth and half notes, therefore 2h+(1/2)e=4 would be the equation, where e is eighth notes. There are many different combinations someone can use when writing music to create a piece that is to be played in front of a live audience. Centuries ago, men like Beethoven and Mozart used this concept every day to create classic pieces such as Beethoven’s Symphony #5 or Mozart’s Moonlight Sonata. This is an excellent example that can be used for classes that include a large number of band students or choir students, to relate the music they are studying in their music classes to their math courses.

With the previous response in mind, a teacher could very well use Youtube as an excellent method to engage their students. A lot of children today are not familiar with how classical music is written or how music is written at all. By playing pieces of music for their students that students are likely to have heard befor, via Youtube or even iTunes, such as Ride of the Valkyries or Beethoven’s Symphony #5, can spark an interest not only musically, but mathematically. A teacher could begin by asking students if they had heard the piece before, then go to the next piece and see who has heard it before. Repeat this process for about 2-4 clips of pieces, then ask which of the students know anything about how music is written. This would lead into what was discussed in the previous response. However, by including the technology as a way for the students to hear the music, and not just see it, can have tremendous effects on their attention.

# 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 Tiffany Jones. Her topic, from Algebra: using the point-slope equation of a line.

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

The topic of using the point-slope equation of a line comes up in some of the early topics of Calculus 1 such as, how to find the equation of the tangent line of a curve at a given point. The slope, ­, of the tangent line of a curve at a given point, , is equal to the instantaneous rate of change or slope of the curve at that given point. The slope is calculated by evaluating the following limit:

$\displaystyle m = \lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$

If the difference quotient has a limit as h approaches zero, then that limit is called the derivative of the function at . Then, values of  and  are substituted into the point-slope equation of a line to determine the equation of the tangent line of a curve at a given point.

$y-y_0 = m(x-x_0)$

C1. How has this topic appeared in pop culture?

On December 31st 1965, Chuck Jones’ released an animated short titled “The Dot and The Line: A Romance in Low Mathematics”. This ten minute, Oscar-winning film explores the complex relationship between lines, dots, and disorganization. The Line as desperately in love with the Dot. Yet, the Dot is currently involved with a chaotic Squiggle. The Dot ignores the Line, disregarding him as boring and predictable. He lacks complexity. Through a montage following this rejection, the line teaches himself to create angles, form curves, and produce close-ended shapes as well. With this new confidence, he then reveals his newfound self to the Dot. The Dot sees that there is no method to the Squiggles madness.

While the topic of using the point slope equation of a line is not an explicit topic of the short, I feel that this video as an engage activity can be great conversation starter about the relationship between a point and a line. From there the lesson can go on to talk about the point-slope equation. Furthermore, this video can open discussions about the slope-intercept and the point-point forms of a line.

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

Explore Learning offers a Gizmo and worksheet on the point-slope form of a line. The Gizmo is an interactive simulator that allows the student to physically move the point around the Cartesian plane or use the sliders to adjust the point values and the slope value. The Gizmo shows the resulting line. I think that the use of such a tool can reinforce the relationship of a particular slope and a particular point to give an equation of a line.

The Gizmo offers to the slope-intercept form of the equation. So this simulator can also be used for a lesson on the slope-intercept form. Also, the Gizmo can place a right triangle along the line with leg lengths to show how the rise and run values change with the overall slope value.

Additionally, I think that this simulator can be used to allow the students to explore the equation. For instance, the students can see why when the graph is shifted to the left 2 units, the resulting equation has (x+2).

References:

http://www.imdb.com/title/tt0059122/?ref_=ttawd_awd_tt

https://www.explorelearning.com/index.cfm?ResourceID=16*4&method=cResource.dspDetail

https://s3.amazonaws.com/el-gizmos/materials/PointSlopeSE.pdf

# 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 Perla Perez. Her topic, from Algebra: multiplying binomials.

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

As students progress through different levels of math, they will continue to utilize tools, such as multiplication of binomials. When I give students the solutions to a quadratic function and ask them to find the equation, I expect for them to know how to multiply the binomials. For example: find the quadratic equation with the solution x=-2,2. The students are to set up as: (x+2)(x-2) and go forth. The students can also be given a quadratic equation, x2+6x+8 and are to find the solutions in representation (x+2)(x+4). In order to arrive at the answer, the students will have to factor the original equation. To check their work, they can just multiply the answer that they get. Multiplying the binomials is a more complex form of the distributive property. It’s a building block for more challenging math concepts. Multiplying binomials essentially does the opposite of factorization, which students will learn later on in their algebra class. Binomials are also used in sciences, such as physics, biology, and computer science, so it helps for students to have a strong foundation on this topic.

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

I’ve seen students panic when a new concept, equation, or definition is introduce. Before they begin thinking again that math is some sort of sorcery, showing them something familiar will help ease the students into a new topic that is an extension of what they previously learned. Students learn about distributive property in their pre-algebra course. In order for students to multiply binomials students need to understand distributive property. Distributive property is a building block that is needed for the multiplication of binomials. It works with singles terms being multiplied, where as binomial multiplication works with two. In a way it is like learning how to add single digits to double digits. In order to teach this, I would first reintroduce 4-5 problems they’ve seen in their previous class using distributive property with single terms such as 4(x+5). Once they begin to recognize and solve the problems, I will begin to introduce two terms rather than just one. When they compare their previous knowledge to this new idea they will see that it is not very different.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Students often find it difficult to understand why we use certain tools, such as the multiplication of binomials. Word problems are a good solution when introducing a new topic. There are many methods for multiplying binomials, such as the FOIL and the CLAW methods, and it is important that student learn them; however, students who struggle with the topic need new information to be presented in a different way. The website mathisfun.com has a great word problem for multiplying binomials.

I like this problem because it divides the topic into separate steps, making it easier for the student to understand what to do. With this particular word problem, the teacher can begin to see where the students are having difficulties. This allows the teacher to see what areas need to be revisited, such as order of operations, the multiplication of a negative or positive number etc. Word problems also help teachers evaluate the critical thinking skills of their students.

My References are:

https://www.mathsisfun.com/algebra/polynomials-multiplying.html

http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.html

# Engaging students: Fractions, decimals, and percents

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 Perla Perez. Her topic, from Algebra: fractions, decimals, and percents.

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

This past summer when I worked as a program assistant for TexPrep, we had the opportunity to have a pizza party. How fun! Well it took longer than we thought to pick out a place and figure out how much we all had to pay. I got to thinking about how this could be a great engaging activity for students to get excited about decimals, fractions, and percents.

The activity will go as follows:

Students are split up into groups of four with each group given a pizza place. Every person has one of the following roles: the researcher, the recorder, the calculator, and the presenter (to compare with other groups). Their goal is to find the pizza place that is the cheapest, gives the most pizza, and figure out how much each individual would have to pay. By comparing each other’s work during presentations, students get to compare, contrast, and see the different methods used to solve the problems. This also gives the teacher an opportunity to understand their comprehension level of the subject and see if converting a percentage is difficult for them or not. When all the groups are finished gathering their information they will present. Afterwards (if allowed), we will reward ourselves with eating pizza! Through this activity students will have to come up their own way to solve these problems. It leads them to work with: Decimals, since they must include every penny (including tax); Fractions, when it comes to figuring out how much each individual owes; and Precents, when asked to compare prices between pizza places.

C3. How has this topic appeared in the news?

Decimals, fractions, and percent are used in media to represent a variety of concepts from the percent of the candidate poll elections to percent chance of rain. Now some of these topics might not sound interesting to most students, but current events such as the movement to raise minimum wage to $15.00 can grab their attention. Students can then be given questions such as: How does that affect the regular worker financially? Are employees working the same hours? Do employees get fewer hours and more pay, or do they keep their regular hours? In the Time article “Here’s Every City in America Getting a$15 Minimum Wage”, it mentions how some restaurants are increasing their prices from 4% to 21% which begs to question, is everything in the market going to increase as well? All the answers to these questions can be found in the news and prompt their interest in actually doing the math to find out the answers. The news also gives them the real world application student’s consistently are trying to find. Engaging students about the news and simply prompting them before the lesson allows students to continue thinking about it as they go forth in the lesson.

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?

As we continue to advance in technology, we begin to see how there are many ways a student can learn. The internet is full of different educational games, activities, calculators, and above all videos that are useful to educators. There are videos basically for everything. So what better way to engage students than with a video that knows exactly how they feel like in this one: https://www.youtube.com/watch?v=cGqQOQavbls. The video is a great representation of how a unique activity such as magic can be used to stimulate students in understanding the idea of how fractions, decimals, and percentages relate to one another. Aside from funny videos students also like to interact in games like: http://www.math-play.com/Fractions-Decimals-Percents-Jeopardy/fractions-decimals-percents-jeopardy.html and http://www.topmarks.co.uk/maths-games/7-11-years/fractions-and-decimals. The first game allows students to practice converting fractions, decimals, and fractions from one to another and shows them how they are related. The last website gives teachers a variety of tools to choose from, all of which can help a lot in the classroom.

References:

# New world record for largest prime number

As of this week, we have a new world record for the largest known prime number:

$2^{74,207,281}-1$

The adjective known is important, because there are an infinite number of prime numbers (but not all of them are known). A good video describing this finding is below.

A good article is here:

# The antiderivative of 1/(x^4+1): Part 3

This antiderivative has arguable the highest ratio of “really hard to compute” to “really easy to write”:

$\displaystyle \int \frac{1}{x^4 + 1} dx$

To compute this integral, I will use the technique of partial fractions. In yesterday’s post, I used De Moivre’s Theorem to factor the denominator over the complex plane, which then led to the factorization of the denominator over the real numbers.

In today’s post, I present an alternative way of factoring the denominator by completing the square. However, unlike the ordinary method of completing the square, I’ll do this by adding and subtracting the middle term and not the final term:

$x^4 + 1= x^4 + 2x^2 + 1 - 2x^2$

$= (x^2 + 1)^2 - (x \sqrt{2})^2$

$= (x^2 + 1 + x\sqrt{2})(x^2 + 1 - x \sqrt{2})$.

The quadratic formula can then be used to confirm that both of these quadratics have complex roots and hence are irreducible over the real numbers, and so I have thus factored the denominator over the real numbers:

$\displaystyle \int \frac{dx}{x^4 + 1} = \displaystyle \int \frac{dx}{\left(x^2 - x \sqrt{2} + 1 \right) \left(x^2 + x \sqrt{2} + 1\right)}$.

and the technique of partial fractions can be applied.

There’s a theorem that says that any polynomial over the real numbers can be factored over the real numbers using linear terms and irreducible quadratic terms. However, as seen in this example, there’s no promise that the terms will have rational coefficients.

I’ll continue the calculation of this integral with tomorrow’s post.

# The antiderivative of 1/(x^4+1): Part 2

This antiderivative has arguable the highest ratio of “really hard to compute” to “really easy to write”:

$\displaystyle \int \frac{1}{x^4 + 1} dx$

To compute this integral, I will use the technique of partial fractions. This requires factoring the denominator over the real numbers, which can be accomplished by finding the roots of the denominator. In other words, I need to solve

$x^4 + 1 = 0$,

or

$z^4 = -1$.

I switched to the letter $z$ since the roots will be complex. The four roots of this quartic equation can be found with De Moivre’s Theorem by writing

$z = r (\cos \theta + i \sin \theta)$,

where $r$ is a real number, and

$-1 + 0i = 1(\cos \pi + \i \sin \pi)$

By De Moivre’s Theorem, I obtain

$r^4 (\cos 4\theta + i \sin 4 \theta) = 1 (\cos \pi + i \sin \pi)$.

Matching terms, I obtain the two equations

$r^4 = 1$ and $4\theta = \pi + 2\pi n$

or

$r = 1$ and $\theta = \displaystyle \frac{\pi}{4} + \displaystyle \frac{\pi n}{2}$

or

$r = 1$ and $\theta = \displaystyle \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$.

This yields the four solutions

$z = 1 \left[ \cos \displaystyle \frac{\pi}{4} + i \sin \frac{\pi}{4} \right] = \displaystyle \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$

$z = 1 \left[ \cos \displaystyle \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right] = -\displaystyle \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$

$z = 1 \left[ \cos \displaystyle \frac{5\pi}{4} + i \sin \frac{5\pi}{4} \right] = -\displaystyle \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}$

$z = 1 \left[ \cos \displaystyle \frac{7\pi}{4} + i \sin \frac{7\pi}{4} \right] = \displaystyle \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}$

Therefore, the denominator $x^4 + 1$ can be written as the following product of linear factors over the complex plane:

$\displaystyle \left(x - \left[ \displaystyle \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right] \right)\left(x - \left[ \displaystyle \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \right] \right) \left(x - \left[ -\displaystyle \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right] \right) \left(x - \left[ - \displaystyle \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \right] \right)$

or

$\displaystyle \left(\left[x - \displaystyle \frac{\sqrt{2}}{2} \right] - i \frac{\sqrt{2}}{2} \right)\left( \left[ x - \displaystyle \frac{\sqrt{2}}{2} \right] + i \frac{\sqrt{2}}{2} \right) \left( \left[ x + \displaystyle \frac{\sqrt{2}}{2} \right] - i \frac{\sqrt{2}}{2} \right) \left( \left[ x + \displaystyle \frac{\sqrt{2}}{2} \right] + i \frac{\sqrt{2}}{2} \right)$

or

$\displaystyle \left(\left[x - \displaystyle \frac{\sqrt{2}}{2} \right]^2 - \left[ i \frac{\sqrt{2}}{2} \right]^2 \right) \left( \left[ x + \displaystyle \frac{\sqrt{2}}{2} \right]^2 - \left[i \frac{\sqrt{2}}{2} \right]^2 \right)$

or

$\displaystyle \left(x^2 - x \sqrt{2} + \displaystyle \frac{1}{2} + \displaystyle \frac{1}{2}\right) \left(x^2 + x \sqrt{2} + \displaystyle \frac{1}{2} + \displaystyle \frac{1}{2}\right)$

or

$\displaystyle \left(x^2 - x \sqrt{2} + 1 \right) \left(x^2 + x \sqrt{2} + 1\right)$.

We have thus factored the denominator over the real numbers:

$\displaystyle \int \frac{dx}{x^4 + 1} = \displaystyle \int \frac{dx}{\left(x^2 - x \sqrt{2} + 1 \right) \left(x^2 + x \sqrt{2} + 1\right)}$,

and the technique of partial fractions can be applied.

I’ll continue the calculation of this integral with tomorrow’s post.

# Engaging students: Factoring 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 Banner Tuerck. His topic, from Algebra: factoring polynomials.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

In relation to a specific case one can generate a word problem well within their students reach by relating the factors of a said quadratic polynomial to the length and width of a rectangle or perfect square. Many online resources, such as http://www.purplemath.com/, offer diverse and elaborate examples one could use in order to facilitate this concept. Nevertheless, this way of viewing a factored polynomial may appear more comfortable to a class because it is applying the students preexisting knowledge of area to the new algebraic expressions and equations. Furthermore, it has been my experience that geometric activities interrelating algebra aid in straying students away from ignoring the variable in an expression as a value.

A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?

The above problem is a prime example pulled from the Purple Math website one could use to illustrate a physical situation in which we need to actually determine the factors in order to formulate a quadratic expression to solve for the width. It should be noted that some of these particular word problems can quickly fall into a lesson relating more towards distributing and foiling factors to form an expanded form equation. However, as an instructor one can easily work backwards from an expanded equation to interpret what the factored form can tell us, say about the garden with respect to the example given above.

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

Factoring polynomials allows students to further comprehend the properties of these expressions before they are later applied as functions in areas such as mathematics and physics. For example, projectile motion stands as a great real world topic capable of enlightening students further on the factors of the polynomial. Specifically, how these factors come about geometrically and how knowing their role will benefit our understanding of the functions potential real world meaning. Lastly, factoring polynomials and evaluating them as roots during middle and high school mathematics will definitely be used when students approach college level calculus courses in relation to indefinite and definite integrals. The previous are just a few examples of how factoring polynomials plays a role in students’ future courses.

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

Doing a simple YouTube search of the phrase “factoring polynomials” allows anyone access to nearly 57,000 videos of various tutors, instructors, and professors discussing factoring and distributing respectfully. I would say that future generations will definitely not be without resources. That is not even to mention the revolutionary computation website that is www.wolframalpha.com. This website in and of itself will allow so many individuals to see various forms of a factored polynomial, as well as the graph, roots (given from factors), domain, range, etc. Essentially, computation websites like Wolfram Alpha are intended to allow students the opportunity to discover properties, relationships, and patterns independently. However, there is a potential risk for such websites to become a crutch the students use in order to get good grades as opposed to furthering their understanding. Similarly, with the advancing technology of graphing calculators students will become more engaged when discussing polynomial factorization for the first time in class. Many modern calculators have the ability to identify roots, give a table of coordinates, trace graphs, etc. Some even have a LCD screen or a backlit display to aid in viewing various graphs. Although, just as with computation engines, calculators could potentially distract students from thinking about their problem solving method by them just letting the calculator take over the calculation process. Therefore, I would suggest using caution regarding how soon calculators are introduced when initially engaging a class in factoring polynomials.

References:

http://www.purplemath.com/

http://www.wolframalpha.com/

# Engaging students: Factoring quadratic 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 Kristin Ambrose. Her topic, from Algebra: factoring quadratic polynomials.

In previous courses, students would have learned how to solve one-variable linear equations. These kinds of equations would involve variables to the power of one. Quadratic equations extend from this since they add a variable to the equation that is to the power of two. Since students learned how to solve linear equations, they may be curious as to how they can solve quadratic equations that extend from this. Factoring is a way for students to solve these kinds of equations.

Also, in previous courses students would have learned about the ‘factors’ of a number. When talking about numbers, the factors are the numbers you multiply to get another number. For example the positive factors of six are one and six, and two and three. Factoring quadratic polynomials follows this logic, except instead finding the factors of a number, students are finding the factors of an expression. For example, the factors of the expression $x^2+4x+3$ are (x+3) and (x+1). Just like how when we multiply two times three we get six, when we multiply (x+3) times (x+1) we get the expression $x^2+4x+3$.

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

There is a popular video game called Angry Birds in which the user launches birds to try and knock down structures built by pigs. This game relates to factoring quadratics because if we were to plot the trajectory of the birds being launched on a graph, the result would be a parabola, in other words the graph of a quadratic function. Factoring quadratic polynomials is a way to find the solutions of a quadratic, and the solutions are where the parabola crosses the x-axis. In Angry Birds, we could set our x-axis to be the ground, and our solutions would correspond with where on the ground the bird would land, if nothing were to block its path. If students were given the quadratic equation for the parabola corresponding with the bird’s trajectory, students would be able to factor the equation to solve for where on the ground the bird would land.

What are the contributions of various cultures to this topic?

Indian and Islamic cultures are two major cultures that have contributed to the topic of factoring quadratic polynomials. In Islamic culture, Al-Khwarizmi contributed to this topic by creating a way to solve quadratic equations by reducing the equations to one of six forms, which were then solvable. He described these forms in terms of squares, roots, and numbers, much like the terms we use today when factoring quadratic polynomials. The ‘squares’ related to what would today be our ‘x2’ term, the ‘roots’ related to the ‘x’ term, and the ‘numbers’ to the ‘c’ constant term. One of the forms he described was “squares and roots equal numbers,” in modern terms, “ax2 + bx = c.” Today, we factor quadratic polynomials of the form “ax2 + bx + c” which is similar to the form Al-Khwarizmi described. (Islamic Mathematics – Al-Khwarizmi)

In Indian culture, Brahmagupta contributed to the concept of factoring quadratics by introducing the idea that a number could be negative. This was significant because it meant a number like 9 could be factored into 32 or (-3)2. Since a number could have a negative factor, it followed that quadratic equations could have two possible solutions, since one solution could be negative. Factoring quadratic polynomials like we do today would be impossible without the knowledge that quadratic expressions can have two solutions. (Indian Mathematics – Brahmagupta)

References:

“Islamic Mathematics – Al-Khwarizmi.” The Story of Mathematics. 2010. Web. 17 Sept. 2014. <http://www.storyofmathematics.com/islamic_alkhwarizmi.html&gt;.

“Indian Mathematics – Brahmagupta.” The Story of Mathematics. 2010. Web. 17 Sept. 2014. <http://www.storyofmathematics.com/indian_brahmagupta.html&gt;.