Engaging students: Synthetic Division

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 Deetria Bowser. Her topic, from Precalculus: synthetic division.

green line

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

It is tough to find websites or technology that help with synthetic division, due to the fact that most websites consists of a long list of instruction, which is not engaging. One website that does seem helpful to a student learning synthetic division is: http://emathlab.com. Under the Algebra tab one can select the option for polynomial functions, then synthetic division. Once selected, a synthetic division activity will pop up. In this activity the student is given a polynomial (of third degree most of the time) divided by a degree one polynomial. The student is then expected to correctly fill the cells with the correct numbers for synthetic division. If they do not get the correct number, the cell turns red and they have to keep trying until they get the answer correct. This activity will be beneficial to students because they will be able to get a feel on the correct placement of numbers when using synthetic division. Additionally, this tool will help them realize what to do when they get polynomials, such as x^3-1. Finally this online tool will allow the students to evaluate themselves.

 

 

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

The idea of synthetic division is used to find the zeros of a function. One may need to find zeros of a function in a variety of mathematics and science courses. For example, in physics, one may need to find the roots of a trajectory equation. To find said roots, one could use synthetic division. Also an example of finding roots could be used to help in computer programming. On math.stackexchange.com a programming student presents the following problem: “I am currently programming a simulation for a pinball game and want to calculate the time when the ball hits a circle (if they collide at some point). For the calculation part, I’m adding the radius of the ball to the radius of the circle, so that i only have to check if the midpoint of the ball collides with the circle. Of course, the circle is displayed with it’s original radius.

Now for the ball’s (midpoint) trajectory i’ve got these two equations who define the movement of the ball on the x- and y-axis (depending on the gravitational acceleration):

x(t)=s_x+v_x t, \quad y(t)=s_y+v_y t− \frac{1}{2} g t^2,

with (s_x,s_y)= starting point of the ball, (v_x,v_y)= initial velocity, g= gravitational acceleration and t= time.

To check for collision, I took these two equations and put them into the equation of a circle. Once multiplied out the student got something of the form: a t^4+bt^3+ct^2+dt+e=0. If the coefficients a,b,c,d,e are rational numbers, then he will be able to use synthetic division to find all of the roots, and successfully create his game.

 

green line

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

In previous courses, students are taught to find zeros by either graphing, guessing and plugging in a number for x and hoping that the result is zero, or using long division. Synthetic division provides a more systematic way of finding zero’s than just guessing, and can prove to be quicker than graphing and using long division. Additionally, synthetic division can expand on the idea of showing something is not a factor. For instance, when one tries to synthetically divide the polynomial x^4-3x^2+5x-7 by x-2 one will get a remainder of 7. This is another way of proving that x-2 is not a factor of x^4-3x^2+5x-7. Also, one now knows what the polynomial x^4-3x2+5x-7 is when x = 2. Synthetic division, extends the idea of finding factors and non-factors of polynomials, as well as solutions to polynomials at a specific x.

References

https://math.stackexchange.com/questions/1462858/how-to-find-the-zeros-of-a-fourth-degree-polynomial-without-integer-coefficient
http://emathlab.com/Algebra/PolyFunctions/SyntheticDiv.php

 

 

 

 

 

Engaging students: Synthetic Division

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 Amber Northcott. Her topic, from Precalculus: synthetic division.

green line

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

 

Synthetic division takes a little to get used to, especially after learning long division with polynomials. One thing is for sure and that is once the students get how to do synthetic division they sometimes prefer it over long division because it is a faster and easier way to divide polynomials. However, the first step is to learn it and there are many different ways to learn it. One way is to create an activity the students can do that will help them learn it.

An activity or project idea is to have the students write their own steps on how to solve synthetic division. Make sure to let the students know that they must put it in their own words. Then put students in groups of three to four and have them share their steps with each other. Let them give each other feedback on their steps and the feedback must be turned in. Once the teacher looks at the feedback, the teacher can give it back to the students and give their feedback to the student as well. Then have the student take the feedback into consideration and change their steps if needed. This activity will allow the student to see how they view synthetic division and what steps they take to solve it. By sharing their steps, they can get an idea of how everyone solves synthetic division and learn from each other.

Other activities or projects also include having the students write down the steps to solving synthetic division. This time though they can use their imagination and get creative. The activity or project can be to make up a poem or acrostic or a story to help them remember how to solve synthetic division. Then have them present their poem or acrostic or story in front of the class, so other students can learn those ideas as well to help them remember how to do synthetic division.

 

green line

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

 

Synthetic division is first seen Algebra II. Students tend to learn it right after learning how to do long division with polynomials. After taking Algebra II students don’t see synthetic division for a while until pre-calculus and calculus. This is because when you hit Pre-Calculus and Calculus you see algebra topics within them a lot more than you would a Geometry and Trigonometry class. This doesn’t mean you can’t see them in Geometry or Trigonometry. This is because like all math subjects and topics they intertwine with each other, so you are bound to see synthetic division in quite a few places in mathematics.

 

green line

How was this topic adopted by the mathematical community?

 

Synthetic division is also called Ruffini’s Rule, but we don’t see this title very often in textbooks. The reason why it was called Ruffini’s Rule is because of the Italian mathematician Paolo Ruffini, who brought synthetic division to life around 1809. Paolo Ruffini, like all mathematicians, wanted to find a simpler way to do a mathematic topic. This can also be because mathematicians are known to be a bit lazy.

The mathematic topic he wanted to find a simpler way to do was dividing polynomials, so by creating this system we all know as synthetic division he found a cleaner, simpler, and faster way to divide polynomials. Of course, it has certain conditions to follow in order to be able to do synthetic division, but it’s the option is there.

 

 

Resources

 

http://www.personal.psu.edu/djh300/cyhs/trig/unit-e-adv-polyn/06-05-02-synth-div.pdf

 

 

 

 

My Favorite One-Liners: Part 90

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a typical problem that arises in Algebra II or Precalculus:

Find all solutions of 2 x^4 + 3 x^3 - 7 x^2 - 35 x -75 =0.

There is a formula for solving such quartic equations, but it’s very long and nasty and hence is not typically taught in high school. Instead, the one trick that’s typically taught is the Rational Root Test: if there’s a rational root of the above equation, then (when written in lowest terms) the numerator must be a factor of -10 (the constant term), while the denominator must be a factor of 2 (the leading coefficient). So, using the rational root test, we conclude

Possible rational roots = \displaystyle \frac{\pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75}{\pm 1, \pm 2}

= \pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75 \displaystyle \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{15}{2}, \pm \frac{25}{2}, \pm \frac{75}{2}.

Before blindly using synthetic division to see if any of these actually work, I’ll try to address a few possible misconceptions that students might have. One misconception is that there’s some kind of guarantee that one of these possible rational roots will actually work. Here’s another: students might think that we haven’t made much progress toward finding the solutions… after all, we might have to try synthetic division 24 times before finding a rational root. So, to convince my students that we actually have made real progress toward finding the answer, I’ll tell them:

Yes, 24 is a lot\dots but it’s better than infinity.

 

My Favorite One-Liners: Part 85

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s one-liner is one that I’ll use when I want to discourage students from using a logically correct and laboriously cumbersome method. For example:

Find a polynomial q(x) and a constant r so that x^3 - 6x^2 + 11x + 6 = (x-1)q(x) + r.

Hypothetically, this can be done by long division:

However, this takes a lot of time and space, and there are ample opportunities to make a careless mistake along the way (particularly when subtracting negative numbers). Since there’s an alternative method that could be used (we’re dividing by something of the form x-c or x+c, I’ll tell my students:

Yes, you could use long division. You could also stick thumbtacks in your eyes; I don’t recommend it.

Instead, when possible, I guide students toward the quicker method of synthetic division:

My Favorite One-Liners: Part 82

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In differential equations, we teach our students that to solve a homogeneous differential equation with constant coefficients, such as

y'''+y''+3y'-5y = 0,

the first step is to construct the characteristic equation

r^3 + r^2 + 3r - 5 = 0

by essentially replacing y' with r, y'' with r^2, and so on. Standard techniques from Algebra II/Precalculus, like the rational root test and synthetic division, are then used to find the roots of this polynomial; in this case, the roots are r=1 and r = -1\pm 2i. Therefore, switching back to the realm of differential equations, the general solution of the differential equation is

y(t) = c_1 e^{t} + c_2 e^{-t} \cos 2t + c_3 e^{-t} \sin 2t.

As t \to \infty, this general solution blows up (unless, by some miracle, c_1 = 0). The last two terms decay to 0, but the first term dominates.

The moral of the story is: if any of the roots have a positive real part, then the solution will blow up to \infty or -\infty. On the other hand, if all of the roots have a negative real part, then the solution will decay to 0 as t \to \infty.

This sets up the following awful math pun, which I first saw in the book Absolute Zero Gravity:

An Aeroflot plan en route to Warsaw ran into heavy turbulence and was in danger of crashing. In desparation, the pilot got on the intercom and asked, “Would everyone with a Polish passport please move to the left side of the aircraft.” The passengers changed seats, and the turbulence ended. Why? The pilot achieved stability by putting all the Poles in the left half-plane.

My Favorite One-Liners: Part 81

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a problem that hypothetically could appear in Algebra II or Precalculus:

Find the solutions of x^4 + 2x^3 + 10 x^2 - 6x + 65 = 0.

While there is a formula for solving quartic equations, it’s extremely long and hence is not typically taught to high school students. Instead, the techniques that are typically taught are the Rational Root Test and (sometimes, depending on the textbook) Descartes’ Rule of Signs. The Rational Root Test constructs a list of possible rational roots (in this case \pm 1, \pm 5, \pm 13, \pm 65) to test… usually with synthetic division to accomplish this as quickly as possible.

The only problem is that there’s no guarantee that any of these possible rational roots will actually work. Indeed, for this particular example, none of them work because all of the solutions are complex (1 \pm 2i and 2 \pm 3i). So the Rational Root Test is of no help for this example — and students have to somehow try to find the complex roots.

So here’s the wisecrack that I use. This wisecrack really only works in Texas and other states in which the state legislature has seen the wisdom of allowing anyone to bring a handgun to class:

What do you do if a problem like this appears on the test? [Murmurs and various suggestions]

Shoot the professor. [Nervous laughter]

It’s OK; campus carry is now in effect. [Full-throated laughter.]

 

My Favorite One-Liners: Part 59

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Often I’ll cover a topic in class that students really should have learned in a previous class but just didn’t. For example, in my experience, a significant fraction of my senior math majors have significant gaps in their backgrounds from Precalculus:

  • About a third have no memory of ever learning the Rational Root Test.
  • About a third have no memory of ever learning synthetic division.
  • About half have no memory of ever learning Descartes’ Rule of Signs.
  • Almost none have learned the Conjugate Root Theorem.

Often, these students will feel somewhat crestfallen about these gaps in their background knowledge… they’re about to graduate from college with a degree in mathematics and are now discovering that they’re missing some pretty basic things that they really should have learned in high school. And I don’t want them to feel crestfallen. Certainly, these gaps need to be addressed, but I don’t want them to feel discouraged.

Hence one of my favorite motivational one-liners:

It’s not your fault if you don’t know what you’ve never been taught.

I think this strikes the appropriate balance between acknowledging that there’s a gap that needs to be addressed and assuring the students that I don’t think they’re stupid for having this gap.

 

My Favorite One-Liners: Part 32

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s story is a continuation of yesterday’s post. I call today’s one-liner “Method #1… Method #2.”

Every once in a while, I want my students to figure out that there’s a clever way to do a problem that will save them a lot of time, and they need to think of it.

For example, in Algebra II, Precalculus, or Probability, I might introduce the binomial coefficients to my students, show them the formula for computing them and how they’re related to combinatorics and to Pascal’s triangle, and then ask them to compute \displaystyle {100 \choose 3}. We write down

\displaystyle {100 \choose 3} = \displaystyle \frac{100!}{3!(100-3)!} = \displaystyle \frac{100!}{3! \times 97!}

So this fraction needs to be simplified. So I’ll dramatically announce:

Method #1: Multiply out the top and the bottom.

This produces the desired groans from my students. If possible, then I list other available but undesirable ways of solving the problem.

Method #2: Figure out the 100th row of Pascal’s triangle.

Method #3: List out all of the ways of getting 3 successes in 100 trials.

All of this gets the point across: there’s got to be an easier way to do this. So, finally, I’ll get to what I really want my students to do:

Method #4: Write 100! = 100 \times 99 \times 98 \times  97!, and cancel.

The point of this bit of showman’s patter is to get my students to think about what they should do next as opposed to blindly embarking in a laborious calculation.

green line

As another example, consider the following problem from Algebra II/Precalculus: “Show that x-1 is a factor of f(x)=x^{78} - 4 x^{37} + 2 x^{15} + 1.”

As I’m writing down the problem on the board, someone will usually call out nervously, “Are you sure you mean x^{78}?” Yes, I’m sure.

“So,” I announce, “how are we going to solve the problem?”

Method #1: Use synthetic division.

Then I’ll make a point of what it would take to write down the procedure of synthetic division for this polynomial of degree 78.

Method #2: (As my students anticipate the real way of doing the problem) Use long division.

Understanding laughter ensures. Eventually, I tell my students — or, sometimes, my students will tell me:

Method #3: Calculate f(1).

 

Engaging students: Synthetic Division

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 Chelsea Hancock. Her topic, from Precalculus: synthetic division.

green line

The method of synthetic division is an alternative version of long division concerning polynomials. Synthetic division uses the basic mathematical skills of addition, subtraction, multiplication, and negative signs. They must also understand the definitions of polynomial, coefficient, and remainder. A polynomial is an expression with multiple terms, poly meaning “many” and nomial meaning “term.” A coefficient is a number used to multiply a variable. The remainder is the amount left over after division. Synthetic division involves multiplying, then adding or subtracting the coefficients of two polynomials. On some occasions, there will be a remainder after dividing the polynomials.

 

green line

Mathematicians are lazy. That is a fact of life. One mathematician understood this, so in 1809 he created a cleaner, faster, and much simpler method for division. His name was Paolo Ruffini. In order to more efficiently divide polynomials, Ruffini invented the Ruffini’s Rule, known more commonly as synthetic division in today’s society. In 1783, he entered the University of Modena and he studied mathematics, medicine, philosophy and literature. Then, in 1798 he began teaching mathematics at the University of Modena. He was required to swear an oath of allegiance to the republic, but due to religious purposes, refused to do so. This resulted in the loss of his professorship and was prevented from teaching.

 

green line

There are several videos on the Internet involving synthetic division, but there are two in particular that I personally think are excellent demonstrations of both the method itself and why it works. I have labeled these clips Video 1 and Video 2. Video 1 is a demonstration of the method in action, using a specific example involving numbers, walking the viewers through the process through the whole video. Video 2 explains why using synthetic division instead of using long division is the more efficient and less complicated method for dividing polynomials. The clip uses the same example used in Video 1, but this time the polynomials are divided using long division, walking the viewers through the process the entire time. As the narrator moves through the process, he makes connections between the synthetic division method and the long division method and draws conclusions between the two. By the end of the video, it is evident which is the cleanest method to use when concerning the division of polynomials. These videos not only give great tutorials on both methods of division, but allows the viewers to see the benefits and uses of synthetic division when it is possible to use it.

 

Video 1:

Video 2:

References

 

http://www.mathsisfun.com/algebra/polynomials.html

 

http://www.mathsisfun.com/definitions/coefficient.html

 

http://www.mathsisfun.com/definitions/remainder.html

 

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ruffini.html

 

http://www.personal.psu.edu/djh300/cyhs/trig/unit-e-adv-polyn/06-05-02-synth-div.pdf

 

https://www.khanacademy.org/math/algebra2/polynomial_and_rational/synthetic-division/v/synthetic-division

 

https://www.khanacademy.org/math/algebra2/polynomial_and_rational/synthetic-division/v/why-synthetic-division-works