# Engaging students: Finding prime factorizations

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 Brittnee Lein. Her topic, from Pre-Algebra: finding prime factorizations.

• How has this topic appeared in the news?

Prime factorization is key to protecting many aspects of modern convenience. The Fundamental Theorem of Arithmetic states that every number can be broken down into a sum of two prime numbers. For relatively small numbers, this is no big deal; but for very large numbers, not even computers can easily break these down. Many online security systems rely on this principle. For example, if you shop online and enter your credit card information, websites protect that information from hackers through a process of encryption.

Something for students to think about in the classroom: Can you come up with any formula to break down numbers into their prime factors?

Answer: No! That’s why encryption is considered a secure form of cryptography. To this date, there is no confirmed algorithm for prime factorization.

Prime factorization is a classic example of a problem in the NP class. An NP class problem can be thought of as a problem whose solution is easily verified once it is found but not necessarily easily or quickly solved by either humans or computers. The P vs. NP problem is one that has perplexed computer scientists and mathematicians since it was first formulated in 1971. Most recently, a German scientist Norbert Blum has claimed to solve the P vs. NP problem in this article: https://motherboard.vice.com/en_us/article/evvp34/p-vs-np-alleged-solution-nortbert-blum

Also in recent years, A Texas student has been featured on Dallas County Community Colleges Blog for his work to find an algorithm for prime numbers: http://blog.dcccd.edu/2015/07/%E2%80%8Btexas-math-student-strives-to-solve-the-unsolvable/

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

An activity for inquiry based learning of prime numbers and prime factorization utilizes pop cubes. Students will start out with a single color-coded cube representative of the number two (the first prime), they will then move up the list of natural numbers with each prime number having its own color of cube. The composite numbers will have the same colors as their prime factors. The idea is that students will visually see that prime numbers are only divisible by themselves (each being a lone cube) and that composite numbers are simply composed of primes (multiple cubes). A good point of discussion is the meaning of the word “composite’. You could ask students what they think the word ‘composite’ means and what word it reminds them of. This leads into the idea that every composite number is composed of prime numbers. This idea comes from online vlogger Thom Gibson and the RL Moore Inquiry Based Learning Conference. Below is a picture demonstrating the cube idea:

This foundational idea can be segued into The Fundamental Theorem of Arithmetic and then into prime factorization.
One of the most practical real-world applications of prime factorization is encryption. This activity I found makes use of prime factorization in a way that is interesting and different from simply making factor trees. This worksheet would be a good assessment and challenge for students and mimics a real –world application.

https://www.tes.com/teaching-resource/prime-factors-cryptography-6145275

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

Though not actually ‘reducing’ the value of a number, prime factorization is the equivalency of numbers broken down into their smallest parts and then multiplied together. The idea of reducing numbers goes all the way back to elementary school when students are learning about fractions. Subconsciously they use a similar process to prime factorization when reducing fractions to simplest form. When reducing fractions to simplest form, the numerators and denominators themselves may not both necessarily be prime, but when put into simplest form, they are relatively prime. Being able to pick out factors of numbers –another relatively early grade school concept (going back to multiplication and division) — plays a huge deal in both fractions and prime factorization.

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

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

The point-slope equation of a line can be used in a variety of different ways in mathematics classes that some students may encounter later on. It is used in Calculus when dealing with polynomials. For instance, “key concepts of calculus: limits, continuity, derivatives, and integrals are all relatively trivial for polynomial functions.” It is also seen when dealing with Linear Approximations. “A differentiable function is one for which there is a tangent line at each point on the graph. In an intuitive sense, the tangent to a curve at a point is the line that looks most like the curve at the point of tangency. Assuming that f is differentiable at a, the tangent line to the graph y = f(x) at the point (a,f(a)) is given by the equation.

y – f(a) = f ‘(a)(x – a)

This equation arises from the point-slope formula for the line passing through (a,f(a)) with slope f ‘(a).” In Pre-Calculus with discussing horizontal and vertical shifts you can easily relate back to point-slope equation of a line. You can relate point-slope equation of a line to the definition of derivative where the equation is rewritten with limits to describe the slope as the derivative. This is just a few ways that point-slope pops up in later mathematics courses. It is important to be able to form the point-slope equations of a line, as well as slope-intercept form, and being about to understand it well enough to build off of it when leading into harder concepts.

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

Point-slope equation of a line is used in movies in a huge way that most people probably never even realize. Point-slope equation of a line is used in pinhole cameras. A pinhole camera “is a simple optical imaging device in the shape of a closed box or chamber. In one of its sides is a small hole which, via the rectilinear propagation of light, creates an image of the outside space on the opposite side of the box.” In other words, let’s say we had an object, there is light constantly bouncing off the object. In the case of a pinhole camera, there is a small hole in the nearest wall/barrier which only allows light to pass through the hole. The light that makes it through the hole then hits the far wall, or image plane, creating a projection of the original image. The way point-slope equation of a line is used is first by adding a coordinate plane that has the origin centered at the pinhole. We can imagine that our scene is off to the right of the origin and the image plane is off to the left of the origin. We can choose some point in our scene to be a coordinate point in our coordinate plane. Some of the light bouncing off of that point in our scene will pass through the pinhole and land somewhere on our image plane. One of the ways we can find where it lands in our image plane is by using slope-intercept equation of a line. There is a really cool video on the khan academy website that talks all about the mathematics behind pinhole cameras. There is actually an entire curriculum called Pixar in a Box that goes through a variety of different topics and subject matter that is involved in the making of Pixar movies.

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

References:

http://matheducators.stackexchange.com/questions/9907/should-i-be-teaching-point-slope-formula-to-high-school-algebra-students

http://calculuswithjulia.github.io/precalc/polynomial.html

http://www.pinhole.cz/en/pinholecameras/whatis.html

https://www2.gcs.k12.in.us/jpeters/slope.htm

http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf

# 15-square puzzle

From the category “This Is Completely Useless”: here’s what a 15-square puzzle looks like when you arrange the tiles in order of how many factors they have.

# Difference of Two Powers (Part 5)

In this series of posts, I’ve explored ways that students can discover the formula for the difference of two squares and the difference of two cubes:

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

$x^3 - y^3 = (x-y)(x^2 + xy + y^2)$.

If students have understood the origins of these two formulas, then it’s not much of a stretch for students to guess the formula for $x^4 -y^4$. A geometric derivation requires four-dimensional visualization which is beyond of what can be reasonably expected of high school students. Still, students can look at the above two formula and guess that $x-y$ is a factor of $x^4-y^4$, and that the second factor would contain $x^3$ and $y^3$:

$x^4 - y^4 = (x-y)(x^3 + \hbox{~~~something~~~} + y^3)$.

From this point forward, it’s a matter of either using long division to find the quotient of $x^4-y^4$ or else just guessing (and confirming) the nature of the $\hbox{something}$.

Once students recognize that the answer is

$x^4 - y^4 = (x-y)(x^3 + x^2 y + x y^2 + y^3)$,

then the factorings of $x^5 - y^5$, $x^6 - y^6$, etc. become obvious.

# Difference of Two Cubes (Part 4)

Here’s the formula for the difference of two cubes:

$x^3 - y^3 = (x-y)(x^2 + xy + y^2)$

The formula isn’t terribly complicated; however, the factoring on the right-hand side is hardly the first thing that a student would guess if only given the left-hand side to simplify. The formula of course can be confirmed by multiplying out the right-hand side, but that’s really cheating. It’d be nice to have a way for students to develop the right-hand side, as opposed to merely confirming that the right-hand side is correct.

To this end, I suggest using base-10 blocks, a common manipulative found in elementary classrooms. The figure below shows a 10×10 cube with a 3×3 cube removed.

A (hopefully interesting) challenge for students would be how to build this figure only using the materials found in a typical base-10 kit, and also building it with as few pieces as possible. I think that most high school students, after some thought, can solve this puzzle by using 7 flats (for the bottom 7 layers), 21 rods, and 63 units. This of course provides the correct answer, as

$7 \times 100 + 21 \times 10 + 63 \times 1 = 963 = 10^3 - 3^3$.

After finding the correct answer, students should give this picture some deeper thought. If we let $x = 10$ and $y = 3$, then

$7 \times 100 = (x-y) x^2$.

This makes sense on physical grounds: the volume of the “base” of 7 layers is $7 \times 10$, and the $7$ came from the fact that the top 3 layers are incomplete.

Likewise, the 21 rods can be thought of as

$21 \times 10 = 7 \times 3 \times 10 = (x-y) y x$.

Again, this makes sense just looking at the picture, as the 21 rods makes a solid that is 3 units high ($y$), 10 units long ($x$), and 7 units wide ($x-y$).

Finally, the 63 units can be thought of as

$63 = 7 \times 3 \times 3 = (x-y) y^2$.

Indeed, these 63 units form a solid with a square base of side 3 and a length of 7.

$7 \times 100 + 21 \times 10 + 63 \times 1 = (x-y) x^2 + (x-y) xy + (x-y) y^2 = (x-y) (x^2 + xy+ y^2)$,

which is indeed the formula for the difference of two cubes. Now that students have discovered the formula for themselves, the formula can then be confirmed using the distributive law.

As a postscript, it should be possible to use two different colors of base-10 blocks (to represent positive and negative numbers) so that students can derive the formula

$x^3 + y^3 = (x+y)(x^2 - xy + y^2)$.

However, I don’t personally own two different colored base 10 kits, so I haven’t had time to think through how to do this.

# Difference of Two Cubes (Part 3)

In my experience, students who have reached the level of calculus or higher have completely mastered the formula for the difference of two squares:

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

However, these same students almost never know that there even is a formula for factoring the difference of two cubes $x^3 -y^3$, and it’s a rare day that I have a student who can actually immediately recall the formula correctly. I suppose that this formula is either never taught in Algebra II or (more likely) students immediately forget the formula after it’s been taught since there’s little opportunity for reinforcing this formula in more advanced courses in mathematics.

I recently came across an interesting pedagogical challenge: Is there an easy way, using commonly used classroom supplies, for teachers to guide students to explore and discover the formula for the difference of two cubes in the same way that they can discover the formula for the difference of two squares? (The cheap way is for students to just multiply out the factored expression to get $x^3 -y^3$, but that’s cheating since they shouldn’t know what the answer is in advance.)

I came up with a way to do this, and I’ll present it in tomorrow’s post. For now, I’ll leave a thought bubble for anyone who’d like to think about it between now and then.

# Difference of Two Squares (Part 2)

In yesterday’s post, I discussed a numerical way for students in Algebra I to guess for themselves the formula for the difference of two squares.

There is a also well-known geometric way of deriving this formula (from http://proofsfromthebook.com/2013/03/20/representing-the-sum-and-difference-of-two-squares/)

The idea is that a square of side $b$ is cut from a corner of a square of side $a$. By cutting the remaining figure in two and rearranging the pieces, a rectangle with side lengths of $a+b$ and $a-b$ can be formed, thus proving that $a^2 - b^2 = (a+b)(a-b)$.

Again, this is a simple construction that only requires paper, scissors, and a little guidance from the teacher so that students can discover this formula for themselves.

# 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:

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

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

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

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

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

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

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

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

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