# 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 Bri Del Pozzo. Her topic, from Pre-Algebra: finding prime factorizations.

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

An activity that I would create for my students involving Prime Factorization is based on an example that I saw on Pinterest. I would prepare an activity where students would be given a picture of a tree and assigned a two-digit number. I would then have students decorate their tree and at the base of the tree, they would write their assigned number. Then, as the roots expand down, students would be able to write the factors of their number as a factor tree until they are left with only prime factors (based on the image from https://www.hmhco.com/blog/teaching-prime-factorization-of-36). In the example from Pinterest, the teacher focused on finding the greatest common divisors between two numbers and used the factors trees as guidance. For my activity, I would assign some students the same number and emphasize that some numbers (such as 24, 36, 72, etc.) can be factored in multiple ways, so the roots of the trees could look different depending on how the student decides to factor their number.

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

There are a few ways that Prime Factorization can be used in my students’ future math courses. Prime Factorization is incredibly useful when learning how to simplify fractions. By practicing Prime Factorization, students become more familiar with the factors of large numbers, which becomes helpful when simplifying fractions. In the instance that a fraction is not in its simplest form, students will have an easier time recognizing such and will feel more confident in simplifying the fraction. Additionally, Prime Factorization prepares students for finding Greatest Common Divisors. Knowing how to find Greatest Common Divisors can be useful when solving real-world problems as well as in simplifying fractions. At a higher level of math, Prime Factorization allows students to practice the skills needed to prepare themselves for factoring things more complicated than numbers. For example, the idea of factoring can be applied to factoring a common factor out of an expression, factoring quadratic equations, and factoring polynomials with complex numbers.

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.

Khanacademy.org would be a fantastic website to engage students in this topic because of the inclusion of multiple representations. This website allows students to work through multiple practice problems where they can find the Prime Factorization of a number. When the student gets the question correct, they can move on to the next question, or they have the option to view a brief explanation on how to arrive at the correct answer. If students get a problem incorrect, they can retry the problem or get help on the question. The “get help” feature also provides students with a brief explanation, with options in video form and picture/written form, of how to solve the problem. Another important feature of this website is the ability for students to write out their thoughts as they work through the problem. Khan Academy allows students the option to use an online “whiteboard” feature that appears directly below the problem. This “whiteboard” feature allows students to write out their work and also offers a walkthrough of how to draw a factor tree.

# 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 Brendan Gunnoe. His topic, from Pre-Algebra: finding prime factorizations.

How can prime factorization be used in curriculum?

The teacher starts the class by asking students how they would find the least common multiple and greatest common divisor for two numbers. For the LCM, the most basic answer is listing the multiples of both denominators until they share a common multiple. For GCD, the most basic answer is listing out the factors of both numerator and denominator and finding the largest one in common.

Both processes can be made faster when using prime factorization, especially for larger numbers. First, do the process of prime factorization for both numbers. Then, for each prime, take the highest power on the lists and multiply everything together.

For example, take 12 and 45.

$12 = 2^2 \times 3^1$

$45 =3^2 \times 5^1$

$\lcm(12,45) = 2^2 \times 3^2 \times 5^1 = 180$

The process for finding the GCF is similar. Start off by doing the prime factorization for both numbers. Then, for each shared prime factor, take the smallest power and multiply everything together.

For example, take 12 and 30.

$12 = 2^2 \times 3^1$

$30 =2^1 \times 3^1 \times 5^1$

$\gcd(12,45) = 2^1 \times 3^1 = 6$

This process generalizes very easily for any amount of input numbers.

GCF and LCM are incredibly important when working with fractions and are used when reducing and adding fractions. Because fractions have loads of misconceptions associated with them, giving students another way to understand fractions can be very beneficial.

Technology

Have you ever wondered why we use 60 seconds in a minute and 60 minutes in an hour? Or why there is 24 hours in a day? What about why there is 360 degrees in a circle? One explanation is because these numbers can be divided evenly by loads of smaller numbers that we use often. In other words, these numbers have lots of factors in them. These kinds of numbers are called highly composite numbers.

A great video showcasing highly composite numbers is Numberphile’s video “5040 and other Anti-Prime Numbers,” hosted by Dr. James Grimes. This video is extremely dense with informative as Dr. Grimes explains what a highly composite number is, shows properties of these numbers, explains why they have these properties, and gives examples of how highly composite numbers are used both in math and in real life. Dr. Grimes also gives a few historical uses of highly composite numbers, which answer some of the questions listed above.

Prime factorization is the foundation of highly composite numbers. Highly composite numbers can be an interesting and exciting application of prime factorization.

Application

Semiprime numbers were also used in the making of the Arecibo message. Because the message is composed of 1679 bits, there is only four ways of decomposing the message into a rectangle. All possible decompositions of 1679 into a rectangle are 1×1679, 73×23, 23×73 and 1679×1. If decoded correctly, then the message forms a picture which contains loads of information about the solar system and life on Earth.

For a way to make semiprime numbers into an engaging activity for students, the teacher could have students create their own mini version of the Arecibo message and show them off in class. Students can be made into groups and each group get assigned a certain semiprime. Then, each group gets to decide what information goes in their mini message and draw their message onto a sheet of poster paper with a grid on it. Finally, they present their message to the class, representing the students sending their message off into space for extraterrestrial life to decode.

References:

https://topdrawer.aamt.edu.au/Fractions/Misunderstandings

https://en.wikipedia.org/wiki/Semiprime

https://en.wikipedia.org/wiki/Arecibo_message

# 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 comes from my former student Chris Brown. His topic, from Algebra: factoring quadratic polynomials.

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

The ability to factor quadratic polynomials is at the essence of many two-dimensional kinematic word problems that students will encounter in the future physics courses. One specific word problem that students can now solve, is, “In a tied game between the Golden State Warriors and the Houston Rockets, Steph Curry has the ball for his team. If Steph Curry is 20ft away from the basketball hoop and throws the basketball up in the air at a velocity of 3 m/s, will he be able to make the shot if 3 seconds is left on the clock and win the game for his team? Consider this to be an isolated system.” This special type of problem gives them initial distance, final distance, initial velocity, and acceleration. He student then needs to solve for time, which turns this into a quadratic scenario that requires factoring. I feel like this problem situation is super relevant to the high school age group as it seems to be popular amongst that age group, and with this problem they can extend it to any real-world scenario that searches for time when given distance and velocity.

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

When factoring quadratic equations, one of the universal methods of factoring is called factoring by grouping. Let’s identify a quadratic equation to be ax2 + bx + c = 0. When factoring by grouping, the students must first multiply ‘a’ and ‘c,’ and then find factors of the product which sum to ‘b’. Let’s call these specific factors ‘n’ and ‘m’. Thus far, this brings in students abilities to create factor trees from 3rd grade mathematics. The next step requires students to replace ‘b’ with the factors ‘n’ and ‘m,’ such that we now have ax2 + nx + mx + c = 0. Now the students have to group the ‘ax2’ term and ‘c’ with either the ‘nx’ and ‘mx’ terms in such a way that when the greatest common divisor is pulled away, what’s left is identical for each group. The ability to identify the greatest common divisor between two terms stems from what they learned in 5th grade mathematics. Then, the last step would be to factor out the common term. This entire process, which was not completed here, has used two very fundamental skills from elementary mathematics.

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

I believe Symbolab is an amazing website, that the students can use to aid them in the understanding of the process of factoring quadratic polynomials. I chose this website, because it focuses on the process of factoring and uses common language to explain their steps which the students should be aware of. Lastly, I love this website because it gives students the option to hide the steps and just see the answer. With this, the students can type in random quadratics and work towards the solution, and if they get stuck, they can see all the steps. All in all, it is an amazing way to practice the skill of factoring quadratic equations for as long as they please!

Here is the link to Symbolab: https://www.symbolab.com/solver/factor-calculator/factor%20x%5E%7B2%7D-4x%2B3%3D0

# 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 Trent Pope. His topic, from Algebra: equations of two variables.

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.)

I found a website that has many word problems where students can solve for two variables. An example of one of these problems is “If a student were to buy a certain number of $5 scarfs and$2 hats for a total amount of \$100, how many scarfs and hats did they buy?”. This example would give students a real world application of how we use two variable equations. It would show students that there are multi variable problems when we go to the store to shop for things, like food or clothing. An instance for food would be when a concession stand sells small and large drinks at a sporting event and want to know how many drinks they have sold at the end of the night. After using a two variable linear equation and knowing the price of the cups, total amount earned, and total cups sold, students would be able to solve for the number of small cups as well as large cups sold.

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

This topic extends on the students’ ability to graph and solve a linear equation, which should have been taught in their previous classes. The only difference is that the variable, y, that you solved for in Pre-Algebra is now on the same side as the other variable. For instance, the equation y =(-1/4) x + 4 is the same as x + 4y = 16. We see that we solve for the same variables, but they are both on the same side. This is because you are solving the same linear equation. A linear equation can be written in multiple forms, as long as the forms have matching solutions. This is something that students could prove to you by graphing and solving the equations. They would solve the equations to see that they have the same variables. This makes students more aware that they need to be able to compute for other variables besides x if the question asks for it.

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? 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.

The most effective way to engage a student about this topic is by using a graphing calculator. This is to help students make the visual connection with the topic and check to see if they have graphed the equations the correct way. Students learn more effectively through visual demonstration. Because students are the ones to solve for the equation and plug it into the calculator to check their work, they are going to be able to make that connection, and we will be able to verify that they understand the material. As teachers, we need to incorporate more technology into the ways of learning because we are surrounded by it daily. Using graphing calculators would be a great way to show and check the work of a two variable equation. This gives students a chance to see what mistakes they have made and what lose ends need to be tied up.

References

Solving Word Problems using a system with 2 variables. n.d. <https://sites.google.com/site/harlandclub/Home/math/algebra/word2var&gt;.

# 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 Brittnee Lein. Her topic, from Algebra: factoring polynomials.

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

There are many great websites that can help to provide students a conceptual framework for factoring polynomials in lieu of simple lecture. This website lets students explore polynomial equations with online algebra tiles.

https://illuminations.nctm.org/activity.aspx?id=3482

Algebra tiles are effective in teaching factoring because they provide a visual representation of abstract concepts and allow students to understand that the symbol “=” in an equation really means equivalence (i.e. what you do to one side of the equation, you must do to the other side). I also think algebra tiles are very beneficial in teaching students about zero pairs. There are other websites –such as wolfram alpha– that are especially great supplements to go alongside topics such as factoring polynomials because students can see the graphical meaning of the roots of a quadratic equation. When combined, these websites can aid students in gaining a both conceptual and procedural understanding of the topic.

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

There is an activity called “Factor Draft” where students set up a ‘playing field’ of cards. In this field, there are factor cards such as (x+2), (x-12), etc. sum (5x), (12x), etc., and product cards (1), (42), and so on. The goal of the game is to draw a winning hand of two factor cards and a corresponding sum and product card. Each card is color coded to their type. Each turn a player draws one card from the field of face up cards. The player must pay mind to not only his/her own cards but also those of their opponent’s –as the first person to get two factor cards and their corresponding sum and product card wins. This activity is beneficial in furthering student understanding between the relationships between each term in a quadratic polynomial. For example $(x+4)(x-3) = x^2 + 1x - 12$ and the corresponding factor cards would be (x+4) and (x-3) the sum card would be (1x) and the product card would be (-12). This activity allows students to intuitively get a sense of the process of factoring and gives them practice multiplying out polynomials.

2. How can this topic be used in your students’ future courses in mathematics or science?
• Factoring polynomials is used in many important future science and mathematics concepts. When a quadratic equation cannot be factored simply, teachers must introduce the quadratic formula. This slides into the introduction of complex roots of an equation and complex numbers. When factoring polynomials of higher degree than 2, synthetic division (another topic in high school mathematics) is useful in finding the roots of the equation. If a student is able to understand the meaning of the roots of an equation, that will aid in solving many interesting physics and mathematics problems. Factoring is used quite often to find the domain of a rational equation such as $f(x) = (x+2)/ (x^2+ 4x+3)$. A student must also have a strong basis in factoring polynomials to learn concepts such as completing the square.

References

• National Library of Virtual Manipulatives, nlvm.usu.edu/en/nav/vlibrary.html.

• Cleveland, James. “The Factor Draft.” The Roots of the Equation, 23 May 2014, rootsoftheequation.wordpress.com/2014/05/22/the-factor-draft/.

# 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.