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.

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• 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/

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• 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

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• 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: Negative and zero exponents

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 Austin DeLoach. His topic, from Algebra: negative and zero exponents.

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

The topic of negative and zero exponents is very important when or if the students get to calculus. Although that will be several years down the line, having a solid fundamental grasp on the idea of negative and zero exponents will help them understand derivatives a lot better. Because derivatives of “simple” functions just multiply the coefficient by the exponent and then subtract one from the exponent, it is important for the students to have a good understanding of what negative and zero exponents are. If they do not understand already, they will be confused about why, for example, the derivative of 3x is just 3. It also greatly simplifies derivatives of things like 4/x2, as the students will simply be able to recognize that that is the same thing as 4x-2 and follow standard rules instead of needing to think about the quotient rule and waste time with that. It will also help them in the more near future when they work with simplifying expressions with the exponents written in different terms (i.e. with a positive exponent or with a negative exponent in the denominator), as it will help them recognize what simplifications mean the same thing. Explaining that understanding negative exponents will thoroughly help them in the future may be enough for some students to want to solidify their grasp on the topic.

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D2. How was this topic adopted by the mathematical community?

Although this is not about the early adoption of negative and zero exponents in the mathematical community, Geoffrey D. Dietz points out more recent bias for or against the use of negative exponents in textbooks in his Journal of Humanistic Mathematics (linked at the bottom of this answer). Dietz brings up the idea of what is considered “simplified” when it comes to negative exponents vs exponents in denominators. He rated over 20 mathematics textbooks from 1825 to 2012 from “very tolerant” of negative denominators in simplified answers to “very intolerant”. Interestingly, his first encounter with an “intolerant” textbook was not until the 20th century, and textbooks began getting more polarized as very tolerant or very intolerant closer to the end of the 20th century and getting closer to today. This is interesting when it comes to adoption by the mathematical community, as there is a significant inconsistency, even today, about whether negative exponents can be considered “simplified” or not. It will be important to point this out to your students so they can be prepared for their future teachers who may have different preferences on simplification from you, as that will help them understand the polarity in the mathematical community on this topic, as well as hopefully make them want to understand what negative exponents really mean. Dietz recommends giving your students practice with not only converting negative exponents to positive exponents, but also from positive to negative, in order to make sure they are prepared for whatever preferences come up as well as solidifying their understanding of what negative exponents mean.

http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1110&context=jhm

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E1. How can technology be used to effectively engage students with this topic?
This video from Khan Academy does a good job at explaining why negative and zero exponents are what they are. Although Khan Academy videos will likely not be the most engaging for all students, this video is short enough to maintain the attention of the class, and it the logic in it is helpful for the students who don’t understand how the definition of negative and zero exponents was decided on. The presenter does well explaining the idea of “going backwards” and dividing by the number when you decrease the exponent. It’s a good way to explain the “why” for students who ask about it, and it also is a good way to change up the pace for students, as playing videos during class could prevent it from becoming stale for the students, keeping them engaged for longer.

https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals/pre-algebra-negative-exponents/v/negative-exponent-intuition

 

 

Hamilton Day

No, not that Hamilton.

Courtesy of Slate magazine and mathematics journalist Katharine Merow: Today is the anniversary of the great insight that led William Rowan Hamilton to the discovery of quaternions. Details can be found here: http://www.slate.com/articles/health_and_science/science/2016/10/we_should_celebrate_hamilton_day_a_mathematical_holiday_on_oct_16.html

Or the day can be celebrated in song:

Thoughts on Silly Viral Math Puzzles

I’ve seen silly math puzzles like this one spawn incredible flame wars on social media, and for months I’ve wanted to write an article about how much I’ve grown to loath these viral math posts.

Of course, after months of dilly-dallying, someone else beat me to it: http://horizonsaftermath.blogspot.com/2017/08/sick-of-viral-math.html. I encourage you to read the whole thing, but here’s the post’s outline of the myths perpetuated by these puzzles:

  1. Math is just a bag of tricks.
  2. Math is memorizing a set of rules.
  3. Math problems have only one right answer.
  4. Being smart means solving problems quickly.
  5. Math is not for you.

 

My Favorite One-Liners: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on my favorite one-liners.

Mathematical Wisecracks for Almost Any Occasion: Part 2Part 7, Part 8, Part 12, Part 21, Part 28, Part 29, Part 41, Part 46, Part 53, Part 60, Part 63, Part 65, Part 71, Part 79, Part 84, Part 85, Part 100, Part 101Part 108

All-Purpose Anecdotes: Part 38, Part 50, Part 64, Part 70, Part 92, Part 94

Addressing Misconceptions: Part 3Part 4Part 11, Part 14, Part 15, Part 18, Part 30, Part 32, Part 33, Part 37, Part 45, Part 59

Tricky Steps in a Calculation: Part 5, Part 6

Greek alphabet and choice of variables: Part 40, Part 43, Part 56

Homework and exams: Part 39Part 47, Part 55, Part 57, Part 58, Part 66, Part 77, Part 78, Part 91, Part 96, Part 97, Part 107

Inequalities: Part 99

Simplification: Part 10, Part 102, Part 103

Polynomials: Part 19, Part 48, Part 49, Part 81, Part 90

Inverses: Part 16

Exponential and Logarithmic Functions: Part 1, Part 42, Part 68, Part 80

Trigonometry: Part 9, Part 69, Part 76, Part 106

Complex numbers: Part 54, Part 67, Part 86

Sequences and Series: Part 20, Part 35

Combinatorics: Part 27

Statistics: Part 22, Part 23, Part 36, Part 51, Part 52, Part 61, Part 95

Probability: Part 26, Part 31, Part 62, Part 93

Calculus: Part 24, Part 25, Part 72, Part 73, Part 74, Part 75, Part 83, Part 87, Part 88, Part 104

Logic and Proofs: Part 13, Part 17Part 34, Part 44, Part 89, Part 98

Differential Equations: Part 82, Part 105

 

 

 

 

 

 

 

 

 

The Continuing Conflict Between Mathematics and Femininity

From a compelling opinion piece from Inside Higher Education:

The Continuing Conflict Between Mathematics and Femininity

Researchers have examined women’s experiences within the classroom and in professional settings in an effort to understand why and how young women become alienated from mathematics. The most interesting manifestation of this work looks specifically at how our culture constructs femininity and mathematics as mutually exclusive — in ways that ensure that girls and women have a difficult time understanding themselves as mathematical knowers.

Young female mathematics students feel forced to choose between their femininity and their identity as mathematicians. In interview transcripts, they either defend their talent as mathematicians in spite of their femininity or claim their identity as women while explaining away their mathematical achievements. But they clearly do not have the cultural tools available to reconcile both aspects of their identity. Some have argued that this may be one reason why young women who have achieved great success in the field nevertheless drop out of mathematics after secondary school.

We need to tell different stories to expand our cultural understanding of who can engage in mathematics.

I recommend reading the entire opinion piece.

Jobs in Mathematics

 

Courtesy of the Mathematical Association of America, here are some resources for finding a career in the mathematical sciences: http://www.maa.org/news/quantitative-careers-get-your-piece-of-the-math-jobs-pie

I’ll also link to the list of resources that my university provides to our math majors: http://math.unt.edu/support-math-department/careers-mathematics

A quick programming note: after 4 years (or roughly 1,500 consecutive days of posts), I’m going to be switching to posting on Mondays and Fridays. I recently moved to an administrative appointment at my university, and found through the school of hard knocks that I’m not going to be able to sustain daily posts while also doing my day job.