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 Michael Dixon. His topic, from Pre-Algebra: prime factorizations.
A1. What word problems can your students do now?
One word problem that is easily relatable would be something involving food!
For instance: “Don loves peanut butter and jelly sandwiches. One day he noticed a jumbo jar of peanut butter has 72 servings and a jar of jam only has 40 servings. If he opened the [first] jars on the same day and used exactly one serving each day, how many days until he emptied a peanut butter jar and a jam jar on the same day? Use prime factorization to solve.”
Obviously, this involves finding the least common multiple of 72 and 40. I would introduce this problem at the beginning of class, after my students have already been introduced to the idea of prime factorizations. I do not expect that my students would know how to calculate the lcm using prime factorizations, rather I would want to strike up a class discussion asking students to explore what they know about factorizations and see if they can find any patterns that would lead to the solution. I want to lead them to the idea that prime factorizations make finding the lcm far easier than listing the multiples of each number, especially when large numbers are involved.
B1. Future Curriculum
As mentioned in the previous paragraph, students can learn to use prime factorizations to calculate the greatest common factor or the least common multiple of numbers easily. To take this quite a bit further, we can introduce students to the idea of using factorizations, gcd, and lcm in formal abstract proofs. We would ask them to actually prove anything, just think about the ideas. Ask students how they know that the math that they use everyday actually works. Why does every number have a unique factorization? Why can I calculate the gcd and lcm of any two numbers, and know that that answer is the only answer? Then explain that later on, in higher level math classes, we actually flawlessly prove why our number system works, and how and why primes are important, such as in the Euler Phi function. Without prime factorizations, we would be unable to prove quite a lot of the math that we take for granted.
E1. How can technology be used to engage students?
After your students have been working with prime factorizations for a while and they are getting more proficient, what’s an obvious escalation? Make the numbers larger! Ask your students to factor numbers like 198 and 456. See how long it takes them to work through these. Then, ask them how long it would take to factor numbers like 2756 or even 12857. How could they do these? Is it even reasonable to try? What about 51,234,587 (this is actually prime)?
Here we can introduce using a computer, and using a computer to do the calculations for us. Just a simple website is adequate to show them just how useful computers are when doing large calculations. A website such as Math is Fun is an excellent tool to demonstrate the magnitude of some prime numbers and composite numbers, and show that even as numbers get very, very large, they are not divisible by any numbers other than themselves and one.
References
www.mathsisfun.com/numbers/prime-factorization-tool.html
http://tulyn.com/wordproblems/prime_factorization-word_problem-7928.html
Mean Green Math 