Can we back to Marco’s story? (That’s a Hamilton reference for any fans out there 😉). Welcome to Episode 4 of our What’s That About? series! In this article we finally take a leap into some more complex mathematical topics that are incredibly vital for not only prealgebra, but all the maths that come next.
What's in this article:
Super Quick Recap
Every action has an equal opposite reaction.
Thanks to Mr. Pikake, numbers are fractured like refraction
Try not crack under the stress, Marco understands counting
But factors and multiples are causing some difficult accounting
We get no satisfaction ripping apart Numberfolk factors
But knowing what they are made of is key, they are trained actors
Thankfully the chapter brings in comic relief
As mosquitoes leave Maggie in disbelief
It must be nice, It must be niiiice, to read Marco the Great and the History of Numberville!
(Sung to the tun of Washington on Your Side from the hit musical Hamilton)
One of the best quotes (that is not included in Marco the Great) is “Never underestimate results that count something” found buried in the pages of A First Course in Abstract Algebra by John B. Fraleigh. Counting is the mathematician’s (and since all humans should be mathematicians also the human’s) most powerful tool.
When we teach courses on counting, we always start by asking students, “Who knows how to count?” Each and every one always excitedly raises their hand and they can relate to Marco when he says, “I already know how to count!”
In fact, Counting is a fast subset of mathematics that goes deeper than most of us would ever expect. If you haven’t explored our course Introduction to Foresight and Prediction, make sure to take a peak. We discuss all the basics of the counting you never you needed!
While counting is the first math we learn, 1, 2, 3,… it is not nearly as easy as we might think. This is the first thing Mr. Pikake tries to impart to Marco. He is asked to count a pile of M&M’s.
So often students become so familiar with strategies, they unfortunately become locked into them. Here’s an example: we asked a group of students to find 4×13+7×4. They know right away what the ‘×’ is telling them, so they start scribbling down their algorithms to complete the multiplication. A Saint (if you are not sure what this means pick up your copy!) could find the value of this expression in only a few seconds and without the use of any paper at all. (It’s 800). This is a key idea in Counting. Mr. Pikake urges Marco to see the number 68 as malleable. To most, 68 is a concrete idea, as Marco says “[it’s] either 68 or it isn’t” but algebra is the art of manipulation. The sooner students can realize that numbers are theirs to control, the stronger their mathematical abilities become. Marco is then able to see 68 as whatever he wants it to be, it could be 69–1 or 25% of 272 or any of the infinite possible ways to represent 68.
From this, we are introduced to the idea of prime and composite. Every number can be constructed using primes. This idea is so important it is called The Fundamental Theorem of Arithmetic (yeah, that sounds pretty crucial)! He explores primes through LEGO and building (we continue this idea in our online game Keep ‘em Out) before ripping numbers apart to find the primes they are made of.
Prime Factorization is a vital tool. Marco the Great presents primes as number-traits. This analogy spans far and wide as it allows us to explore concepts like the GCD and LCM in a much more relatable and easy-to-understand way. We connect numbers to the human experience. It is easy for a student to understand that a person (or number in this case) likes certain things like music or baseball or video games. If we build a number using what they like, now we can easily find what two numbers have in common and other key mathematical ideas that stem from this idea. These topics are explored in our course Behaviorism which contains a plethora of games and activities helping students to master these often difficult skills.
“We learned to be happy, we danced ‘round the hall. And learning to count was the key to it all.”
Marco the Great and the History of Numberville contains some of the most insightful quotes from humankind so of course we included Sesame Street in our list! The idea of this quote was not only to bring some light-hearted fun into the mix but also highlight the importance of counting. After all, it is the first concept we learn!
In the Preface, the Mirror of Wonders tells Zil, “Nothing is powerful, for how could you know if there is something without nothing?” If it is hard to understand that numbers are all around us, counting should at least crack open the door. In everything we do counting comes into play. How long does it take to read this article? How many words are used? How would we know what chapter we are referring to? If we ever want to celebrate a birthday, we must know how long is a year? When were we born? On social media we count followers and likes. In business we count sales, employees, and other key metrics. To participate in society we must count money, votes, population.
Hopefully this helps readers to see how ‘counting is key to it all’.
Mr. Pikake asks Marco to think of a time he was ‘torn’. Can you think of a similar time in your own life?
A big concept introduced in Chapter 4 is the idea of disguises. Marco relates this to Peter. What does he mean by this (page 41)?
At the end of the chapter, Marco thinks he sees a look of disappoint come over his tutor’s face. Why might Mr. Pikake be disappointed?
Marco and Mr. Pikake come up with lots of different ways to represent the number 68. Come up with at least 5 ways to represent every whole number from 1 to 10. Compare your findings with a friend, family member, or classmate. Are they the same or different? Why do you think this is?
We first learn to count by ones (1, 2, 3, …) later, before learning of multiplication we learn to count by other quantities (like counting by 5’s or 10’s). Login to The Kryptografima and play BuZZ (this game is located in Behaviorism and can be played in many different forms including with multiple players) to practice your ability to count by different values. Why might this type of counting be helpful in finding the prime factorization of a number?
Explain in your own words why a number’s prime factorization is ‘finding what it is made of’. How does knowing what a number is ‘made of’ help us to better understand it and to use numbers in mathematics?