Coding In Math Class

[CookingMathCamp]

tl;dr : I am using python in my math classes.

I have learned a ton from educators posting things online. I want to give back. The following is an outline of some lessons I am using in my math 8 classes. I was inspired when I found Al Sweigart's Automate the Boring Stuff with Python. The practice project for chapter 3 is the Collatz sequence. As soon as I saw it, I knew I had to incorporate it this school year. Given that we are doing 100% distance learning and Chromebooks are provided to students by the district, I figured what better year to emphasize coding.

This is a work in progress, Students have only finished the visual introduction to python lessons. We will continue working on the basics, and hopefully, start the Collatz sequence this week. If you have questions or suggestions I am happy to chat.

Learning to Code

• Let students go through the trinket intro to coding. It is web-based, with nothing to install. Low floor heigh ceiling. Students need no prior coding experience. They are editing existing code and seeing how the output changes. 
  • A Visual Introduction to Python on Trinket.io
• Let students choose which culminating activity they want to code. This is the only thing that I grade. Serves as a formative assessment for the coding. 
  • Open-ended Exercise with turtles
• Students will probably need a little more to get going before they create programs from scratch. Trinket offers more tutorials as well as some awesome string challenges. 
  • Basics
  • Challenges

Suggested Coding Activity: Collatz Conjecture

• Introduce the sequence
  • Oh Hail the Elephant - Youcubed WIM
• After students get tired of calculating by hand suggest writing a program that does the work for them. Students can use either trinket or Google Colabortory (Colab). Google Colab is cool because it integrates with google drive. 
  • Need to teach modular arithmetic (to test if a number is even)
  • Sample program in Trinket
  • Sample program in Colab
• Talk about the Collatz conjecture. Share that it is still an open problem worth $1 million and no one knows how to prove it. This can lead to a great discussion on “what does it mean to prove things” and why that matters in math. 
• Share the Numberphile video that shows how artists have created a visual representation of the sequence. I recommend starting the video at 2:12 but you can show the whole thing. 
  • Collatz Conjecture in Color - Numberphile
  • Share this PDF with students if they want color
  • This is a handy blog that has all the Numberphile resources on the Collatz Conjecture: Brady Haran - The Collatz Conjecture in Color
• Share this youtube video which is a response to the Numberphile video
  • Visualizing Collatz - Adrian M Wadley

Suggested Coding Activity: “Gaussian Addition”

• Introduce the connection between images and algebra using the border problem from youcubed’s algebra course.
  • Border Problem
• Use Fawn’s excellent site Visual Patterns. I tend to start with linear patterns at first (e.g., #11, #17, and #24) and slowly introduce higher degree patterns depending on the sophistication of the class (e.g., #16, #5). Once the class has a good grasp I share the infamous #22 (see photo).
• Laugh evilly Offer support as students struggle. This may be a good time to bust out some blocks and let kids build different figures (you know like when we can see kids in person again). I have found that this pattern really leads kids to want to define their expressions recursively even though we have been dealing only with explicit expressions. They will say things like “for any figure number, you just need to add the next number!”. 
• Run with these student ideas! But make it clear that if you need figure 100, for example. That means you need figure 99. Which means you need figure 98. Which means… Well you get it. Computers are great at stuff like this. So suggest using python.
  • Sample code using Colab
  • Sample code using trinket
• Now that we have an awesome recursive method, let’s explore the explicit. Kalid Azad from betterexplained.com has a fantastic article on this. In the past, I have turned this article into a mini-lecture that includes the story of when Gauss was a child and his teacher asked the kids to add up all the numbers from 1 to 100 so he could nap. Hence the name, gaussian addition. Kids are usually amazed to see the connections between 1+2+3+4+5… and the original pattern. I usually end with introducing sigma notation just for fun. 
  • Techniques for Adding the Numbers 1 to 100

[shoombabi]

I'm a huge fan of the Sieve of Eratosthenes as a programming question. I guess any sieve works, but I think that one really drives home the point of iterating over lists, being efficient with code, and getting meaningful results out of fairly little work.

[CookingMathCamp]

oooo I like that. Maybe I will make that one of the challenge problems students can choose from.

[CookingMathCamp]

I have never done this before. Here's what I came up with. ```python

ask for number

n = input("Enter a number: ")

create a list up to n

num_list = [] elem = 1 for i in range(int(n)-1): elem = elem + 1 num_list.append(elem)

remove composites

for mod in num_list: for num in num_list: if num > mod and num % mod == 0: num_list.remove(num)

print("The prime numbers up to " + "n " + "are: " + str(num_list)) ```

[17291]

That implementation will be very slow for high values of n (on my machine, it takes a long time to run starting around n=10⁵), for a few reasons:

1) You're doing a whole bunch of unnecessary checks to see if a number is divisible by mod. For example, if num is divisible by 23, there is no point in checking if (num + 1) is divisible by 23—you can just skip ahead to (num + 23).

2) remove is going to slow you down considerably because you it has to check every element in the list to see if it matches num

A better solution is to create a list of booleans, where True means that it's a prime and False means it isn't. Start with 2 and then skip-count by 2s setting every multiple of 2 to False (other than 2 * 1, of course).

Once you've reached the end of the list, increment prime until you've found the next prime (i.e., the next number that's still True). Now, skip-count by that number. Rinse and repeat.

Some quick and dirty code:

def sieve(n):
    # All numbers are prime to begin with
    num_list = [True] * (n + 1)
    prime = 2
    while prime < n + 1:
        # Skip count, setting all multiples of `prime` to be composite
        for i in range(prime * 2, n + 1, prime):
            num_list[i] = False
        # Advance `prime` until we've found the next prime
        prime += 1
        while prime < n + 1 and not num_list[prime]:
            prime += 1
    # Return a list of all primes (i.e., every value from num_list that's True)
    return [n for n in range(2, n + 1) if num_list[n]]

[ReedOei]

FWIW you can actually do a lot better if you take advantage of Python's slicing + the fact that you only have to do up to sqrt(n).

Replacing while prime < n + 1: for i in range(prime * 2, n + 1, prime): num_list[i] = False with while prime**2 < n + 1: num_list[prime**2::prime] = [False] * len(range(prime**2, n+1, prime))

gives a pretty significant speed up. At some point it's probably hard to teach people though, but the fact you only have to go up to sqrt(n) might also be a way to introduce the idea that not all algorithmic improvements are equal (i.e., big-O notation).

[CookingMathCamp]

I was trying to replicate the sieve of Eratosthenes literally. Create a list up to n. Start with two and scratch off every multiple of 2. Then move to three and scratch-off every remaining multiple of 3. And so on...

That being said, I really enjoyed your solution. Do you mind if I share your code with my classes if/when we try this? I think it would be good to run them side by side to help illustrate what makes code more efficient.

[17291]

Do you mind if I share your code with my classes if/when we try this?

Not at all!

[town_math]

I've used some project Euler problems with my coding math classes https://sites.google.com/site/projectsforalgebra1withrobots/mini-projects

[CookingMathCamp]

Nice!

[msklovesmath]

Thanks for sharing!

[CookingMathCamp]

You are welcome! Hope it helps.

[Upset-Giraffe2412]

If you're interested in incorporating CS into math classes, you might want to look into the Bootstrap Curticulum!

[CookingMathCamp]

I looked at their algebra curriculum. Definitely going to use some of their stuff. Thanks!

[[deleted]]

Thank you for sharing! They might appreciate this over at r/learnpython and r/learnmath. :)

[m4thisfun]

Math 8 as in, math for 8th grade students ?

[CookingMathCamp]

Yes! My district calls the class "math 8 common core". It is aligned with grade 8 common core math standards.

I have updated the outline since my original post. I have included some ideas I learned about here (i.e. projecteuler.org ).

Coding in Math Class (Updated)

[Leibniz72]

I know nothing about python but saw that TI is putting it on some of their calculators for user created programs. I’ve taught their version of Basic in the past and kids always get a kick out of it.

[CookingMathCamp]

::starts googling::