Day 12 - Infinities

The Problem...

How big is infinity?

Why?

• Opportunity for Math/English interdisciplinary study of The Fault in Our Stars by John Green
• Address the common mathematical misconception that you can treat infinity like any other number in an equation
• Present the mind-boggling idea that an infinite series can have a finite sum
• Illustrate the idea that there are unanswerable questions in math.

The other day, I asked @emnose what I should write an entry about and she had the idea to use the novel The Fault in Our Stars by John Green.  

Today I finished the book.  Great read. Crazy sad but crazy good.  I was already a huge fan of John Green's YouTube channels (VlogBrothers, mental_floss and his brother Hank's channel SciShow) but hadn't read any of his books.  @emnose highly recommended I read The Fault in Our Stars.  Her rationale for using the book in a lesson idea was that a recurring theme is infinity. She's a smart one.  I don't want to give away any of the book so I'm mostly keeping details out of this entry.

Mathematical Infinity...

"Some infinities are bigger than other infinities," a character in The Fault in Our Stars states after explaining Zeno's Paradox.  I thought it would be interesting for students to explore what infinity exactly means in math, and compare it to how it is used (literally and metaphorically) in literature.  

In math, infinity gets messy.  In physics, its where black holes (singularities) appear.  Grade 11 and 12 students discover some of these difficulties when they are trying to plot rational functions or taking limits in calculus.  Some of the most interesting examples of these problems are found when trying to solve the indeterminate forms.  There are subtleties with infinity in math as illustrated by this TED-Ed video (and in Fault in Our Stars):

Interdisciplinary Infinity...

After a student had explored some of these mathematical problems with infinity, they could start to explore how the word is used outside of math.  The Fault in Our Stars is an interesting investigation because it bridges the gap between infinity being used metaphorically and mathematically.  Using The Fault in Our Stars as a starting point, students can explore other places infinity is used in literature.  

Interdisciplinary investigations are critical to 21st Century Learning.  According to the 21C framework, Knowledge Construction must be interdisciplinary to be Transformative. To bring it more into the 21st Century, the investigations can be done Collaboratively in groups (maybe pair up a Grade 12 Calculus student with a Grade 12 English Lit student) and the results Communicated in a variety of forms (writing, video, infographic, art).

To Infinity, and Beyond...

Infinity isn't the only interesting math/science connection in the book.  Here's one of my favourite quotes:

“I believe the universe wants to be noticed. I think the universe is inprobably biased toward the consciousness, that it rewards intelligence in part because the universe enjoys its elegance being observed. And who am I, living in the middle of history, to tell the universe that it-or my observation of it-is temporary?” 

This could be the jumping off point for some interesting philosophical conversations about why we do science and math (or really any other subject).  Along with my Number Bases blog entry, this idea helps students see the more human-created foundations of the way we understand Math.  

The TED-Ed video above also exposes a highly important fact:  It has been mathematically proven that there are questions in math that are unanswerable.  I believe this has huge implications for how we should think about math.  To come back around to infinity, I'll leave you with the awesomely beautiful and mathematically interesting quote from the TED-Ed video, 

"Someone one said the rationals (fractions) are like the stars in the night sky.  Then the irrationals are like the blackness."

21C...

1. Collaboration: entry - adoption - adaptation - infusion - transformation
2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
4. Skilled Communication: entry - adoption - adaptation - infusion - transformation

Future Lesson Ideas...

• The Philosophy of Math - see this Numberphile video called Do Numbers EXIST?
• Other cool interdisciplinary combos like the Evolutionary Biology in Galapagos (a novel by Kurt Vonnegut)

Day 7 - Calorie Counting

The Problem...

How do excercise apps/machines determine how many calories you have burned?

Why?

• Critical examination of the algorithms behind ICT
• Make mathematical connections between variables
• Experimentation and control variables

I have often wondered about how exercise bikes and running apps are able to calculate the amount of calories burned.  What assumptions do they make to find that number?  This question is an opportunity to combine math and biology (the science often seen as having to do the least with math), as well as tying both into physics.


These are the stats of one of my runs (from last year when I was in way better shape) collected using the free iPhone app RunKeeper.  Apparently, I burned 403 calories. Not too shabby. 

I think an interesting activity would be to try and reverse-engineer where that 403 calories comes from. 

Collect some data...

Students could form groups and collect data on their own devices on a variety of apps and experiment by  playing with different variables (if you look hard enough on the RunKeeper website, there is actually a list of variables used to calculate the calories, but it doesn't tell you how its calculated).  This is a good introduction to control variables.  Students will have to understand they can only change one variable at a time to get useful results.  For example, they may change the weight that is input and see if the calories burned varies linearly with weight.

**If a school has exercise bikes available, they may be a good starting point because there are usually less variables to deal with and the method they use for calculating calories is much simpler (and usually just linear with distance travelled).  What assumptions does the software in these machines make?

Critique and Create...

After coming to some conclusions about how the apps calculate calories, students should critically evaluate the assumptions in the calculation.  I think an interesting way of evaluating this would be for the students to decide what other variables may be important and come up with their own algorithm for calculating calories burned.  They could then submit their recommendations to the app developers.

Curriculum...

This activity encourages the use of control variables and has students creating their own experiments.  They have to think about the mathematical relationships to establish an understanding of the cause and effect of the variables in the app, a critical part of the Scientific Method.

This activity is also a good introduction to metabolism:

• What are calories?
• Why do we use the unit calories instead of joules?
• To give some scale of energy, how many calories does a lightbulb use in an hour?
• How many calories does a litre of gasoline contain?

We can also tie in physics:

• In physics, we say that the total energy in has to equal the total energy out.  Is this true of humans?  
• What other ways do we use energy besides going for a run or bike ride?

21C...

Students participate in Knowledge Construction by performing their own experiments.  They should have the freedom to choose what apps they want to analyse and work in groups to establish the relationships between the variables.  


1. Collaboration: entry - adoption - adaptation - infusion - transformation
2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
3. Real-World Problem Solving & Innovation: entry - adoption - adaptation - infusion - transformation
6. Use of ICT for Learning: entry - adoption - adaptation - infusion - transformation