Engineering Forum - 3D Imaging

Wow, has it been a while since my last post.  Blogging hasn't been my priority since I have been teaching since the beginning of the semester.  It's March Break now though so I thought I would try to fit in some blogging.

3D Imaging Technology

Last night, I attended this year's Engineering Innovations Forum at the Ontario Science Centre with a few of my Engineering friends.  The topic of discussion was 3D imaging technology.  The presenters talked about 3D scanning and modelling for Construction Design, Forensic and Medical applications.  There is some really cool and interesting tech out there.  Particularly interesting to me was the discussion of Photogrammetry (using photographs to find measurements and model in 3D).  Eugene Liscio, president of AI2 3D talked about how taking pictures with any digital camera can be used to generate a 3D model which he uses in Forensic applications.  He mentioned the Autodesk app 123D Catch, which can create a 3D model from photographs with any Apple mobile device.  Pretty neat.  Now how could I use this in my classes?  Something to think about over the next week...

3D Printing Technology

Turning 3D computer models into tangible objects for the Medical field was the theme for another one of the presenters.  You must check out the pictures at this link for 3D Printed casts.  They look awesome.

Breathing

Anyways, the presenter talked about how he was involved in designing a mask to be used to analyze breathing in sleep studies.  He needed to find the optimal spot for the microphone so it picked up the sound of breathing from the nose and mouth at the same time.  To find where to put it, they went outside when it was cold and snapped a picture of the condensing breath.  They used this to find the ideal location of the microphone.  The picture he showed us looked something like this (red lines indicate direction of breath):

Application problem time!  My Grade 10's are starting the unit on solving linear systems so I could get them to overlay an axis and grid on the picture and determine the location of the ideal microphone spot.  Here is a screencap of a Geogebra model of it that I threw together.  I didn't scale it to the size of the face but that would be necessary to solve the problem:

Also, here's a link to a Desmos graph of the same thing.

This could be a quick activity to show the students how this stuff can actually be used in real-life.

And then they could use their answers to design a mask which could be printed on the school's 3D printer.... ok maybe that's not that realistic yet.  It would be AWESOME if every school had a 3D printer.  Only $2500 each!  Somebody needs to get on that.  Real-world problem solving to the MAX.

This is why I still go to Engineering events when I can.  It gives me some ideas for presenting concepts in new ways and helps me stay current with the technology that's out there.  Engineering PD is teaching PD for me as a math/science teacher.

Eglinton-Scarborough Crosstown

As a teacher, it pays to be opportunistic.  At least in terms of student engagement.  Take Toronto transit planning for example (or more like a ridiculously embarrassing lack of transit planning in the GTA).  Some students are bound to know about the most recent craziness surrounding a subway line in Scarborough.  Bringing it up may get other students interested in current events.  Nothin' like a good ol' subway debate to get a class riled up and excited for some learnin'.

Anyways this week I came across this map:

When a construction company is tendering a job, an estimator (or intern) may need to do take-offs: a fancy way of saying figuring out how much stuff they need to buy so they can put a cost to a job.  Here's a good math question:

Using the map above, how much material will they need to remove to construct the tunnels in the underground section of the LRT?

You may say there's not enough information there, which is probably true unless you make some big assumptions.  You can find the length of the tunnel using Google Earth.  If you explore the Eglinton Crosstown website you can find this rendering of the tunnel boring machine launch site:

From the picture, it's apparent that there are 2 adjacent tunnels.  Based on the size of a person (maybe 1.6 or 1.7 metres), you can estimate the diameter of the tunnels.  Using the length of the tunnel and the diameter, you can calculate a volume (after some unit conversions most likely).

Real-World Problem Solving

There are many other ways to find out the size and lengths of the tunnels from reports and construction drawings.  That's the awesome thing about these kinds of questions.  Plenty of ways to an answer.  Also, the answers may vary depending on what assumptions are made!  You can have a good class debate to try to figure out who is closest and why.  

You could also put some numbers to it.  How much would it cost to remove the rock/dirt?  Where should it go?  How big would a pile of it be?  How do I make sure I cover my assumptions?  These are all questions an Estimator has to deal with when pricing a job.  And it all comes from some simple geometry!  Great for any math class.  A lot of Construction and Engineering problems can be boiled down to simple math.  

This is also a perfect opportunity for some Problem-Based Learning.  The Eglinton LRT could be a theme for a unit.  The problem would be to estimate the cost of the whole rail line.  Several math concepts could be brought in to help solve the problem: areas and volumes, slopes of lines, finance, scale, understanding and creating graphs.  It's a way to give a bunch of disparate and lonely math concepts some common context and interest.  There is also the human factor to consider: the impact on the communities and the environment.  Definitely some interdisciplinary potential there.

I do foresee a problem however.  If I was ever to do this activity with a class I'd probably get so excited I'd end up talking about tunnel boring machines for an hour and put everyone to sleep.  The woes of being a Civil Engineer... not everyone cares about dirt as much as you do.  Vince understands.

Root 2

The other day, I posted this Vine:  

[Note: I just started using Vine last week.  I know I'm late to the party with Vine but it's not too late to state that it's an awesome way for students to document and share their ideas and creations.  Doesn't require any video editing skills.  Just a steady(ish) hand and a smartphone or tablet]

Root 2

You can figure out the relationships between the green, white, yellow, red and grey rods from the Vine using a bit of geometry.  Its a nice exercise that really makes you realize how smart the inventors of K'NEX were.  It also becomes obvious why I titled the Vine 'Root 2'.

There are actually hundreds of math questions that you can generate from the Vine.  Questions involving shapes, proportion, angles, colours, similarity, scale, pattern recognition, length, measurement, area, parallel and perpendicular lines, special triangles, manipulating root expressions, polar coordinates and more.  These topics cover every grade and even some university stuff.  K'NEX is a great tool for teaching a wide variety of math to students of any level or ability.  I can't believe I didn't use it for one of my lesson or unit planning assignments for OISE.

Extend...

Once you figure out the relationships between the lengths of the rods, you can start to ask some more interesting questions.  For example:

• What lengths can be made by combining the pieces and what lengths can't be made
• Is it possible to construct a 3-4-5 triangle with K'NEX?  The math gets interesting because you have to mix rational and irrational numbers.  
• How about this question: Is there a piece or combination of pieces that will perfectly join two opposite corners of this cube (below)?  Prove it.  With math.  Even if you had a piece that was the correct length, would you be able to connect it? 

Answers and more K'NEX to come...

Day 17 - Canoe Trip Physics

The Problem...

How can you use an understanding of physics to make a canoe trip more enjoyable and less work?

Why?

• Opportunity for students to use physics concepts to help plan for a canoe trip
• Opportunity to bring sustainability discussions into a Physics class
• Interdisciplinary math, science, biology, physics, ecology, geography

Check out this blog entry from a few years ago by @emnose.  Canoe tripping is awesome.  The planning, driving, paddling and portaging all pay off when your alone on a lake in the middle of Ontario enjoying the silence and the stars.

In a class...

I would only use this activity in a class if right opportunity presented itself:  if students were actually going on a canoe trip (I know a bunch of high schools that regular do canoe trips). Students working on their Duke of Edinburgh are actually required to do a trip so it would be a good opportunity for them to earn some of the requirements.  

The Physics...

Believe it or not, there is actually a ton of physics that can be found in canoe tripping.  Here are some examples:

1. Buoyancy and hydrodynamics

• What canoe shape, size, depth is appropriate for the trip.  Can study the drag/friction of the canoe in the water.
• Optimal weight and number of people per canoe.  Too heavy and the canoe will sit too low in the water and be inefficient to paddle!
• Number of canoes needed for the trip.
• Optimal paddle length and shape.  There is some statics involved in the best paddle length.  The shape has to do with hydrodynamics.
• This site has a whole bunch of canoe recommendations and images including the one below.  A good question may be which one would be most appropriate canoe profile for the trip.

2. Distance-time relationships

• Best route to a destination (connections to contours in geography and optimization in math).  Students would have to develop some criteria for what 'best' route entails.  Easiest? Quickest? Shortest travel time?  In the picture below, what would be the best route from the Three Legged Lake Access Point to Clear Lake? 

Map is of Massassauga Provincial Park.

Clear Lake is a route we've done a few times.

3. Statics

• What is the best way to pack a hiking pack in terms of physics.  Would it be better to pack the heavier stuff closer or further away from your body?  This can be interpreted as a simple statics problem (see below).  How does your body adapt if the centre of gravity of your pack is further away from your back.

Drawn in Penultimate

• Carrying a canoe - In terms of statics, is it better to carry a canoe with two people or one?  What are some advantages and disadvantages of each.  I've never had a problem with 1 person...

Canoes are designed to be carried by one

 person but at first it doesn't seem to make 

sense from a physics standpoint.

These are just a few examples of how to look at canoe tripping through a physics lens.  I'm sure there are plenty more though.

Transformative Environmental Education

As a Scout leader, I have the opportunity to fairly regularly organize and participate in trips with groups of Scouts.  It's something I think every high school student in Toronto should have a chance to experience.  We talk about Environmental Sustainability but sometimes we forget what we're actually trying to sustain.  Living in the city all year, I sometimes forget about my connection and dependance on nature.  

In an article I read at OISE for my cohort class (Global Citizenship and Sustainable Development), Julie Johnston stresses the importance of stepping outside the curriculum box by practicing Transformative Environmental Education.  She presents education as the primary source of social reproduction and thus where we must start the cycle of understanding, caring for and protecting our environment.  Two examples of Transformative Environmental Education she provides (that I could utilized in this Canoe Tripping activity) are:

• Sky Awareness - promoting the importance of just looking up! Talking about how the sun tracks across the sky and how you could navigate by the stars.  This is easily tied to physics (astronomy).
• Bioregion-Based Education - understanding your connection to your immediate environment.  Understanding the ecosystem in which you live.  This ties nicely into biology and ecology concepts.

21st Century...

Students will be using Real-World Problem Solving and Innovation to help them plan their trip.  Concepts from Physics and other courses can help make an actual trip more enjoyable and less laborious.  

The activity also has the potential to be fairly long term and Self-Regulated by students, culminating in the actual trip.  Journals kept by students during the trip could be used to self-assess the effectiveness of their plans.

Lastly, there would be few more interdependent activities than a canoe trip, which basically forces everyone to do their part in order for the group to get to a destination.  It's an excellent opportunity to demonstrate Collaboration.

1. Collaboration: entry - adoption - adaptation - infusion - transformation
3. Real-World Problem Solving & Innovation: entry - adoption - adaptation - infusion - transformation
5. Self-Regulation: entry - adoption - adaptation - infusion - transformation

Day 15 - Road Trip!

The Problem...

What do you need to know to plan for a road trip?

Why? 

• Bring together many skills from the Grade 9 Essentials Math Curriculum
• Have students learn skills that are directly applicable outside school
• Interdisciplinary opportunities (business and geography)

I'm starting this entry in a van on the way home from Fond du Lac, Wisconsin  This past weekend I was on a road trip with @emnose and her family to pick up her brother from university.  Long, long drive but great trip!  Their highway bridges are a lot nicer looking than ours (these are the things you notice when you are a Civil Engineer).  Maybe we should start tolling more of our highways... another discussion for another day.

View Larger Map

The curriculum...

@emnose had the the idea to do an entry based on the road trip so I started to think about some of the things that you need to research when you are planning a road trip. I quickly realized that planning a road trip brings together many of the skills in the Locally Developed Essentials Math courses.  I actually taught a Grade 9 Essentials Math course with my AT during one of my practica. I actually think the road trip activity would be an excellent culminating activity for the course.

The Essentials course has a heavy focus on real-world, concrete skills. The Grade 9 course for example had 3 units: money sense, measurement and proportional reasoning.  Really, the activity could be modified to fit into any math course but I thought this was a practical activity for the Essentials students.

In the classroom...

I would start the activity by asking the class what kind of things you would need to research before going on a road trip (I may specify USA because it provides the opportunity to discuss exchange rates and different taxes). Most of the following things should come up in a class discussion:

• Shortest travel time
• Shortest travel distance
• Lowest travel cost
• Exchange rate
• Cost of gas (conversions of volume and currency units)
• Taxes and tipping practices
• Time zone/daylight savings
• Hotel bookings and availability
• Driving rules and other laws
• Vehicle needed and packing allowances

Students could then get in groups, choose their destination and start working on what was discussed.  Deliverables should be based around Skilled Communication of their findings and could take the form of an itinerary sheet, a budget sheet including a total cost of trip, a custom google earth map and/or a travel booklet/infographic with all the info in it.

Equity and 21C...

To be a better example of 21st century learning and for the Knowledge Construction to be authentic, the road trip should actually happen. I know this isn't plausible for most high-schools or affordable by many parents but what if there happened to be a school team going on a road trip to a tournament? The class could help with the preparation of that (including creating a budget and booking the hotel). 

Because of the equity issues involved, I would probably only do this activity if there were a school trip actually planned that the students that were planning it were able to attend!  I think the activity is only worth it (from a 21C perspective) if they actually use the knowledge they construct and the products they create.  For a lot of 21st Century Learning (as well as Critical Pedagogy and Global Citizenship Education), I think it is necessary for a teacher to be highly opportunistic based on what is going on in the school and school community.  That's one of the reasons why I keep these entries fairly vague and open-ended.  I'll post a quick blog entry about this in a bit...

1. Collaboration: entry - adoption - adaptation - infusion - transformation
2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
4. Skilled Communication: entry - adoption - adaptation - infusion - transformation
6. Use of ICT for Learning: entry - adoption - adaptation - infusion - transformation

Day 14 - EQAO and Equity

The Problem...

What educational equity issues become apparent when looking at EQAO results?

Why?

• Provides relevant context for statistics concepts
• Opportunity to turn a social justice lens on inequity in education
• Potential for Grade 12 students to help Grade 9 students get prepared for EQAO testing

This idea comes from a lesson example that I developed for a curriculum interrogation assignment at OISE.  It's an interesting lesson idea that I thought would be good to improve by looking at it through a 21st Century lens.  

The Math Curriculum...

I have stated before that the math curriculum is very concept-heavy (at least compared to Science which distinguishes STSE and skills expectations from concepts).  This generally leaves little room for deep investigations about culturally relevant and engaging topics.  

The vast majority of examples of using social justice education, global citizenship education, culturally relevant/responsive pedagogy or critical pedagogy in math class that I have heard have been based on statistics.  Unfortunately, in the high school math curriculum, statistics really only seriously comes into play in Grade 12 Data Management.  I think one way of increasing student engagement in math classes would be to spread out the Data Management concepts throughout the grades (another may be providing teachers with a bit more flexibility with the concepts).

Because Data Management is the obvious math course for the delivery of social justice education, I have stayed away from it in my blog.  Up until this point, my math entries have focused more on being critical of bias in math in general.  This however is an investigation about how we can use math to look at equity in society.  It fits in directly with a bunch of the Data Management expectations.

EQAO and Equity...

The following plot compares Primary EQAO data to Stats Canada Data on average family income.  Each dot represents a school's average result.  The article this plot comes from can be found here (it was published by EQAO in 2008).  This plot alone can be the starting point for a serious discussion on equity in standardized testing and education. 

I thought a good way to start the class would be to hand out a bunch of questions from a past Grade 9 EQAO test.  This will get them in the frame of mind of a Grade 9 student.  They can start to think about what factors may give students students trouble with those questions.  Note: A bunch of past secondary EQAO tests and other EQAO related resources can be found here.  

Find the relationships...

Then they could start looking at data.  The above plot can be found online but to facilitate Knowledge Construction students should be going to the EQAO website to get the raw data.  They can then draw their own relationships.  Some of the simple relationships they can get directly from the EQAO data are gender and stream comparisons.  Questions that may be good discussion starters are:

• does the division between genders seem to be increasing or decreasing over time?
• does the division between academic/applied seem to be increasing or decreasing over time?

Other sources of data students can compare the EQAO results to are Stats Canada data and even the Fraser Institute Rankings (which frankly scare me because of how seriously they seem to be taken).

Critical analysis...

After establishing relationships, students can begin discussing why some of the relationships exist.  As a teacher, this is where things can become difficult and exciting.  The discussion is the part of the investigation that can be most thought provoking and interesting but often gets left out for the sake of saving time.  I believe we're talking about 21st Century Fluencies, that excuse is unacceptable.  Bypassing the discussion doesn't help students develop the Skilled Communication and Real-World Problem Solving skills necessary for the 21st Century Learner.

I think another huge topic of discussion is taking a step back and looking at standardized testing in general.  What is the purpose of EQAO?  Really it is to evaluate teachers and schools.  Is this the best way to do this?  Is it a good representation of the academic strength of the school?  How else should schools be evaluated?  What equity issues may arise because of standardized testing?  What biases are present in both the test itself and the data it produces?  

It would also be interesting to look at the standardized testing situation in the United States, where it is much more pervasive and compare it to Ontario.

Action!

Now, how can these results be used by the Grade 12 Data Management Students?  I would say it's a good opportunity for the Grade 12 students to help the Grade 9 students that are preparing for the EQAO.  If they find relationships in the data that could be beneficial to the younger students then they could share their results.  Most likely, they will not be able to directly influence any of the factors that cause inequity but they can at least advocate for and educate the Grade 9's.  Most students probably never think back to EQAO after they write it but I think this is a good opportunity to build some school community and leadership skills by having Grade 12 students setting up tutoring sessions for Grade 9's.  There's the Collaboration!

21C...


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
4. Skilled Communication: entry - adoption - adaptation - infusion - transformation

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 11 - Hockey Stats

The Problem...

• How are all those hockey stats collected and what do people do with them?

Why?

• Engage students with the data collection and management behind sports
• Expose students to the real-world application of statistics
• Entry point to talking about the cultural impact of hockey in Canada

This entry is coming a day late because I was too upset to write one after the Leafs loss to the Bruins in overtime in Game 7 yesterday.  It's fresh on the minds of Leafs fans so I decided I would blog about it and somehow tie it to math.  Below is an nhl.com screenshot of the top 10 players from the 2012-2013 NHL regular season:

http://www.nhl.com/ice/playerstats.htm?season=20122013&gameType=2&team=&position=S&country=&status=&viewName=summary#?navid=nav-sts-indiv

That is a lot of stats to collect for each player!  In addition to analyzing the data on the site in a Grade 12 Data Management class, the question of how and why they collect these stats at all would be an interesting way of pulling the topic into the 21st Century Classroom.  

Close to home...

I think an awesome activity would be for a class to organize in order to collect an array of statistics of a team's season.  Ideally it would be a season played by their own high-school team so small groups of students could go out to collect stats for each game.  The class could consult with the coach/team at the beginning of the year in order to Collaboratively decide what statistics would be most beneficial for the team. They could analyze the stats over the course of the year and create reports to the coach who could ideally use them to improve the team's performance, supporting both Real-World Problem Solving and Skilled Communication.

In addition to potentially providing the coach with tools for improving the team's performance, the purpose of this activity would be to expose students to the process for the collection of data and some of the complications that may arise.  This Knowledge Construction would not occur if students were just studying stats taken from nhl.com.

Note: this obviously doesn't have to be a for a hockey team.  It could be any sport. Hockey is just what I had on my mind at the time of writing.

Digging deeper...

Once they have collected the data, they could start asking the deeper questions about collecting data in sports.  For example: in what ways are the data collected in hockey games are used?  Below are some things that may come up:

• Help teams improve their performance:  find ways to improve their team or find weaknesses in other teams
• Scouting for the NHL: the site behindthenet has a comprehensive report on how junior player's stats are projected to estimate what their performance in the NHL would be
• Fans love stats!  Stats get fans more engaged.  It is a source of revenue for the teams and leagues.
• Provide research for medical studies (for example: the concussion epidemic)

Here are some other questions that may be good discussion starters: 

• How much money is spent to collect stats in the NHL?
• Is it worth all the money to collect the stats?
• How many people does it take to collect stats for one NHL game?
• How might projecting a Junior players stats into the NHL be problematic?

21C...

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
4. Skilled Communication: entry - adoption - adaptation - infusion - transformation

Future lesson ideas...

• Along the same lines but for Baseball: the math behind Moneyball (the movie)

Day 8 - The Odds

The Problem...

What are the odds of winning the jackpot on a slot machine?

Why?

• Provides context for probability and combinatorics topics
• Engage students in discussion on ethics of gambling and the Toronto Casino Debate

Being in Niagara Falls yesterday got me thinking about casinos and gambling, an interesting topic for debate and a common way of illustrating concepts in probability.  

Also, the debate on whether or not Toronto should allow a casino to built or not provides context for a discussion of the topic in a classroom setting.

My internship supervisor and I were wondering what the actual odds of winning at a slot machine were.  What does an 80% payout mean?  It turns out payouts are deceiving.  A 80% payout doesn't mean you will win 80% of the time but the value you can expect to walk away with is 80% of what you started with.  In probability, an expected value of 100% (or 1) is considered a fair game because the player and the casino have equal chances of winning.  

According to this website, Nevada law states that the minimum payout can be 75%.  It is rarely this low however because casinos compete for players by increasing the payout.  90-95% are more common.  In addition to the games being mathematically unfair, they are built so that it is more likely that the machine will stop in the blank spaces on either side of the jackpot symbol to make it feel like the player is close to winning.

The Curriculum...

As an activity in class, students could calculate the odds of winning a jackpot on a slot machine with (for example) 3 reels and 20 positions (symbols + spaces in between symbols) on each reel.

Then, questions get more complicated when trying to find out the expected value (or payout) of a machine based on the probabilities of landing on each symbol and the payout of each symbol.  This website provides the background for some of that math.

These questions touch on various concepts central to the Grade 12 Data Management course (probability, random variables, combinations, expected values, probability distributions).


21C...

As an extension and problem-solving exercise would be to devise a slot machine that would provide a fair (100%) payout.  Students with experienced in programming could write a program to simulate the machine.

A lot of the time in Data Management, gambling and game related probability problems are given to students.  I think it is important to put these questions in context by critically thinking about and discussing the ethics and psychology of gambling.  The Toronto casino debate is potentially an entry point for the topic.


2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
3. Real-World Problem Solving & Innovation: entry - adoption - adaptation - infusion - transformation

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

Day 6 - The neXt Desk

The Problem...

What is the best way to move the neXt Desk?

Why?

• Opportunity for students to demonstrate real-world problem solving skills
• Illustrate the usefulness of several math concepts

I'm posting this a day late but it's because I was at the Connect 2013 conference yesterday and today. Great few days connecting with representatives from different school boards and vendors.  The neXt Desk is an art installation that is a symbol for the TCDSB's Project neXt and the neXt Lesson.  This is a picture of it set up at the Connect 2013 conference in Niagara Falls (which I'll definitely blog about later).

Ask the questions...

In an art class, students might critique the effectiveness of the piece in challenging how we normally look at desks but I thought it would be cool to turn the piece into a math problem as well. 

I would start by giving the students the picture.  Tell them they are the moving crew responsible for moving the sculpture! What might they want to know about it?  What if the only information they had was from the picture?  What would they want to know before agreeing to move the piece?  Students could work in groups and may come up with entirely different sets of questions.  Some examples may be as follows:

• How tall is the piece? 
• How much does the entire piece weigh? 
• The piece comes apart into sections.  How many sections should it come apart in for two people to be able to carry it? (I came up with this because I did this multiple times in the past few days)
• How much tension are in the cables?
• Would the piece stand if the cables weren't there?

How can we find the answers...

Then, in groups, students can discuss how are they going to solve the problem posed.  This may take some research and data collection.  Some possible methods are described below:

• How tall is the piece? They could scale the picture from the size of one desk that they measure in the school.  If they had access to the piece, they could just measure it.
• How much does the entire piece weigh? Students could estimate/measure the weight of one desk and multiply.  But then what is their method for weighing one desk?  Look it up or find a scale and somehow measure it?  What about hardware and connectors?  Does their contribution to the total weight matter?
• The piece comes apart into sections.  How many sections should it come apart in for two people to be able to carry it? They may ask questions such as: what is a reasonable weight that 2 people are able to carry? What are the size constraints such that it would fit through a regular door?  It takes more time to take it apart into more sections.  What is a good balance between weight and time to construct/deconstruct?
• How much tension are in the cables? This is hard!  At least I think so.  Even with a Civil Engineering degree.  I think it is still an interesting question to pose.  Students could try building a scale model and directly measuring tension (but this  brings into question problems with scaling up loads, one of the many causes of the Quebec Bridge Disaster of 1907).   They could also come up with a range of possible values based on the weight of the structure.  Another method would be to compare the different sets of cables: which ones would have the most/least tension and make some assumptions for them.  Another interesting method would be if the students had access to the sculpture, they could theoretically pluck the cables and measure the frequency of the vibration.  They could then calculate the tension using the length, density and diameter of the cable.  I have done this with my guitar students to find out how much compression their guitar neck is resisting.  Another topic for another day...
• Would the piece stand if the cables weren't there? They might look at the tension in the cables.  They might try (safely) experimenting with a desk in the classroom to see how strong it is.  If you are wondering, the answer is yes it does stand but it ain't pretty!

Educated guessing...

Before actually applying their calculations, students should hypothesize what their answers would be.  This is an important part of solving real-world problems.  Know the answer before you find the answer! This is one of 3 Engineering Tenants I learned in my undergrad (which I will dedicate another blog entry on problem-based learning to).

What I like about questions like this is that I (as the teacher) dont necessarily need to know the answer.  To move it, we took the sculpture apart into 4 segments of 5 desks to move it but maybe there is a better way!  


What if...

Once the students have a good understanding of the sculpture you can take it one step further by asking questions such as:

• Why might the artist have used 20 desks?
• How tall would the piece be if there were 10,25,30,50 desks?  What assumptions would you have to make?  This can turn it into a geometry (something I didn't explore but another good way to go with the sculpture) or even a calculus question (rate of change of perimeter to diameter).
• What are some limitations of increasing the number of desks?

Curriculum...

Depending on the grade level and questions the students ask, the investigation can cover topics such as measurement, scaling, relationships between variables (weight of sculpture vs. number of desks, height of sculpture vs. number of desks).

21C...

Though it may not be a real-world problem that the students themselves face, it is one that we had to face moving it and the artist had to face when designing and building it!  Next time the neXt Desk needs to be moved from its location at the TCDSB headquarters to another conference, have some students test out some of their ideas!  (Only with adult supervision.  The piece is not the easiest or safest thing to move)  


1. Collaboration: entry - adoption - adaptation - infusion - transformation
3. Real-World Problem Solving & Innovation: entry - adoption - adaptation - infusion - transformation




A final point.  Obviously this is a TCDSB specific sculpture but the problems students are solving should be specific!  This problem may engage students from Cardinal Carter who walk by the sculpture every day but may not for a student who hasn't seen the piece before.  Teachers should seek out opportunities for investigations such as these in their own school communities.  Put on those math goggles and see what you can find.


Future Lesson Ideas:

• Guitar String Tension vs Pitch