My Calculus students needed some work with piecewise functions, and seemed to enjoy the slope puzzles we had been doing periodically so I had them create their own as a mini assignment.

The requirements were:

• Functions had to be piecewise with at least 3 pieces
• Functions had to be continuous from x = 0 to 10
• At least 3 different types of functions had to be used.

Here is an example of one of the really interesting submissions:

The rest of the submissions:
Samantha and Katherine (Reciprocal): https://www.desmos.com/calculator/k6ta1hpjr1
​Melvin (Polynomial): https://www.desmos.com/calculator/pcytghpgp2
Malcom and Jonathan (Absolute value): https://www.desmos.com/calculator/s18vpu6eyt
Ahillan (Sine): https://www.desmos.com/calculator/moj40jbkeg
Gerardo and Patrick (Sine): https://www.desmos.com/calculator/02bxuszbjn
Jose (Cosine): https://www.desmos.com/calculator/i9jqcvknsw
Nana S (Sine): https://www.desmos.com/calculator/7aam6jjs0p
Lucas (Polynomial): https://www.desmos.com/calculator/84qr54wycg
Almendra (Sine): https://www.desmos.com/calculator/gkoosz85js
Noyangbe (Radical): https://www.desmos.com/calculator/ycku94ltoe
Monica (Radical): https://www.desmos.com/calculator/xm17lhvipf
Lawrenda (Polynomial): https://www.desmos.com/calculator/x4wxfuoerk
Nardos and Priscilla (Cosine): https://www.desmos.com/calculator/70rrejqt3y
Mirabel (Polynomial): https://www.desmos.com/calculator/ttg4pb1uf9
Brian and Joshua (Reciprocal): https://www.desmos.com/calculator/c0ucoms3w5
Jad and Nathan (Sine): https://www.desmos.com/calculator/h6u1bbz00f
Faith (Sine): https://www.desmos.com/calculator/t9ujrlfflz
Donovan and Ranvir (Reciprocal): https://www.desmos.com/calculator/hz19ixgc0k
Anitha (Sine): https://www.desmos.com/calculator/0kxlczcguw
Jayestha (20th degree polynomial, sine, absolute value): https://www.desmos.com/calculator/xghnljuqcu
Ethan (Reciprocal): https://www.desmos.com/calculator/llggi5zedd
Nana O and Michael (Root): https://www.desmos.com/calculator/dlfqhn8cuu
Kurtis (Reciprocal): https://www.desmos.com/calculator/bnrqwevbne

Course: Grade 10 Applied Math, Topic: Finding surface area

​After seeing Andrew Stadel's File Cabinet 3-Act Lesson, we thought it would be an easy one to do with our 10 applied classes near the beginning of the semester.  We didn't just want to watch the Andrew Stadel video though, we wanted our students to actually cover an iPad cart with post-its.  

​After going through the Andrew Stadel 3-Act lesson as a class, I put a few post-its on the cart.  After recording some initial estimates, they immediately got to work measuring and calculating...

"But it would take a whole period to cover the whole cart with post-its!  How on earth do you have time for that???" 
Not every student needed to be involved.  After a quick lesson on calculating surface area, I asked the 2 or 3 students who finished the surface area practice problems quickly or the students who were finding it too easy to do it.  As the rest of the class practiced calculating surface area, they stuck 'em on.  It did end up taking the better part of a period but those students were happy to do it! 
The anticipation in the class grew as more an more post-its were added.  It was super satisfying for everyone to get a final, definite answer (see video below).  It was also great to see the Grade 9's and 12's who came into the classroom afterwards spontaneously start asking questions about the post-it covered cart.  "Can we do more lessons like that too?" 

​Shamelessly appropriated from Alex Overwijk at Slam Dunk Math by my teaching partner and I, this activity is one of our favourites from the semester.  The kids love it too.  They get to problem solve and eat some candy while doing it.

Aside from the obvious, "Who buys 4 Smarties?"  they come to "How much does one Smartie and one Jujube cost?" fairly quickly.
​
So can we figure it out?  After coming up with a few different possibilities for the costs by manipulating the Smarties, Jujubes and "pennies", the conclusion is nope.  There is more than one possible answer.  
What about if I had another receipt from the same store?  Now we're getting somewhere.  The cost of a Smartie and Jujube has to work for BOTH receipts!

The Activity: After working through the example together, receipts from various stores are distributed randomly to each student in the class.  They then have to find their partner (the person who has a receipt from the same store as them) and come up to the front to get actual Smarties, Jujubes and "Pennies" (plastic squares) and then figure out how much a Smartie and Jujube costs at that store by trying different combinations with the manipulatives.  When they have an answer, they check with the teacher and are given 2 new receipts from a different store. ​​

After they solve their required 4 pairs of receipts, they ask for more.  That's when I throw one at them that doesn't have a integer solution, or one that has multiple solutions and let them debate it out amongst themselves.

After the activity: You folks realize you solved some pretty complicated math problems today?  Awesome work.  What about if I gave you a problem like this?

They realize it's not practical to use physical Smarties and Jujubes to solve the problem.  If only there was an easier way...

​Cue Elimination, with a distinct lack of "Why are we learning this."