Navigating the Course

As I finished up Teach Like a Pirate by Dave Burgess, I found myself evaluating my previous lessons and how I used the hooks he mentioned and how I could improve the ones that needed what he calls "seasoning".

The lesson that I want to share is a topic in PreCalculus (used to be taught in Algebra 2) called Conic Sections. If you've ever taught it, you know it can be a beat down for students, causing (pun intended) circles to go around their head, which I find such a shame because it was my FAVORITE to teach.

All students love Play Doh!
(It really is the little things in life)

I loved how changing just a few numbers could affect the graph so drastically yet they were all related to each other in the smallest ways. I enjoyed that I could find all of them by simply slicing a cone differently at various angles. (The intro lesson I use for this unit involves students molding Play-Doh into a cone and using floss to make the various cross sections...yes, high school students go NUTS for Play-Doh!) The best part about this unit? It's usually BRAND NEW to students and I get to be the one who shares it! To watch them ask questions, and say "Oh don't you worry...that's tomorrow's adventure!" and "I love that you connected those ideas. Share that with your group, and if you'd like, with the whole class!" While the formulas can be a bit dull, it's the relationship between the equations that I find the most intriguing - and more importantly, the hidden applications they offer! One of the more interesting applications is to the medical field - whaaa?!?!

The lesson in particular that I enjoyed the most was about ellipses - the elongated circle if you will. I used a variety of Dave's hooks throughout and added some improvements so that teachers who were 1:1 could see how to integrate more technology! Although the hooks didn't really change the content, it made the presentation stronger and students were eager to learn - isn't that what we want?

Google Slides Exit Ticket

The "Hooks": Because the application I chose involved the medical field, all my hooks revolved around that!

• The Mozart Hook: As students were walking in, I had the theme song for Dr. House playing - "Teardrop" by Massive Attack
• The Picasso Hook: Students were given a half-sheet where they connected the intersection of the curves to make an "oval" shape. When they were finished, they measured various distances to discover the definition of an ellipse. You can also do this with wax paper using these directions.
• The Real World Hook: As an extension for that evening, students read up about one application of ellipses (lithotripsy - the process of dissolving kidney stones), solved the problem using the Chrome extension, and then had to find another application online. Curious about the application in further detail? Check out this site!
• The Costume Hook: A dressed up in purple (my school color) scrubs for the entire day.
• The Techno Whiz Hook (New Addition): As mentioned above in the "Real World" Hook, students solve a problem and did further investigating. Instead of doing this on paper and turning it in, they shared their thinking and solving on a collaborative Google Slides and their applications on a Flipgrid (WHICH IS FREE NOW!) 

The Challenge

Could there be even more hooks to include? Absolutely! I must say that when I was reading his book, I found some hooks were challenging for math, but had to ask myself, isn't that good?! I can re-imagine my lessons, re-think some strategies, and will encourage me to utilize the strengths of my team! How energizing, am I right?! Instead of teaching the same lesson, I get to "take the stage" with the content.

One of my favorite parts of his book was that it was content-neutral. He stresses that ANYONE with ANY CONTENT can teach using the P.I.R.A.T.E method if they just begin with an open-mind and a willingness to step outside of their comfort zone. He emphasizes that not every lesson will be success - some might even FAIL! And the biggest kicker - that it's OKAY! You get to try again the next day and even the day after that. I know I've had lessons that have fallen flat, and while discouraging at first, it lets my students see how I respond and how they too can respond when things don't work the first time. If it doesn't work, then it's feedback time! Isn't that a valuable lesson even if it's not content related?

A New Mindset

In one of my book studies that I'm participating in, I'm forced to reconsider the way mathematics is approached in the classroom. Dr. Jo Boaler is a professor at Stanford University for Mathematics Education. Her passion involves helping math teachers ACTUALLY TEACH mathematics. She strives to encourage teachers to move away from "sit and get" styles to pattern-investigation methods.

"When textbooks introduce only the simplest version of an idea, students are denied the opportunity to learn what the idea really is." Dr. Jo Boaler from Stanford University

What I'm reading

She is breaking down the wall and suggesting new, innovative ways to improve EVERY student's success in maths. She does not support the idea that you have a "maths" brain. She explores various research studies that show any student can learn maths if taught using some of her strategies which can include:

• Show examples and non-examples of definitions
• Rethink homework assignments to be reflection based instead of problem based
• Have students explore different methods and compare and contrast
• To reinforce concepts, have students use the concepts in different ways
• NO memorization - but utilize BOTH sides of the brain
• REMOVE timed-testing and math facts

I know what you're thinking...how can I do this? How can students learn maths if I don't show them the best methods and those precious shortcuts?! 

One of my toughest lessons in Algebra was Completing the Square. Students didn't understand for one thing, why it was even called that! They couldn't remember "all the steps" and couldn't make a connection what was really happening. Students memorized the steps and continued on their year which eventually lead to a brain dump to make room for the next memorization event.

Jo discusses the idea of "compression" in our brains. She explains, "when you learn a new area of mathematics...it takes up a large space in your brain." Once you play with ideas and dig deep, you can "file" them away and "compress" them. My biggest "Ah-ha!" moment was when she states, "Notably, the brain can only compress concepts; it cannot compress rules and methods." 

This allowed me to re-create my lesson on Completing the Square utilizing Algebra Tiles to help students explore the concept and build their own connections. If you've never used Algebra Tiles, I HIGHLY recommend it.

Student Notes (Google Slides)

Students build the polynomial using the tiles to literally make a square and determine how many 1's would it take to "complete" it. Throughout the process, students are reflecting on their learning and creating their own solving steps. For the extension exercise (i.e. homework), they will discover the Quadratic Formula! WHAAAA?!!?! Additionally, they will respond on a Flipgrid about their learning.

Student Reflection assignment

Materials:

• Notes: Complete the Square (1 copy for each student) - Google Slides
• Khan Academy Video (already in slides)
• Flipgrid: Create a Quadratics Topic for students to post their thinking
• Bitmoji: Students add bitmojis on their exercises to show how they feel

My Take-Away: 

Every student felt successful learning this new method. I had students make connections to other topics (graphing quadratics using the vertex) and even preferred this method OVER factoring! I couldn't believe the positive energy that was occurring and for once, the students were the ones doing the thinking! I became a guide for the day and my students didn't feel the need to have 20 identical problems for homework. 

Questions you could ask them for reflection exercises (consider using Flipgrid):

1. What kind of number for "b" makes completing the square easier? Explain your thinking.
2. What do you think would happen if "a" is not 1?
3. Do you think this method could work every single time? Defend your position.
4. Is there a time that this method works better than another? Explain by creating your own example.
5. Compare this method with factoring.

Taking a Chance!

Taking a chance on lessons can bring about anxiety...even more so when it's a concept that students struggle with based on your experience. This is how I felt constantly when teaching probability.

Probability is not just the science of chance but also the science of overthinking - at least in my students' eyes. They could understand visually what was occurring, but throw in expected value, standard deviations, random variables, and the eyes begin to aggrandize.

Instead of fighting the uphill battle of "why use these formulas" and the "why did you subtract from 1?" questions, I chose a different approach and created an activity going back to the probability basics - throwing some die around and making some "money".

Sample Student Work

Activity: Bunko! 

Rules of the Game: Bunko is a game of 3 colored die. Based on what the students roll, you could receive "money". Assume that the die are fair.

• If you roll the dice and end up with exactly one die showing a “1”, you win $1.00.
• If exactly two dice show a “1”, you win $2.00.
• If all three dice show a “1”, you win $21.00.
• If all three dice show the same number (any number from 2 to 6) you win $5.00.
• Any other outcome results in $0.00. 

Materials:

• Activity: Bunko - Google Sheet (1 per student)
• Activity: Bunko - Google Slides (1 per student)
• Use this handout if your devices are limited (1 per student)

Concepts Addressed:

• Expected Value/weighted averages
• Combining Random Variables (multiple games)
• Law of Large Numbers

Student Directions:

1. Play 30 rounds of Bunko, recording your answers on the Google Sheet.
2. Determine how much money you would win and record it. Watch your counts begin to fill in!
3. Open the Google Slides and begin answering your questions.
4. Gather all the CLASS data and enter that on your 2nd Tab. You will watch the class counts fill in!
5. Create a graphical display of the class data for your Google Slides.
6. Go to the Google Slides and answer the remaining questions with a partner.

Teacher Prep: Assign the Google Sheet and Google Slides through Google Classroom where "each student gets a copy."

Student's Google Slides! Don't have Google Classroom? Change the link from ".../edit" to ".../copy" and share it with them.

TakeTake-Aways:

Even though throwing dice can seem trivial, students surprised me by getting so excited! They grasped the concepts at a deeper level and performed stronger on their quiz. Perhaps sharing candy for students who had the most amount of money, least amount of money, and which student had the most significant counts??

At the end, they would want to play more games and determine further probabilities! I used this idea and created a probability game day, where they gathered the experimental probabilities and calculated the theoretical probabilities at home. 

Possible Extension for AP Students: Have students run a chi-squared goodness of fit to determine if their individual data was significant using the theoretical probabilities they made on the handout.

Overall, I was thrilled to take a chance and have the students play a game rather than going through additional examples from their textbook. They were more engaged, required to think critically and make connections, and most importantly, they developed a stronger foundation of probability that made the rest of the unit smoother.