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.

But I "regress"...

There are some specific mathematics courses that I have enjoyed teaching. To provide some background, I have taught the following in the last 6 years:

• ESL Algebra 1 - 1 year
• ESL Geometry - 2 years
• On-Level Geometry - 1 year
• PreAP Geometry - 2 years
• PreAP Algebra 2 - 3 years
• AP Statistics - 2 years

As you can determine, I taught multiple preps in various years. My ultimate favorite class has been AP Statistics. It's a branch of math that is similar to geometry in that there isn't anything like it! For my second post, I want to share out a theme in Statistics and how I have digitized it for a 1:1 classroom environment.

The lessons below are ways that I've allowed students to discover concepts in Statistics that have allowed them to process it in their own way and through the use of collaboration. AP Statistics is a unique course in that their AP exam emphasizes reasoning and communication beyond computation. They're graded on their explanations, whaaaa?!?!

99 percent of all statistics only tell 49 percent of the story.

In my course, I encouraged students to always keep in mind the "why" of the data. What's the story? What is the data trying to tell you, and are you interpreting it in the best way? How can you manipulate it? And even more importantly, how can people misuse this? In one of my units, I expand on their current knowledge of Best-Fit Lines (Least Squared Residual Lines). Students have seen scatterplots in previous courses, but do they really understand how to use them and what can affect their story, like those pesky outliers. 

Essential Question: How can I make predictions using a linear regression model, and how can outliers affect those predictions?

Digital Materials: 

• Activity: Influential Points - link
• Students compare types of outliers and discover what makes them influential. This activity requires a TI-84 Calculator or you could use Desmos.
• There are extra instructions for students located in the "Speaker Notes" on each slide at the bottom.
• Activity: Matching Scatterplots to Descriptions - link
• Students determine the relationship between the strength and direction of a scatterplot and how it affects the correlation coefficient (r).
• There are extra instructions for students located in the "Speaker Notes" on each slide at the bottom.
• Various Activities (through Desmos) - link
• Polygraph: Students partner up for quick 20-Questions style game. Encourages the use of statistical vocabulary.
• Non-Linear: Students use Skittles to discover how to linearize a non-linear relationship. 
• MUCH MORE!

I hope you enjoy the materials and that they find a place in your mathematical world. Follow me to learn more and reach out if I can help.

Booting Up

As a newbie to the Blogger world, I've had trouble deciding how and what to write about, especially considering my mathematic's aptitude. However, I've discovered that I have two ultimate passions - mathematics and technology. Throughout this blog, I've received some strong advice from a friend (who is a more prolific writer than myself) - to create series of blogs!

First Series (and pun): Mathematics (with a technological dash)

This previous year, I was humbled to be named a State Finalist for the PAEMST award. After filling out the application, I realized how much I enjoyed utilizing technology in my classroom to enhance my students' learning, which I suppose leads to my current job as an Instructional Technology Specialist for my district home of 6 years. As a specialist, the hardest part can be aligning a felicitous technology tool with content, in particular mathematics. Therefore, why not include some sample lesson ideas, reflections, and when I picked a massive bouquet of whoopsie-daises?

Sample student work

For my highlighted lesson, I utilized GoFormative to help my students participate. Each pair of students shared a laptop and would take pictures of their work. It was a wonderful way for them to show their work and also display their work in an anonymous fashion (whilst having some "graffiti" fun on their lackluster desks. Kids with EXPO markers, who knew?!).

While technology is important, it's what we do with it that truly matters. Muhammad Yunus 

I realized that their participation could improve as long as I provided an outlet where students felt safe to share. I loved saving their responses and ultimately improving the lesson the following day.

I was fortunate to work with a group of students who not only were fascinated by the new tools but also their willingness to let me use them as my technology guinea pigs. Luckily, 90% of them passed their AP Statistics exam so I had some piggies who didn't waste too much time rolling around in the technology mud.

Interested in my statistical materials? Check out the lesson outline below!
Lesson Outline:
Objective: Students will be able to utilize a chi-squared significance test to determine if using your "gut" is more advantageous in a game of rock-paper-scissors.

Materials: The student materials will allow you to "Force Copy" what I created so that you have your own version.

• Student Handout: CLICK ME!
• Student Spreadsheet: CLICK ME!
• Rock-Paper-Scissors Website (uses "non-Marvel" Flash...iPad-ers, beware!)
• Technology Tool: GoFormative, Google Sheets, Google Docs
• Google Classroom (preferred)
• One device per 2 students