Remote Mathematicians

In this "new world" of remote learning, keeping students engaged in content can seem daunting. That feeling of not just sharing content, but making it easily accessible and engaging enough for students to stay connected, brings its challenges.
As a previous math teacher, it was tough enough helping students enjoy the content, so flexing the "creativity" muscle in the brain was a CONSTANT.

During this time, I thought I'd share my top 3 go-to strategies!

TIP #1: Encourage inquiry and self-discovery!

• Geogebra: This site offers Classroom Resources from elementary up to college courses. Students can explore theorems, visualize multiple cases, and then easily develop their proof. Geogebra even has built in activities with self-checks. Follow Tim or Steve on Twitter to see their creations.
• Desmos: Teachers can create a FREE account and send activities to their students. Use one that's already made or create your own! Explore transformations and how variables can affect equations.
• NEW UPDATE ALERT: Add a co-teacher!
• NEW UPDATE ALERT: Send customized feedback to students.

TIP #2: Increase the open dialogue by using open tasks!

• Which One Doesn't Belong: 4 boxes...which one doesn't belong? The best part - there is a reason EVERY SINGLE ONE doesn't belong. Bring the discussion into math and allow students to see there isn't just "one answer." Do you have gifted students? Encourage them to create their own!

• Copy/paste a puzzle and use THESE SLIDES! Share it with students so everyone can edit the same file. 
• Numbers Example (for Elementary up to High School)
• Open Middle Math: One problem can completely replace a worksheet! Students not only practice, but must use logic and reasoning to finish the puzzle. Available for K-12 students!
• PRO TIP: Share on a Google Slides or Keynote for students to respond. They can drop in recording to explain their thinking.
• PRO TIP: Use master slides to create text and image placeholders for students.
• PRO TIP: Attach an Open Middle or WODB problem to the "Focus" in a Flipgrid Topic. Ss can use the whiteboard mode to share their thinking with their peers! 

TIP #3: Using the tech in your tool belt!

• Keynote (on the iPad): While it is similar to Google Slides, Keynote on the iPad takes the advantage! A fantastic mathematician and edtech enthusiast, Morgan Cave, on the team I work on got me into this app, and there is NO turning back! As an Apple Distinguish Educator, she's continuing to push outside the box for how Keynote can be used to explore all concepts.
• Built in shapes (including 3D ones with the latest iOS update!)
• Drop in just audio to explain thinking.
• Utilize the Drawings option with multiple colors to clarify steps. My favorite? Use the "Line Draw" animation so that students can visualize their work (and steps) come to LIFE!
• Use backgrounds and "locking" to prevent students editing important information.
• Templates from Morgan Cave: Shapes Hunt, Alpha See Say Draw, Numbers See Say Draw
• $FREE Apple Books: 20 Keynote Creations & Keynote in the Classroom
• Google Slides (for Chromebook): 
• Use master slides to create template responses with placeholders.
• Ss can drop in pictures, links, videos, text, animations, and MORE all in one place.
• Collaboration feature for discussions or group work is PERFECT! With the editing history, no work is lost and students are help accountable for their contribution.
• Google Sheets (for Chromebook): Check out these helpful resources and templates using the "ugly step sister" of the G-Suite family!
• Eric Curts' Blog
• Create a digital breakout activity with if-then formulas!
• Math Learning Apps:
• Digital manipulatives for the WIN!
• 100% FREE!
• Includes annotation feature! 
• PRO TIP: Ss can screenshot their creation, and drop it in Seesaw or Keynote to add audio!
• Flipgrid:
• Whiteboard Mode = PERFECT for explaining work.
• Drop in a video, picture, and/or links in the Focus for your Topic.
• Students don't want others to see their face? Use Pixel Mode!
• Check out their Disco Library for Topics ready to go!

Google Slides for ALL

Wow, another year has started already, and I'm trailing behind on some posts! I can't believe it! Coming into the new year, I want to focus on helping transform technology from "what it does" to "what it CAN do."

THIS WEEK - GOOGLE SLIDES!

While on the surface, Google Slides is great for presentations. You can add text, backgrounds, embed videos, add GIFs for a bit of animation, and easily include interactive questions through Add-Ons like Poll Everywhere and Peardeck.

What if you wanted to collaborate across the globe or the room? Google Slides allows many authors and a quick revision history for constant adjustments. So why would this tool not be perfect in the hands of students in classrooms?!

How can we transform this "presentation" tool into a something more?

1. Individual Task - Students utilize copy/paste to create combinations of ice cream scoops. This leads them to discover a pattern. This can be for 5th grade up to high school where the formal combinations formula is introduced. This task is based on Jo Boaler's tasks from YouCubed. This could be shared via Google Classroom where every student gets a copy. This encourages students to quickly visualize math while making observations. Lose the glue and scissors - lose the mess! Provide students individual feedback using the [Comments] feature.

Add a Bitmoji to make your tasks more personal!

2. Whole Class Task - Students look at 4 numbers or expressions to determine which one doesn't fit the pattern. Student select the slide color that matches the one they think doesn't belong, which allows the teacher to quickly view which one(s) the students picked the most. Since all students are editing the same Slides, encourage them to add a comment on another peer's slide to increase collaboration. Check out more puzzles on this site!

Use Master Slides to create templates for students!

3. Discussion Board - Students add a response slide to a prompt. You can add your bitmoji to show what you think of their response or have students do this. Encourage the use of comments to provide feedback from all peers. Use the Grid View to quickly see how many comments each slide has!

This is from an online PD, but same idea!

4. Graphic Organizers - Encourage lots of student reflection by quickly sending a graphic organizer to them. Each student can add a new one every time you need. Creating Master Slides allows them to pick from templates ready to go! This allows students to include images, text, links, and videos to synthesize what they've learned. Share it with students in Google Classroom so you can check progress and provide quick feedback. 

Want your own copy? CLICK THIS LINK!

Don't be afraid to see beyond the tech tool and try something new! Think outside the box, and challenge yourself to see new uses!

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?

"Hype" Up your Docs!

I've always been interested in the transforming powers that technology can provide. It can allow

students to learn at their own pace or even reach beyond the classroom walls. Students have the advantage to work with another student across the room without shouting or even across the world. The big question we ask ourselves is...How is this possible? How much time does this take to set up? What happens if they don't learn what they are "suppose to"?

I can answer honestly, I was worried as well. I was nervous to loosen the reigns, to let them explore at their own pace, to find the answers without me showing them myself.

So what changed?! - Welcome to Hyperdocs!

A Hyperdoc is a transformative, innovative Google Doc/Slide that is student-driven. Think of this as doing the guiding for you to free you up to assist students individually. You don't need to send them links and walk them through each step - let the Hyperdoc do it!

According to the Hyperdoc Handbook, before you start jumping in (or at least dipping your toe in), you should consider the following:

1. Consider your OBJECTIVE -- What do you want students to learn?
2. Select your Learning Cycle -- How do you want students to learn it?
3. Select your packaging -- How can this be presented?
4. Build a workflow -- Will students know what to do?
5. Design your Hyperdoc! -- Do you want a Google Doc or Slides?

What transforms a Google Doc to a Hyperdoc?! Ask yourself these quick questions.

1. Is it interactive or static?
2. Is it student-driven or you-driven?
3. Is it personalized or will each student give the same answer/product?
4. Does it encourage collaboration or is it individual work?

Biology Example (Google Slides)

Creation can be daunting. The "where to start" or the "how do you know what to include"? This is where you, yes YOU, come into play. Grab your content and go beyond what you see. Is there a quick video they could watch? Is there a way for students to collaborate on the notes? Can students create something to show their learning? Feeling stuck - no worries!

Below I have some resources that I have found helpful. Some include templates, example lessons, helpful sites, and even some tutorial guides to help you get started. 

Tutorial Guides/Sites:

• Making Hyperslides - Google Slides
• Using Canva - Google Slides
• The Hyperdoc Handbook - site

Favorite Tech Tools to transform your Doc/Slides:

1. Flipgrid - create a topic where all students can create a video response.
   
2. Padlet - create a digital post-it board where students can type, draw, insert links/pics, add video notes, and much more.
3. EdPuzzle - create an interactive video with questions at any point!
4. Google Forms - create a survey or quiz where data can be instantly analyzed.
5. ClassHook - find educational videos ready to use for any content to get that "hook".
6. InsertLearning - a Chrome Extension that makes any website interactive.

Content Examples:

• Algebra 1 - Complete the Square
• Algebra 2 - Intro to Transformations
• Biology - Mitosis v. Meiosis
• Check out LOTS more on the Hyperdoc Handbook Site

A "functional" Transformation

One of the biggest topics in a secondary mathematics class is function transformations (horizontal/vertical shift, compressions, stretches, etc.) Usually students will memorize the rules rather than understand the reasoning. I needed a way to enhance the lesson to allow students room to investigate these concepts and draw their own connections and conclusions - time to spice things up with technology!

Topic: Intro to Transformations

Essential Question: How can I transform ANY function?

Materials:

• Desmos Activity - Make a copy if you'd like to make some adjustments! (link)
• Student Notes - Google Slides (link) Click [Use Template] to make your own copy in your Google Drive.
• 1 device per 2 students - I recommend a Chromebook/computer.
• Recommendation: a Google Classroom to share the notes with each student.

Student Notes (Google Slides) Preview

Teacher Notes:

1. Assign the notes in Google Classroom where each student gets a copy.
2. Go to teacher.desmos.com. If you don't have one, create an account! I recommend with a Google Account to make the sign-in easier.
3. Open the Desmos Activity (Intro to Transformations) above. 
4. Click on the teal [Create Class Code]. This will need to be copied on Slide 1 in the Google Slides. This is how students will access the activity.

   

   Desmos Activity (Preview)

5. Students will partner up and bring their computer/Chromebook. Each student can login so that everyone answers the activity questions and can take screenshots to add to their Google Slides for their notes.
6. Encourage students to take their time exploring each transformation carefully. They may struggle understanding shifting left and right. I related it back to the distance formula - (x - 3) moves RIGHT 3, not LEFT. 
   
7. The Exit Ticket/Reflection piece is on Flipgrid. Create a TOPIC in Flipgrid where students can respond.
8. TIP: I created a topic for EACH UNIT in my class so that students could see how their questions and reasoning improved as the unit continued. 

My Reflection:

• Since the remaining of the year (and future mathematics courses) depended on this lesson, I felt strongly about student exploring it on their own. I tried teaching the rules one year and it only set them up for failure later. 
• I'm currently reading Shake Up Learning by the infamous Kasey Bell of Texas. In her first couple of chapters, she stresses the importance of integrating 21st century skills into as many lessons as possible - hence, shake things up! Students will be asked to analyze critically and shoving rules in their faces wasn't accomplishing this and was NOT preparing my students for their future careers. Careers where thinking on their own and creating their own connections would be a requirement.
• One of my biggest fears with "shaking up their learning" is that my students won't learn the concepts "correctly". What if they misunderstand and I don't catch it?! In my mind, technology allows me to assess formatively more often, therefore I can check understanding frequently. When my concern of "learning incorrectly" snuck in to my mind, I forced myself to remember Jo Boaler - making mistakes is a learning opportunity. Even more so, making mistakes will make their brains grow. 

A "Random" Post?

What is "random?" What does it mean mathematically to be "random"? Ask any person to pick a "random" number, and you'd most likely discover a pattern...wait, what?!

In Statistics and Probability units, students think they understand how random events work. They think they are sincerely selecting random numbers when asked. Funnily enough, this concept can be quite vague. In a previous post, I shared a lesson to help students understand the Law of Large Numbers.

In this one, I want to share a lesson where students understand when they pick a "random" number, it truly is NOT random. My goal with this lesson was for students to discover this definition by using inquiry. Students discover patterns and come to the conclusion that we don't actually pick "random" numbers - quite hilarious when you watch their eyes enlarge at the end of the activity.

Why I used technology:

• Students are engaged by using their own sense of intuition and curiosity. 
• Google Sheets allows them to quickly graph the the class information and find averages without the hassle of too many steps.
• Students log their observations on Google Slides, which they can reference later. 
• All links and materials students will need are in one location - Google Classroom on the Slides.
• Students choose which graph they want to use and explain their choice.
• Teacher can access all students Slides in Google Classroom and provide comments to students work without taking extra materials home.
• Students create their own examples instead of being handed them. To me, this allows the learning to be their responsibility.
• By sharing out definitions on a Padlet, students can share their thinking using various methods (text, picture, GIF, voice note, hand-drawn picture). 
• Students vote on their favorite definitions to develop the best one.
• Ultimately, this enabled the lesson to be student-centered!!

Topic: Simple Random Samples

Essential Question: How can I create a sample of items that is truly random?

Materials:

• Google Slides - Student Notes (1 per student)
• Google Sheet - Student Notes (1 per student)
• Google Doc - Federalist Paper word breakdown (1 for all classes)
• Padlet - create a padlet using the "Wall"style.
• Forensic Linguistic Article (link)
• Simple Random Sample Site (link)

Lesson Outline:

1. Warm Up: Students read a quick article about forensic linguistics using the link on Slide 1. They write what was surprising to them and how math was helpful in discovering the truth.
2. Activity: What is your Pseudonym? Students investigate the Federalist Papers to show how finding the average word length can determine authors. In this activity, students select their OWN 5 words they believe would best help them find the true average word length of the passage.
3. Students gather all the class data on the Google Sheet. Each student creates a graphical display of the data.
4. Now students use their calculator to select the 5 random words and calculate the average word length. Similarly, students gather all the class data and begin to compare the two rounds.
   
5. Expand: Students expand on their knowledge by investigating what a "simple random" sample is and create their own definitions to share on a class Padlet. 
   
6. Encourage students to vote on the definitions they believe are the best. Ask students to find pictures online as well or draw something they can take a picture of and post. 
7. IDEA: Have students grab a partner and submit one answer per team. If you want to keep their names, the definition that the class likes the most could win some prize?
8. Exit Ticket: As a closer, students fill out a Frayer Model with the formal definition, a picture, an example, and a non-example.

Take-Aways:

• When students developed their own definition, they could remember the concept at a deeper level throughout all the units following.
• Gather their examples (from Google Classroom) for the next day as a practice activity. Have the students organize the examples into 2 categories (Good/Bad) to see what they've learned. Discuss examples that students had difficulty classifying.
• Introduce other sampling methods as a follow-up activity. Students can compare and contrast the various methods and when to use each one.
• Students can find articles where simple random sampling has been used or why it was not used. 
• Check out the Mythbusters clip below where they check if yawning is actually contagious! Did Adam and Jaime use a Simple Random for their experiment?!

So how many licks...?

Everyone knows the famous question that's plagued children and adults for millenia...

How many licks does it take to get to the tootsie center of a Tootsie Pop?

One of my absolute favorite lessons in AP Statistics was the Tootsie Pop Lab. This lab emphasizes how to estimate the ACTUAL average amount of licks it takes to get to the center. Will this question ever be answered!? Non-Statisticians would tell you "nah", but the real Statisticians would use inference mathematics. In this realm, we can get pretty close to the actual answer by providing a range of possible answers that could be true - what Statisticians call a Confidence Interval.  

A big idea running through the veins of Statistics is the idea of an unknown parameter - the ACTUAL population value. Example questions we could answer that have these unknown parameters would be...

1. What proportion of the world is covered in water?
2. What is the average life expectancy in the United States?
3. What is the average number of books teens read?

Clearly they were into it!

Clearly, it would be a challenge to find the ACTUAL percentage of the world covered in water or the actual average life expectancy, but we can get reallllllly close by taking sample measurements and drawing conclusions from there - i.e. finding a statistic! 

In Statistics, it's about getting on the dart board - not the bull's eye!

This lab teaches students this very idea! We may never really know the answer, and THAT'S OKAY! Using inference to draw conclusions is what mathematics can yield - and even better, we can provide plausible answers! It's a rare glimpse into the power and insight that only mathematics can provide. 

Activity: Tootsie Pop Lab

Essential Question: 

How can I estimate the true average amount of licks it takes to get to the center of a Tootsie Pop?

Materials:

• Tootsie Pop Presentation - Google Slides (click Use Template for your own copy!)
• Tootsie Pop Analysis - Padlet (click Remake if you want to use it!)
• Student Notes - Google Sheet (click [Make a Copy] and now it's in your Drive!)
• Flipgrid - create 1 topic in Flipgrid where students can post their reflection asnwers

Possible Student Google Sheet

Provide students with the directions about licking their lollipop and remind them to lick in the SAME SPOT. You'd be surprised how into it your students get. I recommend putting them in partners, especially if you have a student who doesn't like lollipops.

As students begin to find their center, have each student enter in their number of licks on the Google Sheet. I highly recommend using Google Classroom to help you share information with students. You'll be sharing the Sheet, Flipgrid, and Padlet with them.

Teacher Presentation (other slides included)

PRO TIP: In Google Sheets, type "=" to get the formulas to work. For example, "=average(...)" and then students can select all the trials at once. It works similarly for the others.

Take-Aways:

• The "homework" for the evening is not practice problems at all! It's all about reflecting on what they discovered in class. Students recalled previous information to see how it fits in to their new content.
• Students LOVE candy (surprised about that?)
• Students enjoyed working with REAL DATA! It's not made-up numbers from a textbook or internet. It was personal. They were the data.
• Putting them in partners helped them find formulas in Google Sheets quickly and allowed them to process their ideas out loud.
• Why I used Padlet: helps students collaborate all at once in an easy way. Students can add voice notes, drawings, text, links, and photos for everyone in the class to see. This allows students to express their learning in a variety of ways that best fits them.
• Why I use Flipgrid: Students can post their reflection questions after the lessons and hopefully hear from other students. This way, students can respond to each others questions and really utilize peer feedback. This develops a community in the classroom, because we are all learning together.

Interested in the answer my students found? Check it out!

Before my district had Chromebooks for every student, I used Fathom to gather all of the data from my classes last year. Here is what we've discovered!

My students were really surprised that both classes had an average close to 330 licks.

Our Class Average amount of licks: 333.629 licks

My Class Data (2016)

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.

Chocolate-y stress in Math

After attending TCEA in Austin recently, I wanted to create a post about going digital in the math classroom, thanks to Amanda! In this post, I'd like to focus on the stress and anxiety that math can induce in students. Jo Boaler, an amazing educator who focuses on mindsets in mathematics, stresses (pun intended) that...

"When students get the idea they cannot do math, they often maintain a negative relationship with mathematics throughout the rest of their lives...the idea that math is a "gift" is responsible for much of the widespread math failure in the world."

My question is, so how can we encourage students to maintain a growth mindset in maths and how to create a class environment that is geared towards helping them achieve this? One way that I hope to answer this is share some lessons that had students arrive at different answers and that our final conclusion in class was THAT THIS IS OKAY! Jo wants students to see the creative and interpretive nature of mathematics, and I think an example lesson will help.

One of my favorite lesson styles is utilizing the 5E lesson model, where inquiry is at the heart of lesson. I think this is one of the best ways to encourage all students to interact with math and alleviates some of the apprehension students face. The lesson below will outline this style and materials are included at the very bottom if you're interested! You will need a Hershey Kiss for every student (why not help relieve stress in math by using a little chocolate???)

Title: Chocolate-y Odds

Essential Question: How can I determine the odds of a Hershey Kiss landing on it's side?

Audience: Statistics, Probability, Math Models and Applications, Geometry

Engage:

1. Break students into group of 2-3 people. 
2. Have students make predictions about what the odds are of a Hershey Kiss landing on it's side. Record their answers on a Mentimeter to gather the their thoughts quicker. 



Sample Mentimeter

3. Have students share out how they made their prediction. Consider how the Hershey Kiss is shaped and compare it to a coin.

Explore:

1. Provide students with this Google Sheet so that they can record their data. Use a "1" if the Kiss lands on it's side and a "0" if it's on it's bottom. This can be done with their group. 
2. Have students calculate the Cumulative Sum using Google Sheets. Example: if I want to add Cell B2 to B4, type "=sum(B2:B4)". If students don't want to type it, they can click and drag all the cells they want.

An Example of a final product!

1. Have students calculate the probabilities with each step by dividing the cumulative sum by the trial number. Example: "=C3/C1" in the cell. 
2. TIP: Google Sheets will start to PREDICT the formula if you keep using it over and over again! Simply click the "blue box" next to the cell and drag it all the way down! 

   

   The "blue box" is in the lower RIGHT corner

3. Finally, have the students create a LINE GRAPH of their probabilities. Steps are included on the Google Sheet if you have students finish the sums early.

Explanation:

1. Have students open the Google Slides. TIP: Share the Slides in Google Classroom by choosing "Students can Edit". 

   

   "Team Response" Slide

2. Students to add the slide "Team Response" to the presentation. 
3. Each partner/group will add their Probabilities Line Graph to the Google Slide and answer the questions with their team based on their graph.
4. If students are finished early, have them go to another group's slide and provide some feedback, like a Bitmoji. 
5. Click [Present] and go through each group's line graph. As you look through each one, have each team write down some observations on a Mentimeter. All students deserve a VOICE! We are in 2018, people!



Choose Open-Ended on Mentimeter!

6. Hopefully one of the observations between all the graphs is that the more times students tossed the Kiss, the line graph gets closer to a number!

CONGRATS, you've introduced the Law of Large Numbers/Kisses.

Elaborate:

1. Have students interpret what it means when a coin is "50/50" using the Law of Large Numbers.
2. Provide the example of the coin again. Ask, "If I keep getting tails over and over again, am I MORE LIKELY to get heads?" 
3. Introduce the idea of independence - each toss doesn't affect the next one. The odds are ALWAYS 50/50 with each toss.
4. Introduce that we are never "due" to get an outcome. This is called the Law of Averages and IS NOT TRUE.

Evaluate: 

Students find an object at home and toss it (not breakable, duh!) and create another line graph and share it on another Google Slides presentation. Open these up the next day and students can share their findings and how the Law of Large Numbers relates.

Extra Materials:

• Teaching AP Statistics or Science: Designing Chocolate (Google Slides) - Experimental Design
• TIP: If you're sharing in Google Classroom, choose "Students can Edit" so that everyone is on the same slides.
• This is PERFECT if you teach AP Statistics to help students understand how to set up a proper experimental design.
• Aren't 1:1 yet or have the devices you need? Here is a worksheet using the TI-84 Calculator instead!
• Teaching AP Statistics? Use this data again to have students create confidence intervals! Use this worksheet if you'd like as a resource! I haven't made this digital...yet ;)

Interested in making Math more digital? Check out my Session Notes from the TCEA conference in Austin, Texas.