# Amazing Race Hay Bale Roll 3-Act Math

I just came back from the Teaching Math through Problem Solving 2 Conference which featured keynote presenters Jon Orr and Kyle Pearce (which was awesome!), and I am feeling newly inspired to do new and exciting things in my classroom in September, including spiralling a course for the first time (but more on that later).

This past semester I was happy to be teaching Grade 12 Data Management (MDM4U) for the second time. I focused as much as I could on using rich math tasks to help my students become critical thinkers. One of the best things I stumbled upon was Bob Lochel’s blog. While there are many resources out there for Grade 9 and 10 math concepts, it was harder for me to find good posts about higher-level discrete math, statistics and probability, and it was a data management teacher’s dream to find a stats teacher with blog posts for days!

This 3-act task comes from the Amazing Race problem in this post by Bob Lochel – scroll to the heading “Statistical tales of the improbable”. This lesson focuses on the hypergeometric distribution. My students had already finished a unit on probability, and in previous lessons learned about the binomial and geometric distributions. Students should have some prior knowledge of probability and a good foundation on counting techniques (combinations in particular).

Here’s how this problem would look if it was in the textbook:

How many of your students have tuned out?

Now here’s the 3-act version:

### ACT 1

First, I played this video up until about 1:14 and had students write down: what do you notice? What do you wonder?

Here’s what one of my students came up with:

I wrote down some of my class’s ideas on the board:

### ACT 2

Next, I revealed some information to confirm what my students saw and heard in the video, and posed a challenge for them to solve:

My students had seen questions like this before in the Probability unit, but if you’re spiralling, all you really need is an understanding of combinations and some basic probability to be able to solve the problem. For those who finished quickly, you can offer follow-up questions:

• If a team unrolls 6 bales, what is the probability of finding exactly 2 clues?
• If a team unrolls 6 bales, what’s the probability of them finding at least one clue?

I then had students make a probability distribution table and graph:

Students then answered some questions about the problem they had just solved. You can do this as a handout, but my school does not use handouts so I displayed it on the board:

I used variables r, n, and a because this is what the textbook uses. In other places I’ve seen variables n, N and M used instead.

We took this up together and I classified these characteristics as defining features of the Hypergeometric Distribution. We talked about the importance of the trials being dependent, which was different from other distributions we had seen (binomial, geometric, uniform). Next, I challenged students to use the variables we defined to come up with a general probability formula P(X=x) for the Hypergeometric Distribution.

### ACT 3

I didn’t have an Act 3 for this problem at the time, but if I were to do it again, I would print off 270 game cards which include 20 clues and 250 blanks and have a student draw 6 cards without replacement.

To wrap up my class, I had students check out this simulation:

Data Management is one of my favourite courses and I was a bit disappointed to find out I won’t be teaching it again next year, but I’m excited for new challenges. I will be teaching Grade 10 applied math (MFM2P) for the first time, and I will be spiralling! Will post more about it as I go!

Thanks again to Bob Lochel for the idea and problem for this post!

Oh and here’s my friend Julia (@MsFolino) and I at the conference with Jon Orr and Kyle Pearce!

EDIT: Meg Craig has corrected me on the rules of the Amazing Race and also suggested an idea for an extension to the problem, which I made into a Would You Rather math warm up:

# Introducing Probability Distributions with “Clear the Board” in MDM4U

Time is flying by in my Grade 12 Data Management class! I can’t believe we are already on our last unit, which is also my favourite unit: Probability Distributions.

I draw a lot of inspiration from other math teachers I’ve discovered on Twitter and through the MTBoS database. One of them is Sarah Carter, better known as Math Equals Love. If you’ve never heard of her, go check out her blog now! Sarah has lots of great lesson and classroom ideas and I borrow from her regularly (thanks Sarah!).

Last year was my first year of teaching and also my first time teaching Data Management (a discrete math, statistics, and probability course in Ontario, Canada). When I was first teaching the Probability unit, I used one of Sarah Carter’s lessons, Blocko! That day one of the teachers in my department and her student teacher decided to stop by my classroom to see what I was up to with the linking cubes. I had them join in the lesson and gave them some linking cubes to play the game with. After I explained the instructions, I was walking around the room to make sure everyone knew what to do, and my colleague asked me, “This is about Probability Distributions, right?” Actually it wasn’t. It was about theoretical and experimental probability. That comment got me thinking, and the more I thought about it, the more it made sense in the context of probability distributions. This semester, I am teaching Data Management again, and I decided to try out Blocko! for probability distributions! Here’s what my lesson looked like:

We started with this warm up question:

We had done questions like this before in the probability unit. Most students used a tree diagram. I encouraged students to think about how they could use counting techniques to answer the question (combinations and permutations).

This was the first day of a new unit, so I reminded students of some terminology and introduced the idea of a random variable: a variable that has a single value for each outcome of an experiment. I had students create a Probability Distribution Table and Probability Distribution Graph for the event of rolling one fair six-sided die.

After that I gave students a modified version of the Blocko game board with spaces for numbers 1 through 6. Each group of 2 or 3 got one game board and 12 linking cubes.

The modified Blocko! game board.

I called the game “Clear the Board” and told my students to place the 12 cubes wherever they wanted on the board, so long as each cube falls in only one section of the board. I explained that I would be rolling one die 12 times, and whichever group is left with the fewest cubes on the board at the end of the game would win. Now that they knew what the game was about, some groups quickly decided to rearrange the cubes. I instructed each group to take a picture of their original game board so that if they won, we would know what the winning game board looked like.

After playing a couple of rounds, I chose one of the winning game boards and drew it on the board at the front of the room (I purposely chose a game board where the cubes were distributed fairly evenly).

Next I introduced the full version: 2 dice, 12 rolls, new board for the sums of the rolls:

This game ran the same way as Sarah Carter’s. Experimental probability is unpredictable. In the first round I didn’t roll any 7s! After a few rounds of playing I chose a winning board and drew it at the front of the room underneath the drawing of the winning board for one die. I then asked students to discuss with a partner: what are some of the differences between the winning boards in the one-die and two-die games? They quickly realized that, as they remembered from the previous unit, when you roll one die all outcomes are equally likely, but when you roll two dice, the sums are not all equally likely.

The winning game boards, overlaid with the probabilities of rolling each value or sum.

I then had students create a probability distribution table and graph for the two-dice game. We compared the probability distribution graphs for one die and two dice: in the first graph all outcomes had the same probability, while in the second they didn’t. I introduced the terminology of a Uniform Distribution and Non-Uniform Distribution to describe the 2 different patterns.

We then did a similar example to the warm up where students practiced creating a probability distribution table and graph.

I like to end my lessons with a summary of the learning goals we covered. I find that announcing the learning goals at the beginning ruins the fun!

Questions? Feedback? Hit up the comments!

# MDM4U: Online Resources by Unit, and Plans for Probability

Happy long weekend!

So far I’ve really been enjoying my 2 classes of Grade 12 Data Management (MDM4U in Ontario). We’ve done several fun activities that involved critical thinking and rich learning. One of the goals I’ve had this year is to really work on making sure my lessons are not only engaging because they’re fun, but also provide good learning opportunities. I’d rather have a less “fun” lesson that involves a lot of rich learning than a super engaging lesson that has very little learning and critical thinking. Some of the lesson activities I’ve done so far:

Combinations and Permutations:

• for nPr and factorial problems (order matters): Dan Meyer’s Door Lock
• for combinations “some of” or “up to” problems: my take on Robert Kaplinsky’s Coke Freestyle lesson

Noticings and wonderings from the Coke Freestyle video.

One-variable Statistics:

• for measures of central tendency: Crazy Math Lady’s mean, median, mode
• for sampling techniques: Mr. Waddall’s take on the Jelly Blubbers Colony experiment
•  for measurement bias: Bob Lochel’s opener on the effect of a leading question.
I did this as a Google Form: I assigned half the class a Google Form with one of the questions to fill out, and the other half of the class the other question (I split the class in half by their student numbers). I did the demo in real-time, which was terrifying, but it more or less worked! (I showed my class Bob Lochel’s data as well.)
• for standard deviation: John Scammell’s celebrity guessing game. Here is my celebrity slide deck (as of March 9, 2018)

Celebrity guessing game!

Two-variable Statistics:

• for linear correlation: Bob Lochel’s Friendship Compatibility Test.
I loved this one. I wrote a Python program to match students up randomly (I try to get my students to get to know each other outside of their chosen table groups). This lesson was scaffolded really well and it went smoothly.
• for least-squares regression: I roughly based my lesson on Fawn Nguyen’s Vroom Vroom and Jon Orr’s take, Vroom. I modified it to teach Least Squares Regression to find a good line of best fit for the linear data.
Comment from one of my students about this lesson: “Miss, class was so fun today! I mean, it wasn’t that fun because we had to do math, but it was still kinda fun.”

Collecting data from the pullback cars in “Vroom Vroom”.

As usual, all the credit goes to the incredible teachers who take the time to share their resources on the internet FOR FREE! Thanks to all of you, me and my students both win.

Now we have reached what I consider to be “my least favourite unit”: Probability.
But why?! Probability is so fun! I agree. Probability is fun. We get to play with dice and cards and spinners. When I taught the unit last year, students had a lot of fun. My problem with the Probability unit is that, although students were engaged, I didn’t feel like the critical thinking aspect was very strong. Even if we have to skip out on some of the fun, this year I am determined to make sure that a lot of deep thinking happens and that my students remember the learning and not just the fun.

Here is my tentative plan for the unit. This is a work in progress and I am still looking for ideas. If you have interesting inquiry-based lesson ideas, please comment or message me and I will happily add them and give you credit! Here goes:

Probability:

• DAY 1: Intro to Probability: I’m actually going to be away that day, so my plan is to leave students with a handout of probability definitions, including introducing the idea of “odds”, and some practice questions. (Boring, I know, but I won’t be there so I don’t want to leave anything too complicated.)
• DAY 2: consolidate intro to probability and odds:
Warm up: jumbled note review of the definitions from the previous day.
Lesson: play Bob Lochel’s Jolly Ranchers game. Tie in connections to probability, sample space, favourable outcomes, and odds.
• DAY 3: review of permutations and combinations:
Warm up: ticket (formative quiz) on previous two days’ material.
Lesson: we will review permutations and combinations as prior knowledge for probability with counting techniques, coming up tomorrow. I’m still not sure what I’m going to do for this lesson yet, so I’m open to ideas. I’d like to do something involving a lot of problem solving on whiteboards.
• DAY 4: probability with counting techniques:
Warm up: one of these from Would You Rather Math.
Lesson: going to tweak Dan Meyer’s Starburst 3-act math to be about counting techniques. Maybe change the problem around a few times to address various concepts of the permutations/combinations unit.
• DAY 5: independent and dependent events:
Warm up:
this.
Lesson: play Skunk Redux. Key takeaway: no matter how many times you roll the dice and do not roll a 1, the probability of rolling a 1 on the next roll is still 11/36.
• DAY 6: conditional probability:
Warm up:
not sure yet. Maybe a formative quiz.
Lesson: Bob Lochel’s Egg Roulette lesson. Focus on tie-ins to conditional probability. Might come back to this for the hypergeometric distribution in the next unit.
• DAY 7: mutually exclusive and non-mutually exclusive events:
• DAY 8: wrap-ups:
Warm up: the famous Monty Hall problem.
Lesson: tie up loose ends, start test review.
• DAY 9: test review
• DAY 10: unit test

That’s pretty much all I have for now. After this is my favourite unit, Probability Distributions. I love this unit because there are many interesting problems that involve higher-level thinking and are also really fun! I will hopefully find time to post about that later on. In the meantime, I’m turning the floor over to you:

What lesson ideas and activities do you have for Probability?
What would you add to my unit plan?
What would changes would you make to my unit plan?

I’m on Twitter! Come say hi! #mtbos

# Coke Freestyle Soda Combos: Secondary Edition

Hello! We are just wrapping up week 3 of the second semester. This semester comes with all kinds of exciting things for me. For the first time in my teaching career, I have the amazing opportunity of getting to teach a course I’ve taught before! The course I’m teaching is Grade 12 Data Management (MDM4U in Ontario). Having taught the course before makes planning a lot easier! Here’s what I’ve been doing to plan each lesson:

1. Look at my Long-Range Plans and the curriculum expectations
2. Find my lesson on the same concept from last year
3. Search the #mtbos search engine and use my own ideas to make my lesson better
4. Keep existing parts of my old lesson that I liked, and use what I’ve learned in the past year about differentiation for students with IEPs and English Language Learners to make my lesson easier for my students to understand

Recently, my class had started the unit on Combinations: choosing r items from a group of n items without replacement, where the order doesn’t matter (“n choose r”). The next topic was Combinations “some of” or “up to” questions: how many ways there are to choose at least 1 item from a group (up to n items)? I searched the #mtbos search engine and found Robert Kaplinsky‘s (of Open Middle Fame) Soda Combos Coke Freestyle lesson. This fit perfectly with what I wanted to do. I modified the lesson to fit the “some of” questions in the curriculum. I sent Robert Kaplinsky this tweet:

He replied back asking me to share my lesson, so here it is!

I started off the lesson by giving a mini whiteboard and marker to each pair or small group of students, and told them to divide their whiteboard into two sections: I notice and I wonder. I played the Coke Freestyle video a couple of times and had students write down their observations. Then I told them to switch the marker to their partner:

Some of my students’ notice and wonder about the video.

Then I asked students to estimate: how many different drinks are possible, if you can have as many different flavours in your cup as you want? You must have at least one flavour. For simplicity, I told them to disregard the second step and only use the flavours from the original panel (not the 7 variations per flavour that the machine offers). I did this to make the numbers a bit less overwhelming, although the problem could have worked with all of the sub-flavours as well. I encouraged students to use “too high, too low, best guess” to help them estimate.

I asked each group to share either their too high, too low or best guess. Most of the groups were hesitant to share their best guess (we’re working on that!), but they shared some great “too high” answers:

(Note: “Raspberry-Lime” is an entirely different flavour and is not “Raspberry” and “Lime” mixed together.)

At first I let the students struggle a bit to figure out a strategy and gave them a couple of minutes to talk about it with their groups. Then I revealed the hint. As mathematicians, we are constantly looking for patterns. We did the first line in the table together, and then I let them do the rest in their groups:

Some student work.

Most students were able to figure out the pattern: the number of choices for up to n flavours is 2^n – 1. This would make the total number of possible combinations for the coke machine 2^14 – 1 = 16 383 different drinks!

Then we talked about what the formula meant:

We did a simple practice question, and then moved on to the formula for “some of” problems with some identical elements. Like in the previous example, I encouraged students to make a table of values, and gradually add items to their pool of objects to choose from.

For the green shirt, you have two choices: donate, or don’t donate it.
For the blue shirt, there are three choices: donate 2 shirts, donate 1 shirt, or donate none.
For the red shirt, there are four choices: donate 3, 2, 1 or no shirts.
We have to donate at least one shirt, so we subtract 1 from the total to eliminate the option of not donating any shirts at all. So the formula in general becomes (p+1)(q+1)(r+1) – 1.

After that, we did a similar practice question and then I gave students some time to get started on their homework and ask questions about the homework from yesterday. Thanks Robert Kaplinsky for the original problem, and for asking me to share my lesson!

Questions? Feedback? Tried this in your own class and want to let me know how it went? Hit me up in the comments!

# End-of-Year Check-in on My Goals for this Year

The first semester is winding down, and 2 of my 3 classes have started final projects. After the break we only have a couple weeks to go before we start exams. I felt like this was a good time to reflect on some of the goals I had for this semester and share how they worked out.

### Processing

Recap: Processing is a platform that combines coding with art to create animated sketches. My main reasons for using it in my computer science classes were that, from an educator’s perspective, it’s very gratifying for students who are learning to see the results of their code visually on the screen, plus it gets students to think more creatively in a subject that is usually viewed as very mechanical and rigid when you don’t know a lot about it.

How it turned out: I think overall, Processing was a success. As I had hoped, it was a good learning tool. I felt like students could really see the usefulness of variables, conditionals and loops. In terms of creative thinking, I was amazed at some of the programs my students created. I also got to see them admiring each other’s work in class, both through their blogs and by looking at what other people around the room were working on. I thought that was pretty cool – definitely not something that would happen if we were using another platform that didn’t have a visual aspect.

Some of the challenges I had with Processing:

1. The school computers did not have Processing installed, and I was told it would take about 6 months for it to get approved (if it got approved). I thought I could work around this by having each student bring in a USB and run Processing off a USB. This worked for my laptop, but it didn’t work on the school computers. In the end I managed to solve it by using P5.js – Processing together with JavaScript. No extra installations needed. It was not as flexible or as good for debugging as Processing in Python, but in the end it was still effective.
2. Processing does not have an easy way to do keyboard input. After some research online, I ended up abandoning Processing for a few weeks to teach keyboard input.

Some student work from one of the practice exercises in Processing.

### Assessment through Conferencing

Recap: in addition to or in place of another summative assessment such as a test, have a conversation with each student about what they learned. They would answer questions about the project they worked on, as well as overall reflections on the unit as a whole (what they found interesting, what they found challenging, etc).

How it turned out: as planned, I did a conference together with a project for the first summative assessment in each of my computer science courses. I quickly found that this strategy was not the most effective – or at least, it didn’t work very well for my classes. One of the biggest problems I found was that it takes foreverSince I was teaching lessons for at least part of every class, with both of my classes at capacity, it took me weeks to get to every student, Student absences, my own absences and late assignments all made the process take even longer. For the next assignment, I adjusted my strategy: instead of a conference, this time each student would write a short reflection on their blog about the project that would answer the same questions (what did you find interesting? What did you find challenging?). I found that the post-unit reflection worked better as a written activity. Much less time-consuming, although I missed the one-on-one time I got with each student in the first assignment conference.

What I found worked the best was formative conferencing. In my Grade 9 class, students first did a formative activity on whatever skill we were working on. They worked on that for a couple days, then I would post the summative. While the students worked on the summative in class, I went around the room and conferenced with each one about their formative work. I gave them feedback on what they did well and what they needed to improve. This saved me a lot of marking at home, and the students found the in-person feedback helpful.

The next time I teach data management (MDM4U), I am planning to conference with each student about their ideas for the final project. Last year, I found that some students constantly had to revise their ideas, and I found it worked best to come up with a plan as a conversation rather than putting it into words in a structure that didn’t really fit.

Overall, I’m glad that conferencing was something I tried, even though it didn’t turn out exactly as I thought it would.

### #OntarioClassMatch

Recap: connect with another class somewhere in Ontario (or around the world) and collaborate in some way. I learned about it from Heather’s post here – check it out!

How it turned out: I reached out on Twitter to anyone I could find who had tweeted with the hashtags #ICS2O, #ICS3U or #ICS3C recently (the courses that I am currently teaching). I managed to connect with a few teachers who shared some great resources with me, although unfortunately none of them were teaching computer science this semester. Then, one of my friends from teachers college told me he would be teaching Grade 10 computer science this semester, so we decided to connect our classes through class blogs. In the end we didn’t communicate with each other’s classes as much as I would have liked. We did manage to connect our classes once. Both of us are new teachers teaching three different courses for the first time, and we were both too busy to really invest in it. I think it was still a good experience for our students to see that there are other classes out there, and that we can learn from each other. Next time, I would aim for one blog a month for the semester – a total of five posts. I think having a concrete goal would help keep the initiative on track, despite how busy life is otherwise.

In the future, I’m excited about connecting my math classes with other classes around Ontario in some way. Computer science is an elective and math is for the most part a mandatory course, so I think it would be easier to find another class to connect with – or even multiple classes. For data management, I think it would be cool to have my students and another class fill out each other’s surveys and analyze the data. In any class I teach, we could make up and share questions with another class. Lots of different options. Next semester I am hoping to try #OntarioClassMatch with another class.

### High School Genius Hour

Recap: students get to work on a passion project throughout the semester that is connected to the course that I am teaching – in this case, BTT1O, Grade 9 Business Technology Today.

How it turned out: this one didn’t end up happening at all, for a couple of reasons:

• no time – we were a bit pressed for time to finish the curriculum and didn’t have time for any extra projects
• the business department at my school does a similar project in the Grade 10 intro to business course, and in the interest of students continuing to take business courses, I didn’t want the courses to be too similar

### Collaborating with Primary Students

Recap: do something together with elementary students. In my case, I wanted my students to do the Hour of Code with an elementary class.

How it turned out: this actually happened, and I think it was successful! I am planning to write another post about how it went in detail later on (but no promises). If I do, I will link it here.

Happy holidays!

# Binomial Distribution: A Lesson I Found on the Internet

This will be a fairly short post (for me anyway), but this was something I felt needed to be shared. It was the end of the year and I was exhausted. My data management students were in the final stages of the final project for the course, a large research-based project involving statistical analysis, which meant a lot of marking for me and I was crunched for time. Out of desperation, one night I Googled “binomial distribution lesson plan” and this is what I found:

https://blog.mathteachersresource.com/?tag=fun-way-to-teach-the-binomial-probability-distribution

I was a bit surprised because I had never seen this website before, and I follow a lot of math teacher blogs (and I mean a lot). I used the lesson almost exactly as it was shown in the post above. We started with this warm up question from Would You Rather Math:

After taking it up, I announced that we would be having a formative pop quiz to test my students’ prior knowledge about a topic we hadn’t talked about before – kind of like a diagnostic:

For the purposes of the Binomial Distribution, it was important that the students answer every question, even if they have no idea what the answer was. The students were very confused and they wanted to know what the Battle of Gettysburg had to do with math. I encouraged them to focus on their quizzes and try their best.

When we took up the questions, it was hilariously clear that my students knew very little about the Battle of Gettysburg. The students made a frequency table and frequnecy diagram of the number of correct answers, out of a possible total of 20:

As you can see, no one in my class got a score above 6 out of 20. (One of my students in my other class got 8/20 and his friends loudly accused him of cheating!)

I then explained how the pop quiz modelled the Binomial Distribution – I won’t repeat the explanation because the article where I found the lesson explains it very clearly. I loved this lesson because it was easy for the students to follow and understand, there was very little set-up required, and the students were curious and engaged. So thank you to Math Teacher’s Resource for this fantastic lesson – all the credit is yours. If you are on Twitter, please feel free to get in touch with me.  Happy Friday!