Are you game?

Spiralling in MDM 4U continues and we've reached the end of the second spiral. With a great focus on probability and counting techniques, students have had the opportunity to experience and demonstrate in these two spirals knowledge of permutations, combinations, factorials, binomial and geometric distributions, probability and odds.

As a summative assessment, students were asked to create an unfair game. They could use any materials available in the classroom or provide supplies from home as needed. In addition to creating the game, students were also asked to analyze the game - determine the probability of winning, show that their game was unfair, along with a few other questions.

Most students chose to revolve their games around dice and cards. These were manipulatives that we had used repeatedly in class and they were comfortable determining probability with. 

Some examples included:

Game #1: 
A player pays 3 tickets for the dealer to roll a 20-sided die on a flat table.

- If the die lands on an even number (red) excluding 20, the player loses the 3 tickets they bet with
- If the die lands on an odd number (black) excluding 1, the player loses 1 ticket they bet with and 2 tickets are given back to them
- If the die lands on either a 1 or 20 (green), the player wins 10 tickets, but paid 3 to play so overall gains 7 tickets

Game #2:

The name of the game is Cross the River, this is because the pathway takes the students across a beautiful river. It costs the student 2 tickets each time they play this game. It guarantees that the student wins every time! He or she will start on the start sign and will start off by rolling the two dice once. If he/she rolls an odd sum then he/she will move up two spaces, but if he/she rolls an even sum then he/she will move up one space. They will get four rolls in total. This game guarantees that they win because if they roll four even sums, which is the minimum, they will at least get to the fourth space, and this will give them one ticket back. Technically they are winning, but in reality they are losing one ticket in the sense that they pay 2 tickets to play, but only get 1 ticket back.

Game #3: 

The name of my game is “The Role Of Fortune”. I choose this name because I was hoped that the familiarity of the name, derived from “The Wheel Of Fortune” would grab the students attention and draw them towards my game. My game is a very simple game which will captivate and interest younger students because of its ease to play and enjoyment from playing. My game costs 3 tickets and involves rolling 2 die. The rules are simple, your objective is to roll a sum of 2, 3, 4, 7, 10, 11, or 12. Any other role will result in the loss of your 3 tickets. A sum of 2 or 12 results in the grand prize of 3 tickets. A roll of 3 or 11 results in a prize of 2 tickets and finally a roll of 4 or 10 results in the prize of 1 ticket.

We then had the opportunity to host the grade 8 math class for a Games Day. Students used the opportunity to collect experimental data for their games. Following the Games Day, students were asked to analyze their experimental data and compare it with their theoretical calculations.

When students began planning their game, they focused primarily on making it unfair - as stated in the expectations of the project. After the Games Day, students were then contemplating how to make their games more "interesting" and more fair. As part of their final reflection about the project, students commented on improving their game by making it straight forward (not too many steps) but also fun and not too obvious.

Finally, students were asked to create a sign for their game so that players knew the rules of the game, the cost of the game, and the probability of winning the game. They struggled with how to state the probability of winning their game without giving away that it was unfair. It was another example of a reassuring theme in the course that we must be analytical with data to ensure we have the whole picture and do not simply believe all that we read. 

A sweet 3-Act Math Task

The other day we had the opportunity to host our Grade 8 students in the Senior School to showcase what Grade 9 math is all about. To give them a taste of linear systems, John Doma (@Domanator19) and I created a sweet 3-Act Math Task.

Act 1:
Act 2:
Act 3:

Though I have used several 3-Act Math tasks in my MPM1D class, this seems to have been one that created the most opportunities for different solutions. Though unintentional, the data that we provided in Act 2 allowed for multiple solutions. This lead to the majority of students being engaged in the activity and willing to contribute their ideas (maybe it was also because we gave them a small box of Smarties to use in their problem solving!).

Here are some of the ideas that students brought forward:

a) 2 cm = 215 Smarties. Therefore 1 cm = 107.5 and so 15 cm = 15(107.5) = 1612.5 Smarties
b) 5 cm = 485 Smarties. Therefore 15 cm = 3(485) = 1455 Smarties
c) 11 cm = 1147 Smarties. Therefore 15 cm = 11 cm + 2 cm + 2 cm = 1577 Smarties

My take away from this activity was making sure to truly think about what data to provided. Going forward, I will definitely be reconsidering the data I do provide in Act 2 taking into consideration multiple pathways to creating an answer.

Developing a classroom of inquiry

My MDM 4U course has been an adventure this year. My goal, as mentioned in previous posts, has been to spiral the course or at the very least deliver the content through activities. It's now been two months and I thought I would share some highlights thus far.

1. Students are more engaged and willing to participate.

• They absolutely love playing games - even if its as simple as "pick a number from 1 - 6 and once your number is rolled, sit down". They are starting to ask questions that help lead the lesson as opposed to being told what they need to know.

2. Students regularly contribute and offer answers, even if they are unsure if they are correct.

• Most activities require students to contribute their findings or results of an activity. I found that by having each begin the course by contributing in this way, I have created a space where students are more willing to contribute.
• Its okay to say "I don't know". It's even okay to offer an answer and when asked why you chose that number to say "I don't know" but I thought...

3. Students offer solutions without definitely knowing the correct formula.

• I have students who have prior knowledge and are stuck on having a formula. It's been great getting them away from the need for a formula and focusing more on explaining why something is happening. Students in my class are getting better at communicating their understanding.
• Students are making connections between activities and again are not so focused on just memorizing formulas.

So until the next post, here is to creating more activities to discover statistics.

Favourable outcomes

We have spent a lot of time rolling dice in my MDM 4U classes so far this year. One day we played 3 different short games all involving being in pairs and rolling a pair of dice.

Here are the details of the three games.

Game #1:Roll 2 dice and consider the sum of the dice.
Player A wins 1 point if the sum is EVEN. Player B wins 1 point if the sum is ODD.

Game #2: Roll 2 dice and consider the product of the dice.
Player A wins 1 point if the product is EVEN. Player B wins 1 point if the product is ODD.

Game #3: Roll 2 dice and consider the sum of the dice.
Player A wins 2 point if the sum is EVEN. Player B wins 1 point if the sum is greater than or equal to 7.

This lead to a discussion of which player would you choose to be in the game and why.

For Game #1, I had a justification for 3 different outcomes:
1. Choose Player B because the mode sum is 7 so there would be more odd outcomes.
2. Choose either one because Even and Odd are equally likely (but not sure why they are equally likely)
3. Choose Player A because Even outcome more frequent
Even # + Even # = Even #
Even # + Odd # = Odd #
Odd # + Odd # = Even #

It was great that students were willing to share their thought process and we had supporters and opponents of each theory. We then developed a chart to show all possible outcomes of the sums and saw that both Even and Odd outcomes are equally likely. Probability was introduced.

In Game #2, based on their results of playing the game, each student would choose to be player A. Similar to game #1, by examining a chart of the possible products, it was clear that Even outcomes are more likely.

We started to discuss the idea of fair games and mutually exclusive events.

Homework for the class was to determine how to chance Game #2 so that it was equally likely for Player A or Player B to win the game. Students came up with some alternate version of one of the following:

- Player A gets 1 point for an EVEN product and Player B gets 3 points for an ODD product (changing the point value of each player)
- Player A gets 1 point for a product of 1 and Player B gets 1 point for a product of 36. All other products result in no points for either player. (only selecting some of the products)

Further discussions in upcoming classes will focus on fair games.

Spiralling MDM 4U - Day 1

I've made it through my first day adventuring down the path of pseudo-spiraling with MDM 4U. And it was actually quite successful and fun! My first disclaimer is that what I am doing may not be true spiraling but I can definitely say that it is activity-based learning. The students have been warned that each class may look disorganized but there is a master plan! My biggest leap of faith was leading a lesson without having a nice handout for students to fill in as we went through the activity.

Today's lesson began with each student being given a dice and asked to roll it 30 times. They were asked to keep a tally of each value rolled in a chart. Students were then asked to chat about their outcomes where they discussed why a few of one number and more of another arose and the discussion led to each value should be represented equally.

We then tallied everyone's responses in one large table and discussed these findings. There was some excitement when the first two columns (# of 1s and # of 2s) were the same and then further intrigue when there were almost double the number of 3s. Again, the discussion led to why each value was not represented equally.

A comment was made about the size of our sample (287 total rolls) and if this was large enough. This is where we distinguished between a population and a sample and started to lead to the idea that the closer the sample represents the population, the better representation it is of the population.

The data was then graphed in a histogram (first individually by each student and then together with the large group). One question that arose was "why can't we draw a scatterplot?" I was really excited that this came from the group and I wasn't telling them right from the start that a histogram was the way to go because of ...

Again, we discussed what should the histogram look like in an ideal situation and why use a histogram over other graphs. The terms discrete random variable and continuous random variable were introduced. Outcomes and events were discussed.

Students were then given a second die that was different than 6-sided and asked how their results would differ if the same activity were repeated with the new die. The idea that the larger the number of sides on the die, the less uniform the histogram would be for 30 rolls. Furthermore, the larger the number of sides in the die, the more rolls would be required to see a uniform distribution.

We conlcuded the lesson by reading a current article and discussing the need to critically assess the numbers we read. In the future, I will give the article for homework before discussing it to ensure all students have enough time to fully read the article.

Key Terms Introduced in the Lesson:
Population; Sample; Discrete Random Variable; Continuous Random Variable; Event; Outcome

Plinko: A summer project

This upcoming year, I will be teaching MDM 4U for the first time in quite a while. My ultimate goal is to have students explore most of the concepts through activities before practicing key skills.

After perusing the textbook and doing a quick Internet search, I came up with the idea of building my own Plinko Board to help with some of the probability concepts. This became one of my summer projects.

What you will need (see below for a photo of the final product):

·            2’ by 4’ sheet of peg board

·            2’ by 4’ sheet of plywood

·            5/16^th dowels

o   We purchased 3 feet long dowels and cut them into 3’’ pieces. I did see you could have bought craft dowels that were precut and about the same size.

o   My board has a total of 300 dowels (216 (18 x 12) white dowels and 84 (7 x 12) black dowels). 

·            Wood glue

·            1/2’’ plywood cut 3.5’’ wide for the sides and bottom edges

·            2 hinges to secure the stand

·            stand (2x3’s)

·            hooks to secure the stand when in use

·            ribbon (to create the zig zag walls)

o   I sized out how much ribbon I needed and then sewed loops at each end. I then slipped one loop at one end over a peg in the first row and then looped it down along the sides and slipped the other loop on the last peg in the last row. This was done on both sides.

·            paint

Every summer, I spend a few weeks up in Northern Ontario visiting my parents and I am very fortunate to have a retired father with a creative mind and a workshop that he lets me invade over the summer for a project or two. One of the first days that I was home, I was watching “The Price is Right” with my dad after lunch and I said “Dad, I am going to build a Plinko Board this summer. You can do it using peg board and some dowels.” This now became a joint project – me for education purposes and my dad for interest sake.

Disclaimer: my dad did the building of the board and I did most of the aesthetics of the board. My dad is great at being given a description and he magically creates the idea with the materials he has lying around his workshop. So I know what materials were used but have little details to provide on how all the materials came together (especially with the stand – it seem to magically appear over night and I have no idea how it was created).

Some things I did note during the building process was that it is a great idea to paint the board and the pegs before it is all put together.  Also, you will need to create a barrier along the sides so that the chip doesn’t get stuck in the sides. My first thought was to use elastics but could not find elastics that were wide enough. My final decision was to use ribbon. See the list of materials for more details on how I put that together.

I decided to leave the bottom values blank so that I could adjust them as I saw fit. My idea is to begin the highest score in the middle of the board and working outwards. I would create small paper slips that I would place at the bottom and could replace if need be.

It was now play time and after some trial and error it was discovered a single poker chip was a bit too light and sometimes stopped along the drop. The weight of 2 poker chips seemed like a better tool to drop. I also found foam golf balls and they seem to work the best – they did not get stuck along the way. My current thought is to have a few different items to use to drop in my class.

My thought at the moment is to have the board in my class and use a few minutes each class for the first few weeks to collect data. I would have the students play and record their result on a common chart. They would keep track of what location they dropped from and where it landed. We would then use this data to drive the discussion in a future lesson.

Stay tuned for how this all unfolds in my class in the upcoming school year.

My year in review

As we approach the end of another school year, I take this opportunity to reflect on my teaching over the past 10 months. Though this was my 12th year of teaching, I realized that I had many firsts in the 2015-16 school year.

Primarily, this year I
1. Tweeted for the first time (I'm sure this isn't the correct terminology - I'm still learning this Twitter world)
2. Created my first blog post.
3. Attended my first EdCamp in Windsor (I found out about this on Twitter)
4. Used 2 Truths and 1 Lie thinking activity (again, a Twitter find)
5. Used VNPS
6. Participated in my first TwitterChat

Yes, I guess you could summarize this by saying I finally discovered Twitter this year (Though I created my account in the last school year, I became a more active user of Twitter this year - not just reading items but also posting and using what I read). I'm still amazed at Twitter's power to universally connect teachers to collaborate and easily share ideas.

Though I may not always walk away from browsing Twitter (yup, pretty sure all the Twitter pros are laughing at my incorrect terminology) with an activity that I can use in my classroom, I readily find statements or articles that make me think - my bookmarks have grown greatly this year. This thinking often leads to a discussion within my department. I know I have greatly appreciated these department discussions - they can sometimes be lengthy but passing ideas back and forth and having colleagues that challenge my ideas has been extremely valuable.

I'm not exactly sure what my specific teaching goals will be for next year but I know that Twitter will continue to be part of my daily routine! Looking forward to more learning and more discussions next year.