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:

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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?

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

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

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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:

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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:

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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:

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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:

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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:

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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
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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)
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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.”
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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:
    Not sure about this one yet. Maybe something with John Scammell’s free throws for the win.
  • 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?

Write me in the comments!


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:

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

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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:

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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:

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Then I gave them some more information:

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(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:

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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:

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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.

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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!

Grade 10 Math – Resources by Course Expectation

Last year in second semester I taught 1 section of Grade 10 academic math (MPM2D), and 2 sections of Grade 12 data management (MDM4U). Grade 10 math was one of my favourite courses to teach as a student teacher, although I had only taught it at the applied level (MFM2P). The curriculum is so relatable, and it’s concrete enough to constantly have fun 3-act math and other similar activities that I could draw from or make in order to make the math come alive. My class was super eager and into it, and there were many “aha” moments. As a teacher, it was exciting to see my students experience math in a new way.

In order to keep myself organized, I use Google Sheets to keep track of the online resources I use for future reference and to pass on to other teachers looking for ideas.  I made one spreadsheet for each course, and within that, I have one sheet for general resources and information, and one sheet for warm up activities, blogs, and other things like spiralling resources, pedagogy and assessment ideas. I then added a new sheet for each unit: linear systems, analytic geometry, quadratics I, II and III, and trigonometry. I was not spiralling the curriculum this time, but someday I’d like to. Although I don’t like to see math as a bunch of isolated topics, this format helps keep me organized to make sure I am meeting all of the curriculum expectations.

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Below are my Grade 10 math resources, organized by unit and topic. Some of the activities I did for one unit I ended up coming back to later in order to teach a different concept. In a spiralled course, these could be blended together over a few days. My resources are geared towards Grade 10 academic (MPM2D) but many can also be done in an applied class and adjusted to meet the expectations, many of which are similar or the same. All the credit goes to the original creators of each activity, lesson or idea. If I forgot to give anyone credit for something, my sincere apologies – email me or comment and I will add it in.

Note that this is not a comprehensive list. Some of the topics I didn’t use an online resource for or haven’t found one yet. I will update this post as I find new information.

Unit 1: Linear Systems

  • Review: slope and y-intercept. Lots of activities for this – here are a couple:
    -Jon Orr’s Reading Relationships
    -Mr. Hogg’s Fast Walker
  • Review: independent/dependent variables
  • Review: graphing linear equations – Jon Orr’s Crazy Taxi (4 representations)
  • Solving systems – substitution and elimination
    -Kyle Pearce pile ups
    -Alex Overwijk – solving systems with manipulatives
  • Solving systems – word problems
    -there are a few types. One is the distance/speed/time problems – I use Jon Orr’s Two Trains

Unit 2: Analytic Geometry

  • Distance formula: Jon Orr’s distance formula without the distance formula (love this one!)
    -review of Pythagorean Theorem – Andrew Stadel’s basketball travel 
    -consolidation: Desmos Zombie Apocalypse by Andrew Stadel
  •  Equation for a circle: no link, but developed my own lesson in a similar style to the “distance formula without the formula” lesson
  • midpoint: Dan Meyer’s best midpoint (similar lessons available for best ________, but I found that too many similar type lessons gets repetitive)
  • review of equation of a line: Kyle Pearce’s paper stacks 
  • equation of the median, altitude, perpendicular bisector
  • characteristics and properties of triangles
  • verifying characteristics and properties of quadrilaterals, circles, triangles, other shapes

Quadratics I: Factoring

  • expanding and simplifying expressions: box method (no link right now)
  • review of algebra tiles: warm up – #29 on Which One Doesn’t Belong
  • common factoring
  • factoring (simple): Jon Orr’s algebra tiles for factoring
  • difference of squares and perfect squares
  • factoring complex trinomials

Quadratics II: Zeros of Quadratics

  • difference between linear and non-linear: find the pattern/group similar things together (is there a Desmos activity for this?)
  • first and second differences: I used Tips4Math’s “Going Around the Curve” activities
  • solving quadratics by factoring: used a video from Legendary Shots with an estimated distance and equation (that factors perfectly) to solve – I will blog this eventually
  • quadratic formula: Dan Meyer’s will it hit the hoop?
  • read and non-real roots
  • problem solving (word problems)

Quadratics III: Transformations and Completing the Square

  • modelling quadratics: my own performance task – see my post here
  • transformations a, h and k: Laura Wheeler’s Desmos pattern finding
  • comparing y=x^2 and y=2^x: Mary Bourassa
  • graphing y =a(x-h)^2 + k
  • determine the equation of a parabola from the graph
  • transformations activity: Desmos Faceketball
  • transformations activity: Desmos Marbleslides (so much fun!)
  • vertex midvalue method
  • completing the square
    -mathcoachblog: the box method
    -completing the square visual representation
    -whenmathhappens: why completing the square works
  • max area given perimeter: Jon Orr perimeter jumble (tweaked to focus on the Grade 10 expectation)
    -a teacher at my school has his students do the max area given set perimeter by using toothpicks to make the biggest possible area – as a square, using only 3 sides, etc.
  • other max/min problems

Trigonometry

  • review – rates, ratios, proportions, angles, congruence/similarity
    -see my post on reviewing angles here
  • comparing congruent and similar triangles: Desmos card sort – congruent triangles
  • similar triangles word problems (shadows, reflection, etc.)
  • tangent – Jon Orr’s trig through slope (love this one)
  • primary trig ratios (SOH CAH TOA): see my tweet here
  • Sine Law: see my post here
  • Cosine Law (and Sine Law): I didn’t use this personally, but maybe you will be able to use some of these visuals
  • problem solving: Dan Meyer’s marine ramp makeover (one of my favourites)

 

That’s all I’ve got. If you know of any online resources that I missed, please comment and I’ll add it to the list! Happy Friday!