Spiralling MFM2P: The First Day

What a busy week! For those who are new here or didn’t read my last couple of posts, I am spiralling Grade 10 Applied Math (MFM2P in Ontario) for the first time. (I am also teaching this course for the first time in my career.) I have a new classroom with vertical chalkboards on most of the walls, plus some portable vertical whiteboards. My school is one of four “early start” schools in my board, which means we start a week before Labour Day (and we get a week off in late October/early November). Here’s how my first day went down:

Planning

Originally I used Kyle Pearce’s spiralling guide to create my Spiralled Long Range Plans. Later in August, I was at a board workshop, and I met a history teacher in my board who told me that there was a school a few minutes away from mine that was doing, some cool new thing in their math department that I might be interested in, he thinks they called it… “spiralling”? I got the department head’s email and got in touch with him. The department head told me that they spiralled their courses by spending 4 days teaching, with the Friday being review and a quiz. Then on Monday they would start a new strand. I really liked the idea of having a routine that students can rely on so that if they are frustrated or struggling with one concept, they know that they can start fresh on Monday with a new topic. Hopefully the spiralling and constant review will help students remember the concepts for the exam as well. I updated my long range plans to fit this model a bit better (more or less).

My updated Long Range Plans are here. My plan is to continue to add links to resources for each topic throughout the semester. (Eventually I will probably fix up a nice version in Google Sheets, but I work best with a calendar-type LRP, so sticking with that for now.) Review days that seem random are days when I will be away.

The First Day

There were a lot of things I wanted to do on the first day and I knew I wouldn’t have time for all of them, so I made a list of everything I wanted to do and then chose the most important ones. Talking Points is a favourite of mine that I’ve used in all of my math classes for the past 2 years. I think it helps establish my classroom norms about growth mindset in math, but this year I decided to change things up a bit. I really hope I get a chance to squeeze in a short lesson about growth mindset later sometime this month. Here’s what I did instead:

I started the class with this warm up from Which One Doesn’t Belong:

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I had students discuss it at their tables, then I had each student vote on which number didn’t belong, and explain why. Most of the students agreed that 43 didn’t belong, either because it was prime or because it wasn’t a perfect square (they needed some help with the correct mathematical terminology). The students who didn’t choose 43 were feeling a bit uneasy. I then told the students to choose another number and tell me why it didn’t belong. I tried to enforce the idea that there is no wrong answer here. One student pointed out that the digits of 16, 25 and 43 all add up to 7, but 9 doesn’t, which I had never noticed before.

I then took attendance. A lot of teachers on Twitter suggested forgoing all of the typical “first day” syllabus stuff, but I felt that establishing my expectations and possible consequences was important to do from day 1. I found this article about classroom management from Cult of Pedagogy to be really helpful. One of the strongest points in the article was that for 90% of students, making your class engaging and interesting for them will help deter any classroom management challenges. But for those 10%, you need to have clear expectations and consequences so that students know exactly what is expected of them. As the article recommended, after going over the serious stuff, I told my students that all that being said, I plan to do a lot of fun things this year and I’m looking forward to getting to know them.

I then jumped into Jon Orr’s game of NIM:

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The students loved this. I challenged the class to play against me, and then we worked out some of the mechanics of the game. I talked about why I chose to play this game and how it relates to learning math.

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Then we did another activity I got from Jon Orr, Graphing your Subjects. We started with the first quadrant only graphing as a review, and went over success criteria for graphing.

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We ended up using paper and pencil for this activity instead.

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I’m hoping the math point will change by the end of the semester!

Finally, it seems like every teacher swears by Sara Van Der Werf’s name tents. I wasn’t sure if the kids would go for it in high school, but I figured I’d give it a try this semester, and if it doesn’t work, I won’t do it again. Although some students didn’t take them seriously, I found they were very effective for most. Some students told me things that I never would have known otherwise about how they felt about the lesson, questions they had that they didn’t want to ask out loud, things like that. I don’t know if I would do them in a Grade 12 class, but they were great for my 2Ps and my Grade 10 open computer engineering class.

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I was also planning to have students fill out a short survey and join our Google Classroom, but we ran out of time, so I decided to end there and do the rest tomorrow. Over the next few days, I continued to introduce a bit of syllabus-type information each day for the rest of the week (materials to bring to class, when to get extra help, etc).

Overall, I think day 1 went well, and I would definitely do all of the activities again. Many thanks as always to all of the teachers whose ideas and activities I used.

If you have any feedback for me, hit up the comments! Welcome back to school!

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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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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:

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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.

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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.

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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.

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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:

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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.

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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.

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

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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!

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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
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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!

Introducing Algorithms in ICS3U/ICS3C

Today marks the end of Computer Science Education Week, the initiative for teaching kids and teens how to code. Schools across Canada and the US, including many schools in my board, are doing the Hour of Code on code.org. Lucky for me, I get to teach kids to code every day! In my Grade 11 U/C computer science class, we have been working on writing code in JavaScript to solve problems and perform simple tasks. For example,

Write a guessing game program that prompts the user to guess a number between 1 and 10. The program should say “too high” or “too low” and prompt the user to guess again until they guess correctly.

Yesterday I introduced the idea of algorithms to solve problems. In order to write more complex programs, you need to have a strategy and come up with an algorithm: a series of steps needed to solve the problem.

We started the lesson with a bit of an unusual warm up. This idea was borrowed from a friend of mine from my computer science curriculum class in teacher’s college:

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I got a few funny looks from students when they walked in and saw the board! I walked around to help the students, and encouraged them to write detailed, specific steps. Instructions like “butter the bread” are too vague: they need to be concrete and detailed, such as, “Use the knife to spread butter on each side of both slices of bread”. One student protested, “But Miss, these instructions are already detailed! A little kid could do this!” I told her that I still don’t understand – she needs to be more specific.

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One student’s grilled cheese instructions.

Then we took up the warm up. It took us 5 or 6 steps just to open the fridge, take out all of the ingredients and close the fridge! In total we came up with 26 steps to make a grilled cheese sandwich.

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The last of the 26 steps it takes to make a grilled cheese sandwich.

 

Then I got to the heart of the lesson: why am I doing this?

“I’ll bet some of you are wondering: what does this have to do with computer science?” I got some nods. I then explained that computer science is not about programming any more than biology is about microscopes. Computer science is about using programming as a way of solving problems. This is what computer scientists do.

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When you tell a computer to do something, you have to be very specific and detailed. You can’t just say, “Draw a rectangle” – you have to be specific: where on the screen do you want to draw the rectangle? What is the width and the height of the rectangle?

We then jumped back to some code that we had worked with earlier in the course in order when we learned about functions. What were the steps in the algorithm used to make that program?

The next thing I had planned was to come up with an algorithm to solve a common programming problem: find the biggest value in an array or list of numbers:

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At the last second I decided to change the example to something that would be more familiar to them. The biggest number problem is a bit less intuitive and I knew that some of the students might not even know where to start. While I do believe in struggle in order to learn, this question was not the focus of the lesson, so I decided to start with a more familiar problem instead: find the average of an array of numbers. I liked this problem because finding the average is something everyone knew how to do. At this point it felt easier for most students to start coding and then take a step back and determine the steps that make up the algorithm. Eventually it will become easier for them to come up with an algorithm first, and then translate it into code. Here’s what we came up with:

(Please excuse my messy whiteboard writing.)

We then did a shortened version of this activity from code.org on real-life algorithms. We spent the rest of the lesson doing some of the Hour of Code activities on code.org to prepare for an upcoming field trip (details TBA at a later date!). Overall it was a good day: students were engaged and some of the usually more reluctant students were raising their hands and contributing to the discussion! Oh, and we may have made some paper airplanes, too. Happy Code Week 2017!

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If you want to see the full PDF of my lesson, shoot me an email in the “Contact me” section of my blog.

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!

Reviewing Angle Theorems in Grade 10

As a new teacher, two things that I don’t feel the most comfortable with in my teaching are:

  • teaching theorems by name. For example, opposite angle theorem (OAT), corresponding angles, etc. I know how they work, but I’m not very good at remembering the names.
  • review. I am still fairly new to teaching, and it’s been difficult figuring out exactly how much detail I need and how fast I should be going when I review material from previous years. I’m always a bit anxious and eager to start the new material (in this case, one of my favourite units, trigonometry.) Usually I end up doing too much or too little. This is something that will hopefully get easier for me with time.

On this particular day, I ended up doing both of those things.

Personally, I don’t think it’s even really necessary to teach all of the angle theorems along with all of their names. You really only need 1 or 2 rules. The rest can be derived from the first 2 to solve for any missing angle. But because my class was one of seven Grade 10 academic math classes running that semester, I thought it was important for consistency that they learn the names. We also have the same tests for all 7 classes on the same day, so I needed to make sure that if knowing theorems by name was on the test, students would know the names.

Based on what other teachers had done in the past, I began with a warm up review of some common terminology that students would be using later on in the trig unit:

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The students didn’t seem to have too much trouble with it, and we were able to take it up quickly.

We did a bit more practice with ratios and other odds and ends:

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(If none of this seems like it really goes together, please forgive me. New teacher syndrome.)

Then we got to the main part of the lesson: the angle theorems. Instead of going over each rule individually (all 9 of them! So many names!), I wanted to see if my students could derive the theorems themselves. So I put up this diagram and told my students that all 9 angle rules appear in the picture:

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We did the first one together:

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I then gave the students some time to work with a partner and told them to find all 8 other rules:

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When we took it up, my students got all 9 rules on their own! Some of them remembered the names of the rules, the others they told me which angles made the rule and I helped them with the names (I may or may not have had the names written on a sticky note on my laptop to remind me of what each theorem was called). I took a picture of the board and posted it on our Google Classroom as a summary of what we learned:

(Usually I have another column for who came up with the rule, but I cropped it out for the sake of my students’ anonymity online.)

The rest of the lesson we spent doing practice questions.

This was one of my favourite lessons of the year: student-focused and student-driven. Bring on the trig!

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:

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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:

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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:

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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!

A New Approach to Sine Law

It’s Monday evening of Labour Day, which for most students and teachers in Ontario means tomorrow is the start of a new school year. My school is one of four schools in my board that start a week early, so tomorrow is just another day for us.

I wanted to write about a lesson I’ve taught twice already introducing students to Sine Law. The first time I taught this lesson was as a student teacher, to a Grade 12 college prep math class (MAP4C). The second time I taught it was last year, to a Grade 10 academic math class (MPM2D). For context, we had just finished the primary trig ratios (SOH CAH TOA) and “solved problems involving real-life situations using the primary trig ratios” (also known as word problems).

The lesson went like this:

We did a quick warm up of “find the fake”, an idea I got from my host teacher back when I was a teacher candidate. The students have to figure out which triangle is possible, and which isn’t.

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I told students to think of it like shining a flashlight through a hole at each of the angles. At each angle, which light from the flashlight would create the biggest spot of light on the opposite wall? The students quickly realized that the biggest angle would allow the most light to get through – meaning the longest side must be opposite the largest angle. Triangle B is the fake!

We did a brief review of what angle of elevation and angle of depression mean (to help with some of the homework questions). Then we got to the main part of the lesson: developing Sine Law. (We took a minute to talk about how to label sides and their corresponding opposite angles.) Similarly to the lesson I taught when I was a student teacher, I had students work with a partner to complete this table:

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By now the students were used to looking for patterns in order to learn new concepts, and I think this format works really well with the Ontario Grade 10 curriculum, where almost everything is concrete enough to be able to learn through patterns.

Within a few minutes, my students figured it out:

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Here’s our complete table when we took it up as a class:

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The students were able to deduce the 2 forms of Sine Law (sides on top vs. angles on top), and talked about when to use which form. The rest of the lesson we spent going over a few examples, and doing practice questions.

Overall, the lesson went okay and my students understood the idea, but I felt like there was something bothering me about it. Later in the week, I was talking with another teacher in my department, and she told me how she taught Sine Law. The first thing she did was put up a non-right triangle on the board (one that can be solved using Sine Law):

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(Note: the above triangle is a random example found online – not necessarily the one my colleague used.)

And all she said was, “Solve it”. At first the students were stumped, but most of them quickly figured out that if they draw a perpendicular line from angle A to the opposite side, they had 2 right triangles, and could therefore use SOH CAH TOA:

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Once they had solved all of the angles and sides, she gave the students another non-right triangle, which they were able to solve confidently. She then asked them to look back on what they did and determine how they solved for each missing value in a more general way. They were able to deduce the formula for Sine Law, as proven through the primary trig ratios.

I realized that I liked my colleague’s idea better: while there is definitely value to students figuring out rules by finding patterns, a few examples that fit a pattern is not a proof. As a math student, I remember my university professors stressing that just because a rule works in a few cases, it doesn’t mean it will always work. My colleague’s technique was stronger because the students derived Sine Law using formulas that they already knew – a true proof. Because they derived it themselves, these students had a deeper understanding of why Sine Law works, and if they ever forget it, they know how to derive it again. This is how I plan to teach Sine Law next time around. It’s always better that students have a deeper understanding of why formulas work rather than simply memorizing them.