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

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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!

NTIP Learning Tour: Modern Learning and Ed Tech

Recently, I attended a PD session run by my board for new teachers called the NTIP Learning Tour. During the workshop I got to observe two teachers team-teaching a class and learn more about modern learning and education technology.

One of our board’s initiatives this year is modern learning. The school I went to visit had a modern learning classroom, with whiteboard surfaces on large shared desks and a wall between the two classrooms that could be opened for team teaching. One thing I took away from the experience was that you don’t need to have any specific furniture or materials in order to make learning relevant and skills-driven (as opposed to content-driven).

The lesson I got to observe was a Grade 11 university prep English class (ENG3U). Before the workshop I was a bit skeptical about how I would apply what I learned from an English class to my teaching as a math and computer science teacher, but the ideas were very applicable and transferable to other subjects. As teachers, it’s rare that we get to sit and observe a colleague teaching, and I think this is something that needs to be done more often because we can really learn a lot from each other. In this post I’m going to share some of the things I learned.

Know your students

The session started out with me and the other new teachers learning about the background of the class we were going to see: who the students are, what their strengths and weaknesses are, what their needs are, the prior knowledge they had coming into the lesson. I think most of us do this without realizing it, but when we plan lessons, we always have to think about whether this lesson idea would work for this particular group of students. Jon Orr once mentioned somewhere that he’s never taught the same course the same way twice – how can you, when you don’t have the same students?

Whiteboard surfaces for better problem solving

I’ve said this before and many others have too, but having students work on erasable surfaces – whether it’s jotting down ideas in an English class, problem solving, or writing code – opens up student thinking in a way that writing on paper doesn’t. Students are much more willing to try new things and write down whatever ideas they have when it can easily be erased, modified, changed in one swipe. In my computer science class we don’t have any whiteboard surfaces or markers, so my students do rough problem solving on scrap paper, and I’m finding that the students don’t get the same value or deep thinking out of problem solving as I’ve seen in my math classes where we had whiteboards. (In Ms. Folino’s tweet here we had a chat about DIY whiteboards.)

Bad Idea Factory

The teachers I observed had a policy that no idea is a bad idea. They encouraged students to write down whatever they were thinking or feeling, even if they weren’t sure if it was “right”. As I observed the class, I found that the students were very open about sharing their ideas – even in front of a group twice the size of a regular class! The kind of openness that this class had is what I aspire to do in my math and computer science classes: whenever I can, I’ll do a warm up from Would You Rather Math, Which One Doesn’t Belong, or another source where there’s no single correct answer: it gets students to think about things in different ways and to think critically and creatively.

Student reflections on feedback to improve

One of the ideas we talked about after the lesson observation was how to get students to use feedback to inform their learning. I sometimes have students fill out a reflection about an assignment, test or ticket out of class answering the questions:

  • What did I do well?
  • What do I need to improve?
  • What did I learn?

In the PD session, one of the teachers leading suggested incorporating the reflection into the assessment: after the assessment has been returned with feedback, students use teacher feedback to answer these questions and hand these in as part of their mark for the assignment. I love this idea because it motivates students to take reflection seriously and really think about where they can improve and what they learned. This is kind of similar to Jon Orr’s take on tests. I used this strategy in my data management classes last year and my students loved it. One told me that if it weren’t for the mark upgrading, he never would have understood a certain concept. Sometimes the learning happens after the test, and there’s nothing wrong with that.

Since attending the workshop, I had a chance to try the post-assignment reflection with my Grade 10 and 11 computer science classes. I was impressed with their insightful reflections! Here are some examples of the comments they made (each from a different student):

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Ed Tech Stuff

At the end of the Learning Tour, the teachers showed us some tech resources we can use in our classes. Here are some of the ones I haven’t heard of and am looking forward to trying:

Screencastify: an extension to Chrome that allows you to record yourself speaking along with a video of your screen. Great for anything from giving video feedback on student work to flipping the classroom to anything else.

Hyperdocsfree resources for all subjects that open as a Google Drive folder that you can copy to your own drive. They only thing they ask is that when you make a copy, each resource says “Copyright _______” in the footer. When you use it, add “adapted by ________” to the footer.

4C’s for Ted-Talks: works with any Ted-Talk as a way for students to analyze the message. You can split up students into 4 different groups, one for each “C” to focus on in this Ted-Talk, and then share thoughts as a class.

DocAppender: an add-on to Google Forms that allows you to grab all the information from all the Google Forms assessments that you’ve done and create one Google Doc per student with all of the information about that student stored in the same place.

Have feedback for me? Post it 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.
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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.


Questions? Comments? Ideas? Feedback is always welcome. Hit up the comments!

Happy holidays!

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!

Modelling Real-Life Quadratics in Grade 10 Math

Happy first day of October! This post is about an assignment (or “performance task”) I did in my Grade 10 math class last year. It could be done as an in-class activity without evaluation as well. It is based on the following expectation from the Ontario Grade 10 math curriculum:

Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology or from secondary sources; graph the data and draw a curve of best fit, if appropriate, with or without the use of technology.

Like with most lessons, I started off by looking at what other teachers who had taught the course had done before. In the past, students used imprints of their molars, which fit a quadratic model quite well. They used TI-84 graphing calculators to model the set of the points created by the imprints, which were done on graph paper.

I wanted the task to be a bit more open-ended. I am not a huge fan of TI-84s, so I decided to use Desmos instead. I decided to combine the quadratic modelling with a task my host teacher told me about when I was a student teacher, where students used iPads to do a parabola scavenger hunt around their school and online. Technology-wise, I was teaching at a 1:1 school, so each student had a laptop they could use for the task.

Onto the assignment: students were asked to find or take a picture of a parabola and upload the picture to Desmos:

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I demonstrated how to enlarge or shrink a photo on the screen, and how to move it so that the parabola is symmetric about the y-axis. The next instruction was to estimate 10 points on the parabola as accurately as possible.

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Then comes the cool part: Desmos has its own quadratic regression model.
Yes, it really is so cool. With this one line of “code”, you can easily do your own quadratic regression on any set of points:

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I had students fill out a table to find similarities and differences between their model and two friends’, instructing them to use parabola vocabulary that we had talked about in class (opens up/down, vertical stretch/compression, etc).

Here are some examples of what my students came up with:

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Overall, it was a good day: it was a light, fun activity for the students the day after a test, and it was easy and fun to mark.

If I were to make any changes the next time around, I would have the rubric reflected as a mark out of 10 for each achievement category, where each level is converted to a percentage. I would also try to match up the rubric language with the language that the math curriculum uses for assessment to make it more student-friendly (page 20-21 here).

The full assignment is available here: QuadraticsIIIPerformanceTask

(If you don’t feel comfortable downloading from online, shoot me a message in the “contact me” section and I’ll send it to you by email.)

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.