Equations of Circles

In grade 10 academic math we look at midpoint of a line segment, distance between two points and then equations of circles. But only equations of circles centered at the origin (sigh). This is a one day thing, that has been quite confusing for some students. I should note that these students have worked with linear equations a lot, but not much else, so these are very different looking equations. In past years, there has confusion despite my best efforts to connect the equation to the Pythagorean theorem/distance formula. I decided to add a little intro activity this time around:

They worked in groups of 4 and got a quick refresher on finding the length of a line segment. They also quickly figured out that the points collectively were leading to a circle. We then defined a circle. They came up with all kinds of properties of circles. When I could, I would provide a counter-example, like a shape that is round, but not a circle. We honed in on "all the points are the same distance from the middle" which we turned into a mathematical definition. Then we "developed" the equation of any circle centered at the origin.

Next, I hopped on to Desmos and asked them them what to do with the equation to make my circle have a radius of 6, or 3, or 8, or 3.5. They got it. Have I mentioned lately how much I love Desmos? I also showed them how to make the circle "move". We had looked at linear equations in the form y = a(x - h) + k, so it wasn't a huge stretch (no stretches involved, actually!) to perform horizontal and vertical translations on our circles.

When we talked about how to tell if a given point is inside, on or outside a circle and they really got it. I love how making a little change can make the rest of the class become seemless.

Oreos, Candies, Chocolate and Chocolate Milk

Sometimes life gets in the way of blogging so this post has morphed over the past week. One of my goals this semester is to add activities to my grade 10 academic class. Here is how things have played out for the first unit on solving systems of linear equations.

I wrote about how I changed my introduction to solving systems by elimination here. The next day they worked on the Oreo problem. I stole this from Nathan Kraft (here) which is why I haven't blogged about it myself. I introduced this activity last year and since it is awesome, I continue to use it. Students have to figure out whether the wafers or cream centre of an Oreo has more calories. (But really, you must read Nathan's intro to this.) Here is the information they are presented with in order to solve:

They worked on the big whiteboards and some even presented their solutions to the class. Note that Canadian Oreo packages have 2 cookies per serving for both regular cookies and Double Stuf cookies, which is not nearly as interesting as the ones shown. I tell the kids this once they have answered the question. And, yes, I do bring in Oreos for them.

I also added in a candy lab when we got to solving word problems. I stole this from someone who posted on Twitter. I sent myself the link but it didn't have a name associated with it, nor does the Google doc. If someone knows to whom I should attribute credit, please let me know! I started the class like this:

They were fairly (!) excited when they saw the word candy last period on a Friday afternoon. I randomly picked names from my tin of popsicle sticks to make groups of 4. Each group received a brown paper bag upon which I had written a letter and a number. The letter identified the bag and the number indicated the total number of candies and chocolates in the bag. I borrowed kitchen scales from our science department (thank you!) so that they could weigh their bag of candy. I wrote the mass of 1 candy (7 g), 1 chocolate (13 g) and an empty bag (8 g) on the board. And they were off! They did good work and once they had an answer we did the reveal - I opened the bag and counted out the candies and chocolates. They weren't perfect, but they were close and their work was excellent. I also had a few more challenging bags that had candies, chocolates and granola bars in them with an additional hint written on the bag. I will add pictures...sometime! As a side note, a greater total mass in the bag led to a better result. For next time...

Next in the word problem collection were mixture problems. My students have always found mixture problems to be particularly confounding. I thought I would use a demo to help (the idea, again, stolen from the Internet). A litre of milk appeared along with chocolate syrup, and the excitement was palpable! 

I told them that Noah only put 1 tablespoon (15 ml) of chocolate syrup with 250 ml of milk, while Isabelle puts 4 Tbs of chocolate syrup for the same volume of milk. (This is a lie. They would both put as much chocolate syrup as they could.) I made their respective chocolate milks for the class and showed them to all the students. They could see the difference between the two as one was much darker than the other. We calculated the percent of chocolate syrup for each. Then the question:

Fake context for sure. Jacob would want more chocolate syrup than Isabelle and Noah put together! But my students were engaged. They were paying attention. They, for the most part, wanted to know how to do this. And we did. Then, they worked through one more on their own:

I find that I am not worried about time, despite having to stick to a pretty regimented schedule. I believe that it is better to do one example where are all students engaged rather than 3 traditional ones where several (many?) of the class is not really paying attention.

What else have I changed? Homework. I do give homework in this class (but not in grade 10 applied math), which I check daily for completion. I believe that some practice is important in consolidating the material we have (un)covered and hopefully a deeper understanding can be developed, at least some of the time. But while looking for midpoint activities I found an old post from Dan Meyer about homework (or not giving homework) which included a suggestion from someone about modifying how homework is assigned. I liked the idea so implemented it the following morning. I am breaking up homework into basic, regular and challenge questions and have asked students to do 2 of the sets. The students who are confident with what was done in class can start with the regular set and move on to the challenge questions. Those who are a little less solid on the material start with the basic questions and then do the regular. It looks something like this:

I still think there is too much homework there, but I am working on that. The feedback I have received from students has been positive. I did feel the need to point out that after doing the regular set, students should not do the basic set in order to avoid the challenge questions!

Overall, I think I have made some positive changes. Not a huge overhaul like I did with grade 10 applied last year, but a move in the right direction nonetheless.

Solving a System of Linear Equations

This post is a bit of a repeat as I took something I did with my grade 10 applied students last year and am using it for my grade 10 academic students this year. We have solved systems of linear equations by graphing and by substitution. Before doing elimination, they normally do this investigation on equivalent systems that takes a very abstract look at why you can multiply equations by a constant and add or subtract equations. In the past students have done the investigation but really did not get anything out of it (other than confusion). I have never liked it, but it was one of those things that all the other teachers were doing, so I assigned it too. No more! This is what we did instead:

The pictures were key to some students' understanding. There were clearly 2 more coffees on the top line and that was the ONLY difference, so those 2 coffees had to account for the difference in price. From there they could work out the price of 1 coffee, then of one doughnut. Then we "translated" it into something more algebraic:

It took a little prompting for them to come up with the idea of subtracting. I asked what they had done with the costs to get them going down the right path.

Next:

I loved hearing a student come up with the idea of doubling the first order. This now gives us an equivalent system (that has meaning) where the number of cookies is the same in both orders so the difference in price is due to the extra latte.

Their homework was to finish this question and do one more that involved multiplying both equations. I really think this method brings meaning to the whole process and will make tomorrow go really smoothly.

Good Things from Day 3

In Advanced Functions today, I used the Popsicle sticks for the first time. I have been reading Embedded Formative Assessment for a while now, and this is a suggested strategy when questioning students. Whenever I had a question, I drew a name written on a Popsicle stick from my Starbucks tea tin (I finished the tea long ago) and that student had to provide an answer. I did try to give leeway for some questions to help them feel less anxious about the process. I think it went well - I even said that they could only raise their hand when they had a question. I also found myself saying "Convince me" many times throughout the day. I may not be reinventing the wheel in this class, but I am still trying to make important changes.

My grade 10 academic crew wrote a quiz at the beginning of class. I wanted to see if they were solid on graphing lines (most are) as we head into solving systems. I added a question, though. I wrote something like "Convince me that your graphs are correct.". I wanted them to use another method to check their work. If they used slope and y-intercept to graph, then they could check a couple of points or find the x-intercept, if they found the x- and y-intercepts to graph, then they could rewrite the equation and check the slope, etc. I am trying to get them to reflect on their work and try to think of multiple ways of solving problems. Anyway, I liked the results and will continue to ask similar questions.

The good thing about my grade 10 applied class is that I have some idea what I'm doing this time around (!) and know where I'm heading. I have a better sense of where I can push them a little more, and where I need to give them more time to absorb concepts.

And there was a double dose of Desmos along the way, which is always a good thing.

Here we Go!

Today was the first day of the new school year. Last semester I blogged daily about my grade 10 applied experience as I was spiralling through the curriculum with activities for the first time. I don't plan on blogging daily again, but really do feel that it helped me reflect about my practice, which I believe is really important and often neglected aspect of our profession. So my plan is to blog when I do something that I think is interesting, however often that is.

In Ontario, most schools are semestered so we teach the same students every day for half the year, then get a new crop of kids in February. We see each of 3 classes for 75 minutes and have a 75 minute prep period each day (when we are not supervising or covering someone else's class). This semester I have MHF4U (grade 12 advanced functions), MPM2D (grade 10 academic math) and MFM2P (grade 10 applied math) first semester.

At the end of last year, one of the cards I got from a graduating student said something to the effect of "I will never forget the first day of math class in September". At the time I read it, I had no idea what we had done back on September 3rd! I did figure it out though - we did the Marshmallow Challenge. And we did it again today in MHF4U! I stole this from someone (thanks and I'm sorry I don't remember who you are). I put students in random groups of (mostly) 4 and gave each group a large whiteboard.

I actually gave them 20 minutes to build as I remember time being really tight last year. Here are some of the final products. These are the sturdiest ones:

and here is the winner:

It was really interesting to watch them interact and I loved hearing the numerous "What if we ...". I was impressed with their efforts and collaboration. We also talked about mindset and learning from mistakes. It was a good first class.

Next up, grade 10 academic. I have done the "Don't Lose Your Marble" activity on the first day of MPM2D for ever (well, almost). Students have to find the relationship between the height of a ramp and the distance a marble will roll. I like this activity because I think it gives students a chance to remember something about linear relationships in a low pressure situation. (Note: I know that it is actually a quadratic relationship and I will tell them that tomorrow. For the small data set that they collect, a linear model works quite well.) They work in groups and help each other remember things like dependent & independent variables, slope, etc. And I get to observe them which is a great way to start to get to know them. I plan on doing many more activities with this class this year so we are off to a good start.

Last period of the (very hot) day was grade 10 applied. The girls all sat together and the boys did the same. I moved a few people around to even out the groups. We talked a bit about spiralling and the fact that we will be working through a lot of activities, which was met with a positive response. Then I gave them Fawn's Noah's Ark problem to work on in groups. I think my next comment is a reflection of why many of these kids are in the applied (vs. academic) math. It is like they have have the curiosity zapped out of them. It is sad that they gave up so easily, or tried to. That they did not want to solve the problem. That they thought they were not smart enough to solve the problem. I continued to encourage them and kept saying "Convince me!" when they came up with an answer. I tried to point out some of the good strategies they were using and nudged them in the right direction if they were really stuck. No one finished the problem in class so I said there would be a prize if anyone comes in with a correct, well written up solution tomorrow. We'll see. There is always a lot of work to build up the confidence in many of the students in MFM2P. I believe they can all succeed and will work toward having them believe the same.

I did accomplish my day 1 goal: learn the names of ALL my students. Now to finish planning for day 2!