MPM2D - Day 16: Taco Cart --> Distance Between Two Points

Today's goal was to come up with a way of finding the distance between two points. I didn't know if we would actually get to the distance formula itself, but wanted to make sure my students had a solid understanding of how we find these distances. We started  with Dan Meyer's Taco Cart problem:

I showed the Act I video (the link above will get you to all the videos) and before I could even ask, a student said "But we don't know how fast they are going." Great! What else do we need to know? Distances. Here you go:

And they got to work. There were good conversations about rates and the Pythagorean theorem. I asked who was on team Dan and who was on team Ben and Dan won by a landslide! At this point one student was about to leave the room and said "I'll wait because I need to know how this turns out!" - how cool is that? This is where we ended up:

with a little detour to talk about the relationship between speed, distance and time:

And then I played the Act III video (I won't spoil it for you).

My next question for my students was "How can we find the distance between two points if we know the coordinates of the points?". We started with what I thought was a straightforward example:

A number of students had no idea what to do until I suggested plotting the points. I think this was the first time I had seen them just sitting there, not knowing what to do and not trying something. I am trying to think of what I could have asked them to help them get going without explicitly saying plot the points... Any ideas? Once they had the points on a grid, they quickly saw that they could make a right triangle and then use the Pythagorean theorem.

They worked through two more examples after which I asked them if they could figure out a way of getting the distance without drawing the triangle.

A student solved (b) and you can see that the side lengths had been labelled as 6^2, etc. so we addressed that along the way. I let them struggle for a while, trying to find a relationship between the points and the side lengths before we consolidated that piece.

We then began to generalize but didn't complete that part so I will pick it up there tomorrow. Here is today's homework. One of the benefits of doing lagging homework is that there are only 3 questions asking for the distance between two points. So even if they create a right triangle for each one, it should not take too long. I am being mindful to keep homework to one page and to ask meaningful questions, not the same one 20 times over.

I wanted to share the nice email that I received from the parent of one of the students in this class:

This is one reason that I wanted to spiral this course. I hope I can make math accessible to all my students and make them like it along the way.

MPM2D - Day 15: Oreos & SolveMe Mobiles

Today was Oreo day. 

I told my students all about Mr. Kraft and Mrs. Runkle, whose story can be found on Nathan's blog here. They were just as grossed out as me that Nathan would eat cookies that Mrs. Runkle had licked. Once the story was set up, I asked the question: who was eating more calories? (this is all stolen from Nathan's blog)

I gave each group a large whiteboard to work on and the conversations naturally flowed. Along the way I gave each group cookies to keep them inspired. 

Only there were more cookies than people and some were regular Oreos and others were double-stuf Oreos - distributing them fairly among the group members became a whole different problem to solve!

Before circulating to see how they were doing, I checked yesterday's homework (set 12). I was delighted to see that they really seem to understand how to solve (this particular type of) word problems using elimination. They rocked that homework!

Here are their solutions to the Oreo question:

Lots of great, very similar solutions (we did talk about the one with the correct work, but incorrect conclusion). This next one started with a slightly different strategy, which was the catalyst for some good discussion.

I then showed my students Nathan's blog post and they saw the various ways the question had been answered. I am trying to encourage my students to look for different ways of approaching a problem and this did a great job of demonstrating just that.

As some groups had worked through the Oreo question faster than others, I returned the homework that had been handed in yesterday (set 11) for them to look over and correct. Trying to ensure that they take the time to learn from their mistakes means that I will devote some class time to this practice. 

Next we solved some puzzles. I introduced them to SolveMe Mobiles, which, sadly, do not work on mobile phones (but I still love them). We did a couple of simple ones together to make sure everyone understood how they worked and then tackled #63, then #64, shown here:

Followed by #67 (below) and #71.

I find it really interesting that students who can solve these "in their head" tell me that they can't solve them algebraically. However, if they tell me step by step what they did to solve it, and I write down each step, they see that it all matches. Working on that perseverance...

Today's homework is a mixture of questions which can be found here.

MPM2D - Day 14: Solving Systems by Elimination

Today's goal was to learn how to solve linear systems by elimination. We jumped right in with the first example:

Side note: I do not actually drink coffee, unless you put a little chocolate or caramel in it. Yes, I'm one of "those" Starbucks coffee drinkers. I actually prefer tea. And I don't remember when I last ate a doughnut. I'd much rather have some dark chocolate.

A couple of years ago Sheri Walker showed me how she teaches solving systems by elimination and I kicked myself for not realizing that I needed to teach it this way sooner. I teach both my applied and academic classes using these examples as they make it all make sense. I use pictures and I find this helps certain students make the link between the words and the equations. I also wrote the equations for this first example on the white board.

We compared what was the same in both orders and concluded that the extra 2 coffees had to account for the extra $4.50. From there we could find the price of 1 coffee and then the price of 1 doughnut.

Next, a slightly more difficult example:

 

I was impressed by the number of students who thought to double the order. I really liked it when one student suggested adding 2 cookies to the first order as it let us discuss why we can't do that.

They worked through the 3rd example and noticed that there was more than one correct way of solving it. Some doubled the first order and tripled the second, while others did the opposite. One student multiplied the first order by 1.5. We talked about this not working in context (you can't buy half a muffin...), but clearly this student was not having any trouble making this more abstract.

Then we went over the process of solving by elimination, emphasizing that we are creating an equivalent system by multiplying one or more equations by a constant. They then practiced solving more systems which no longer had a context, and where they needed to make a choice between adding or subtracting.

I had hoped to have them work on the Oreo problem also, but that will have to be saved for tomorrow. Here is the homework I gave today.

My Spiralling Process

Before I actually begin this post, I will link to some resources about spiralling as my intent here is not to explain what it is or its benefits. Suffice it to say that I am really enjoying teaching this way. If you have no idea what spiralling is, you could read my blog post recapping a session that I co-presented at TMC14, found here. There are also some interesting articles/blog posts found here, here, here and here.

I am often asked how I spiral (interleave) a course. Thus far, I have not had particularly insightful answers, as the only course I had spiralled prior to this semester was grade 10 applied (MFM2P). I stole everything my first time spiralling that course from Alex Overwijk, with whom I collaborated throughout that semester. I cannot thank him enough for taking the time to work with me and Sheri Walker, and for generously sharing all his resources. 

What has changed? This semester I am spiralling the grade 10 academic course (MPM2D) for the first time. I am doing this alone, but feel confident having spiralled MFM2P for the past two years (and continuing to do so), and having taught MPM2D dozens of times (so I know the content inside and out). By the end of TMC15 this past summer, I had envisioned how I could make spiralling this course work for me. Although I found it very difficult to do a lot of planning before actually being in the trenches of teaching the course, I worked out a draft of my plan. I first stated the overall curriculum expectations for the course. I then worked out what I wanted the first cycle to look like and got less and less specific with each subsequent cycle. It looked like this:

I then made a gdoc for my first cycle. Choosing which activities fit in with my goals takes a little time, but I try to make each day as meaningful as possible. I also want my students to enjoy math class so choosing good activities that will engage them is key.

Soon after the semester began, I added columns for what I actually did each day as things always take longer than planned and my plans are always flexible. I adjust what I do based on how the previous day went. Here is what that doc looks like now:

I am recording what I do each day, along with what homework I assigned. Speaking of homework, I also decided to create my own homework sets. I am doing lagging homework some days. For example, homework set 11 had one question where students had to find the equation of a line given two points, one question where they had to sketch a parabola given the equation in factored form, one question with binomial multiplications and one question where they had to solve systems of linear equations using substitution. This way the content is always fresh and students who struggled with a concept will get more opportunities to practice and improve. I try to give feedback on homework (it never counts toward the grade) several times a week, and check only for completion occasionally. I feel that this has helped me really know where each student is in the course. Lagging homework also doesn't let students "escape" any of the material. If they didn't understand something they soon learn that it's going to keep coming back and that they really do need to take advantage of some extra help. I post full solutions to all the homework after it has been handed in, allowing students to figure out some of their own errors and how to correct them.

In addition to the gdoc above, I have gone through the curriculum document and highlighted all the specific expectations that I will hit during cycle 1, and pencilled in cycle 2, 3 or 4 for each of the others. I hope to create a digital version of this at some point.

Evaluations in a spiralled course look a little different. They have multiple curriculum expectations, making them more like a little exam than a standard unit test. I mark on levels (R, 1, 2, 3, 4) by expectation. This makes it clear to each student how they are doing on each expectation. Should they get a level R, they will be given an opportunity to reassess before the next evaluation, after getting some help from me. I record results using a locally-developed program called MaMa which allows me to see progress throughout the semester. Here is a screenshot from a previous year:

The curriculum expectations are along the left and each test is a different colour, getting darker as the semester progresses.

I am also blogging my way through this course. I write a post each day where I explain what I did and reflect on how things went. I also post link to all the handouts. If you are interested in reading more, my first post for this semester can be found here. I also blogged my way through the MFM2P course last year and the one before.

Do you have questions about spiralling that I have not addressed? Please fire them my way in the comments.

MPM2D - Day 13: Solving Systems by Substitution (continued)

After finishing the last example from yesterday and going over how to do a formal check, we looked at a couple of special cases as we continued to practice solving systems by substitution.

I gave them the rest of the class to work on the homework set, which you can find here. I am trying to create lagging homework sets, so in addition to solving by substitution, this set has finding the equation of a line given two points, sketching a quadratic in factored form and multiplying binomials.

Side Note: Tomorrow is Carp Fair Friday which means that I will likely have no students and will therefore not have a blog post either.

MPM2D - Day 12: Solving Linear Systems by Substitution

We started today right where we had left off yesterday. Frankly, today was a lot of notes and examples, which I try to avoid, but I am not sure how else to get all my students to point where they can reliably solve a system by substitution. I will ponder this some more, for next time...

We began with the stacking cups data from yesterday. There were many more graphs that had been completed than what I saw yesterday and several students tried to solve algebraically. Attempting to solve that particular system graphically did make it clear that algebra would make life better.

At the end of class we did actually test out the solutions with cups, but sadly, I did not get a picture. This solution of 201 cups was way off, but many groups got around 160 cups, which was definitely closer.

Then we did some examples as a class.

We went over some more notes, before working through another example. This time the equations did not present themselves in y = ... form.

In case you think that Canadians actually eat the tails of beavers, here is a picture of a beavertail. It is a lot like a doughnut that has been stretched out before being deep fried. They are served hot with various toppings, my favourite being lemon and sugar.

Back to the question...

And then I found myself looking at the clock and trying to fit everything into that last 6 minutes... Another example, returning yesterday's quizzes and stacking two sets of 160+ cups. Dumb, I know. As a result, I did a lousy job at two of the three things. Here is the no-longer-given-in-context next example:

As much as I would like all questions to be in context, I realize that my academic students need to be able to work in the abstract. We will finish this one tomorrow. Homework was the back of the homework set from yesterday.

It's funny that the way I ran my class today is how I taught for years and years. It was my comfort zone and I think I do it decently. But I find it so boring now and can only assume that at least some of my students also find it boring (some love to sit and take notes so I likely made them happy today). I have developed good methods of not teaching like this in my applied classes, but am struggling with how to deal with a lesson like today's without direct instruction. I could create a very scaffolded handout for them to work through and help those that get stuck, but my experience around teaching this topic tells me that it would be a disaster, as our students have virtually no experience substituting anything other than a number for a variable. I realize that there may not be a better way for some topics, but welcome feedback. Elimination is coming!

MPM2D - Day 11: Quiz + Cup Stacking Systems (day 2)

Today started with a quiz on what we have done so far with quadratics. While making the quiz yesterday, I was searching Fawn's fabulous visual patterns site for a good quadratic pattern, but was having trouble finding one that would not be too trivial for my students. I didn't want the pattern to just be the step size multiplied by the step size plus 1 or doubled. I had a little epiphany when I realized that if I found a pattern that I liked, I could offset the step number (step 2 becomes step 1, step 3 becomes step 2, and so on) and make the pattern just that little bit more challenging. Probably obvious to everyone else, but it was a happy moment in my day.

Update: here is the quiz.

I told my students that they could take as much time as they needed for the quiz as I don't want them to get stressed out about having to do math quickly. When they finished, they handed in the quiz and worked on their stacking cup systems graphs from yesterday. Once they solved the system graphically, which took more than one attempt for many as their scale did not allow them to see the solution, I wanted them to think about how to solve it algebraically. Unfortunately, many did not relish the idea of graphing so that ended up being assigned as homework, if it had not been completed in class. We did talk about the equations representing the height of the cups and what purpose graphing the relationships will serve.

Here is today's homework set (they only have to work on the first side tonight).