MPM2D - Day 50: Distance, Rate, Time Problems

For day 3 of linear system word problems, we tackled distance, rate, time questions. We talked about the relationship between the three and a students shared the "triangle" method (which I have issues with only because I have seen many students mess it up). Still, I like that they want to share ideas and feel comfortable doing so.

The first question was not too challenging, but reinforced that organizing the information in a table can be helpful.

The second question is the type that they generally dislike the most. However, I read Cheesemonkey's very timely post this morning which gave me a new approach for this type of question. I really liked the idea of "helping" or "hurting" in terms of wind/current/etc. It just makes so much sense.

For the third question I had two students volunteer to be planes. 

The door was Montreal and my desk was Calgary. One needed to go faster than the other and they stopped when they met. They even put their arms out to "be" planes and made fun noises! I asked a lot of questions - who went faster? who went further? who took more time? how far did they travel? Then they had to try to fill in the table. They got the speeds in and the total distance was not too bad, but then many were stuck. They talked in their groups for a bit before someone suggested putting variables in for the distances and calculating the times. This created fractions, which they don't particularly like, so we looked at a different approach - using the variables to represent time. They preferred this and came up with the first equation easily. The second equation required me asking "Who took longer?" again for it to click into place.

We didn't actually solve any of these questions as I know they can all do that. They will do two full questions like these for homework, so they are still getting the practice in. 

We spent the rest of the period doing a little self-reflection. Well, they did. I had them enter their results from quizzes and tests on to their (paper) evidence record. This picture is not the best but the columns are the curriculum expectations and each row is for an evaluation. So test 1 hit 5 expectations, test 2 hit 6 of them. This (hopefully) gives them a better picture of how they are doing and specifically where they need to improve.

I enter all of this into a locally-developed program. Here is how one evidence record looks so far:

They also started filling in this self-reflection sheet. I made a column for each strand and a general column for things like the way they communicate their solutions, common errors, and so on. They have a row for each test so I hope this will help them focus on the areas that need the most work.

Here is today's homework set.

MPM2D - Day 49: Candy Lab & Chocolate Milk

Today was fun. We started with the Candy Lab. I did this with my class last year and wrote a short blog post about it which I'm really glad I read over before doing it this year. Sadly I still don't know to whom I owe credit for the original idea. If you recognize this, please let me know!

I use this software to create the random groups. I showed them what they would receive - a bag indicating the number of items contained inside. There were large items (full size Rolo) and small items (Halloween size KitKat) and their job was to figure out the number of each. The only tool at their disposal was a kitchen scale.

I asked what information they wanted from me. They asked for the the mass of each item and (this made me very happy), the mass of an empty bag.

Given that, they got to work. Here is an example:

They recognized that they needed to round their answers as they couldn't have part of an item. When they finished I took the staple out of the bag and let them do the reveal to see if they were right. Every single group was! And I let them choose to either share a Rolo amongst their group or each have a KitKat.

A couple of groups got stuck early on because they let their variables represent the masses of the items, instead of the number of items. Their system of equations broke down pretty quickly. One group was stumped for longer (they calculated a number of items greater than that indicated on the bag) and I asked others to go help them. It was really nice to see the collaboration taking place - many students looking at one whiteboard trying to find the mistake(s).

If you try this, remember that working with heavier objects gives much more successful results.

And if that wasn't enough fun for one day, we moved on to making chocolate milk. 

I brought in milk, chocolate syrup, a measuring cup and glasses and actually made chocolate milk according to the information in the table.

It was evident that one was far more concentrated than the other. We filled in the last two columns of the table together. I was surprised at how much difficulty many students had coming up with the percentage of chocolate syrup. I had to say "If you got a test back and got 10 out of 110, how would you work out your mark as a percent?" to get them to make the connection. Next, I said that Jacob wanted chocolate milk with 15% syrup, but that mom took away the chocolate syrup so he had to make it using the pitchers of Noah's and Isabelle's chocolate milk.

I was really impressed that my students (collectively) found these equations. It was not straightforward - we went around the room and bounced a bunch of ideas around before landing on these. To see if they really got it, Chloë got in on the chocolate milk action:

They did great! We set up one more, not actually with the detail shown below, but well enough to say that they understood.

Here is today's homework. I hope the work we did today helps them understand mixture problems which students always find tricky.

MPM2D - Day 48: Solving Systems

I decided to begin cycle 3 with solving systems. Back in cycle 1, my students learned how to solve systems by substitution and elimination. Today we began looking at different types of questions involving systems.

We started with a quick refresher:

I randomly chose a student to write their solution on the board which we then went through, both in terms of content and presentation. We again spent a considerable amount of time working with fractions as that is a skill that they can't seem to practice enough.

We then talked briefly about problem solving steps (identify variables, write 2 equations, solve, check, conclude) before looking at our first one:

We talked about how to parse that first sentence to turn it into an equation. Finding an equation to represent perimeter was more straightforward. And then we solved.

Next up:

I asked how many students read that and just wanted to quit. Several hands went up. I told them they could think about it like a puzzle. I asked what they did first when they were working on a jigsaw puzzle. They said the corners, followed by the edges. I said that they were organizing the puzzle by doing that, and that we needed to do the same. So we made a table:

They did well filling it in once we set Katie's age now as 'k' and Anne's age now as 'a'. But we needed to take a side step to figure out the first equation. We looked at an example - if 2 years ago Anne was 10, how old was Katie? Twice as old, so Katie was 20. Underneath those ages I wrote the expressions for their ages 2 years ago and asked what we would have to do to make them equal. From that, came our first equation. The second equation came a little more easily.

And then we solved.

We set up two more questions, without solving them.

Here is today's homework.

MPM2D - Day 47: Test (Day 2)

Nothing too much to report today except that even after teaching this course every year since its inception in 2000, I continue to learn. I am getting better at making notes to my future self about how to improve test questions. Better questions is a focus of mine this year; one that I haven't focused on enough so far. It's important to take a few minutes after correcting tests to evaluate the questions that were asked and think about whether they truly drew out students' understanding. So often when we finish marking a set of tests the relief of the job being done is all that we care about. However, it's when you are right there, in that moment of seeing your students' thinking, that you need to reflect on what small tweak or large change you could make to push them further or to allow them to better show what they know.

On a completely different note, I am glad to report that many of my students realize that it's okay to ask a question during a test. I know this is frowned upon by some, but if they don't understand the question, then I'm not going to get anything good from them. Lots of my students struggled with the same question today. If they couldn't draw the diagram correctly, they were stuck. My mini Golden Gate Bridge helped tremendously!

I also had several students ask for help when they were stuck. Some questions were as simple as "I don't remember what standard form looks like.", while others didn't know how to begin a section of the test. As I have done for a few years with my applied classes, I write what I tell them in orange or pink pen so that I can take it into consideration when I mark their paper. I worry about the inequity toward those who don't say anything and I don't notice until it is too late. But that is just something else I need to work on.

Oh - and another little plug for spiraling... Part of an email from one of my MPM2D parents: "*** is amazing us with her math study discipline!   You seem to have turned her mind to math!   ‎Way to go! ".

MPM2D - Day 46: Test (day 1)

You know how you can spend hours making a test so that the questions are exactly what you want? How you choose the points of a quadrilateral to form the right shape without being obvious, and with coordinates large enough that an algebraic solution is really the only way to go (without the use of technology, that is)? And you are so sure that everything is perfect, having triple checked all the values, having checked it all in GeoGebra, that you don't write up solutions before the test? Sigh... Somehow when I typed 30 a gremlin made it 20 making my perfect quadrilateral, very imperfect. I noticed in time to help some students, but will deal with the messy results of those who had already worked through the problem. Ironically, I had written up the solutions to tomorrow's part of the test, and it is all fine. The kicker is that I know better, mostly because I know I make a lot of silly errors (hence my love of Frixion erasable pens). However, I simply made it clear that it was my mistake and that no one should erase any of the work they had already done.

Log Transformations - Desmos Activity

Earlier this week I made a log transformations activity using Desmos Activity Builder. You can find it here (I also submitted it to the Desmos Bank). I made it for my MHF4U students (that's grade 12 Advanced Functions) who, at this point, should understand function transformations inside and out. I greatly appreciate the feedback I received from Kara Gilby, @sheriwalker72, and my students. If you try this activity with you class, please let me know how it went and anything you think would make it better. Thanks!

MPM2D - Days 44 & 45: Review

Today and tomorrow are reserved for doing review for the upcoming test. I asked them to go over all of their past homework and figure out the parts with which they had struggled - this usually meant looking for where I had written "try again". It was encouraging to see many of them actually doing this and showing me their new solutions. They also received this review sheet. I will be splitting up the test into two parts. On Monday they will work on analytic geometry and on Tuesday it will be quadratics and trig.