MPM2D - Day 56: More Factoring

I felt the most unsure that I have this entire semester going into today's class. I had a really hard time predicting where my students would take the lesson. I presented them with all the quadratics they factored using algebra tiles on Friday. I asked them what they noticed and whether they could organize the quadratics in some way. Had I given them more time and given them the quadratics on slips of paper they could move around, they may have done more toward organizing them, but I decided against that as I didn't think it was worth the time. I don't think I would do this differently next time, but I will have to think on it some more.

They suggested separating them based on the signs within the factors and they said that some had a coefficient in front of the x^2, and others didn't (we clarified that they all had a coefficient). We went with the second suggestion. Here is the organization that we came up with, as a class, with definite prompting from me:

This is what they looked like moved into groups:

We then focused on monic trinomials. What was special about these?

It did not take long for them to say that the two numbers in the factors multiplied to something in the trinomial and added to something else in the trinomial. This would have gone more smoothly had I factored the first one correctly (ugh!). We fixed it and moved on, tying this back to how they have been factoring in homework for weeks and to the box method.

Then we looked at complex trinomials using the box method and decomposition.

We had time for a few examples and got help from Desmos for the one that did not factor.

We will need to continue to explore factoring different types of quadratics tomorrow and consolidate what we did today. I don't think I did a particularly good job today, so I will have to figure out what to do differently next time.

Here is today's homework.

MPM2D - Day 55: Factoring with Algebra Tiles

Today's entire class was devoted to factoring with algebra tiles. My students had two pages of expressions to factor and they did a really good job.

In some cases they ran out of the right colour tile, so they improved instead of getting another set.

Here's a difference of squares:

The goal today was to begin noticing some patterns emerging. We will talk about those on Monday.

Here is today's homework.

MPM2D - Day 54: Factoring, Day 2

The algebra tiles were waiting on students' desks as they walked in this morning, but as we had not finished what we started yesterday, the tiles had to wait. My students were not impressed with common factoring binomials. Even with pumpkins and happy faces.

And then I really made things worse with this:

We did a little work on the whiteboard to try to figure out how to deal with this case. They game me numbers, we worked out x - y and y - x. We talked about what was the same and what was different and then I asked how I could get the value of x - y given the value of y - x.

Of course multiplying by -1 will change the value, so how can we overcome that? Multiply by -1 again! Yeah, I lost a whole lot of them at this point. I know I did. I decided to let it sit with them for now and come back to it next week for a second try...

But then (I am so cruel) I made them factor by grouping. This is all leading somewhere, I told them. I'm not sure they believe me!

I think they were all really thankful to move on to "playing" with the tiles. We expanded, we factored; all was good again. I emphasized that they need to make a rectangle and that the x^2 tiles will be in one corner, while the unit tiles will be in the opposite corner.

A couple of examples for them to try:

Then we added negatives to the mix. Here is the link to the Gizmos activity. I click on "new" until I get a trinomial with negatives. I find this quicker for introducing the concept of adding zero pairs. And then they tried a few that were not monic.

I think they understand the how and the why of factoring trinomials. They will spend tomorrow practicing many more questions with the tiles and (hopefully) will start to notice some patterns emerging.

Here is today's homework.

MPM2D - Day 53: Common Factoring

Students arrived to find this on their desks.

A version of this has been floating around Twitter lately and I thought it would be a good way to get them thinking about common factoring and would lead us to finding the greatest common factor. Here are some of what they came up with for parts (h), (e) and (g) (ignore the simplification of x^5 over x^2 in the middle!). I asked them why my version was "better" (I showed them all my questions which are also at the top of each column, below). Someone said that what was left was in lowest terms. Loved that!

So then we moved into the lesson itself.

I emphasized the why, giving examples of quadratics in factored and standard forms and asking which was more helpful for graphing.

I tied factoring to area and showed an example with algebra tiles.

I break up the skills, to make sure that students can do each part correctly. Many don't remember the exponent rules well (hence the example on the picture of the whiteboard above). The answers below may look messy, but are all correct. We haven't worked with negative exponents, so that was fairly new to most.

Next, we looked at finding the greatest common factor.

I randomly drew names to get six students up writing answers and 5 of the 6 answers were incorrect. I told them that and asked them to talk in their groups to see if they could figure out what was wrong. They had only written the GCF (in one case not even the greatest) of the coefficients. Not one student had a variable in the GCF. I found this really interesting as there was no connection made to today's initial activity. Once they figured out that we needed some variables in the mix, corrections were made pretty quickly.

Having both pieces in place, we now moved on to actually factoring.

We did the first few examples explicitly showing the process then continued and only explained the calculations that were being done in our heads instead of writing them all out. At this point I saw one of my student acting like her brain was exploding so I went to ask what was up and she said that it was miraculous because she understood it all!

Here is today's homework.

MPM2D - Day 52: Back to Word Problems

I had today's lesson all planned out. We were going to formally start factoring (they have been factoring in their homework for weeks) and then I checked yesterday's homework on solving distance-rate-time systems questions. Only 13 out of 27 students handed their homework in and, of those, only 6 seemed to truly understand it. The previous day's part of the homework dealing with mixture problems had similar results. It was clear that it was important to stop and help them better understand these questions. So I made up teams, with those 6 students as team "captains". Each team's task was to work through the word problems from homework sets 31 and 32. I thought that since me teaching these lessons only reached a few of my students, I wouldn't try to teach again, but instead would let them teach each other. I told them that I would be circulating and asking questions to random people within each group. The team would score 1 point for each correctly answered question. I tried to emphasize that the goal was for EACH team member to understand all parts of each question.

I was really impressed with their work. Each group had a big whiteboard and they tackled the questions that had stumped so many of them. They did a good job explaining to each other and it was really interesting to see the dynamics within each group. Some team captains were in full "teacher mode", while others hung back and let everyone contribute more equally. I really liked the diagrams from this group:

I walked around observing and asking questions. "Can you explain your speed column?" "Why are you multiplying those?" "What does that number represent?" I put an emphasis on them being able to explain what each part of an equation represented. They were pretty grossed out when they realized that the 3 litres was 3 litres of butterfat in the 20 litres of cream!

Once they had finished working on the homework sets, I handed individual students within a team new questions and I told them that each correctly answered question earned the team 5 points. If there were errors, the team could correct them for a maximum of 3 points. Not all teams got to this stage, but those that did kept racking up the 5 points. They knew what they were doing. Yay!

In terms of logistics for the second part, I had found 17 questions which fit on one page and I labelled them A - Q. I cut out the little strips of paper and students did their work in their notebooks. I worked out all the answers quickly this morning so that I could check whether they were right. Here is the file with those questions.

I wasn't sure about the idea of having them competing for points, but it really wasn't an issue. They wanted to learn how to do these questions and took the opportunity to work together as teams. It was good.

MPM2D - Day 51: Shortest Distance

Today's class started with this:

I asked them to draw a point and a line and to figure out what the shortest distance would be and what was special about it. I soon began hearing "... perpendicular..." which was my cue to move on.

We consolidated before tackling our first example.

Okay, we actually didn't do that! I wanted them to struggle through the first example so I didn't give them steps, but included these in the notes I posted. Here is the question they worked on - I didn't show the picture or give any hints initially:

They worked in their groups to come up with a strategy and I suggested that they check in with each other after each step to ensure that any errors were caught quickly.

We went over the solution together, spending a little time talking about what to do with those square roots (they don't really come up in our curriculum until grade 11).

They worked on one more example (I notice now that my examples are numbered incorrectly - oops!):

When we took up the solution, we talked about working with fractions - again. I said that I'd rather they use decimals than give up on the rest of the solution (some are that fraction-phobic), but that being able to work with fractions is important.

Here is today's homework.

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.