Linking Cube Towers

I am not doing a daily blog about my grade 10 applied class this semester. This is not because everything is the same as last year or eve last semester. In fact, I have changed almost everything so far in this first cycle. I have different warm-ups, am doing topics in a different order and have made some new resources, too. If I'm not happy with things, I cannot leave them alone. I have thought about sharing what I have done at the end of each cycle - if that's of interest to you, please let me know.

Last year during a lesson study process we looked at an activity that involved students creating towers out of linking cubes and competing to see who could get the tallest tower. This is what I mean by linking cubes (also known as cube-a-links and unifix cubes):

I believe the students all had cards that told them how many cubes to begin with and how many to add each time. It went fairly well, but once a student was "out" (because their tower fell), they were no longer really engaged. Anyway, that is what inspired today's activity which will be my students first look at solving systems of linear equations. I am trying to have them really understand how the starting value and rate of change will affect their towers (and corresponding graphs). We have done a few visual patterns and solved some equations with a variable on both sides, so I think they will be ready for this.

Here is the file. I would love feedback. I will add a postscript if it's a disaster ;)

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 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 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.

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.