Tuesday, September 19, 2017

MPM1D1 - Day 11 Fractions

We started with this visual pattern:

I asked them to find three rules for the pattern: the number of squares, the perimeter and the area. All groups were very quick to find the rate, most using a table and looking at the first differences even though we haven't talked about them yet. Most groups determined that they needed to multiply the rate by the step number but the answer was off so they adjusted, adding the appropriate amount. I love how much of this they're figuring out on their own.

For groups that finished early I asked them how things would change if we said that each square was 2 unit by 2 units. Their first reaction was that they just needed to double their previous answer. I asked them to prove it to me and they soon discovered that the area of the new pattern was actually four times the area of the original. Such great thinking by a great group of students.

The other day one of my students asked why the rule for dividing fractions worked. I was so happy to hear this. We didn't have time to get into that day so we had a look today.

I started with this visual, then moved into looking at dividing with a common denominator.

Clearly the picture above wasn't going to cut it so we split things up a little differently.

From here it was easy to see that the green would fit into the red once (the green rectangles would fit over 8 of the red) leaving one rectangle. So we would need 1 out the 8 green ones. The solution then was that the green fit into the red once plus an eighth. I think many students appreciated the visual nature of this approach. However, when I asked them to try it some of them just wanted to use 'the rule'. We talked a little about how you could do this without drawing a picture. You could find a common denominator then divide the numerators by each other and do the same for the denominators. I love this approach.

Next we talked about adding and subtracting fractions, again starting visually, then becoming more abstract. We were running out of time so I had to forgo doing a problem today. I gave them the last 8 minutes to practice operations with fractions.

One of the students asked today if we were going to be doing any textbook work this year. Since we don't have any books for this course my reply was no. He seemed happy, which seemed odd given that the homework I give is the kind of work you find in a textbook. The boy sitting beside him was the boy that approached me last week saying he wasn't sure what was going on. He said that he really liked all the group work, problem solving and working at the board.

It seems that my daily routine in this class, generally, consists of a warm up, a problem and some skill work. I really like the balance but it's tight to fit it all in everyday. It would be perfect if our classes were 20 minutes longer!

Monday, September 18, 2017

MPM1D1 - Day 10 Finishing Up The Giant Toonie

The warm-up for today was this Would You Rather problem:

I had students work in groups but at their desks rather than the whiteboards since their work from Friday was still up on the board. It was much harder to see what was going on when students were seated compared to what they would do at the board.

One student said he'd prefer the 40' pool because it was longer. He seemed happy to stop their so I asked if he could find out how much water in the pools. Some groups converted the feet to yards, others the yards to feet. One student wanted to use the formula for surface area but fortunately his group convinced him that he was calculating the wrong thing. All the groups had some good success with this problem.

We then went on to find out how many toonies would fit in the giant toonie. This problem had multiple steps to it and many students struggled with those steps. This reinforced the importance of being organized and methodical. All groups were able to find the volume of the actual coin, after asking for a formula. Determining the volume of the giant coin proved to be a challenge. They had a hard time figuring out that they needed to find the scale factor, then use the scale factor to find the thickness of  the giant toonie. There were a few unit conversion errors but all of the groups knew that they needed to divide the volume of the big coin by the volume of the little coin. Of course this assumes that the coin can be packed in (melted down?) without any space for air.

This was a fun problem to watch groups struggle through. They really had to think about what the problem was asking and come up with a plan. Some students were quite frustrated by it, but hopefully this process will get easier for them as we do it more often.

Once we were done we did a quick not on calculating area and perimeter and I gave them some questions to practice.

Saturday, September 16, 2017

MPM1D1 - Day 9 Fractions and Measurement

I came across this problem last night so I thought I try it in class since we had been discussing fractions.

Some groups were quick to recognize that the fractions on both sides were equivalent, others performed the calculations and didn't really notice anything that was the same. I asked how they could compare the fractions. That was enough to get them to thing about reducing their answers. Some groups chose to scale the smaller numbers up. All the groups seemed to do just fine with the division, which was a bonus.

We moved on to Jon Orr's R2D2. Which didn't take long to solve. It was interesting to see that the half of the class on the west of the room did it one way (divide the width of the bulletin borad by the width of the sticky note to find how many would be needed, do the same for the width then multiply) and the groups on the east of the room all found the area of the board and the area of a sticky and divided them. Perhaps knowledge was moving around the room but not crossing the centre line.

Next up we were going to look Kyle Pearce's Big Nickel. I showed the video and asked how many of them had been to Sudbury and how many had seen the Big Nickel. I was surprised to see that only two of my students had seen the Nickel.  I then asked how many of them had been to Campbellford to see the Giant Toonie. It was about half the class so I decided to proceed with the toonie.
I showed a picture of the toonie. 

They asked some questions such as "How big is it?" and as a class we decided to find out how many real toonies would fit inside. I gave them the following information about the giant toonie and we headed to Wikipedia for details about the actual toonie.

I told them to assume that the giant toonie was built to scale and then set them loose. We haven't talked about measurement so I was keen to see where this would go. One group started by figuring how many toonies would fit across the diameter of the monument, another calculated the circumference of both. One group wanted to work on volume but couldn't remember how to find the volume of a cylinder so we talked about how to find the volume of a prism and I let them sort things out from there. We ran out of time so we'll have to pick up where we left off Monday.

Thursday, September 14, 2017

MPM1D1 - Day 8 Fractions and Proportions

We started with this Open Middle Problem:

The idea is to fill the squares using only the digits from 1 to 9, at most one time each. This was fairly easy for the class. Somebody asked if they were allowed to have improper fractions. Some groups drew pictures, while others just worked with the numbers.

Since the students were so quick to solve the first one we moved onto this one:

Some groups started with pictures but realized that it was tough to tell the difference between 4/9 and 5/11 when they were roughly drawn on the board. Most groups eventually came to the conclusion that they needed a good way to compare the fractions. Some discussed common denominators but most decided to convert the fractions to decimals or percents.

We then talked about equivalent fractions, mixed numbers and improper fractions. I mentioned that at some point we'd have to go over performing operations with fractions. Lot of them were keen to do it right then, so we did talk about it.

Next up was Fast Clapper. All but two of my students were pretty excited to be clapping and counting. They found unit rates and solved proportions by considering them equivalent fractions. I was hoping to do Smart Car Smash but we were running out of time. Instead I handed back the mastery test from yesterday (overall the results were quite good) and then I handed out a page on equivalent fractions, mixed numbers, improper fractions and multiplying and dividing integers. We'll redo the master test tomorrow.

Wednesday, September 13, 2017

MPM1D1 - Day 7 Mastery Test, Variation & Slope

I've never done three Visual Patterns in one week, but they seemed to tie in nicely with what we were doing this week. A couple of days ago we did this one:

Then yesterday we did this one:
Today we did this one:

All groups found the equation and the number of squares in the forty-third step easily. I wanted to show this one because we talking about direct and partial variation. We talked about how many squares the 0th step would have. We discussed what the graphs of the three patterns would look like (number of squares vs. step number) and connected an initial value of 0 to direct variations. We also talked about what was the same in all four tables and all four graphs. Somebody mentioned that the values in all the tables were going up by the same amount. Almost all groups had created a column for the first differences, even though we've never talked about it. Somebody else realized that the graphs would be going up at the same angle. We took a few minutes to get some information about direct and partial variations along with some information about slope into their notes.

I was happy to get through this when I did. Today was picture day and shortly after I finished about half a dozen students had to leave and get their photos taken.

The rest of the class did a mastery test on integers. Our department uses mastery tests to get at key skills in a course. They are short ten mark quizzes that focus on very specific but important skills. The idea is that we write the mastery test in class. The teacher marks them then hands them back (usually the next day) and go over any trouble spots. We rewrite a similar mastery test which get marked again. After the second attempt students can rewrite as many times as they want (outside of class time) until they get a mark that they are happy with. In this way the assessment is formative until the student decides it should be summative.

We finished up the mastery test and I handed out a set of data and asked them to create a scatter plot. They had to choose which variables were dependent and independent, create a scale, draw a line of best fit and list the characteristics of the graph (discrete/continuous, partial/direct, positive/negative slope).

Tuesday, September 12, 2017

MPM1D1 - Day 6 More Cup Stacking

I modified the Visual Pattern we did yesterday slightly to see how quickly my students would be able to spot the differences and come up with a solution. Again I asked them to find the forty-third step and a rule/equation to find the number of squares in any step. Here's what we started with:
Most groups started by making a table of values. Those that didn't weren't really sure where to start and so I suggested a table. I was happy to see many groups showing the first differences (though we didn't call them that) in their tables.

 I asked what was the same and what was different compared to yesterday's pattern. It was great hear things like "The constant is different" or "It's going up by the same amount". We talked a little about how these showed up in the equations.

Next we moved onto more cup stacking. The goal for today was to change how we stacked the cups and how that changed the equation and graph. I asked how many cups would be needed (stacked inside one another) to reach R's height. All groups saw that the height of the stack was changing by the lip of the cup for each additional cup. A couple of groups struggled with the initial value. They thought it should be the height of a cup rather than the body of the cup. Every group did manage to come up with an equation but struggled to solve the 2-step equation needed to find the number of cups (not surprising since we have done much equation solving yet).
Once they were done finding the number of cups needed to get to R's height they went back to their seats and plotted Height vs. Number of Cups. We talked about how the graph was different from the one they made yesterday. This led to a need for some terminology (partial vs. direct variation) so we wrote a note about graphs. The note included dependent vs. independent variables, continuous vs. discrete data, lines of best fit and interpolation vs. extrapolation. I was hoping to get into partial and direct variation and slope but we ran out of time.

I gave some homework on plotting points on the Cartesian Plane and identifying whether variables were dependent or independent.

At the end of class I had a student come to me and tell me that he was feeling lost. He said he was able to follow what his group members were saying but he wasn't sure he'd be able to come up with the numbers on his own. He told me that he did well in math last year but wasn't feeling very confident. We chatted for a bit and he agreed to come in tomorrow at lunch so we can go over a few things. I'm curious to see if his issue is related to skill or confidence.

Monday, September 11, 2017

MPM1D1 - Day 5 Cup Stacking

We did our first visual pattern today. I started with this one:

I told them that the images represent the first three steps in a pattern then asked if they could find the number of squares in the forty third step. I also asked if they could they find a rule or equation to represent the number of squares in any step (the nth step). Normally I ask for an equation and I think that is often intimidating at first. This time I focused on the rule, which we could then be turned into an equation. They worked at the problem in groups at board. They did a great job. We spent a bit of time talking about what makes an equation.

Then we moved on to Cup Stacking. I held up a styrofoam cup and asked what they noticed and what they wondered. There were some great observations but they were fairly quiet when I asked what they wondered. So I posed the question "How many cups would be needed to make a stack to my height?". This led to a discussion about how the cups were going to be stacked. Normally when I do this activity I tell the class that I want the cups stacked inside of one another. The class really wanted to stack the cups one on top of the other as shown below, so we started there. I figured we could do it both ways and discuss direct vs. partial variations.

Most groups came up with a solution pretty quickly. Once they finished I asked if they could come up with an equation that related the number of cups to the height of the stack. I had a few blank looks and reminded them of the visual pattern we did at the beginning of the class. That was enough to get them going.

We had a bit time left so they created graphs showing the height of a stack of cups vs. the number of cups. Tomorrow we'll see how stacking the cups inside one another compares.