I decided to switch warm-ups this morning. Instead of doing a Would You Rather, I handed out the small whiteboards and pulled up this website. Someone from my lesson study group told us about it and I saw it in action last week. Here is the first one we did:
I had each student write their answer on their whiteboard and then asked them all to hold them up. I chose one student to explain their reasoning to the class. They did it well. We continued.
They were liking this. They were talking to each other - about math! They were all engaged! We continued with a few more (I skipped some) and a tutorial popup showed us that you can drag the beam at the top to show the equation:
We were now starting to see the tie-in to algebra. We kept going (and I was hearing "Can we do this all period?") until this one stumped some students:
The "I quit!" quickly turned to "I can do this." after I asked about the 28 at the top. What did it mean? What did that mean for each side?
They could see the patterns in solving these and were keen to keep going. But I wanted to start making the connection to an algebraic representation stronger (side note: they did not feel this need at all).
(I was tempted to solve this by elimination, but that was not my goal for today and didn't want to lose them.)
We did one more before moving on.
I really like these puzzles and will include them in my warm-ups from the beginning next time. I think it will really help build up students' equation solving skills over the semester.
They each received this handout, which was created by Alanna Street (thanks, Alanna!). They zipped through the front but then I asked them to write algebraic solutions, too. It is hard to convince them to do this as they are really good at solving in their heads. As we worked through them together, they started to realize that they knew what to do. The hurdle here was writing it down.
They continued to work on the questions on the back side of the handout while I circulated and helped some get going. Those who finished early got to go to the Solve Me site (on their phones) and I showed them the higher levels (Puzzler & Master).
We finished off by tying this back to what we were doing yesterday. I decided that I didn't want to interrupt their work and collected new data for the red cups myself. We found the rate of change, starting value and equation to relate height to the number of cups. Then we worked on finding the number of cups that would give the same height - this is just what we had been practicing!
And look:
6 cups, same height : )
Today's warm-up was this visual pattern:
Most students saw this growing the same way:
We discussed the work they had done yesterday and they seemed to be on the right track. So we returned to this question:
We reviewed the equation relating the number of cups to the height for the Styrofoam cups, emphasizing what each variable represented (I actually wrote out the words and did not use x and y). I then gave each group 5 red cups and asked them to come up with a model. It took some work but they started measuring and figuring out the rate of change. Once most groups had an equation I took one group's data (they had a constant rate of change) and we analyzed it together.
We then set up the system to find out when we would have the same number of cups and they would reach the same height. The question they were supposed to be answering had the red cups starting on the desk, but I didn't think their models would good enough for that. We got a result of 10 cups that should give the same height and this is what it looked like:
Hmmm. Clearly not the same height. What could have caused this? The algebra we did to solve the system of equations was correct, so the model for the red cups must be the culprit (we were consistent with our model for the Styrofoam cups on Monday). I asked for other groups' models for the red cups(written in brown, above):
h = 0.3n + 10.5
h = 1.2n + 2
h = 0.5n + 10.5
So many issues of values that were simply not reasonable. Could the red cup really be 2 cm without the lip? Was the lip only 0.3 cm? Back to the drawing board, as they say. With only 3 minutes left in class I asked them to try collecting data again. I am not giving up on this so there will be more tomorrow!
I was away in Kingston (Ontario) yesterday to participate in a "Think Tank" - we were sharing our experiences around spiraling with activities and seeing what teachers in other school boards in eastern Ontario have been doing.
My fantastic substitute teacher did her best with my 2P crew, but they were not very cooperative. They did an estimation 180 as a warm-up and then started on some more cup stacking questions, found here. You can tell that I stole much of this from Dan Meyer - here is his blog post about it.
Next they had to figure this out:
And they were going to test out their answers, but I think that's when things fell apart. Sometimes working with concrete materials becomes a distraction. However, I will have a conversation with the class this morning and we will get things back on track - cups and all!
In Ontario there is a strange grade 12 course called Calculus & Vectors. The two components are completely separate and we are on day 4 of vectors now. Part of today's work involved plotting points in 3D, which they have never had to do before. I start by getting them make their own set of 3D axes using straws (and tape/pipe cleaners) which they can then use whenever they like.
I have my own made of wood thanks to a fabulous former student.
I had them use their models to plot points (in the air) and then showed them the points plotted on this cool little website found on the Solve My Maths site. They then drew the points by hand.
I really liked this and my students appreciated how it showed the point starting at the origin and moving along each plane.
At the end of the lesson I got them to do this treasure hunt. I told them the treasure was a lollipop and they were (strangely) very motivated by this. They had to use the clues given to determine the location of the treasure (such a fake context!) and they really seemed to enjoy it.
And they also enjoyed their "treasure"!
Today's warm-up was a counting circle. We started at 27x - 52 and added -2x + 5. Some students were intimidated at first, but quickly realized that they needed to deal with like terms separately.
The picture you see is of Vector from Despicable Me. One of my Calculus & Vectors students drew it, framed it and wants it to stay permanently on the whiteboard!
I started today's activity by standing, holding a stack of Styrofoam cups and asked my students what questions came to mind. It did not take long before someone asked "How many cups would it take to reach your height?". Boom! I displayed their random groups for today and explained their task to them. Each group got 10 cups and had to determine how many cups it would take to reach my height. They picked up the handout and started measuring.
Someone asked how tall I am - I said they could measure me. Two groups did, but were not very accurate, as it turns out. Most students were not being at all precise with their measurements. They said the lip was about 1 cm and 10 cups were 20 cm tall and thought that was good enough. It took some prompting to get them to be more accurate and reason through whether the rate of change made sense (some had RoC of 1 then 1.6, then 1.2...). They all eventually got to an equation, but then all but one group whose equation was correct put my height in for the number of cups! I asked them to talk about what each variable represented, but they did not see any issues with their work. Here are their results:
The closest group was 5 cups off - in previous years they were much closer. We talked about how they should have been using their equation to calculate the number of cups versus what they had actually calculated. We also discussed the sources of error.
To close out the class I showed them a short stack of red cups that look something like this
and asked if they would need more or fewer of these cups to reach my height.They reasoned through this really well and even showed that with only 7 cups, the Styrofoam cup stack would already be taller.
I thought today's warm-up would be quick, but we spent a little while digging deeper. This was the Always-Sometimes-Never statement and what they came up:
They were quite sure we were done and then I asked about 1/2 squared. Reluctantly, they figured it out and noticed that the result was smaller than the number squared. Hmmm. We tried 3/2. That one was bigger. So I asked them to try any fraction and see what they got. Two ideas came out of that: the result is smaller if
- the number is less than one
- the numerator of the fraction is less than the denominator.
We explored these ideas and refined the first one a little and agreed that they both represented the same thing.
I really liked the thinking and noticing involved in this A-S-N.
Next, they glued two factoring examples in their exercise books. Here is the file. One required them to add some zero pairs.
Once they completed that, they got back into the same groups as yesterday and contined to work on the Shell matching activity. I circulated helping each group understand what to do with the tables of values. I picked one of the expressions and we worked out the table together then found the matching piece. In most cases I chose 2n + 6. I then explained that some of the expressions were equivalent to others, like 2(n +3) is to 2n + 6 so they share a table and an area model. A couple of groups almost finished:
I normally have them figure out the equivalent expressions before they start gluing anything down, but instead they drew lines to make the connections. I need to think about how to make this activity work better. I think I need to make sure they have an entire period to do it - perhaps have them cut up the pieces the day before, even. I think I would also encourage them to organize them into "linear" and "quadratic" and work out the equivalent expressions as a first step so that they are making clusters, instead of lines of matching pieces.
As it turned out we spent the entire week doing factoring and this matching activity. Longer than I had planned, but that is the nice thing about spiraling. I can adjust the timing as needed and tailor it to my current students' needs.
Today was another lesson study observation day so I did not get to teacher my 2P class. My colleague did, but sadly, as the wifi was down, they did not do the planned Would You Rather. They moved on to talking about different representations of algebraic expressions and started working on this matching activity from the Shell Centre. I LOVE this activity - it is great. There are linear and quadratic expressions, they need to expand and factor, write mathematical expressions in words, match and fill in tables of values and match area models.
We will finish it up tomorrow and I will post some pictures then.
Today was my favourite visual pattern:
I asked them to show me how they see the pattern growing which many find really difficult. They count the squares and see how many are being added but don't relate it to the pattern. A few finally coloured in the corner squares and explained to me that the remaining number of tiles on each side was equal to the step (pool) number. Here is the consolidation:
I love how you can see the pattern in many ways AND show that all the associated rules are equivalent.
We returned to factoring after the visual pattern and tackled negatives. We did two examples together - they had algebra tiles out and I drew on the whiteboard. The second example required the addition of zero pairs of x-tiles.
Some were still struggling but they were doing better than yesterday. So on to speed dating! They each took one question on a brightly coloured sheet of paper and factored it.
I asked them to write the result, fold the paper just above the result and trade papers. They each then had to multiply the binomials to see if the factoring was done correctly. The goal here is that they become the expert at factoring their question and will be able to help anyone struggling as they do the speed dating. I gave them 5 minutes for this - it was not enough, but I found it helpful to have the timer so I could gauge how they were doing. Once they were all done we arranged the desks into two long rows of desks facing one another. They folded their papers along the line I had made below the question and set it to face the person opposite them.
And then we began! I gave them 3 minutes for the first couple and then 2:30 and then 2:00. Some were getting really good at factoring, others were likely still learning how to factor. I think it was more engaging than sitting with a worksheet. Oh - here is the file with the equations to be factored (I printed 2 per page).
Today's warm up was this Estimation 180:
And here are their estimates:
There was only one student whose answer was clearly way off and he freely admitted that he just guessed without really looking at the question.
We consolidated yesterday's work, but staying on task was very challenging for some today (we went into a "secure school" first period today, which got extended so things were just "off"). We went over the first page together, emphasizing the need to make sense of the numbers.
Then we moved on to factoring. Out came the algebra tiles and we struggled through the first two pages of this handout. I teach factoring by asking them to create a rectangle (as close to a square, if there are multiple options) and read off the length and width to get the factors. I used my document camera to show what I am doing with the tiles. I was very pleased when one student asked to come up and show how to make a rectangle using the document camera.
This is what we got done today:
I left that class fairly certain that about half the class still has no idea how to factor so we will be working on that some more tomorrow. I will see if I can come up with a way of making this more engaging...
I decided to make today's counting circle a little more interesting by adding an expression instead of just a number. We started at 5x + 20 and added 2x - 3. This was a really good way of reviewing how to add like terms, which was very timely given that it is a skill needed when multiplying binomials.
After the counting circle I handed out the matching sheets from Friday and they continued to work on them. Almost all finished (they should all have finished but a couple were having trouble staying on-task) and I posted them on the back blackboard of the classroom.
The next handout, found here, connected the skills they had just practiced within a (fake) context. I had several students ask me what the vertex was and what the axis of symmetry was. I referred them to their exercise books to look up the information they needed, which they did. They seemed to be working through the handout fairly well, but I think we will consolidate it together at the beginning of tomorrow's class.
