I chose an Easter themed Estimation 180 for today's warm up:
I have had students write their estimates on small whiteboards and hold them up, but this time I went around the classroom with a big whiteboard and recorded their estimates. This ensured that everyone was participating and let me see how close they were to the answer. Once I had everyone's estimate written down I asked for reasoning. One student said that there were "9-ish" layers and then we counted around 9 or 10 eggs per layer.
I know that some students just guess so I hope that showing them how to reason through to an answer will help them with future estimations. Two students had the exact answer and got chocolate Easter eggs as a reward!
We continued with Speedy Squares from yesterday. I have updated the handout (there were a few "oops" on my part). I collected the times that they calculated to build a 26 by 26 square:
My students did a really good job determining the relationship between time and number of blocks based on their data. I helped them by asking how many cubes they had in their 2 by 2 square and what the associated time was. They ran with it from there.
They needed a little guidance to "design" a house - really just a top view of the floor plan with approximate, but realistic, dimensions. I drew an example to help guide them. Most designs were simple but good. There were, of course, a few crazy ones that were ridiculously large or small.
Once they had calculated their square footage, they used the Lego My House site to determine how many blocks their house would need.
Then they used their equation relating time and number of blocks to find the time it would take to build their house out of blocks. The time they calculated was in seconds so they had to finish by converting it to days so that it was a more meaningful answer.
I have been involved in a lesson study group for the past few months and today was the day that my class was being observed. Monday, 8 am, grade 10 applied class with an extra 4 teachers and the principal - how could that not be fun?!? I joke, but it was fun!
We began with our counting circle. Today we started at 62 and counted down by 13. We stopped after the first student to talk strategy. I thought it was really interesting that he said he just knew the answer, but when pushed a little more he said that he subtracted 10 and then subtracted 3. Bingo! Anyone who didn't initially have a strategy (beyond counting on fingers) now had something to grasp on to. We continued for a few students and I stopped them at student J. and asked what number we would be at when we reached student S. I let them think for a bit then asked for strategy. Someone said that they could multiply. I asked what they would multiply and they said 13 times the number of people. I switched to another student and asked how many students there were from J. to S. and we counted 10. We worked out 13 times 10 (sometime students are reluctant to participate but I wait them out) and I loved that someone said that we could just add the 130 to 42 because they were both negative and we should be at -172 when we reached S. We continued and got there (yay!). Some students are still learning about how to break up numbers to make adding or subtracting easier, but I believe that they are developing this important skill with each week's counting circle.
On to the main activity: Speedy Squares. A little background... another teacher and I created this activity at our first lesson study meeting. I had not used it yet, but really wanted to so our group of four teachers co-planned it last week. Here is the setup:
The answers they had for how long it would take were very interesting. Hours, no, days! In hindsight, I should have had them all estimate how long it would take so that we could compare at the end. Asking how many cubes they would need led to some good discussion. One student said that since the perimeter was about 100, the area would definitely be more than 100. We now had an estimate that was too low. Another said that we would need 2600 squares. I asked what dimensions would give an area of 2600 and we came up with 26 by 100 - so now we had an estimate that was too high. One student was convinced that it was 236 (he did 10 x 10 twice plus 6 x 6). I let another student use a calculator to find 26 x 26 so we now knew that we would need 676 cubes. Actually, each group would need 676 cubes. How many is that? "A lot!" they all said. I showed them my 7 neat stacks of 200 cubes and said that we didn't have enough. What could we do? It took a little bit of circling around ideas to get to a student saying we could build smaller squares and then multiply. The idea here was that we could collect data on the time needed to build smaller squares and then extrapolate (yes, they did use that word!). Okay - now they were ready to go. I showed them the random groups for today, explained the role of each person in the group and set them to work. Here is the handout.
I circulated and made sure that each person had a job and that they understood how to build their squares. The more consistent they were, the better their data was likely to be (I emphasized that they could not switch builders along the way).
The data was really good. Here is one example:
There were good conversations about which variable was independent and which was dependent. They are so accustomed to time being the independent variable that they had to think to ensure they understood that in this case time was dependent. They also talked about whether this was linear or quadratic data and were able to justify to each other why it was quadratic.
They used graphing calculators to do a quadratic regression and used their equation to find the time it would take to build a 26 by 26 square. Their results were interesting - ranging from about 20 minutes to about 40 minutes. We could have talked more about whether this meant one group was building twice as fast as the other and how a small change could have a big impact when you got to a side length of 26 - perhaps next time. Instead, I set up part 2:
I told them about James May who, along with around 1000 volunteers, built a real house out of Lego. Here is the link with lots of fun pictures. That got everyone's attention. We talked a bit about the factors that went into determining how long it would take to build a house out of Lego - they brought up the number of people and the length of the work day. I said that they were going to use the data they had already collected to help them figure out how long it would take them to build a house out of blocks. We will pursue that more tomorrow. I did also show them the Lego My House website (link) which determines how many blocks you need based on square footage. We briefly discussed what square footage means and did one example using the website. The results are quite something!
That was all the time we had for today... more tomorrow. I would like to say that my students were FANTASTIC! They were engaged, on-task and really implementing the mathematics they have been learning. It was truly a pleasure to have visitors.
We did two warm ups today since we didn't have class yesterday. This is the Would You Rather we did:
I wrote some conversions on the board for them as they seemed to need a push to get started. Some of my students spent the entire day yesterday writing the literacy test and were wiped out today...
Here is what they came up with:
They still have some work to do to write solutions in such a way that others can understand what they are doing, but I was pleased that so many groups were able to justify their choice mathematically. The big takeaway here was that in order to compare, the two quantities have to be in the same units. If I had more time I may have asked them to convert the $1.23/l to cost in US $ per gallon.
We did the second warm up quickly by a show of hands. Here was what we looked at:
No hands went up for "always", about half for "sometimes" and none for "never". It was one of those days. We talked about why and that you only need one counter-example to disprove something.
Then it was on to finishing the spaghettini bridges handout. I gave them a few minutes to ensure that they all had an equation to represent their data. They had trouble explaining the numbers in their equation so we discussed it together. I wrote down a few of the equation they found and we went over the meaning of the slope and of the y-intercept and we talked about the need to round values to a whole number.
They continued with the questions on the handout and those who finished started with this handout intended to reinforce what they know about slope and y-intercept. Thanks to my colleague, Michelle, for creating this one.
It felt weird that I hadn't blogged today. My students wrote the OSSLT today so we didn't have math class. I did supervise many of them though. I feel really fortunate that I teach such great kids. There have been times and classes when I would be very happy to have a day away from my students. Today, I was happy to see them even if we didn't get to do math together. It certainly makes it easier to be excited about teaching and feel energized at the end of the day (despite also feeling exhausted!) when you have great classes. That's how I feel at the end of almost every day this semester. That's a good thing.
Today's warm up was this visual pattern:
I explicitly asked them to colour in how they saw the pattern growing. So many of my students just count the blocks and don't even look at the pattern itself that I felt that I needed to make sure the connection is being made. Here is where we went with it:
I love how this is like a little spiral within the cycle as we talk about rate of change/slope and y-intercept every week, even if our focus at the time is not linear relationships. There is also the possibility of doing a little algebra when you can write the pattern in different ways, as shown above in green.
My class spend the remainder of our class time continuing the spaghettini bridges work from yesterday. They got their scatter plots done by hand and put the data in their graphing calculators which they used to generate the equation of a line of best fit. Some got as far as the question asking them to interpret their equation at which point they got stuck. I will pick up from there on Friday. Yes, Friday. I lost 25 minutes of today's class for OSSLT (Ontario Secondary School Literacy Test) prep and will lose my whole class tomorrow when they actually write the OSSLT.
Tuesday means Estimation 180 for our warm up. We did a belated St. Patrick's day one:
The estimates were good and all between 31 and 50. This tied in nicely with the volume of a cylinder work we have been doing. I like that you can see the layers inside the glass.
My students then spent a little time working through some surface area of prisms and cylinder questions. They had to do some unit conversions along the way. I like to incorporate these where it makes sense along the way, not do conversions for the sake of doing conversions (they are in the 2P curriculum).
Next, spaghetti bridges. I actually chose to use spaghettini instead as it breaks a little more easily. Spaghetti is remarkably strong! Students were randomly grouped into teams of three. They made "bridges" by holding the ends of the spaghettini through which they had threaded a small Dixie cup. I had already holepunched the cups for them.
They started with 1 piece of spaghettini and added pennies to the cup until the spaghettini broke. They recorded the number of pennies then started again with 2 pieces of spaghettini, and so on. They stopped when they could no longer add pennies to the cup. I loved hearing them sharing how many pennies they got for a particular number of spaghettini. By sharing I do mean shouting across the room, but when they are shouting about the math activity they are working on, that's okay!
Once they have collected their data they plot it by hand first and then using a graphing calculator. It's not the "cleanest" data so they will likely use the calculators to get the equation of the line of best fit. They didn't get that far today, so they will continue tomorrow.
I would strongly recommend that you have a broom in the room if you do this activity as it gets a little messy with broken spaghettini all over the floor. In my experience some students like to sweep up and are more than happy to pitch in.
Here are the handouts: spaghetti bridges, spaghettini bridges and the extension with credit to Alex Overwijk.
To be continued tomorrow...
Today is Monday so our warm up was a counting circle. We started at 47 and counted down by 3.
When we got to student B we stopped and I asked what number we would be at when we reached student C. Some counted down 7 times to get to 8 and then others shared their strategy of multiplying the number of students to get from B to C (7 students) by 4 (because we were counting down by 4) and subtracting the result from the number that student B said. I hope that exposing students who would count down for each student to an alternate strategy will help them add this to their tool kit.
Next up, Which One Doesn't Belong? I showed them the logo first and asked them which one they thought didn't belong.
The first student said the bottom left because it was pink. The next student said the top right because it was a circle, not a square. The next one said the bottom right because the font was white and the top right also because the font is different. I wanted to make sure they understood that there was a reason for each one to not belong.
Then I showed them their task.
We talked about using volume and surface area as criteria and looked at the volumes of the prisms shown:
They told me that using a cylinder would be a bad idea as both the surface area and volume involve pi which means that it would be impossible to get either surface area of volume to match that of a prims. They said that they could use triangular prisms. I mentioned that they could draw pictures if they chose to use triangular prisms. And then I let them go.
Well, the Monday morning after March break was perhaps not the best time to give a wide-open task. We did solidify understanding of volume. When they had created one rectangular prism with a particular volume at least one member of each group understood that to get another with the same volume they needed to take the same number of blocks and arrange them differently. That was good. Some groups calculated surface area and found that they were all different. Several groups came up with 3 prisms with equal volume but struggled to move forward from there. One group was doing odd shapes and could have matched volume and surface area. Here are some examples:
One group did come up with a proper WODB? set, but I didn't get a picture. This is the "almost there" picture which I will replace with a good one when I recreate it:
I believe this activity could be really good but I would scaffold it more next time. I would have groups randomly choose volume or surface area and work with that as a required criteria. I will work on a handout to go with this where they can list the criteria they are choosing and show their attempts - more of the record of their learning and progress. I also continue to learn how to better serve their needs and bring out the e.
Four days ago I wrote this blog post about using Which One Doesn't Belong? as an activity for my calculus classes. I have had a lot of positive feedback about it and along the way Pam Wilson tweeted this:
and that also got a good response. So I decided to take this on. It's a good thing that I'm on March break as this has taken a few (!) more hours than I normally have "free" during the week.
<insert drum roll>
Announcing the Which One Doesn't Belong? website at http://wodb.ca/
I really hope that many of you will contribute to the site (some of my children already have!) as that is the only way it will grow. I plan on having my students create their own and will add them - please do the same with your students.
You can also follow @WODB?Math on Twitter.
2015-03-20 - Update
In order to help you use and create WODB?, here are some links.
To help you create:
WODB? Template (SMART Notebook)
Graph software that I use (produces very clean graphs)
To help you use (especially if your students have trouble with right & left):
WODB? ABCD Template (SMART Notebook)
WODB? ABCD Template - white background (.jpg)
WODB? ABCD Template - blue background (.jpg)
Please let me know if there is anything else that would be helpful. Also - I am happy to format submissions if you just send in content.
In February, Christopher Danielson released his shapes book: Which One Doesn't Belong? It is fantastic for all ages and if you haven't downloaded it yet, do that now. I'll wait... I put the link up there...
More recently, Chris Hunter wrote this blog post after tweeting out some really cool Which One Doesn't Belong? (henceforth known as WODB) graphs like this:
and this:
He also had graphs of functions and that made me think that I could create some of my own for my calculus classes. We are doing curve sketching right now so it would be perfect! The more time I spent on this, the more value I could see in having students do this so here is what I have so far.
1. WODB? Student have to figure out which graph is the odd one out using characteristics of graphs that we study in calculus like asympototes, max/min, points of inflection, non-differentiable points, etc. The key here is that there is a correct answer for each of the four graphs in each set. I have, so far, made two of these. Making them has been so much fun, but also quite challenging.
2. I have them work in groups to create one or more sets from these graphs. Many of these are the "crazy" ones I found along the way when I was trying to create something specific. They will cut them out and see if they can make it work.
3. They will create their own WODB? from scratch, in groups. This adds another level as they have to come up with the equation for each graph (and there will be heavy use of Desmos, of course). The level of thinking required to get a particular characteristic given the other constraints is great! If they are like me, they will encounter a lot of cool graphs along the way that they may not have ever seen before. They will undoubtedly be making or deepening connections between graphs and their equations in such a cool, different way. Instead of starting with the equation and finding out what the graph will look like, they have to create the equation based on very specific characteristics.
I am so excited to try this out with my students after March break! The discussions should be really rich and interesting. A huge thank you to Chris and Christopher for inspiring me. This is why I love the MTBoS!
