If one buys a bag of Sour Patch Kids, will there be an equal distribution of the good colours and the gross colours? Because if I could buy a bag of just red and avoid the gastly green, and blue, I’d be happy about that! The children in my class disagreed and hoped that there would be an abundance of blue. But we all agreed that it should be equal. Time to test it out!
I put a bag of Sour Patch kids on every table, and told the students they could pick their own groups. This doesn’t happen often for us, but I wanted to see if they would distribute themselves evenly. One group of 3 got a whole bag to themselves because nobody else wanted to work with them. That left one big group of 6 to share a bag. I think the kids in that group will think twice before they settle on a group next time (cause you know there will be a next time!)
I was very interested in the strategies students would use. I had predicted that there would be some organizing into groups by colour, and I was right for all but one table. It took them a few minutes of debate before they all agreed to do this. At first, each was starting his/her own groups, stealing from the others to try and create one pile for each colour, except they were all trying to create the pile right in front of themselves. They had 4 red piles, 4 blue, etc. Finally they realized, with a tiny bit of prompting, that one pile for each colour would suffice.
Since September, we have been talking about how organizing into groups of 5 makes counting a lot easier. But…still…lots of kids were counting by 1’s. *sigh*
Over time, however, they switched to grouping, usually by 2’s, but at least it was grouping. One group, the group I would least expect to struggle with this, organized each colour by a different number. Then they couldn’t figure out how to make that into a graph. We had a very interesting conversation about this, so I’m counting it as a win.
The graphing was fun to watch too. We’d talked about how we don’t always have to count by ones on a graph, but we clearly have some growth to do in this understanding. Though I’d given them a task that required more than the number of rows I had given them on the graph template, they still thought they could just fudge it.
Let me interpret this for you: this group had 21 Red, 17 Green, 15 Orange, 13 yellow, 22 Blue, and 1 split (I have no idea why, but one child was obsessed with the possibility that 2 colours might have melted together and therefore 1 SP kid fell in to 2 categories AT THE SAME TIME! This did not actually happen, but this student really wanted me to believe it was not only a possibility, but an inevitability!) I have just one picture of this to share, but it happened all over the place. I eventually pulled them together and reminded them about choosing a scale. The grade 3’s got it after that, but the grade 2’s not so much. It’s not one of their expectations anyway, so I’m not worrying about it. They could read the graphs the grade 3’s created, so we’re good!
This week, assuming the one day I have to do some actual teaching actually doesn’t get interrupted by the unexpected, we are going to find out if we get more caramel popcorn or more cheese popcorn in a bag of Chicago Mix. We are going to compare the store brand to the Orville Redenbacher brand and see if one is more even than the other. We are also going to compare the Humpty Dumpty brand “party mix” and the Doritos brand “party mix” to see if I am truly being ripped of, as I suspect, and getting more than half a bag of pretzels. I firmly believe there should be a “No Pretzels!” option here, just like there is a “No peanuts!” option for a can of mixed nuts. Am I alone in this?
I used to do food math all the time. Froot Loops, Smarties, M & Ms – they all make great math manipulative. But for several years I have had students with food allergies and bring any sort of food into class was so stressful for me that I avoided it at all costs. This year I have very little of that to contend with, so I’ve been going for it.
Oh…almost forgot! One group got two half Sour Patch Kids. They weren’t sure what to do about it. We had a great conversation about the 2 halves making a whole. They were reluctant to believe me, which just shows that I didn’t do quite enough with fractions this year. I’ll have to rectify that next year.
Believe it or not, playground equity is something my students have talked a lot about. At my school, we have a “Primary” playground and a “Junior” playground. The Primary students think the junior side is better, and the junior students agree, but sometimes wish they could go back to the Primary side to play. I mention this because the context of a community playground and having equal play space is something I know they could relate to. This week’s problem can be found in detail here.
Trying to solve this problem reminded me how much I still struggle to see and draw a model of my thinking. I really didn’t have any trouble with those rectangular subs! But 3/4 of 2/5 and 2/5 of 3/4 sort of messed with my mind. I wanted to work it out with some manipulatives at school today. I was going to use colour tiles. Unfortunately, I have this thing called a “full time job” that tends to interfere with my blogging life and I didn’t get to it. Add to that the fact that we are right at the beginning of the “Measuring for the Art Show” unit and any spare time I have to work things out for my own understanding is being devoted to that right now. (I’m going to blog about that unit at the end of the week! Or I’m going to work on report cards. Or maybe both.)
I was able to work this one out with the numbers though. This is my first attempt at drawing a model:
I was confusing myself, so I went back to numbers. After, tried to model it again (on the top sheet.)
Working this problem took me right back to school. I was forever answering the question I thought was being asked instead of making sure I understood the problem. Time and again I would have a test returned to, read a problem to see if I could figure out what I had done incorrectly, and discover I simply hadn’t answered the right question. With age comes wisdom and this time I went back and listened to the teacher set up the problem a second time before I tried to solve it. The first time I thought I was to find 3/4 of the whole and 2/5 of the whole and compare to see if the playground and pavement were equal. Now I know what I have to ask myself if the question makes sense, and this one did not. After listening the second time I knew I was to find 2/5 of 3/4, and 3/4 of 2/5 and compare which is a different question!
Without going into too much detail, I’ll tell you that this turns out to be a common concern amongst my colleagues. Students answer the question (in reading, in writing, in math, on the provincial assessment) they think was asked, not the question that was asked. I love the way putting the problem in an understandable context, and presenting it in a way that encourages students to really discuss the problem until they understand what they need to figure out, helps us make sure we are really assessing what the students know about math, not about following directions. And I also love that the conferring process can focus on straightening out misconceptions so the student can go forward with a set of skills that help them make sense of problems, not just find answers.
And I love that if I was actually going to teach this unit (I won’t because I don’t teach the grades these units are recommended for) there are some models modelled in the books and I could learn how to draw them before I teach the unit to my class.
Found this on Twitter the other day, and decided to give it a try. We are “finished” with some learning about adding and subtracting and unitizing and counting. We need to move on – both for my sanity and because there are other things that need to be covered in the curriculum. We will circle back to this in the new year, and we’ll keep practicing when we do a daily Number Talk at the beginning of our day.
I’d decided to move on the geometry, mostly because I’m a sucker for all the Christmas tie-ins. I love making 3D shapes into ornaments and decorations! But then this fun fraction activity popped into my life, and I thought it would make a nice transition from Number Sense into Geometry.
On day 1, a shortened math period due to irrelevant circumstances, everybody cut out their pieces. (As an aside, I know that cutting isn’t math, but we have some fine motor issues, and it’s really good to for us to cut as often as possible. Also, I have a life to live and it’s takes 20 kids 15 minutes to cut these out so I let them!) I started by holding up the “whole” and one of the fourths, and then asked the recommended question: What do you see?
They named a bunch of stuff, unrelated to math: I see white and yellow, I see a fried egg. Then they moved on: I see a small square and a large square; they both have 4 sides, and 4 corners; they are the same but different. Then I let them go play with their shapes. We came back and talked about what we saw: the different shapes in different sizes, teh same shape in different colours, etc. Then we had lunch.
On the second day, I asked them if they could use the paper to show half. I was interested in their understanding of half, and also, would they see that you could make half of all the shapes, not just the whole. While I walked around and talked to different groups, I was so glad that I know that I should ask them to explain their thinking. Some of them had very unexpected answers that showed partial or complete understanding in a different way than I was originally thinking.
All of these came with justifications for how it shows half. One is half yellow (front) and half white (back), one has purple lines dividing it into sections – half the sections are yellow and half are white, and one uses the purple “lines” to divide a design in half.
We talked about symmetry, which I hadn’t intended to do, but it naturally fit. I really thought I could do this activity and move on to fourths and thirds. It was clear we weren’t ready. Luckily I am SO organized (haha) I was able to grab this book off my shelf:
It’s about a brother and sister who do not want to share fairly, but an unspecified adult (mom? dad? babysitter??) tells them to share. They each get half of the pizza, the dessert, the juice. Then I sent everyone back to show half again.
We had better results, and each came with a really good justification. Then, just to make sure we all understood that each shape could be divided in half, I sent everyone to fold each piece in half. Which they did.
My favourite was this one:
He explained that both show half because the one folded twice shows half of half. 🙂 That’s going to get us started on fourths!
On Monday we are going to find other things in the room and in the school that show half, and then we’ll move on to fourths.
Wednesday is pizza day at our school. As I prepared to hand pizza out to my student, a little voice whispered: “#mathematizethis!” So of course I took a picture before passing it around. (To clarify, taking pictures of food is not a thing I usually do!)
Just for good measure, I took this picture too, and I was glad for it.
Today, Thursday, I used these pictures for a Number Talk. I put up the pizza picture and said, “What do you see?”
There are 2 pieces of pizza missing. (This might not seem like a big deal, but it showed me that they could extrapolate the information by comparing the 2 pizzas. This came about as people said, “There is one whole pizza and one part of a pizza.” And “There are 10 pieces in a whole pizza.” And “I can see 5 cheese, and 3 ham, so two must be missing because we need 10 for a whole pizza.”
At first, they were all about the counting. (We have done only 1 other fraction Number Talk.) One girl said “I saw 1,2,3,4,5 on one half, so that means 5 on the other half, then 5 on the other half, so that’s 15. Then 3 more, so there are 18 pieces of pizza.” UNITIZING!!! AND SUBITIZING!!!
Soon after that, the fractions started rolling in.
3/18 are ham
15/18 are cheese
2/18 are missing. (We talked about how it was really 2/20 missing.)
I asked, “What fraction of the pizza would a person have if they only had 1 piece?” I am super excited that 1 person said 1/10 since we’d established that 10/10 is a whole pizza. Nobody argued for 1/18, and nobody looked confused, so I call that a win.
There is one milk missing. 9 can fit in the box.
1/8 of the milk are plain. 7/8 are chocolate. They went to fractions much quicker the second time.
I asked what they would do if a class ordered more than 9 milks. Most agreed you could stack the milk on top, but some thought a second box would be required.
Overall, I am super-nerdy excited about how this went. It’s a contextualized problem, and one that they will encounter again because pizza comes every week. I am also thinking this could be a Guided Math centre question at the beginning of every month. The number of children who order pizza tends to change a bit every month. AND, we could take pictures of the pizza in other classes.
I am going to teach fractions earlier next year just so I can do this! And then I am going to teach graphing so we can graph our results. I just decided that right now, so if you don’t see this reflected in my long range plans, please hold me to it!
I taught 2 fractions lessons, and they went really well. I came away from them thinking, “This is going so well I could probably stop right here! They get this!!!”
And then we did a few things, like had a long weekend, and when we got back to our math I realized they didn’t have it after all. I mean, they probably could have done a few worksheets, and probably would have coloured them in correctly, but I am not much of a worksheet teacher.
Instead, I asked them to do some drawing. I wrote this on the board. A decided to just give them paper and set them to it for two reasons: 1) I thought they thought fractions were easy-peasy. 2) I wanted them to do more of the thinking that I did.
I wrote this on the board:
The bottom line, circled in green, was there as a challenge for anyone who actually did find the work easy-peasy and needed a challenge.
The little people were my feeble attempt to get them to focus on the math, not the art. (When will I learn?! Everything is art, right?)
Here are some examples of what happened:
Most did not take the challenge. It was too challenging. Fine. I can live with that. We haven’t done that much work with equivalent fractions, so it makes sense.
The picture with the pink dresses belongs to a child who wasn’t sure how to sort out the eye issue. He kept repeating, “I just don’t get it.” and he kept repeating the question: 1/4 blue eyes. Finally I said, “Well, you know, 1/4 have blue eyes, and the remaining 3/4 have eyes that are a different colour. Like green eyes, or brown eyes.”
That was all it took. I felt like that was a good prompt that didn’t give away any of the answer. I know him as a mathematician, so I knew that what he was really trying to say was, “This question doesn’t make sense to me.” And I knew that he probably didn’t need a math hint. Luckily, the first thing I said to him, that green and brown are also eye colours, happened to be the thing he needed. If only I was always this lucky!
As a follow up, I sat everyone on the carpet the next day. We were in a big circle, and I put 4 people in the middle. I started describing them using fractions. “Half of these people are boys. 1/4 of them have a pink shirt. 2/4 of them have long pants.” Then I invited others to come up with fractions to describe the group. After a few, we switched to new people in the middle. After 4 or 5 things were said about this group, we spotlighted a different group, and so on until the whole class had been in the middle once. They started to get more creative with their noticing as we went along, and after the first group, I didn’t give any more answers.
I sent them back for more drawings. I gave them three options, and told them to pick two. Some picked all three.
Draw a group of people.
1/2 are tall.
1/3 have hats.
3/4 have long hair.
They did OK.
Here are the grade 2 math expectations that mention fractions:
– determine, through investigation using concrete materials, the relationship between the number of fractional parts of a whole and the size of the fractional parts (e.g., a paper plate divided into fourths has larger parts than a paper plate divided into eighths) (Sample problem: Use paper squares to show which is bigger, one half of a square or one fourth of a square.);
– regroup fractional parts into wholes, using concrete materials (e.g., combine nine fourths to form two wholes and one fourth);
– compare fractions using concrete materials, without using standard fractional notation (e.g., use fraction pieces to show that three fourths are bigger than one half, but smaller than one whole);
I felt like we had done that on the first day. I felt like all of my grade 2s and most of my grade 3s were good with this. Except when I looked closer, especially after the second time around, I realized they didn’t understand “the relationship between the number of parts of a while and the size of the parts.” I mean, they seemed to get that when it came to fractions of a whole, but NOT for fractions of a set.
Here are the grade 3 expectations that mention fractions: (I am looking at number sense only. There is related stuff when you look at time and money.)
divide whole objects and sets of objects into equal parts, and identify the parts using fractional names (e.g., one half; three thirds; two fourths or two quarters), without using numbers in standard fractional notation;
I feel like we have this now. If I took them outside and said, “Fill this bucket half way up with water.” They’d do it. And if I said, “Do you want 1/4 of the pizza, or 2/4 of the pizza?” They could give me a reasonable answer like, “2/4. I am starving!” or “1/4. I have other stuff in my lunch.”
OK you guys! I am so excited about my math class on Friday. Perhaps too excited?? IMPOSSIBLE!
First, I’ve been thinking a lot more about the learning that takes place when students develop a model on their own.
Second, last year I worked a lot on the open number line with my grade 3/4 class. I thought we were doing so well! Then we started working on fractions and they didn’t know where 1/2 fit on the number line. They could identify 1/2 of a set, or half a whole, but 1/2 of a distance was too much. I was going to say “at first”, but I never really felt like they owned that piece of information. They just sort of borrowed it from me for a while and then silently put it back on the shelf, I think. Though this is giving them some credit for putting things back where they found them, and that might also be misplaced. So…this year I wanted to add fraction of a distance to our study of fractions.
Today I wanted to start talking about fractions. Yes, this is a bit last minute. (I’m not going to tell you the whole story of why this happened.)
So how to develop the context and ask everyone to develop their own model?
Typically, I would gather the class, show them a half, and them let them go make some halves. Then we’d do fourths and thirds, and then we’d try out other things that make kids happy like 5ths, 6ths, etc. I would do fractions of a set one day, then fractions of a whole another day, then practice them a variety of ways – sometimes alone and sometimes side by side. Then we’d make some connections to half of dollar, or half of an hour, and then we’d be done with fractions. We are a grade 2/3, so equivalent fractions would show up in there, and so would mixed numbers, but that is not our focus.
But new-me wanted them to develop more of this on their own. All of it, I hope.
Students in my class sit at tables. There are 4 of tables, so 4 groups. I put colour tiles on one table, pattern blocks on another, LEGO on the third, and snap cubes on the last. I told them to go play with the blocks for 5 minutes, and then we’d start. It’s May, and we’ve used the math tools a lot, but they seem to still need to just play with them for a few minutes every time. If you can’t beat them, meet them where they are, I always say. Or at least I say that sometimes.
After 5 minutes, I had them all join me on the carpet. I wrote 1/2 on the board. (I had 1 on top of 2) “What is that?” I asked. I got a variety of answers. The 4th or 5th student called it “half” but others called it 1 divided by 2, 1 out of 2, a 1 and a 2, and 1 = 2. I decided to give it to them. I wrote “half” under the fraction. “Oh, half, right,” a few responded. “Now, go to your table and use the tools I put out for you to show me half.” Away they went.
Here are some responses:
I was surprised at the number of students who weren’t sure what to do.
One student had this:
I asked, “Is this half?”
She said, “I’m not sure.”
I turned to her neighbour who had this:
I asked, “Is this half?”
He said, “Yes.”
I asked, “Can you explain to (her) how you know it is half?”
This was really interesting. He kept saying, “Well, it’s half. See? It’s half.” Then after a few repetitions for this, he used his hands to separate the two halves and said, “See? Half are here and half are here.” He’d separated them by colour. He wasn’t sure how to explain it, even though he knew he had half.
I asked her if she could see it, and she could. So I challenged her to make her own.
At the pattern block table, I took this picture:
“No,” the student said. “You have to take the picture from above. Then people can see that half is yellow and half is red.”
“What about this one?” I pointed to 2 red trapezoids.
“It’s cut in half by that line.”
Here was a very interesting one:
When I asked him to explain to the class, he said, “Well I have 4 blocks. 2 are over here, 2 are over here, so half are in each place.” I was really surprised by this because I thought he was showing me two models: one half orange, half red, and another that was 2 halves making a whole. I explained this to the class and they got it both ways.
His way: half the blocks in one place, half in the other.
My way: 2 models, each show half.
One of the people at the connector cubes table had this example:
He had set it up two ways. First he said, “Half are here and half are there.” Then he changed it and said, “I mean half are yellow and half are something else.” I think he was surprised when I said that each was a different way to show half and he’d been write two times with two different answers. They are used to me saying this kind of thing, but I think he thought he was wrong the first time.
After we congressed this, I sent them back to their tables. “Now that you have seen some other models, go show me 5 ways to make half.” And they did. I mean, AND THEY DID!!!!!
We had to skip math on Thursday because of an assembly, so I decided to do double math today. I told them that after recess I’d have different blocks on their tables and they could use a new tool to show me half. They came in and got right to work. That’s a lie. They got to work eventually. But they all showed me half in a few ways with a new tool. Photographic evidence:
One child had this, which she explained like this, “When my mom says, ‘Eat half.’ this is how much I would eat.” I was happy to see that connection to fractions in her real life, even though she didn’t quite understand that each of her piles should have an equal amount in them. In reality, who counts every exact pea? But I think that might start happening, and I should probably call her mom and apologize in advance.
I then read them a great book called, “The Cookie Fiasco” by Dan Santant. It’s part of the “Elephant and Piggie Read” series. It’s funny! Four friends, 3 cookies, how can this be solved? I read up to the part where the friends are completely befuddled about how to solve this problem. Then I gave them 3 square cookies made of paper, and asked them to figure out what the friends could do. In the book, the nervous hippo had already broken the cookies in half, but there still weren’t enough cookies for everyone. So, all of my people cut their “cookies” in half. Most of them cut them again, and within a few minutes everyone had figure this out.
Congress: “What did you find?”
Everyone agreed that everyone got 3 pieces of cookie. I asked, “Ok, so how much cookie did they get.” This took some discussion. I had to reassemble them into wholes, then move them around until someone said, “Well, they are quarters.” and then we counted by quarters, “1/4 + 1/4 + 1/4= what?” Finally someone said, “Well, it’s 3 quarters.” and the lightbulbs that flashed on above all the heads nearly blinded me! It was awesome! They got it! I mean THEY GOT IT!!!
Then I showed them how some of our classmates had cut the “cookies” into squares and some into triangles and some into rectangles. It took a bit more discussion, but we all saw how they all ended up with 3/4 as the answer to how much cookie each friend got. I told them they could put the cookies in the trash, and about half of them put the in their backpacks instead because sometimes trash = more important to take home than our actual work or, you know, report cards. You’re welcome, Moms!
Finally, fraction of a distance Last week in gym I asked everyone to run halfway across the floor and stop. It was trickier than you might think, especially given the fact that there is a giant black line halfway across the gym. But then I said, “From there run halfway to the wall.” and this time there was no line to signal halfway. Some of them did it, and some of them didn’t. We ended our super day of math by going outside into the glorious sunshine. I had two hula hoops and asked kids to stop halfway between the two. More of them did than didn’t. Of those that didn’t, most overshot the mark by a good amount. We’ve obviously got some work to do around fractions of a distance. I’m thinking they need to do it in a place where they can’t really run. I think they are going too fast and 1 meter into the full out sprint they have forgotten they are supposed to stop halfway, then one classmate stops, or they see the end goal, and something deep inside them says, “Wait? Was I supposed to stop somewhere?” and by then they’ve gone more than half way.
Tuesday we’ll do some work with writing down fractions, looking at the models and pictures and writing the number. Then on Wednesday we start EQAO with the math section and I am super paranoid about being accused of cheating so I don’t teach math during the 2 days when we are doing the math sections (which is where we begin this year.) That gives me some time to think about a few more ways to connect their fraction learning to their past learning.