math

“Hey! You taught us that!”

I wrote most of this post about 2 months ago.  It’s been on my mind ever since, even though I decided not to publish it originally.  I’m still thinking about it, so I decided to do some revision and post it today.I taught a lesson this morning on perimeter and area.  We’d talked about area before but perimeter was new.  Provincial testing is starting next week and there is often a perimeter and/or area question – usually a big one – so I always like to spend some time on it really close to the dates of the test.  (Writing that makes me feel like a bit of a jerk for not teaching this because it is interesting and important.  I do think they are (interesting and important) but the timing is due to the test.  I still feel like a jerk but it’s reality, eh?)After doing a whole group activity with colour tiles, and discovering that everyone remembers that 5 of them makes a pentomino AND after discovering that I may have been mispronouncing that word (saying it the American way) for the past 15 years, I sent them off to do some work on their own.  Nothing spectacular.As I helped one student, I asked her, “OK, you’ve made the shape.  We’ve coloured it on the grid paper.  Now, what is the area?”  She quickly figured it out.  And then she said, “Hey, you taught us this!”  Why yes, yes I did.  Approximately 5 minutes ago in fact.  “No, I’m sure it was a while ago,” she replied. She was still on the verge of realizing the lesson and activity were connected.”No, it was today’s lesson. See?  It’s still up on the board.”She looked at me suspiciously, clearly not believing me. She shrugged. “Alright.”I pointed to the next one.  “Can you figure this out for me?”  She did.”Hey,” she said in amazement. “This is getting easier!”I’ve been debating for a few minutes, and I’m going to count this one as a win.  I was this kind of student, except I always knew that the day’s lesson would be connected to the day’s activity. It took me until I was in the last year of high school to realize that the month’s lessons were going to be on the test, and that I could go back through a chapter in the textbook and do a problem or two from each page to help me study for a math test.   I was just trying to get through each class period, never looking forward or back.  I hope that this year I helped this student see that it’s all connected, and that she remembers it in future years!My grade 3 students went to another location to write their EQAO test.  After the first math test in particular, they returned and said something like, “There was a bunch of stuff on the test that you taught us!”   Again…amazing!  I was glad math was coming early on in the schedule so I could reiterate: “That is going to happen over and over on this test.  The whole point is to test you on the things we were supposed to learn.”Sometimes I wonder what children think is the purpose of school.  I know they think about the social aspects, even in the early grades.  I can’t remember that far back so I don’t know what I thought the purpose was.  I do know I was supposed to learn how to read and write and do math.  I know that by the time I was in high school I thought the whole point of high school was to help me get into a good college, and since that wasn’t really something I thought I would do (until I got to the 11th grade and realized I had to go to some sort of post secondary school if I wanted to teach!) I didn’t worry about putting forth too much effort.  I was there for the assemblies and the socializing.I have, for a few years, working hard to help students see the connections between everything we do. I try to be very explicit with them – telling them exactly how things are connected, and asking them how they think things are connected.  Maybe I need to spend  more time talking about the purpose of school.

Data Management, math, Math Workshop, Number Sense & Numeration

Candy Math

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.

 

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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.

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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.

 

Data Management, math

Slice of (Math) Life

Way back in September, I had read about this neat activity from Marilyn Burns called “The Door Project“and had decided to use it to start my school year.  On day one I always like to take the students for a walk around the building.  If anyone is new, or even if they are just coming to my end of the school for the first time, I want to make sure they have a chance to orient themselves, find the washroom and drinking fountain – that sort of thing. On the first day of school, I combined that with some math.  It was our first provocation, if you will.

When we got back to class, I asked, “Did you notice how many different types of doors we have in our school?”  We took a look around the class, and started thinking of ways to sort the doors in our school. Some have windows, some don’t. Some are metal, and some are wood.  Some go to the outside, and some into closets.  I gave everyone paper, asked them to think of 2 or 3 categories to compare, and away we went.  (I’ve condensed it here – we actually spent a whole class period on this!)

It was a disaster. I just looked back to link you to my post, and found I didn’t write about it at all!  That’s how bad it was.  As I recall, we were having a lot of trouble managing our data.  The categories were all mixed up, the tallies were not organized, and nobody could do anything with the information we collected. I had intended to graph our data, but, alas, it was not meant to be.

Today we tried again.

I reminded everyone about this activity, and some sort of remembered it.  We quickly reviewed our door types.  I gave everyone a choice of paper – plain, lined, or graph.  They grabbed clipboards and pencils, and lined up like pros!

We made it all the way around the school, gathering all of our data, in very little time.  Nobody had to shut their door when we stopped to count!  When we got back to class, everyone was able to count their tally marks by 5’s and find their totals in record time!

I reminded them how far we had come.  I reminded them that in September this activity had been really hard, but now it was barely a challenge.  Everyone collected the information independently, found their totals, and is now ready (and able!) to graph it tomorrow.  Now I am sorry I didn’t keep our first disastrous attempt so they could see how far they’ve come! (Who am I kidding…I probably do have it and will find it on the last day of school when I am finishing my clean up!)

What a difference a year makes!

slice-of-life_individual
Just about every Tuesday I blog for the Slice of Life challenge over at Two Writing Teachers. You can read posts from others on that blog.
Geometry, math

Mapping

I sat down with the social studies curriculum the other day, double checking to make sure I hadn’t forgotten anything.  As I looked at all of the expectations, I realized I hadn’t done that much with mapping this year.  We did a bit, but not much.  Then I looked over the geometry things I knew I had left and realized that I really hadn’t done enough mapping.  So last week, we worked on mapping!

After talking about maps and their need to make things really clear and help people find things, I gave everyone a piece of graph paper.  They all drew a map of our school yard.  They did a pretty good job!

Then when we went on a walk for science, I asked everyone to pick up a small rock.  Back at the class, we painted them different colours.  The next day, I asked everyone to think of a place where they could hide their rock, then mark that spot with an X.   We were going outside to hide our rocks in that spot, then trade maps and see if someone could find our rock. Everyone was very excited to head outside and get started.

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I hadn’t, of course, anticipated that the kindergarten would be outside at this exact time. It’s ridiculous because they go outside at the same time every day, but I was so excited about my own thing that I forgot about their thing.  Soon, as you may have guessed, there were 10 or so kindergarten completely over the moon because they’d found a gorgeous rock unexpectedly in the yard.

New plan:  The next day, I drew a map.  We were talking about symmetry, and I had shown a picture of school on Google Earth.  It’s a completely symmetrical building!  My map was an ariel view of the space just outside the fence.  We’ve walked there a million times.  During recess I ran back to a less than secret spot, and hid a bag of candy.

When the bell rang, I met everyone outside.  I had them join their May Learning Partner and share a copy of my map while they searched for the treasure.  Everyone wandered around for a about 10 minutes before someone said, “This doesn’t make sense.  The X is behind the school, not in the play yard.  It doesn’t make sense!”

“Why not?” I asked.  He just kept repeating “It doesn’t make sense.”  Finally I prompted, “Well, are you trying to say we need to leave the play yard and go behind the school?”

He gave me a blank look and then said, “Yeah.  I think we do.”  He then went about trying to see if anyone agreed with him and soon we were headed to the back yard.

“Pick a landmark,” I told them. “Someplace where you want me to stand.” They picked a spot; I reminded them that if they found the treasure they were to keep it a secret and come sit beside me.  Two people actually did that before another person was overcome with excitement and gave the hiding place away with a loud, “Here it is!!”  But they were all in the general vicinity when he did that, so I’m calling this a success.

For the past few weeks I have been experimenting with using the outdoors as the classroom.  We go out to do work we could be doing indoors, but I am also trying to do things that teach about the outdoors, and use the outdoors as a resource, not just a work space.  I feel like this activity would have been less successful inside because we would have all been a bit more stressed about the noise we generated.  (Ok, mostly that would be me.) I also feel like it would have been less successful if we hadn’t been outdoors so much lately, taking the time to notice the yard and the trees, and exploring the landmarks (natural and human made) around us. I feel like putting the math and mapping skills into this context helped everyone see meaning in the activity.  (Another example of how Cathy Fosnot is right about everything!)

Technically I have now accomplished what I wanted to accomplish.  But there are only a few weeks left until the summer break and that is when the really good stuff happens if you ask me.  I am quite sure that I could do this activity a few more times.  My map drawing skills will greatly improve!  I also think I can get some members of the class to draw the maps for other members to follow.  I have 6 grade 3s and the drawing is more for them anyway.  They will love drawing a map to help their classmates get to the ingredients for an ice cream party on the last day!

 

math, Patterning & Algebra

Patterning

So – Patterning.  I’m thinking a lot about this skill and how to make it meaningful for my mathematicians. I’m thinking a lot about its connection to algebra and how to set my grade 2’s & 3’s up for success and start them on the road to algebraic thinking.

I put them to work on Monday.  I put baskets of math manipulatives out and told them to go make patterns.  As predicted, they made a bunch of repeating patterns.  They were quite proud of them in fact.  On Tuesday, we talked about growing patterns.  They weren’t really showing an understanding of reading the pattern left to right, so we had a bit of a chat about that on Wednesday when we talked about shrinking patterns and about how the direction matters. As seems to happen often this year, they were amazed by this knowledge.  I think it will stick!  Here is one of the examples I built to show them that direction matters:

Today, Thursday, I asked everyone to actually put their pattern on a number line.  We have done a lot of work with number lines this year, and with the 100 chart.  I feel like it is really paying off!  I started with some guided inquiry.  What, I asked, would my pattern look like on a number line?

Together we constructed a few:

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Really glad I put this on top of the gross old tape line that children have been slowly picking off for 2 years!

Then I sent them to make some patterns of their own, and map them on number lines.  I didn’t hand them the paper until they had their patterns made and could talk to me about how the pattern was growing and shrinking (by ones, by 3’s, etc.  Actually, no “etcetera” because everyone either did ones or threes, like our example.  I’m not worried though because tomorrow I can tell them they are too good to stick with ones and threes and they need to choose something else!)

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This one led to a conversation about how they actually have 2 different patterns: add 3 tiles to each term, OR add one column to each term. They hadn’t noticed the second and were trying to figure out why I was so excited about it! I feel like there is a double number line opportunity here, but we aren’t ready for that!

I know it might not be right to have favourites, but this is my favourite conversation:

First, there was this:

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The child who made this pattern was insistent that it was a growing and shrinking pattern.  His partners were not convinced.  In fact, they were downright mad because he was so sure and they couldn’t see it. I couldn’t see it either, to be honest.  I wanted so badly to tell him that this was not going to work!  But Cathy Fosnot’s voice echoed in my head, “Productive struggle…productive struggle…”  so I handed him the strip of paper and a marker, and walked away.  A few minutes later, I returned to this:

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He’d figured out on his own that to make a number line his “special stones” needed to be laid out in a straight line.  He was also able to finally show us that the green stones aren’t actually part of the pattern.  They just mark the end/beginning of each set of clear stones.  As soon as it was straight, he could help his partners see his thinking – he could explain it so much more easily.  He’d made it through the struggle and came out successful on the other side.  (He did write in the numbers and finish the number line – I didn’t get a picture though.)

Two others made this pattern. (I’ll add the picture later!)

When we chatted about it, they told me that they knew 22 should come next, but didn’t have enough special stones. This was a huge piece of info for me! I thought they’d just been rote counting, but are perhaps ready to make a line without having to build the concrete pattern first.

I am, however, left with one question:  How does one put a repeating pattern on a number line?  ABABABA patterns, or ABCABCABC patterns – can they be put on a number line?

*update* Today I challenged everyone to try something besides 1 & 3.

They tried 2, 4, 5 & 10.

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…backwards…. We still have work to do!
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This one went on and on, but there are kids in the picture near the end.
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They made a “counting by 1” pattern and had to be convinced to add the second row and then to understand the second row. I had to start the number line for them, but it felt like an appropriate scaffold.
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I can’t show the “100” at the end. But I can say there was a great conversation about why 10 and 50 were so close, but 50 and 100 were so far away! They decided they needed a “refresh” and drew a new underline with perfectly iterated lengths between the numbers. Also, they realized they didn’t need to build every term in the pattern to draw the number line.
math, Number Sense & Numeration

Proportional Reasoning is so cool!

This image is from the “Grocery Store, Stamps and Measuring Strips” Context for Learning unit.  I love this unit and think it is a great way to introduce multiplication to students.  At the end of the unit, students are asked to look at this image.  (Note:  I am only including part of the image because this is not my work and I don’t want to violate copyrights!)

We had done all the proportional reasoning work before this:  figuring out how tall or long everything on a city street (trees, a bus, a few buildings) might be in relationship to 4 foot tall Antonio.  It was time for some final assessment.

“How many design elements are on the curtain?” I asked. One image shows a full extended shade, with 16 (maybe 20?  I forget)  oranges in an array.  The grade 3’s easily told me the total, and explained how they had counted.  Lots of multiplying I was very happy to note.  But they said there were 14 stars on this shade,  and 12 or maybe 18 diamonds* on the other curtains. They debated it for a while.  Then I said, “What if I stretched that shade so it covered the whole window, just like this one with the oranges?”

Blank stares.  The 5 of them looked at each other.  They looked at me.  They recounted the 14 stars and 18 diamonds*.  They  weren’t sure what I was asking.  “Well, we can see part of the window in each of these, and there is light coming through.  But what if the curtain was closed?  What if we could see the whole shade?”

They thought some more.  They used their fingers to measure.  They finally decided that if there were 12 on one curtain, there must be 12 on the other, and 2 groups of 12 = 24. . The roller shade wasn’t as easy, but once they figured out the curtain they had a strategy. “I think,” said one, “that there would be 3 more rows of stars.  So thats 7+7 doubled, plus 7 more.” OH MY GOODNESS!  Proportional reasoning AND partial products???  I could not have been happier.  Everyone agreed, then took turns explaining how they’d counted the stars and diamonds.

I’m calling this unit a success!

 

*We are calling them diamonds instead of rhombuses because they don’t have parallel sides that are straight, and the angles aren’t right for rhombuses.

slice-of-life_individual
Just about every Tuesday I blog for the Slice of Life challenge over at Two Writing Teachers. You can read more posts on that blog.
math, Number Sense & Numeration

We can make our own number lines!

I’m not going to lie:  making the jump to drawing number lines independently has taken a while!  All the grade 2s can explain what I am doing on a number line, and all of them (ok, most of them) can describe a strategy and when I draw it on a number line they confirm I have drawn what they were doing in their heads.  But to make their own?  That’s been hard.

We had completed all of the activities in “Ages and Timelines”, one of the Context for Learning Units, and people were still referring to tools (hundred chart mostly, they they tried to use math racks unsuccessfully) so I wanted to spend an extra week just talking about how to use that number line AND draw it independently.

One day last week, I created some Smart Notebook slides and we all sat down the chalkboards.  Here’s the first one,along with some notation to show that people were flexible with strategies…they used both addition and subtraction to find the answer. I will say that those who added were surprised some had used subtraction, and those who subtracted were surprised that addition could be used, so we had a great conversation about this slide! (Oh, and we are collecting paper towel tubes for science! 🙂 )

Here is another of our questions:

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I wandered around and captured some number lines.  Now, this might not be beautiful to you, but darn it!  It is gorgeous to me!  Look at the line, the iterated jumps, the acknowledgment that 4 jumps of 1 is the same as 1 jump of 4…*sigh*  I’m smiling again just thinking about it!

 

I caught one person who was struggling, and handed that child a hundred chart, with a 25 chart on the back.  For this problem, this child was able to use the 25 chart, but for later problems, had to use the 100 chart, and did! img_0677.jpg

Another beautiful number line… Another great demonstration of the iterated units drawn evenly and there even arrows on the end!

 

One of the most amazing things that happened is shown here.  One child had a 100 chart and was using it well.  Another had no strategy and was looking around the room to see if she was alone in this dilemma.  As soon as she spotted the 100 chart, she scooted over to it.  However, upon arrival, she realized she wasn’t sure what to do.  The other child showed her!!!  (There is a sock on one child’s hand because we use them to erase the chalkboards!)

So, it took us an extra week, and I am quite sure the number of grey hairs on my head has doubled since March Break. Next week, mixed in with some probability to math workshop centres, I am going to be sitting at a table interviewing these lovely grade 2’s to see what they can really do all on their own.  Can’t wait!