math, Number Sense & Numeration, Number Talks

Estimating and Number Lines

This week we were focused on two things:  estimating stuff and counting to see if our estimate was close.  I’m feeling really good about it!

There were some fun activities we did that I think really helped.

First, I had some small jars full of stuff.  We started the week by reading a book about estimation.  Then I held up one jar at a time and asked everyone to estimate how many were in the jar.  After the second one, I sent them to their tables to practice.  They had a great time estimating how many paper clips, beads, erasers or rocks were in each jar.  We did this on Tuesday and on Wednesday.  (We didn’t have school on Monday.)

We also played a game that I first learned this summer during a free week of online PD offered by Christine Tondevold.  There were new webinars every day, and one of them featured Graham Fletcher.  He dropped counters into a container but students couldn’t see what he was doing.  They had to rely on their hearing to count along and then identify how many were in the container.  We did this each day last week as a counting routine at the end of the lesson.  On Thursday, I started with this activity.  We had estimated enough times this week that I was ready to take it to the next level.  I pulled out one handful of counters. I asked the class to estimate how many I had.  They turned to a partner and discussed, then I constructed a number line as we went along to show where everyone’s estimate fit on the number line.  They were all convinced that I had no more than 12, so that was the last number on my line.  Then, I dropped them while they counted.  I had 17, so we had to stretch out the number line.  Next, I took 2 handfuls and asked them to estimate.  They did a quick turn and talk.  The first child I called on said, “Well how much is 17 and 17?  Cause if you can fit 17 in one hand then you probably have double that amount.” I was excited about this response!  The child is in grade 3, and I thought this was prefect reasoning. I annotated his explanation as he explained how he added 17+17 (sorry…had to erase that before I got a picture.)  We all agreed that it was pretty likely that I had 34 in my hand.  We started to count.  I had 37, which we all agreed is pretty close to 34, so 34 was a good estimate.

After we had counted them, one of my friends suggested that maybe I had 47.  Win some, lose some, right?  But I put that on the number line and we discussed our answer of 37 again and I think that friend understood that I had 37 and how far away from our estimate 47 is.

An interesting thing happened while we were counting.  Thirty-seven is a high number for some kids to hold in their head so they were using fingers and counting out loud to aide their working memory. I wanted to talk about this strategy so that those who hadn’t done it would know it’s a strategy they could use.  One friend said that he had actually only been able to count to 10 on his fingers at first so each time I got to ” a group of 10″ (“Like 10, 20, 30…like that!”) counters he held up 1 finger. He knew he had 3 fingers and that is 30 counters, then he just had the 7 to go with the 30.  I tried to draw that thinking too.  This strategy actually lead really nicely into our lesson.  We are working on the “Collecting and Organizing” Context for Learning unit next, and counting stuff is the beginning of that unit.  He introduced to us the idea that things can be put into groups of 10 to help with organizing and counting.

We did a bit more counting on Friday.  Everyone tried to make groups of ten, but many aren’t yet convinced that this will help.  We’ll dive deep into this unit whenever we go back to school (hopefully Monday!) and I feel confident they will have it by the end.

We finished on Friday with the “Flying Cars” Esti-Mystery from Steve Wyborney’s new Esti-Mystery set.  It was a huge success and the students were so excited that their estimate was so close to the real answer.  I was so excited that their ability to both reason and explain their reasoning had come so far in just one week.

Up next on the spiralling document I have been following is more counting (forward to 100 for grade 2 and 200 for grade 3).  This week we did some hundred chart puzzles.  I had some made with 101-200 charts for the grade 3s to work on.  They all did pretty well.  They can now become a centre when I need everyone to do independent activities while I run Guided Math groups.  This will become really important in about 2 weeks (depending on if/how long schools are closed for the strike) when I want my grade 3s and grade 2s working on some different units. We also need to be able to count backward (from 50 for grade 2 & 3, and from 500 by 100s for grade 3s) so that will be the focus of our counting routines next week.

And look….nobody went to the washroom during our Number Talk that day!  Interpret that as you will.

 

Data Management, math, Number Talks

Sorting and organizing

I am using the spiralling document found on EduGains to work through my math program this year. The first week is meant to be devoted to sorting and organizing skills from the Data Management strand of our math curriculum. I decided to get started on Monday even though we were, ironically, waiting for information about re-organizing classes because of our enrolment numbers.  Because I will be coming back to sorting and organizing many times, I didn’t worry about doing this without a few students who will be joining our class on Monday.

I started with some “Guess My Rule” slides I made on PowerPoint. I had enough to do 3-4 each day this week.  As we discussed them, everyone tried to “guess my rule” and we discovered that there could be 5 or 6 different guesses and all could be correct.  What mattered here was the ability to justify one’s “guess” about my rule. This is a really important skill that everyone needs early on!

Next, out came our math tools.  Everyone worked on sorting the tool of their choice.  I haven’t started with a “guess my rule” game before and I was pleasantly surprised to find that my students did not focus on sorting by colour.  This is often a problem.  I have to spend a lot of time getting them to think about other attributes.  I feel like the “guess my rule” activity set them up for success because they were already trying to be very clever and “trick” their friends.  Nobody is tricked by colour, so we (most of us) tried to think more deeply about our tools.  Those who needed prompting quickly moved on past colours. The work we did last week on how to use the math tools properly also paid off!

Some of the tools I chose for them to use included the necklaces, coloured glass marbles from the dollar store, attribute blocks, and base ten blocks.

The necklaces were sorted by colour, but also by bead type.

Attribute blocks were sorted by colour, shape and thickness.

These glass beads are the best money I’ve ever spent at the dollar store.  They are very versatile: bingo chips, counters, sorting tool, and some kids just love playing with them for no reason at all!  They can only be sorted by colour…or so I thought!  Turns out they are not all exactly the same size.  The  new ones I bought this summer, 10 years newer than most of the others, are slightly larger and a slightly different shade of blue and green.  I’ve lost a whole bag worth of red over time.  They are very popular!

We did this sorting activity for 2 days because we had some other interruptions that shortened our math class.  On the third day, I asked them to use any material in the room to create their own “Guess My Rule” page.  Here are a few.  Can you guess the rule?  Once again, there were many possibilities guessed, so I know students were looking at many attributes of the materials they chose.

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Bright colours vs. dark colours

 

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None of us got this one! “I will not buy containers without matching lids!”
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Some guesses: flexible and not flexible, longer than a metre & shorter than a metre, colourful & wooden, inches and centimetres & only centimetres.

This activity was a lot of fun.  We had a field trip on Thursday, so we were doing math in the real world instead of in the classroom.  On Friday, the students had all been re-sorted into the class they will have for the rest of the year.  I elected not to do the math I had planned because I want to do it with my “new” class (about 1/3 are new to me since the first day.) So here I am, going into the 4th week of school and already “behind” where I thought I’d be.  Typical!  But I am not worried because I will come back to those activities later when sorting comes up again.

Next week:  Counting! This is where I would typically start.  I’m glad I started somewhere else because I have seen everyone as mathematicians aside from their ability to count. I need to start my math interviews as well, so some of the centres I had created for organizing will work as activities  to keep people productively engaged while I am doing individual assessments.

math, Number Sense & Numeration, Number Talks, Problem Solving

This past week at OAME, I was pretty focused on spiralling the math curriculum and on finding more problem solving tasks to use with my class.  I find that a lot of the tasks are a bit beyond our reach, which is frustrating.

One of the things I was introduced to was Graham Fletchy’s 3 Act Math Tasks. I so appreciate when a person is willing to create a resource like this and then share it with the wide world!  While planning my week, I picked out a few in particular that I thought would engage my students, while also spiralling us back to some Big Ideas we haven’t worked with for a while.

Today we did this task, called Snack Machine.  We have had a lot of practice working with each other.  We have had a lot of practice thinking about a strategy to use to solve a problem.  But this task, and others on the site, really allow for a lot of divergent thinking.   There are multiple entry points, and multiple paths to a solution.  It’s great!

In the Snack Machine, a video shows a girl buying something from a vending machine.  We watched, then talked about it, then watched again, then talked again.

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At this point, the children didn’t know what the problem would be.  They were simply looking at the video and mathematizing it. The discussion started off with someone suggesting that the girl in the video looked at her change and was disappointed.  That definitely had people thinking about why.  I got a kick (as my grandma would say) out of one of them suggesting that the machine scammed her.

After the second viewing, we had things to add.  We heard 4 coins fall, so which coins might they have been?  That lead to a long conversation, mainly because 4 toonies would make that sound, but would be an awful lot of money for a bag of chips, but 4 nickels wouldn’t really make sense either.  In act 2, there is a picture of the vending machine showing us that the chips actually cost 60 cents. Then another video shows the machine counting up the money.  We added that to our board:

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Sorry about the cropping – I have written the initials of the person who contributed the idea and don’t want to publish them. Also, SO THAT’S WHERE MY ERASER AND RED MARKERS HAVE BEEN ALL DAY!

After this, I sent them off to figure out the coins she must have used.  Amazing things happened!  After everyone had a pretty good shot at solving the problem, I showed the final video.  In that video we see that the change was 2 dimes.  They used this to confirm that 80 cents had gone in, 20 cents had come out + 60 cents worth of chips, so it all made sense. No scam!

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This friend needed help putting in the + sign, and also knowing where to put the $ sign.

 

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This friend needed help knowing that she’d arrived at the answer. Annotating our thinking continues to be a skill we need to practice.

The money used was American money, and of course a little bag of chips would cost more than 60 cents in a Canadian vending machine. But I told them the two coins we saw were dimes, and that was good enough for them.

Yesterday we worked on Sliced Up, which had us estimating, thinking backward from oranges cut into wedges to whole oranges, and finally multiplying (5 whole oranges, 4 wedges from each orange so how many wedges in all?) For tomorrow, I am debating between It All Adds Up which is a nice money connection to Snack Machine, and The Whopper Jar   which is a nice follow up to the estimating we did in Sliced Up.  Whichever problem doesn’t make the cut tomorrow will our Monday task.  I’m learning toward the money problem because I have a bunch of activities we could do as Number Talks to stretch that learning all week.

It’s EQAO week at our school and I like having some fun, confidence building task for my students to work on.

math, Number Sense & Numeration, Number Talks

Slice of Life: Multiply the Money

I wrote yesterday about a Number Talk I had worked on with my class.

Today I extended that activity by making an array using money.  I used the Mathies money tool to display an array made of nickels, then I asked, “What do you see?”

We built this display of our thinking a bit at a time, so I am sure it made more sense to us than it might to you!  Someone saw 45 cents.  Then someone else saw 15 cents (3x5cents) and then 3 groups of 15 = 45.  Some saw the array 3 x 3 = 9 nickels all together. I pointed out 3 x 15 and 9 x 5.  Thanks to our work with fact families, a few realized 45 “shared by” 3 people means they get 15 cents each.  Seriously!  They didn’t just point out that 3×15=45 so 45 divided by 3 = 15.  They actually explained what was happening! (Full disclosure: not all of them.)

After this conversation, I cleared the board and made an array using toonies. Now, you might have been expecting me to use dimes, but I thought the $2 coin was better considering that we haven’t talked about multiplying by 10 and my grade 2s would get a lot more out of the conversation if we were thinking about multiples of 2 rather than 10. I will probably give 10s a try tomorrow on their own.  After our success today it seems like a good way to get everyone to start thinking about multiples of 10. I’m looking forward to it!

 

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, Number Talks

What you see isn’t what I see

For Number Talks this past week I made some arrays on Smart Notebook and projected them on the board. We spent time each day talking about what we noticed and thought. Early on the fact families emerged. I’m glad because we’ve just finished up some multiplication and division learning and I was glad to see this being put into practice.

On Friday I displayed the picture below:

As you can see there were multiple ways to see this picture. Immediately people saw the array of 4 rows with 3 bikes in each.

One child kept insisting it was 2 groups of 6. It took a minute for him to convince his peers, and I had to help by circling the 2 groups. I’m glad we took the time to let him explain! He was clearly showing some beginning “partial products” thinking and I wouldn’t have known this if I hadn’t probed for an explanation.

Finally someone started talking about the bike tires. I don’t have a photo of that annotation, but it was interesting to see how the students went in to figure out that 12 groups of 2 = 12+12 = 24. Of course they were able to complete the fact family. At this point very few were counting by one’s. I was quite happy about that for sure!

It took me very little time to use the tools in Smart Notebook to make these arrays. I’ve definitely used Mathies for this as well, but the pictures in Smart Notebook led to a deeper conversation. This week I need to find some cars with 4 wheels to expand on our conversation.

This work also builds on the “Eyes on Math” number talks and picture-based number talks we’ve been doing all year. Tomorrow we are going on an array hunt around the school with our iPads. That activity has been preempted a few times but I think it will end up being a better activity now that we’ve discussed the picture arrays a few times.

math, Number Sense & Numeration, Number Talks, Patterning & Algebra

Counting

Years ago I bought this treasure at a yard sale for $1:

There are well over 100 beaded necklaces in that bin!  I use them exclusively for math, though I definitely have had some children in the past 10 years who would have loved to wear them, or just run their hands through them over and over.  (It does feel nice!)

I bought them to use for a specific counting game.  I didn’t know about this game until I came to Canada.  Seemed every Core French teacher I ever worked with loved this game, though now that I am in an Immersion/English dual track school it isn’t as popular.  In French, this game is called “Dix” or Ten. The class sits in a circle and counts to 10, each saying one number.  Whoever says ten gets to sit down, and the game is played until there is just one person left.  I bought these necklaces when I was teaching kindergarten.  I didn’t want anyone to get out because the “out” people aren’t getting any practice.  I feel like I may have read about this in the Effective Guide to Instruction in Mathematics, but I can’t be sure.

Over the years, this game has evolved. I now use it for skip counting by all sorts of numbers: count by 10s and whoever says 100 gets a necklace, count by 5s and whoever says 50 gets a necklace, and so on.  I am getting ready to start some multiplication with my class after the March Break, so last week I pulled out the necklaces and we started using them every day for a few minutes before the mini-lesson.

On Friday, I asked everyone to count by 10s, and whoever said 30 got a necklace.  After we’d made it around the circle once, I asked them to talk about the pattern they could see.  Several realized there was a pattern.  It was identified as a “no, no, yes” pattern an “ABBABB” pattern, and a “skip, skip, yes” pattern.  Finally someone said, “It goes, 1, 2, 3! 1, 2, 3!” (emphasis on the 3!) I asked what would happen if we counted by ones.  Sure enough, every time someone said 3 s/he was wearing a necklace.  Then we counted past 3 to see if the pattern would continue.  I scribed on the board for them so everyone could see the numbers while we counted, and then I circled the numbers that corresponded with a person wearing a necklace.

Sure enough!  The pattern continued.

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We talked about how we could use what we had learned to count by threes, just like when we count by 5s or 10s or 2s.  Everyone was amazed, and several were happy to share their strategy: say the numbers you are skipping quietly to yourself then say the third number loud and proud.

I’ve been reading the book “Number Routines” by Jessica Shumway, and this activity shows up in that book too.  She recommends that the class start with one of her many number routines, then Number Talk, and then the mini lesson.  I’ve been giving that a try this week and I like the way the counting routine lead into the lesson, which is going to lead into our next unit of study.

Well, not exactly “next”.  We’re going to spend a bit of time on time and temperature.  But then it’s off to multiplication we go!

math, Math Workshop, Number Sense & Numeration, Number Strings, Number Talks

Early Algebra

We started the “Trades, Jumps and Stops” Context for Learning unit quite some time ago. A variety of inclement weather days has interrupted us a lot, so we are behind where I thought we’d be at this point. That might be part of the issue I’m having with this unit.  I’ve not taught it before, so that has to take a bit of the blame as well.  And finally, I’m thinking I might have misjudged our general readiness for this unit.  But I talked to my down-the-hall neighbour who is also doing the unit and she concurred:  It seems that some kids are easily getting it, and some are really, really having to work hard to get it.  There’s not a lot of in-between here.

Day 4 of the unit begins with a mini-lesson that we needed quite a bit of time with.  I was to fill 2 separate bags with a certain amount of coins in each, plus a “mystery” coin.  We have had a bit of trouble adding up coins, so I decided to stretch this out a bit instead of trying to get through it in 15 minutes.  Instead of putting the bags out of reach, I gave the coins to kids.  I started by explaining, “X and Y have some coins.  They have an equal amount of money, but they have different coins.  I want to see if you can figure out which coins they have.” On the first day only one child had a “mystery” coin (a poker chip!), but on the second day both kids had one.  The children holding the coins were very excited to be given this job. It gave them practice identifying the coin by name and value.  Have you ever thought about how we interchangeable use “dime” & “10 cents” when talking about coins? Some of the kids are still calling this “The Boat”.

We unpacked the bags a bit at a time.  I don’t have pictures of the whole process, but this is how it looked in the end.  Obviously our mystery coin was worth 10 cents this time.  This is the beginning of the children learning about a variable, and I think they did OK!

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On the second day, we talked more about the signs < > and =.  We added up the coins in chunks, as suggested in the lesson, and then decided if we needed to have a < or a > or if we finally needed the =.  This time, instead of adding them up as a group, I had the students work with partners.  We knew that X had 40 cents, and Y had 5o cents, but how much would they have once they each got another quarter?  This partnership was trying to make a number line across the bottom end of this photo, without actually making the number line.  It’s a step in the right direction for them!

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They wanted to have the numbers in a long line, but couldn’t hold all those totals in their head. Writing them above helped them work on the math and compensated for the stress load on their working memory.

We had lots of people able to do this:

Finally we made it to the mystery coin.  We knew how much money everyone had in actual coins, but what was that mystery coin worth? This led to one of those really cool moments when I felt like (most) everyone was excited about the math.  You can see here where I recorded some of the different responses to the value of the mystery coin.

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I was even able to use a double number line to show two of the answers, and since that is the goal of this unit (developing an understanding of the double number line is in fact the very next lesson we will do!) I felt very good about that.

The next day I needed a Number Talk to reinforce the understanding of the variables.  I didn’t really NEED to do it, since this doesn’t come into the curriculum until a later grade, but they were excited about it, and it certainly can’t hurt them so we did it anyway. I found some images on Math Before Bed and use them for our Number Talk.  I feel like they really help to reinforce the student’s number sense because there is more than one way to make 10, or 12, or any number really.

The weather here is terrible today (Sunday) but I’m hopeful there will be plenty of actual school days this week when we can move forward with this unit.  I don’t want to lose our momentum!  Next time I do this unit, however, I am going to maybe wait a bit.  Of course, the thing that keeps getting me through is that I have to trust Cathy Fosnot! She says this will work, and she has seen it work with many, many children, so I am going to go ahead and finish the unit.  The students are mostly getting the ideas behind the math. Some of them are actually in need of more practice with adding up money.  I am going to make sure we have a day this week when we have some activities that require counting money and we’ll rotate through those to give everyone lots of practice. Often these blogs help me think through what has happened and what needs to happen next!  Time to stop writing and plan my money counting activities.