math, Number Sense & Numeration

Summer Math: Math Before Bed

We love doing “Math Before Bed” as part of our “read at bedtime” routine.  We get out of the habit sometimes though because we also love to play card games (UNO, Go Fish, Old Maid, Memory) before bed. Last night I pulled up this picture:

I quickly counted them:  10 per column, 4 in each row. 40.

My 8 year old started counting by ones.  She said, “I think there are 38.”  Knowing she was not correct, I asked, “Besides counting by ones, how else could you find the answer?”  At the same time, I started counting the far right column. Only 9.  Hmmm. I counted again.  Yup.  I had assumed all 4 columns had 10.  We chatted about this.  We talked about how we could use my original answer of 40 to find the answer.  “There are 2 missing from the last row!  How can we use that?”  She had a bit of trouble figuring this out.  She kept saying, “10, 20, 30, 40.” over and over. I said, “Well, 40 but two are missing.  Maybe someone ate them!”  She counted backward to 38 and we were done.

Then she asked, “Can I make my own picture like this tomorrow?”  So that is what we have just finished.  She decided to use plasticine.  I was recruited to mix colours together and help her make tiny balls.  She decided she needed 60 of them. She also decided she wanted to do rows of three because 2’s and 5’s are too easy and she likes a challenge. (HOORAY!!!)  After counting over and over by 3’s, making a few mistakes along the way, I prompted her to notice that there were 10 in a column.  “10, 20, 30.  Oh.  Halfway there.”  🙂

In the end, we had more than we needed. She put those into groups of 5 (and one group of 4) to figure out how many were left. “5, 10, 14,” she said.  It’s so interesting to me that she can skip count, but often counts by ones.  She says this is because “ones is more easier.” She only switches to larger numbers and skip counting when she has a lot of things to count. I suppose this makes sense.

 

 

math, Number Sense & Numeration

Summer Math:Counting and Subitizing

I’ve been in a “blogging about math” funk for a few months.  One of my summer goals is to write more, so I thought I’d start a series of blog posts about the math that I am doing with my children at home this summer.  To be clear, this is not a series that is planned.  Instead, I am going to try to be very mindful of the times we do math together formally or informally.  My children, who just finished grade 1 and 2, are probably involved in informal math conversations about the same as many children of teachers. Both of them are pretty good mathematicians, and by that I mean they use flexible strategies to do mental math calculations, they notice math in the world around them, and come up with strategies for solving math problems that naturally occur around them.  I’m a firm believer that this happened because I am intentional about helping them mathematize their world, just as I am intentional about making sure they learned to read by reading to them at least every night before bed (and usually more often!)

Earlier in the week I received an e-mail about the latest Mathies resources.  This morning we finally had time to sit down and explore a bit.  I picked a game for my 6 year old called “Representation Match” and if it wasn’t for his Minecraft addiction I think we’d still be playing it (he only gets to play on Saturdays and when he sneaks my phone into his closet unnoticed so it’s tough competition!)

I chose the numbers 0-20 for him, and I chose all the representations of those numbers.  He had to find matches – two ways to make 14, or 19, or 17, or any number between 0-20. These are all the choices available.

Screen Shot 2019-06-29 at 10.13.03 AM He had to work at this!  He was not able to subitize all the numbers so we had a few conversations about how to figure out the number represented.  For example,  there were 3 dice, two showing 6 and one showing 5. I prompted him to think about 6+6 which he knows is 12.  Then I pointed to the 5. He counted on by 1’s to get to 17, then chose the numeral 17 as it’s match. Sometimes he had to match two picture representations.

When he played again, he chose 2 or 3 representations for himself, always a different combination.  My daughter did the same.  She played with the 0-20 cards, even though she is in grade 2.  She likes to get answers fast, so this appealed to her. She was also playing with the cards hidden, more like a traditional memory game and said she had a lot to keep track of in her mind if she was playing with the higher numbers.

We use Dreambox a lot at school.  I love it!  But I also like to have students doing some targeted math activities that keep them immersed in a specific skill for a while.  Dream box allows them to pause a game and move on to something else, which is fine, but also lets them give up too easily sometimes.  I think this Mathies game would make a great supplemental activity for us during the first month or more of school when we are talking about counting strategies, as well as for practice throughout the year.  Did I tell you I’m scheduled to teach a grade 1, 2, 3 split next year? I haven’t taught grade 1 before so I am anticipating how that will look.  The “Representation Match” game will let me set them up to match numbers 0-5, 0-10, 0-20 and 20-50, I think it will be good for the whole class.

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

Math at Home

My son, who is in grade 1, has really good number sense.  He has a lot of mental math strategies that he uses efficiently and flexibly.  He adds on, he counts back, he finds landmark numbers, he even splits numbers!  And no, this is not because we spend a bunch of time every day drilling math.  It’s because we play lots of games and have math conversations that pop up throughout our day.

As I watched him play “Sorry” I was surprised that he was having some counting trouble.  He has been able to count in sequential order with one-to-one tagging for quite some time. He can count a variety of object by ones, more than 100, and when he makes a mistake he notices it on his own and fixes it.  He subitizes, and I feel like this what he is doing  while he counts and that his how he notices his own mistakes.  But that’s a tangent I won’t go on right now.

What surprised me as we were playing “Sorry” this week was the trouble he was having  moving his pawn the correct number of spaces on the board.  He recognizes every number in this game, and connects the number symbol with the amount. He’s done this with other games many times, such as when we play other games and he has to compare which of two numbers is larger. (I had a hard time writing that sentence because I kept thinking about how we haven’t played War in a long time!)  When he drew 5, for example, I know he knows that is 1, 2, 3, 4, 5.

When he would draw a number he would count to that number as he bounced his pawn around the board, but invariably any time he had a number higher than 3 he would bounce a different number of spaces.  Sometimes he would go fewer than he was allowed, and sometimes he would go farther than he was allowed.  If you draw a 4 in this game, you have to go backward, and he did OK with that but he would count slower than usual, so I built that into my intervention. I told him about the problem.  “Just like when you are counting things, your pawn has to touch each square when you count it.” I started by putting his hand in mine, and making sure that every bounce had his pawn landing in just one box without skipping any boxes.  After several rounds of this, he started doing it on his own.  He would slow down his counting and he’d land in the right spot.

The next day we played again, and the problem resurfaced.  This time I explained the problem to  him, then instead of holding his hand I put a finger on the square as he counted.  If he got ahead of me, or skipped a square, he would recognize this on his own and correct himself (and sometimes his big sister had to butt in and point out his mistake, but that’s a different post altogether!)

The third time we played the game, he needed a verbal reminder, but that was it.  And the fourth time he needed the verbal reminder.  And if we have time to play it again tomorrow, which I hope we will, I expect he’ll need the reminder again, but I’ll wait and see.

This whole thing has surprised me some, mainly because as I said before he knows how to count with one-to-one tagging and has for a while.  So why was he having trouble? This is what I think: there was a little pressure on him this time that isn’t normally there. First, he loves to win and he knew that winning in this game requires getting around the board quickly.  That was a distraction and a stressor when he was trying to count. Second, besides just counting, there was some other thinking that had to happen.  If you land on a square with a triangle you get to slide, and if you land on a square that already has a pawn on it then you say “Sorry!” and bump that pawn back to start, and sometimes I could see that he was making a move with one pawn while also thinking about how maybe he should actually be moving a different pawn to get a better outcome. He’d be in the middle of a move, suddenly stop, put the pawn back where it was and move a different one instead.  Third, …I don’t actually have a third.  I think those two things are enough to explain why he was having some trouble. I did double check to make sure he was wearing his glasses the first time I noticed it, and he was, so we can’t blame the vision.  And his coordination is such that moving a pawn around the board is not a physical difficulty for him.

Counting is such an interesting thing, isn’t it? I feel like I have some new insight into him as a mathematician.  I have since noticed that he also needs reminders to slow down when he is doing calculations.  He also does a better job when it is just me and him and he doesn’t have to worry about his sister butting in with answers. (Are you noticing a theme here?  It’s hard to be the little brother!) Finally, he does a much better job and enjoys the whole thing more when he can do single step problems. I feel like that last part is developmental and will work itself out over time.

My diagnosis is that there is an executive functioning thing going on.  He is using his working memory to do multiple tasks each time he takes a turn, not the least of which is to manage his emotions around the fact that his big sister is always butting in.

I am, of course, thinking about how to help my son with this particular thing.  But what does this look like in a classroom?  I’m thinking it would be useful to sit down with a few of my students and play a round of “Sorry” or “Trouble” or even “Snakes and Ladders” and really play with them.  They do these sort of things sometimes during indoor recess, but if I were to set this as an activity during class it would be so a group of children would be busy while I work on the real math with other kids.

Time to rethink that practice.

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!