math

## Summer math: money and even& odd

What does 50 + boat + beaver = ?

If you are Canadian, you may have known that it equals 65 cents. “The one with the boat” is what my 6 year old often calls a dime, and of course the nickel is the beaver coin. Not sure why the names of these coins elude him. He has no trouble remembering the value and I suppose that’s what matters now.

This conversation came up because we saw yard sale signs. Last summer we started letting the kids do their own yard sale shopping. It really helped them start to understand the value of money. I don’t mean the actual value of the coins and bills, but the whole concept of working hard to earn (or find!) the cash and then having to decide if the desired item was worth that amount. Of course they have to count it themselves, and they are both getting pretty good at it. I’m getting pretty good at making them think it was their idea to not buy the junkiest item on the table.

Money came up again today at Canadian Tire. I received 40 cents in Canadian Tire money after my transaction (I haven’t embraced the electronic version of this.) The self-checkout (which I only used because the boy was not getting a new bike helmet like his sister and needed a job to distract him from the injustice of it all) gave us eight 5 cent CT dollars. These had to be equally shared. There was a “some for you, some for me exchange”, some negotiation and finally each was convinced they had an equal amount. It fit nicely into an ongoing conversation we are having about even and odd numbers as well.

Speaking of even and odd, did you know that 13 is odd? 6+6=12, so if you have one more than that it’s not even because there is one extra. (Explanation courtesy of the 8 year old!)

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.

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:

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!

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.

## 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.

## Slice of (Cooking) Life

I couldn’t help myself.  I mean, it IS cooking, and people can’t cook without doing math.  So even though I had signed up to lead a cooking elective group, and even though the 14 children who signed up to be in the group were expecting cooking, we were actually mathematizing as much as we were cooking.

There was all the standard math you are expecting, like measuring and running the timer.  But at one point in the lesson, after the first batch of cookies had come out of the oven, it came time to see if we were going to have enough.  Plans were being made to take some home, of course.  “Wait,”  I said. “If you want 2 cookies each to make your ice cream sandwich, we have to make sure we have enough for that before you start making plans to take some home.  So..do we have enough for that?”  And I walked away.  Everyone, grade 1-6, started counting each other and counting cookies. A few kids jumped up and ran over to the oven to see how many cookies were in the oven.  A few others were checking the bowls to see if we had enough dough for  more cookies.

“Well,” I asked again, “do we have enough cookies so that everyone can make an ice cream sandwich?”  They agreed we did, and several spoke over the top of each other because they were so excited to justify their answers.  These aren’t my regular students, so I have no idea what sort of work they usually do.  However, their explanations were great!  And I loved that some were counting by 1s and some by 2s and some counted all of the m by 1s or 2s but then said, “We have 14 here, and 13 in the oven, so we need 1 more cookie from the next batch before we have enough for each of us to make a sandwich.”

Next week we’re making pizza.  I don’t really intend to turn this into a math club, but we’re probably going to have lots of chances to talk about fractions. Hearing all of the awesome mathematizing was almost as great as my oatmeal cookie + homemade ice cream sandwich!  Almost.

## #mathematizethis: fractions

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?”

Pizza:

• 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.

Milk:

• 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!