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!)

math, Number Strings

Minus/Subtraction/Take-Away

Minus…subtraction…take-away.  Do these all mean the same thing?  They are certainly all represented by the same symbol.

Last week the math coordinator was in  my class for a few days.  (Here name is Melissa and she blogs here!)  After watching me do a number string related to subtraction, she encouraged me to always say “subtract” when I am reading the problem to the class, rather than “take-away.”  Some kids will actually do some adding to solve these types of problems, and by always saying “take-away” I would be restricting their thinking and maybe even imply that they need to use a certain strategy, namely that they need to remove.

I know that there are different ways to solve a subtraction problem:  add on, count back, think of it as a part of a fact family and figure out the addition problem.  But I hadn’t really been intentional about my language when discussing subtraction with the class.  I was more focused on the answer!  (I’m hanging my head in shame!) (not really…but you know what I mean!)

On Thursday and Friday we had bus cancellations, so I didn’t really get a chance to try this out until today.  We were working a Number String from Cathy Fosnot’s mini-lesson book.  We talked about 14+1, then 14-1 (Did you read that 14 subtract 1? or 14 minus 1?)   Then I gave them 14-13.  You can’t see them in this picture, but I had the maths.ca relational rods going in the background, and had build 14+1, and 14-1, and those were still visible to the students. I saw lots of kids with their fingers out counting back.  It’s an okay way to get  correct answer, but very inefficient.  However, I then asked my favourite student (my daughter!!)  to tell me how she solved it.  I’d seen her working away on those fingers, and I know that if she spent a tiny bit of time thinking before she started that, the answer would have been obvious to her.  Knowing this, I had to ask her about her strategy.  “Well, I thought about having 14 cookies, and then I ate 13 of them, so yeah…one is left.”  This is not totally unreasonable for her (don’t judge my parenting!) especially if they are Viva Puffs!   I annotated her thinking like this:

I pointed out to everyone that CC was thinking of subtraction as “taking away” something.  And then asked others what they thought about when they saw a subtraction sign.  Someone else said, “Well, I knew you would only need one more to get from 13 to 14, so I knew it would be 1.”  I talked about how that child was thinking about the difference between 14 and 13, which was different from CC’s but they both still got the same answer.  Then we did 2 more problems from the string, and talked about the “take-away” strategy and the “find the difference” strategy.

img_3213-1.jpg

Someone even mentioned that they thought about 9+2=11, which is a great connection to some work we did a week or so ago, so that was awesome too.

It’s funny how being intentional about how I was reading that symbol to the class changed the strategies they used.  This wasn’t truly the goal of the Number String, but “m delighted by the results.  I am hoping the forecasted 25-35 cm of snow (and 80 Km/h winds!) hold off until late on Tuesday so we can work on this again tomorrow.  I feel like we are developing a really big understanding about subtraction!

Guided Math, Number Sense & Numeration, Number Strings

Addition of double digit numbers

There were about 50 of these on our whiteboard at different times over the last few weeks. We’ve gotten pretty good at adding the tens using mental math strategies. 20+20 -> 2+2=4, so 20+20=40…no problem! But it’s was time to move on!

I really wanted everyone to learn how to effectively use a number line. We’ve been working our way through a Context for Learning kit called “Measuring for the Art Show”. I demonstrated it about 1 billion times. First we used cubes to make a line, then we annotated this on some cash register tape, and then we moved to the whiteboard. Finally, I gave everyone some problems (from the kit) and some paper they could use for drawing the number lines.

As I walked around I could see lots of kids with lots of right answers but no number lines. “How are they doing this??” I wondered. So I asked. And I was amazed! So many of them were using the mental math strategy of splitting. They thought about how many ones there were in each number, and how many tens were in each number, then they found a total.

But the number line isn’t the go-to strategy yet. So I’m annotating the problems two ways now. The way lots of them are doing it, and the way some of them are doing it. Both are ways I hope all of them can do these problems eventually.

Our next step…my next step…is to organize into small groups (I know…back to some guided math. It keeps coming up!) I need to help the kids who are splitting learn the number line, and help the number line kids do some splitting and help the “I have no idea what to do kids” get some ideas.

I’m ALWAYS happy to reach Winter Break, but it always comes about a week before I’m ready. I don’t want to interrupt our math learning. But I’m confident the stuff we’re doing now will stick. I won’t have to start from scratch on January 7.

Here is more work from today:

math, Math Workshop, Number Sense & Numeration

Counting

I never used to worry too much about teaching students to count. I mean, the year I taught kindergarten we did a lot of counting, but in grade 2…or 3…or 4? Nope.

In the last few years I’ve become aware of how important counting is, and the layers of skills that are involved.

After interviewing my people, I discovered most can start at 30 and get to 100 without difficulty. Some had trouble not starting at 1 – and they also had trouble assembling a 100 chart. This is not, I think, a coincidence.

One of the classes figured out we need 12,000 laps around our track to equal Terry Fox’s journey across Canada.  We’re keeping track of our contributions with tally marks!

Some students were very organized with their counting…

And some students were not organized with their counting.

After we congressed these photos, I sent everyone off the do more counting. I asked them to count out 17, or 24, or 52, or 65 of the math tool they wanted to work with. Everyone tried an organization strategy of some sort! Even when they sorted into rows or groups of 5, many were counting by tens.

I loved this one because this child said, “10, 11, 12, 13, 14, 15, 16, 17!” It was great to see her counting on!

I think when we look at these images and congress them today, I need to make sure I talk about how the organizing helped. One of the things I know I need to do better is point out these things that seem so obvious to me.  Some students will have already realized the advantage of using groups, but some will not have.  I need to help them with that.

I used to always start the year with addition. This made more sense I guess when I was teaching grade 3/4 classes. But starting with counting and making sure everyone has a strong foundation of number sense to build on has truly made my addition and subtraction units go more quickly. Everyone seems more prepared for the addition and subtraction work once I know they are solid with counting skills.

math, Measurement, Number Talks

Who is the tallest?

Every June I wish I had measured everyone’s height in September so we can see how much everyone has physically grown. Every September I forget. But not this year!

On the third day of school, we started talking about measuring things. Grade 2 is the first year students use standard units of measurement instead of investigating things like “how many markers tall are you?” I know the grade 1 teacher was working on this in May and June, so measurement seems like a good place for us to start. It’s a quick thing we can work on after spending some time each day setting up Number Talk routines.

It was really interesting to note that the grade 3 students in the class aren’t necessarily the tallest, and the tallest grade 2 is not the oldest grade 2.

After measuring our height, we brainstormed other things we can measure and compare – who has the longest feet, the biggest hands, longest hair, and biggest eyebrows? We don’t have answers to these questions yet, but we will by mid-week.

Changes in season make interesting times to measure temperature too. I’ve got my thermometer ready to go, and we’ll be tracking the temperature each day as we move from “It’s so hot we shouldn’t be keeping schools open” to “Sorry I was late. I had to scrape ice off my windshield.”

Grade 3 students study plants in science, and this is a great opportunity to integrate math into science, or science into math if you prefer. We’ll be planting some plants for our windowsill soon, and measuring their growth.

Most exciting of all is that when the final days of this year arrive, we’ll have both the skills and the data to determine exactly how many centimetres taller everyone has grown.

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.
Guided Math, math, Number Strings, Number Talks

Guided Math: part X (I’ve lost track)

This past week, predictably, was crazy.  Halloween in the middle of the week?  Seriously.  Why even bother having school that day?  I know people think it’s important for kids to have good memories from their childhood associated with fun things, like a costume parade on Halloween at school.  But I think we can all agree it’s gone too far.

It’s also been a weird time for our math class.  As you may recall, 4 of my grade 3 students were my students last year.  Two of my grade three students were in a 1/2 split, and the rest of my class are grade 2 students who are all new to me.

My grade 3 students are solidly moving along as a group.  They make a beautiful cohort – teaching important things to their younger, less experienced classmates. Up until now, I was satisfied with how they were helping to scaffold the class through Number Talks and Number Strings.  I was happy with what they were teaching the grade 2’s about communicating their mathematical thinking.  About mid-week last week, the tide shifted.  I started to feel that the grade 3’s were dominating the conversation too often.  They were figuring everything out way before the grade 2’s. If I plotted the 2’s and 3’s on a Landscape of Learning, they were in two very different spots.  So different in fact, that I felt I had to do something about it.  That something, I decided, would be to split the class into two entirely different Context for Learning units.

Now, I have taught split grade classes for most of my career. I have, many times, had the kids in one grade working on something different from the kids in the other grade.  In math, this usually looks like one grade continuing on in a Context unit that we started together, while the kids who are not ready to go on work on something else to help them solidify the part they are A) ready for, and B) required to learn thanks to the curriculum. You’d see this, for example, when it comes to multiplication and division.  3’s and 4’s have a similar starting point, typically, based on their needs.  But 4’s need exposure and practice with dividing that 3’s don’t.  To be clear, if I have some 3’s who are developmentally ready to move forward, and keen to move forward, I would take them along on the trip.  But if they need to hang out at “multiplication up to 7×7” for a while, I let them.  This would probably include them repeating some games we had played, or something like that.

But this time, I was feeling strongly that I needed to be pushing both groups, not just letting one group sit in one place for a while.  Here are two of the Number Sense and Numeration Big Ideas for Grade 2:

  • demonstrate an understanding of magnitude by counting forward to 200 and backwards from 50, using multiples of various numbers as starting points;
  • solve problems involving the addition and subtraction of one- and two-digit whole num- bers, using a variety of strategies, and investigate multiplication and division.

And for Grade 3:

  • demonstrate an understanding of magnitude by counting forward and backwards by various numbers and from various starting points;
  • solve problems involving the addition and subtraction of single- and multi-digit whole numbers, using a variety of strategies, and demonstrate an understanding of multiplication and division.

My grade 3s have mastered the grade 2 concepts.  (As they should have last year by June.)  My grade 2’s, however, are still needing a bunch of work on this. (As they should until this year in June.)

So what was I going to do??

Well, I started two Context units at the same time. Seriously.  In the same week when Halloween was on a Tuesday.

As part of my evolving thinking about Guided Math, I thought I could have one group working independently each day, while another group was mainly spending time with me and getting my attention.  So far, it’s been chaotic.  But I feel like we are accomplishing what I want to accomplish.  I’m really wishing I could do more observing and conferring, so I will be addressing that in my planning this week.  I don’t want to be occupied with whole-group teaching and missing out on the conversations kids are having while they do their work. I did throw in the towel and have a “Fun Friday” during math because I felt that instead of moving on I needed to regroup.

Grade two’s are still working on unitizing single digit numbers and using the 5 structure to help them add.  They are working on the “Double Decker Bus” unit.  We ran into some problems because I am a paper-saver and had given them one day’s work on one side of a page, and the next day’s work on the other side.  I know: rookie mistake.  Even though I clearly told them and showed them were to start, 3 out 5 groups tried to do the wrong side.  Learning experience for all of us!

Grade three’s are working on the same things, but using the “T-Shirt Factory” unit to move into hundreds, using the 10-structure, and building a deeper understanding of place value into the hundreds.  They need more help with the use of a T-Chart to organize information.

I congressed with both groups on “Not-So-Fun Thursday!” as I am now calling it.  One group was working  on something while I congressed with the other.  I think I’ll keep this.

Moving forward, I am going to continue in both of these units.  After Fun Friday, I discovered that there are still some counting issues for grade 2s. I think they can have a “Count-Everything-in-Sight Monday” or maybe a “”Put-All-These-Numbers-in-Order Monday” while I get the 3’s started on the next part of their unit.  Then on Tuesday Grade 3’s will be able to work independently while I get the two’s started on their unit, and then I can wander and confer.

And it is going to take me the rest of the day to figure out how I can work Number Strings and Talks into it all.  Cause we have different needs there, as you probably guessed.

Thankfully I have lots of Halloween candy to get me through!

 

It’s such a big week in math that I’m blogging twice!! See the other post, about a Number String, here.