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.

math, Number Talks

Collaboration

We’ve taken a little break from partner work in our class.  Increasingly our struggles with problem solving were getting all tangled up.  Instead of feeling frustrated because working with partners is hard, I felt too many of the students were starting to blame math for the problem and I didn’t want that feeling to perpetuate.  We took some time to do some worksheets *GASP* alone – mostly because I needed a bit more data to feel confident about assigning letter grades on report cards.  Geometry has been the focus of these.  We also took some time to loop back to patterning because this seems to be an area where lots of people hadn’t made a connection between patterning and using an open number line to add numbers.  I think we are there now!

Before we dive into our next unit (Trades, Jumps and Stops from the Context for Learning kits by Cathy Fosnot), I’m going to take some time next week to do some work on the collaboration part.  I have been assigning partners all year.  We’ve talked a lot about why I am choosing those particular partners for everyone.  Now it is time for them to make some choices of their own and I will also be asking them to justify those choices and articulate what makes a good partners.

Once partnerships are established, partnerships last for a month.  They stick together for every part of our day when they might need a partner – writing, reading, science, math and anything else. We will be working on building our collaboration skills all day long.  Specifically in math, I am going to ask everyone to do a “turn and talk” with their partner during each Number Talk.  Usually we do what I think most people do:  I put up a problem, kids work them out alone, then we discuss them together.  I think the turn and talk time will help them practice actually talking to their partner about how to solve the problems.  They will be empty handed, so they can focus on talking about the math instead of arguing about who will be using the marker to write it down.

The second thing I am going to do is create some problems for everyone to solve.  Today we are going to do an activity from The Super Source where partners work together on some describing and listening skills.  One builds a design using no more than pattern blocks. The second partner is not allowed to see this.  The first partner describes the design that was built so the second partner can recreate it.  It’s a tricky exercise for 7 year olds, believe it or not.  Positional language,  attributes of geometric shapes, and expanding on one’s own words are all practiced.  I find that the person describing often reverts to giving directions such as “get a triangle and put it on top of the square…no that way…no that way…no down…YES!”    The other problems are going to involve some addition, maybe some subtraction and will be put in a context they can work with.

The final thing I really need to work on is how to respectfully disagree, and how to accept that “No, I don’t think so” isn’t the same as “I hate your guts and will never speak to you again!”  It’s a hard one, but necessary.

I had initially planned for next week to be the start of my next unit.  But I’m feeling better about this plan of action.  It’s going to help us have a smoother run through the unit, and it is going too help me set up the Math Workshop groups we’ll need during the unit.

Geometry, math, Math Workshop, Number Talks

Real Geoboards vs. Virtual Geoboards

This past week we’ve been doing some geometry work in class.  The grade 2 curriculum expectations for geometry are fairly simple:  name, sort and make 2D and 3D shapes.  In general, children arrive in grade 2 already knowing most of these.  The more common the shapes are in the natural environment, the more likely this is true.  Octagon and hexagon usually give everyone a hard time with their tricky names, but by the end of grade 2 few children can’t recall these names.  In grade 3, we have to do a few more things. The vocabulary is increased (quadrilateral, angles) but again other than folding and unfolding nets of 3D solids, it’s nothing too complex.  Of course, I say that from this point of view – some kids do find it a bit tricky.  In all, however, it’s about 1 week’s worth of expectations.  I like to teach them early on because there are a lot of problem solving opportunities that can involve geometry and once we have the vocabulary learned the problem solving comes more easily.

This week I had rubber bands on hand.  That’s not actually something that happens all the time.  Since we had them, I pulled out the old geoboards.  Lack of rubber bands is actually one of the main reasons I don’t always pull them out.  The virtual geoboards, available here, here and here, are so much more reliable.  And nobody can shoot a virtual geoband across the room at somebody.

In the activity shown below, our Friday lesson, students were asked to make some shapes according to a rule.  Then their classmates had to figure out the rule.  Was the rule: shapes that have 3 sides?  Shapes with 4 corners?  Shapes I enjoy making because they create cool patterns? Here are some of our results:

A few years ago I remember reading an article about how important it is for students to have a real experience with a manipulative before they move to the virtual version.  I tried to find that this morning and couldn’t.  My brain doesn’t remember the source!  So I put it out to my virtual PLN (Professional Learning Network) on Twitter, and found a lot of teachers agreeing with my thinking. Reflecting on our week today, I was so glad that I had used the real geoboards.  There was some really interesting stuff that happened.

First, students were making shapes of different sizes over and over in different ways.  On the apps, they tend to get busy playing with other tools – like changing the colour of their geobands, or colouring the shapes in with a variety of colours.  They get focused on the wrong things and come up with rules like:  Shapes that are orange.  And I’m sorry, but orange is not a geometric attribute.

Second, I noticed that some children were struggling to stretch the bands across the pegs.  Some of the rubber bands are smaller than others, so this became a problem solving challenge.  I feel like they were motivated by the task to work through the challenge and find a new rubber band or change the perimeter of their shape.  This simply doesn’t present as a challenge to be worked through in the virtual environment. I had forgotten about this part!  Developing learning skills needs to be embedded in every part of our day, and I’m glad they got this chance to work on some problem solving skills.

Finally, there were some social things that we could work on.  Sharing is always an issue for 6 and 7-year-olds. They had to cooperate and collaborate to share the rubber bands on their table, and to decide who would get which geoboard.  I tried to make sure that every board at least matched the other boards in a group, but I didn’t always make that happen.  Again, students had to talk through this because everyone wants the one that is different, and therefore special.

Now that we have spent some time with the geoboards,  they can become one of the activities students can do during a Math Workshop session.   I can put them on a table with some task cards, or the students can request them to help solve a problem.  When we move on to perimeter and area (after we spend some time working with number lines for the next 3 weeks!) I can incorporate questions about shapes and feel confident that everyone knows the shape, and can work with the shape.  And I can add some

 

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 Talks

Fractions

Found this on Twitter the other day, and decided to give it a try.  We are “finished” with some learning about adding and subtracting and unitizing and counting.  We need to move on – both for my sanity and because there are other things that need to be covered in the curriculum.  We will circle back to this in the new year, and we’ll keep practicing when we do a daily Number Talk at the beginning of our day.

I’d decided to move on the geometry, mostly because I’m a sucker for all the Christmas tie-ins.  I love making 3D shapes into ornaments and decorations! But then this fun fraction activity popped into my life, and I thought it would make a nice transition from Number Sense into Geometry.

On day 1, a shortened math period due to irrelevant circumstances, everybody cut out their pieces.  (As an aside, I know that cutting isn’t math, but we have some fine motor issues, and it’s really good to for us to cut as often as possible.  Also, I have a life to live and it’s takes 20 kids 15 minutes to cut these out so I let them!)  I started by holding up the “whole” and one of the fourths, and then asked the recommended question:  What do you see?

They named a bunch of stuff, unrelated to math:  I see white and yellow, I see a fried egg.  Then they moved on:  I see a small square and a large square; they both have 4 sides, and 4 corners; they are the same but different.  Then I let them go play with their shapes.  We came back and talked about what we saw:  the different shapes in different sizes, teh same shape in different colours, etc.  Then we had lunch.

On the second day, I asked them if they could use the paper to show half.  I was interested in their understanding of half, and also, would they see that you could make half of all the shapes, not just the whole.  While I walked around and talked to different groups, I was so glad that I know that I should ask them to explain their thinking.  Some of them had very unexpected answers that showed partial or complete understanding in a different way than I was originally thinking.

All of these came with justifications for how it shows half. One is half yellow (front) and half white (back), one has purple lines dividing it into sections – half the sections are yellow and half are white, and one uses the purple “lines” to divide a design in half.

We talked about symmetry, which I hadn’t intended to do, but it naturally fit.  I really thought I could do this activity and move on to fourths and thirds.  It was clear we weren’t ready.  Luckily I am SO organized (haha) I was able to grab this book off my shelf:

Give Me half by Stuart J. Murphy

It’s about a brother and sister who do not want to share fairly, but an unspecified adult (mom?  dad?  babysitter??) tells them to share.  They each get half of the pizza, the dessert, the juice.  Then I sent everyone back to show half again.

We had better results, and each came with a really good justification.  Then, just to make sure we all understood that each shape could be divided in half, I sent everyone to fold each piece in  half.  Which they did.

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My favourite was this one:

 

He explained that both show half because the one folded twice shows half of half. 🙂 That’s going to get us started on fourths!

On Monday we are going to find other things in the room and in the school that show half, and then we’ll move on to fourths.

 

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.