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. 

 

math, Number Strings, Number Talks

WWCFD

Today was the day that I was going to “do something different” based on all the learning I did with Cathy Fosnot last week.  I sat down carefully on Sunday night and planned out the Number Strings I wanted to do each day this week.

I started with this goal:  Guide my students to discover the Big Idea of “Part/Whole relationships”.  I feel like this is one thing we can work on in our Number Talk time that will help everyone understand numbers a little bit better, and help us understanding adding and subtracting.  I also feel that too many of my grade 2s are still having trouble counting on from a whole number, and are struggling with hierarchical inclusion.  There’s is too much “counting 3 times” and starting at 1 every time something is counted.

I started with 4+3 = on the board.  Through conversations, we ended up with this:

img_7773

So far, so good, right?  One of my grade 3 students said he’d found the double then added in the “extra” one.  Right there I thought this is the difference between Number Talks and Number Strings.  I could have written more problems that let everyone practice the “find the double” strategy.  But I was really looking for the “part/whole relationship”.   We started talking about other ways we could make four.  I made a list.

img_7774

I was so determined to stay focused on my goal!!  But then someone mentioned 3 + 1, even though we had 1+3 and we had to talk about that.  HAD TO!  I mean, really, I thought, “What would Cathy Fosnot do??”  and I decided she’d explore that AMAZING discovery that someone had just made.  I had to ask, “Wait…you mean I can just reverse the two numbers I’m adding and still get the same answer?!”  I added that thinking to the board, complete with a name label, and then had to ask the next question:  “Will that work every time??”  We tried out a few examples and decided that yes, it would for addition, but no, it would not for subtraction.

Here’s something I have decided:  The goal doesn’t have to be completed in just one day. I have the whole week to explore part/whole relationships.  And maybe it’s stretching it out over a few days that really helps the kids to see some ideas.  I could have easily turned this into a full 1 hour conversation.  But after 10 minutes people start to get restless and stop really listening, so I ended it.  We’ll pick it up again tomorrow.

One of the things that really struck me in Cathy’s lesson on Saturday was that she displays all three parts of her conversations (the question, the model and the conjectures) at once.  One is not behind the other.  She said that this way kids can see all three and each helps to strengthen the other.  This is one of the main things I want to really do this week (and beyond.)  I am going to start to do these talks on the large white board instead of the small one so I have room for everything.  I also, as you may have already thought when you looked at my pictures, need to make sure I am organizing this better as I go.  But I’m okay with admitting this is going to take some practice.

 

math, Number Talks

Number Talks: Week 1

I hadn’t decided yet if I wanted to start off by making this sound like I am a genius (which I totally am!) or if I was going to list all the sources of this idea and talk about how they converged into this seemingly-original idea.  But then I listened one morning this week to a podcast from Derek Rodenizer on voice.ed radio.  (You can listen to it here.  It’s only 5 minutes!)  He posted this graphic later on Twitter:

I think you can see the original Tweet by clicking on the picture.  Sorry, I can’t figure out how to post it!  

That podcast made me realize that this Number Talk idea was just that – an example of how sharpening my pedagogical sword gave me an idea that wasn’t/isn’t truly mine but is me knowing a bunch of stuff and then putting it all together and coming up with something different.

The “Introducing Multiplication” unit by Marilyn Burns starts with an activity where students are asked to work with partners to make posters about numbers.  Specifically, students are supposed to write down things that come in 2’s, or 4’s or 7’s, etc.  So this is partly that idea.  If you look on Pinterest under “math activities” you’re sure to see art projects students have made where they have written down all the ways to write a number, or what the number means to them. (If you don’t know what I am talking about, look at this.) This is partly that idea too.

On the first day of school, I told my new students I was going to write a number on the  board and they were going to tell me everything that number made them think or remember.  I then talked to them about how we put our thumb on our chest to show we have an idea, and that they could take as much time as they needed because math is not a race.  I wrote a 2 on the board.

I am not exaggerating when I say we talked about this for probably 15 minutes. Here is our result:

 

img_4869-1
I have erased student names before posting here!  “(Blank) has 2 kittens.” used to be attributed to a specific student.  

It was so interesting and, I thought, successful that I couldn’t wait to do it for the rest of the week.  Here are our results:

I do not think it is a coincidence that we got more information for 2 and 5 than we did for 23.  This is a grade 2/3 class and most of the ideas on 23 came from the 3’s. Also, there aren’t a lot of every day things, like bike wheels and toes, that come in 12’s and 23’s.

There were some common themes:  each day they wanted to talk about someone who was the age that matched the number.  They know people who are 2, 5 and 12 – mostly siblings.  But 23 is too young to be one of their parents, and too old to be a friend or sibling (for most.)  When we talked about 23 as an age, it was clear they don’t really know what 23 years-old looks like. They guessed that I am 23, which I totally look like I am but I am not. They then guessed that our principal is 23, which is what she actually looks like (no this is not an evaluation year for me.) We settled on “Justin Bieber and college students” after some discussion.  Everyone finally agreed that a 23 year-old is a “young grown-up”.  But I think many of them are still not 100% clear on this idea.

My original goal was to start using Number Talk routines – mainly the one with the thumbs up instead of hands up.  I also wanted to start talking about them building on each other’s ideas, which they did.  I was busy building routines, not taking notes.  But I still feel like I got some important information from some of my new students.  This will help me going forward.  I had originally planned on doing dot-number talks for the week.  I have already prepared slides to use for this using Smart Notebook software, even though I do not have a SmartBoard in my class. I kind of chickened out because the tech-gremlins have been busy at my school over the summer and things were randomly not working.  I didn’t want to start off my school year muttering curses at them while repeatedly jabbing buttons.  I’ve saved that for the coming week just to give myself something to look forward to!

To read more about how a dot number talk can be used to start the year, check out this great blog post!  https://heidiallum.wordpress.com/2017/09/09/the-power-of-dot-number-talks/