Last week I was hanging out in a kindergarten during Counting Collections. It was amazing and beautiful. Afterwards, the kindergarten teacher, Becky Wright, and I were talking about the challenge of switching from counting by tens to counting by ones. These three students will give you a sense of what happens at the transition.

We’ve all been thinking about what might help students get more comfortable switching back and forth between counting by tens and counting by ones. Today, Becky and I were talking with Debbie Nichols, who teaches 1st and 2nd grade. Together, we landed on the idea of passing out 10s and 1s – connected sticks of ten cubes and single cubes, base 10 rods and units, etc. – and then having a counting circle.

In kindergarten in the late fall/winter, Becky would have the kids holding tens positioned at the beginning of the circle. As kids counted around, adding what they have, they’d keep a running total of the cubes. So a count with 20 kids might sound like “10, 20, 30, 40, 50, 60, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83” if the first 7 kids were holding 10s and the rest had 1s.

What about having one of the kids with tens switch places with one of the kids with ones? Now the count might be, “10, 20, 30, 40, 41, 51, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 81, 82, 83.” Ooh!

Or, what about going around the other way, starting with the ones and ending with the tens? “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 23, 33, 43, 53, 63, 73, 83.” Hard!

With 1st and 2nd graders, Debbie wanted to pass out tens and ones at random. Her kids came back from recess and we gave it a go right away:

As kids counted around, Debbie kept track by pointing on the hundreds chart. After one count around, we found we had 77 cubes.

The kids asked if we could count around the other way – counterclockwise instead of clockwise – and we had a great discussion about whether we would still land on 77 or not. I suggested Deb keep track of the running total by tracing its path on the hundreds chart, using a different color for each count:

We tried clockwise and counterclockwise. We tried rearranging the kids by switching every third spot. For our last count, everyone stood up and traded cubes with somebody else.

What do you notice? What do you wonder?

Some of the kids’ claims:

“It will always be 77 unless we add some cubes or take some away.”

“It doesn’t matter what order we add in. It will always work out the same!”

There’s so much potential here.

Debbie’s planning to do the same thing with dimes and pennies on a different day. And, of course, we could give older kids multiples of ten and/or multiples of one.

After doing it just four times, we noticed an increasing smoothness for some of the kids. They were noticing that they’d either move over or down on the hundreds chart.

I’m excited to see what other versions we might come up with. Have an idea? Please put it in the comments!

And let me know if you’ve read about this idea somewhere else. Happy to cite it if we reinvented the wheel.

What a great idea. Can’t wait to share with other teachers!

I hope you’ll let me know how it goes!

Developing Number Knowledge Chapter 5 – Wright, Ellemor-Collins, Tabor

Developing Conceptual Place Value: Instructional Design for intensive intervention – Ellemor-Collins & Wright

Which mental strategies in the early number curriculum? A comparison of British ideas and Dutch views – Beishuizen & Anghileri

Jumping Ahead: An innovative teaching program.-http://www.fi.uu.nl/publicaties/literatuur/6319.pdf Menne

Thanks, Kristin! I’ll check them out.

I’m wondering if any of the kids can write the equations for any of the paths we took today using the 100’s chart as a reference. Worth it or not?

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Great question, Deb. I think it’s definitely worth a shot. If I understand you right, you’re not talking about taking the time to record the equations live, right? I think that’s a good call. It would really disrupt the flow of the counting circle. But I think it would be a great challenge for some of your kids, especially your 2nds, to use the 100s chart to try to write the series of equations. And it’s a great chance to teach them the habit of new equation, new line:

1 + 1 = 2

1 + 10 = 12

12 + 10 = 22

etc.

I expect at least some kids will do the 1 + 1 = 2 + 10 = 12 thing. If they do, more equality chats!

Once they write it one per line, I’d love to ask what patterns the class notices and wonders. It won’t be obvious to some kids where that long list of +1 and +10 addends comes from, so that’s a great opportunity for them to connect the equations to the path on the hundreds chart to the cubes in their hands. How nice to have a “connect and compare” discussion about these different representations of the count!

I think my first choice way to do it would be to do a new count with a new number. Maybe just do a clockwise and a counterclockwise count and stop, so there are two (beautifully symmetric) paths. Have a few kids do equations for one and a few do the other. Then what do they notice or wonder? There will be lots of patterns there.

And then maybe do it again after a more random reshuffle from trading, maybe later in the day?

Or some other idea?

I’m so lucky to work with you!!

This ties right in to showing addition on a number line that I am working on with my second grade teachers right now. We’re using meter sticks as number lines and lining up tens and ones and counting them but they’re having trouble knowing when to count by tens and when to count by ones. I’m going to add this activity as a warm-up to these lessons. Later, when we’re adding three digit numbers I’ll throw hundreds blocks into the mix.

Sounds great! Please keep me posted, Jo!

I tried this today in two second grade classes–it went slowly today but on their own they realized that they could count a lot faster when they grouped the tens and the ones. Some of them are still wanting to start counting by hundreds when they go above 100 or so (so they count 84, 94 104, 204,304) which they have been doing consistently when counting by tens. We started recording their counts and having them pay attention to what place was changing. I notice this with rounding too–they can easily round to the largest place but struggle with places in the middle of numbers.

My teachers loved this activity today and plan on including it in the warmup until the kids can do it fluently.

I noticed in the picture one child had the cube on the back of the hand, while the others were holding them in their palms. Maybe for an extension activity, that way of holding could designate a negative/anti-cube or stick? For example, starting from the 10:30 position in that photo and counting clockwise, it would go:

10, 20, 30, 31, 30, 31

Now, will we still get to the same place no matter which direction we go? where we start? Whether we mix it up (this might need colored cubes to remove the choice each child has about how to hold the material)?

Since each kid has a choice about how they hold their cube/stick, you could build some games around this: how can we all hold our material to get closest to some target, say 20 to start, 0 a bit more advanced, 24 (or some other non-multiple of 10)?

I also noticed the paths on the 100 board. So, what are some things we can do with paths?

(1) count them: given the number of 10 sticks and 1 cubes, how many different paths are there to our final destination?

(2) count attributes of the paths: how many corners are there? Which of our paths has the most/least corners?

(3) block them: again, a game version. Randomly choose [5?] numbers on the 100 board (I would roll 2d10) and then these numbers are landmines. When we draw a path, we can’t go through those squares. Can we still get from 0 to our target?

This is solid Tracy! Jumping back and forth between tens and ones is a monster and this shows the power of counting circles and their high scalability. I could even see this being extended into the 2nd and 3rd grade using the hundreds flat.

I really like the connection back to the hundreds board to record student thinking. Nice way for students to recognize the patterns and foster more classroom discussion.

Thanks for sharing!

Thanks, Graham! I love the hundreds flat idea, although that makes tracking on the hundreds chart a no-go. How could we record/represent it? In Investigations, we had 101 – 200, 201 – 300 charts, etc., but most teachers don’t have something like that posted in their room. Hmm…

This is a great idea. I think I am going to try this next week. I have done choral counting before when we switch what we are counting by, but it was sometimes too abstract for kids. This is a great way to ground our thinking using models. I wonder if I could use digiblocks to count by whole numbers and decimals with some fourth graders.

Oh! And I wonder if you could do it with money?

For sure! Let me know how it goes.