Teaching Place Value on a Shoestring -
or a Toothpick

Despite the fact that I am an unschooler by nature, my daughter Noa has several math workbooks. This is at her own request; her friends frequently want to play school and she has realized that she doesn't have homework like other children. So she asked me for math homework! I went to Target and picked out several different workbooks, mostly on the second-grade level even though she was only five at the time, as there was nothing in the preschool and first-grade books she didn't already know. We work a page or two when she asks for her math books, as she did last night. 

After a page or two of fill-in-the-missing-number and counting-by-2s-and-5s and like exercises, we came to a page on place value. Being a fun little workbook and not a math text, the explanation was more than a little sketchy, but she seemed interested. As she's been exhibiting confusion over place value for a while now ("Mom, when I write 15, which side does the 5 go on?") I thought this merited a little extra help, since it was being carried on her own interest.

So this morning, I pulled down three paper plates, the toothpicks, a marker, and some tape. Having been burned before, I didn't exhibit any unusual enthusiasm or try to get her interested in any way. Instead, I sat down beside the baby on the floor, near where she and her little brother Colter were playing, and just began counting out toothpicks into piles of ten, then bundling and taping them together. By the third bundle, both Noa and Colter were right next to me, asking to help, and eagerly began counting out tens of toothpicks. After instructing Colter that they had to be counted and placed away from the large pile (instead of counted back into the large pile), we were off and running. They didn't even ask what we were doing until we had bundled more than 100 toothpicks, and I replied, "Math!"

[I'll note here that my daughter is one of "those" children that homeschool-wanna-be's often complain about: "If I tell her to do something, she'll never do it! I can't teach her anything!" My daughter is just like that and if she feels manipulated in the slightest, won't comply. However, she generally enjoys doing what I am doing - as long as I'm enjoying myself - and if I can catch her enthusiasm about something and couple it with the opportunity to "do something fun with mom" then usually she gets excited about it. Does it always work? Of course not, but frequently enough it does. Frankly, I'd never be worried about teaching place value, or really any math, to a six-year-old, but she loves it! So since she wants to know more, I give her what she wants, in a way that she'll both enjoy it and, I hope, retain it.]

I asked her to count out ten bundles of ten toothpicks and asked her how many that was. Once I reminded her to count by tens (since there were ten toothpicks in each bundle), she caught on. I then taped ten bundles together for 100. We ended up with two bundles of 100, ten bundles of ten, and ten individual toothpicks. If you have more than that left over, put them away; have no more than ten singles and ten bundles of ten, or the "ah-ha" light won't go on as quickly. Likewise if you're working with 1000s, have no more than ten bundles of 100.

I took the marker and wrote a great big "1" on one plate, a "10" on the next, and "100" on the third. On a piece of paper I drew a graph like the one below. (Note: her handwriting will be in green, mine in purple.)

Then I sat the three plates in front of her with the 100-plate on the left and the 1-plate on the right, and the toothpicks beside the plates. On the graph I pointed to the "8" in the far left column and said, "Make this with the toothpicks."

"I can't! The toothpicks are straight and the eight is round!" Once we got over that hump, she counted out eight bundles of ten, and put them in the 1-plate.

I said, "Good! Now, what is the number I wrote?"

"Eight."

"Right. How many toothpicks do we have here?"

She looked down at the toothpicks and back up at me. I picked up one bundle. "How many in this bundle?"

"Ten."

"Right, so if there are ten in each bundle, how many is eight bundles?"

She counted by tens right up to eighty.

"Is eighty the same as eight? No? How do we get eight toothpicks?"

She grinned and quickly counted eight single toothpicks and dropped them in the 10-plate.

"Okay, you've got eight toothpicks, but this plate is for bundles of ten, see?" She moved them to the 1-plate.

"Now let's write on this page what you did. We have single toothpicks in the 1-plate, so let's write in the 1-column. How many toothpicks in the 1-plate?"

She wrote "8" in the 1-column.

"Now we'll write it over here on the right again. I know it doesn't make sense, but it will in just a minute. Okay, now we'll try something a little harder." Under the "8" in the left column, I wrote a "12" as she dumped all the toothpicks off the plates.

"Now, make the twelve with your toothpicks."

She picked up all of the single toothpicks and counted them out into the 1-plate. Reaching ten, she looked around for more singles. "Hey, I can't make twelve! I don't have enough toothpicks!"

"Really? How many toothpicks do you have?"

"Ten."

"Oh. How many toothpicks are in this bundle over here?"

"Ten."

"Yeah, ten! What if you traded these ten single toothpicks for one bundle of ten?"

Wow, did she catch on fast! "I only need two more!" We straightened out where the bundles went: "Bundles of ten go on the 10-plate, right?" and had one bundle on the 10-plate and two singles on the 1-plate.

"Now, let's write what you did. What have you got on the 10-plate, and what is on the 1-plate?" She wrote them in, then I said, "Look, what is 10 plus 2? Write it over here in the right column. That's the number we started out with!"

We did a couple more double-digit numbers, then moved to 133. Her eyes got big and round, then thoughtful. She looked over at the fat bundle of 100 toothpicks, and plopped it right down on the 100-plate. Grinning, she quickly counted out three ten-bundles and three singles.

We had some confusion with 210, as she thought she didn't know how to skip-count by 100s, and wanted 100 + 100 to be a million.

She asked, "What's 100 plus 100?"

I answered, "TWO hundred."

"What's 200 plus 200?"

"FOUR hundred."

She caught on.

Then just for fun, I wrote "99" on the chart. "Oh, that's easy," she said scornfully.

"Yeah, I know. Just do it!"

She quickly counted out nine bundles and nine singles, and wrote the numbers on the chart. Underneath the "99" I then wrote "199." When she started to dump all the toothpicks off the plates, I stopped her. "Wait a minute. Look at the number." She looked at the number, back at the plates, back at the number. Click!

She grabbed the 100-bundle and plopped it down. "Done!"

And we were.

Copyright (c) 2002 Carma Paden. All rights reserved. No portion may be reproduced in any fashion without express permission.

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