## Wednesday, October 23, 2013

### Instructional Conundrum: 100 Board or 0-99 Chart?

In my work with preservice teachers I find myself thinking a lot about the language of math and the models we use. Today I want to discuss the differences between the 100 board and 0-99 chart.

In some circles, the 0-99 chart is also called a 100 board. In old-school language it is called a counting chart. I don't particularly like either of these names. Here's why.
1. Even though 100 numbers are represented on the 0-99 chart, it does not extend to 100. To be called a 100 board, it should include this number.
2. A counting chart should begin with the first counting number, which is 1. Zero is not a counting number. Therefore, a chart that begins with 0 should not be called a counting chart.
I know these are really minor points, but they speak to the issue of precision and the language we use to describe mathematical tools.

Beyond the semantic issues of what we call these charts, I have greater concerns about the use of 100 board (and the 120 board we now see to support Common Core standards).

Our system of numeration is a base-10 system. This means we have 10-digits (0-9) and that every number can be made using one or more of these digits in combination. Ten and powers of ten are used to construct the system.  Larger numbers are built by repeatedly bundling ten: 10 ones make one ten, 10 tens make one hundred, 10 hundreds make one thousand, and so on. In simpler language, every time we reach ten of a particular unit, it is regrouped and renamed. Here's what the tens column looks like on these charts. Note the placement of each.

This is where my problem with the 100 board comes in. When we reach 10 ones on a hundred board, they still remain in the ones row (or column, depending on how the chart is arranged), but they belong at the beginning of the next decade. On a 0-99 chart all the numbers in a decade appear in the same row. For example, on a 100 board, the decade row for thirty begins with 31 and ends with 40. On a 0-99 chart, the decade for the thirty row begins with 30 and ends with 39.

This representation on the 0-99 chart is much clearer and more accurately represents the way our number system operates.

The other idea the 0-99 chart makes explicit is that zero is an even number. Even though we don't begin counting with 0, placing it on the board shows students that one less than 1 is 0 and that zero IS a number! On a 100 board we recognize numbers ending in 0 as even, but because zero does not appear, students don't view it as a number (just a placeholder) and often question its classification, wondering if it is even, odd, or neither. The 0-99 chart can help students overcome these misconceptions.

In the final analysis, both charts can be used for developing skills in counting up and counting back, skip counting, finding one more/one less and ten more/ten less than a number, recognizing patterns, place value, addition, subtraction, finding multiples, prime numbers, and more. However, the 0-99 chart does this while helping students work within the structure of our base-10 system.

Now that I've articulated the reasons that I feel the 0-99 chart is a better tool than the 100 board, here's a set of charts for you to use. One is a traditional 0-99 chart (oriented horizontally), while the other displays the numbers vertically.

What are your thoughts about the difference between the 100 board and the 0-99 chart? Which do you prefer and why?

1. Thank you for the download. I agree with you, the 0-99 chart is a better tool.

2. I also agree that the 0-99 chart is a better visual and instructional tool! Unfortunately, the hundred chart is used in our curriculum and reporting (Ontario, Canada), and I don't want to confuse the children by using both... I teach a Grade 1/2 class -- not sure if they could handle using them both? Do you have an opinion on this?

3. I am so glad you brought this up. I thought I was the only one that thought this way.

4. Personally, I like the 1-100 chart because it seems more natural to me to start counting at one (I use the chart in lots of different counting games), because it emphasizes the n-ty nine to (n+1)-ty pattern, and especially because it shows plainly that 100 is 10 tens.

I also like the inverted hundred chart. You can use either version, just start counting at the bottom and put the greater numbers up higher.

For more fun with a hundred chart, check my blog post 30+ Things to Do with a Hundred Chart.

5. I'd like to question what exactly a math specialist was thinking when he or she came up with the 100s chart (and then got it added to a provincial curriculum?!?). The whole value of a positional system (ones column, tens column etc.) is overlooked in this presentation and the value and utility of zero is painfully underplayed. From a first look, at least, this seems like a simple case of organizational inertia.

Even Tricia's much improved 0-99 chart has the numbers justified, specifically, centred within each box. This makes good artistic sense, but numbers are not justified this way, because it obscures the number of hundreds, tens, and ones in a 3 digit number. The numbers should be (as numbers always are) right justified, as in
http://static.eastpole.ca/patterns-100.pdf

Thanks for bringing the issue up, Tricia!

1. Tai,
I never gave any thought to the justification of the numbers. Excellent point! I will have to work on a new version. Thanks so much for pointing this out.
Tricia

6. We just adopted a new math series that uses the vertical number chart. Thoughts/comments about why vertical chart is better than horizontal chart?

1. I've been ruminating on a vertical chart but I think we need to come to terms that our numbers build from the right (counter to English language arts) and so the smallest numbers should be in the lower right and, after the ones column, only the top square of each column/round decades (10, 20, 30, 40, etc) should have numerals in them as the squares on the chart, below them, are simply part of the collective group of ten, without their own specific identity.

7. I use the 0-99 chart for number sequence but don't spend much time with it.

I use a totally different system for teaching place-value (on which I spend the bulk of early math learning). Place value is much more important and includes learning the counting sequence. After many days using the usuall manipulatives, I print a sheet with three columns and 10 rows. the columns are colored left-right red-blue and green. the kids use unit blocks, tenbars, and hundredflats to build the quantity then write the number in the correct column. We do start with zero because we can't count nothing but we need a way to indicate that there is nothing and that 10 means one set/bar of ten and nothing left over.

It's hard to explain in writing but 4yr olds grasp the concept of place value quickly and can't wait to complete their own "booklet of numbers".

then I present another sheet with six columns and 10 rows repeating the original three colors and separating the six columns into two sets of three columns with a good size comma. There is discussion about the new sheet and how it differs from the first. Then I use models of the new larger quantities. The kids are fasinated by the new sizes to represent the new place values. After exploring numbers from one unit through 999,999 I encourage the kids to tell me what the next sheet would look like and how big a one million unit block would be etc. We do not write these new numbers at this point.

Using just three colors (and commas) to represent place value is much easier for the kids to grasp and remember rather than using the rainbow as some do. with older kids that can draw a cube, keep the colors till place value is firm then use only the cube, flat, and bar if still needed

1. thanks this is so helpful! would you be able to share pictures of the sheets or pdfs? currently working with my 5 year old

8. Thanks so much for the vertical chart! I could not find any others on-line!

9. Thanks! The 0-99 chart feels like a great way to further clarify number with a concise visual of the base 10 system.