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together so that students can track the pattern of 0, 2, 4, 6, 8 in one decade before moving to the next. In a way it is an extension of counting by 10s, in that if students can remember the decades in order, they can apply the pattern of counting by 2s within the decade. As mentioned earlier, having 0 in the first row makes the pattern explicit. As students move from counting and writing num bers, to learning place value, the hundreds chart has pros and cons. Although the numbers 1 - 100 are shown and discussions can be had about how thir ty, for example, is composed of three tens and zero ones, the pattern that is shown on the chart neglects 0, except at the end of each group of ten. The hun dreds chart shows groups of 10 on each line, which corresponds to a rod, which is likely one of the rea sons that the hundreds chart was constructed in this way. However, using this construction it is not clear that each decade starts with zero ones. One activity I find very useful is to have students place a base - 10 unit on every square in the 99s chart. We first talk about why we do not put a cube on 0. Then students can count nine cubes and we talk about what happens when we get to ten. We can trade the ten cubes that have been accumulated for one rod. This rod can lay horizontally next to the 10, showing that we have one group of 10, and then we count to 19. Again, we group our units into a rod as we exchange ten units for a rod. We contin ue this until we get to 99, when we talk about how many rods we will have if we add one more. I draw attention to the fact that we have used all of the digits again, this time in the tens place. We look at how the number 100 shows that there are ten groups of ten, but it is also one group of one hun dred. Many students struggle with the concept of zero (Vacca, 1995) and will often dismiss zero as unimportant or unnecessary, particularly in larger numbers. If they are able to see that 0 is an integral part of building numbers both visually and as a val ue, I believe it helps stave off those misconcep tions. The 99s chart is able to tie counting, the visu al of how numbers are built from digits, and how that connects to place value all together.

Figure 6. Example of a rounding chart. An extra column is added so that each row shows the begin ning and ending ten for that group of numbers.

modified to help students as they are introduced to rounding (Figure 5). The difference is that students who are used to the 99s chart, and are aware of ze ro as integral part of the number system have an easier time rounding numbers 1 - 4. I work with stu dents in 3 rd , 4 th , and even 5 th grade who struggle to round these numbers without a chart because they don ’ t see 0 - as part of the first group of ten or as a valid amount to which to round. They will round those numbers to 5 or to 1. When we look at a 99s chart together and identify that the two groups of ten are the numbers that are between them, then the note of disbelief and amazement in their voices that 0 is included is both entertaining and heartbreak ing . The final use of these tools for which I believe the 99s chart is preferable, is for any type of calcula tion. Many students benefit from the ability to see the numbers they are using, and their relationship to each other, when adding, subtracting, and learn ing to multiply and divide. The problem with the hundreds chart is, it does not start at 0. There is no way for students to see what happens when num bers are added to 0, although they can do the in verse and add 0 to a number. They can subtract 0 from a number, but they cannot subtract a number from itself and see that the answer is 0. The hun dreds chart works well for multiplication, but does not allow students to demonstrate division as re peated subtraction. Of course, number lines are available for these demonstrations and calculations,

Both the 99s chart and the hundreds chart can be

Virginia Mathematics Teacher vol. 47, no. 2

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