M4N2. Students will understand and apply the concept of rounding numbers.

a. Round numbers to the nearest ten, hundred, or thousand.

Rounding is a specific technique to approximate numbers. Some teachers in primary grades actually teach their students rounding when they want students to "estimate." However, "estimation" and "rounding" aren't the same idea. In fact, as an approximation technique, it is probably better to teach rounding when students are working with larger numbers.

As you know, rounding a number to the nearest designated place means to look at the numeral to the right of the place to which we are rounding. If the numeral is 4 or less, we will round down (i.e., simply change all places to the right of the designated place 0's) and if it is 5 or above, we round up (i.e., increase the numeral in the designated place by 1 and change all numerals to the right 0's). So, when 45,542 is rounded to the nearest thousands place, it will be 46,000, and when it is rounded to the nearest hundreds, it will be 45,500.

One question students sometime ask is why we round up with a "5" even though 5 is right in the middle (of 0, 1, ..., 9). Some teachers will simply say it's just a rule. But, is it?

Let's consider 45,542. If we want to round this number to the nearest thousands place, we are really asking is it closer to 45,000 or 46,000. According to the procedure, we will be checking the numeral in the hundreds place. So, what numbers between 45,000 and 46,000 have a 5 in the hundreds place? Well, 45,500 is definitely one. But there are a lot more: 45,501, 45,502, 45,503, ... 45, 598, 45,599. Altogether there are actually 100 numbers in this range with a 5 in the hundreds place? So, which of these numbers are closer to 45,000? 46,000? Right in the middle? Well, it's obvious that all but one of these numbers are actually closer to 46,000, and the one exception is right in the middle. If that's the case, in general, does it make sense to round a number with a 5 in the hundreds place up or down?

The problem with "5 is right in the middle" comes up only when you are rounding to the nearest tens (and only if we are looking at whole numbers). Since approximate numbers are used when we have very large numbers of very small numbers, perhaps trying to teach rounding, a specific approximation procedure, with such small numbers may not make any sense.

## Friday, June 19, 2009

## Tuesday, June 9, 2009

### M7n1 a - Meaning of 0

M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.

a. Find the absolute value of a number and understand it as the distance from zero on a number line.

I usually don't get many comments on my blog entry (and I would be happy to hear from more of you), but on May 30, PJGould said that he had come across a child who started his counting with zero. Of course, he noted, that made his counting always off by one. After all, zero is not a counting (natural) number. But what does zero mean?

In elementary (K-5) curriculum, there are 3 meanings of zero - perhaps it is more accurate to say 3 ways zero is used. First, zero indicates the cardinality of an empty set - that is, zero means 'nothing.' This is probably the most commonly used meaning of zero in elementary school. Another place zero is used is as a place holder in a written numbers, such as 3042. Of course, this is a slight extension of the first meaning in that there is no unit of one-hundred in this written number. So, it is still pretty close to the first meaning.

The third usage of zero in elementary schools is the starting point of a number line. In some textbooks, a number line actually starts with zero as shown below.

In other textbooks, the tick mark for zero is not at the end of a number line, implying that there may be something to the left of zero as well.

As students study positive and negative number, one of the important understanding students have to make is the meaning of zero as a referent point, or the origin, on the number line. As long as students are stuck with the idea that zero means 'nothing,' some will have difficulty making sense of numbers that is 'less than nothing.' Rather, students must look at zero as a referent point on a number line, and those number to the right of zero are positive and those on the left are negative. The distance from zero, whether on the right or left is the absolute value of the number.

For those of us who already understand positive and negative numbers, this way of looking at zero is not a major issue. However, we should be aware that this meaning of zero isn't something students are familiar with. In most elementary curriculum, very little explicit discussion takes place about the role of zero on a number line. Thus, when we introduce positive and negative numbers, we do have to keep this shift in understanding of zero in our mind.

a. Find the absolute value of a number and understand it as the distance from zero on a number line.

I usually don't get many comments on my blog entry (and I would be happy to hear from more of you), but on May 30, PJGould said that he had come across a child who started his counting with zero. Of course, he noted, that made his counting always off by one. After all, zero is not a counting (natural) number. But what does zero mean?

In elementary (K-5) curriculum, there are 3 meanings of zero - perhaps it is more accurate to say 3 ways zero is used. First, zero indicates the cardinality of an empty set - that is, zero means 'nothing.' This is probably the most commonly used meaning of zero in elementary school. Another place zero is used is as a place holder in a written numbers, such as 3042. Of course, this is a slight extension of the first meaning in that there is no unit of one-hundred in this written number. So, it is still pretty close to the first meaning.

The third usage of zero in elementary schools is the starting point of a number line. In some textbooks, a number line actually starts with zero as shown below.

In other textbooks, the tick mark for zero is not at the end of a number line, implying that there may be something to the left of zero as well.

As students study positive and negative number, one of the important understanding students have to make is the meaning of zero as a referent point, or the origin, on the number line. As long as students are stuck with the idea that zero means 'nothing,' some will have difficulty making sense of numbers that is 'less than nothing.' Rather, students must look at zero as a referent point on a number line, and those number to the right of zero are positive and those on the left are negative. The distance from zero, whether on the right or left is the absolute value of the number.

For those of us who already understand positive and negative numbers, this way of looking at zero is not a major issue. However, we should be aware that this meaning of zero isn't something students are familiar with. In most elementary curriculum, very little explicit discussion takes place about the role of zero on a number line. Thus, when we introduce positive and negative numbers, we do have to keep this shift in understanding of zero in our mind.

## Tuesday, June 2, 2009

### M3N5 a - Modeling decimal numbers

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.

a. Understand a decimal fraction (i.e., 0.1) and a common fraction (i.e., 1/10 represent parts of a whole.

According to the GPS, decimal numbers are introduced in Grade 3. In Grade 3, though, students only consider decimal numbers only in the first decimal place (or the tenths place, if students have already learned fraction terminology). Decimal numbers to the 2nd decimal place and beyond are studied in Grade 4 and above.

I have heard many teachers who say they use money as the model for decimal numbers. However, if we are limiting decimal numbers to be discussed in Grade 3 to the first decimal place, we cannot use money as an appropriate model as money amounts are shown to the second decimal place. Thus, money as a model for decimal numbers is appropriate only starting in Grade 4, and only with decimal numbers with 2 decimal places. Some people argue, and I agree, that money is not a good model for decimal numbers.

So, why isn't money a good model for decimal numbers? First, as we saw above, it is a very limited utility as a model - only in Grade 4 (and above) and only when we are dealing with decimal numbers with 2 decimal places. Although it is true that a 0 may be annexed to a decimal number with only 1 decimal place, e.g., 0.4 = 0.40, looking at all decimal number as 2-digit decimal numbers may not be the most helpful habit to develop.

Perhaps more serious problem with money as a model is that students (and adults) don't really have to think about money amounts as decimal numbers. Rather, they are really combinations of two monetary units, dollars and cents. By using two different units, we can simply work with two whole numbers. For example, we don't consider $2.35 as two and 45 hundredths dollars. Rather, it is TWO dollars and THIRTY-FIVE cents. If we get additional $3.18, we simply add TWO and THREE dollars and THIRTY-FIVE and EIGHTEEN cents. Therefore, we are not really considering those numbers as decimal numbers - they only use notations similar to decimal numbers. Mathematically speaking, there isn't really that much difference between monetary amounts and durations expressed in hours and minutes. If you spend 2 hours and 35 minutes watching TV and 3 hours and 18 minutes playing computer games, then you wasted 2+3=5 hours and 35+18=53 minutes!

So, if money isn't a good model, what other models can we use? Base-10 blocks are always an option - we just have to designate something other than unit cubes as "1." We can also use paper strips, too, just as you might use them to model fractions. No matter what model you decide to use, an important idea we want children to develop is the unitary perspective of decimal numbers. For example, 0.4 is made up of 4 0.1-units. This is very similar to the unitary perspective of fractions I discussed in the last post. This way of looking at decimal numbers will allow students to bridge decimal numbers to whole numbers. So, it is very important for us to think about models to use, but we should also keep in mind the goal understanding we want our students to develop.

a. Understand a decimal fraction (i.e., 0.1) and a common fraction (i.e., 1/10 represent parts of a whole.

According to the GPS, decimal numbers are introduced in Grade 3. In Grade 3, though, students only consider decimal numbers only in the first decimal place (or the tenths place, if students have already learned fraction terminology). Decimal numbers to the 2nd decimal place and beyond are studied in Grade 4 and above.

I have heard many teachers who say they use money as the model for decimal numbers. However, if we are limiting decimal numbers to be discussed in Grade 3 to the first decimal place, we cannot use money as an appropriate model as money amounts are shown to the second decimal place. Thus, money as a model for decimal numbers is appropriate only starting in Grade 4, and only with decimal numbers with 2 decimal places. Some people argue, and I agree, that money is not a good model for decimal numbers.

So, why isn't money a good model for decimal numbers? First, as we saw above, it is a very limited utility as a model - only in Grade 4 (and above) and only when we are dealing with decimal numbers with 2 decimal places. Although it is true that a 0 may be annexed to a decimal number with only 1 decimal place, e.g., 0.4 = 0.40, looking at all decimal number as 2-digit decimal numbers may not be the most helpful habit to develop.

Perhaps more serious problem with money as a model is that students (and adults) don't really have to think about money amounts as decimal numbers. Rather, they are really combinations of two monetary units, dollars and cents. By using two different units, we can simply work with two whole numbers. For example, we don't consider $2.35 as two and 45 hundredths dollars. Rather, it is TWO dollars and THIRTY-FIVE cents. If we get additional $3.18, we simply add TWO and THREE dollars and THIRTY-FIVE and EIGHTEEN cents. Therefore, we are not really considering those numbers as decimal numbers - they only use notations similar to decimal numbers. Mathematically speaking, there isn't really that much difference between monetary amounts and durations expressed in hours and minutes. If you spend 2 hours and 35 minutes watching TV and 3 hours and 18 minutes playing computer games, then you wasted 2+3=5 hours and 35+18=53 minutes!

So, if money isn't a good model, what other models can we use? Base-10 blocks are always an option - we just have to designate something other than unit cubes as "1." We can also use paper strips, too, just as you might use them to model fractions. No matter what model you decide to use, an important idea we want children to develop is the unitary perspective of decimal numbers. For example, 0.4 is made up of 4 0.1-units. This is very similar to the unitary perspective of fractions I discussed in the last post. This way of looking at decimal numbers will allow students to bridge decimal numbers to whole numbers. So, it is very important for us to think about models to use, but we should also keep in mind the goal understanding we want our students to develop.

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## Creative Commons

Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.