Thursday, February 25, 2010

M2N2a Addition/Subtraction Algorithms

M2N2. Students will build fluency with multi-digit addition and subtraction.
a) Correctly add and subtract two whole numbers up to three digits each with regrouping.

Understanding the addition and subtraction algorithms is an important goal for Grade 2 GPS. However, what is the point in teaching algorithms? For that matter, what does it mean to teach algorithms? Some believe that we are to teach children the algorithms that are currently conventional in the United States. Others believe that teaching algorithms mean to help students make their own methods into a written procedures. This perspective doesn't mean teach children different algorithms. Many of today's textbooks include "alternative" algorithms, and some teachers will teach their students each one of them, thinking children can pick what is the easiest to them. However, the source of the algorithms is still outside of the children. Thus, although they are given some choices, algorithms are still imposed on them. Helping children make their own methods into a written procedure means the source of the algorithm is within children.

Take, for example, subtraction. Ask children to make 85 with base-10 blocks. They have no problem representing this number with 8 longs and 5 units. If you then ask them to give you 32 from what they have, they will give you 3 longs and 2 units, in that order. Ask them to make 82. Then, ask them to give you 46, they often will give you 4 longs and then pause. They see that they don't have enough units to give you. At that point some will ask if they can trade one of the remaining longs with 10 units. Once they trade the long for 10 units, they can give you 6 units, leaving them 3 longs and 6 units.

This process can be made into a written procedure like this:

This "algorithm" will work with any numbers, however long they are. If the purpose of teaching algorithms is for students to have a reliable and efficient computational method, there is nothing wrong with this algorithm. So, is that the only reason we teach algorithms?

Actually, I think there is another very important point we should help students understand when we teach addition and subtraction algorithms. That idea starts when we study simple addition and subtraction problems like 40 + 30 or 80 - 20 in Grade 1 as I alluded in the last entry. Addition problems like 40 + 30, 400 + 300, 4000 + 3000 are all related to 3 + 4 because in each case we are putting together 3 of something with 4 of the same thing. On the other hand, 300 + 40 does not relate to 3 + 4 because even though we may have 3 of 100 and 4 of 10, 3 and 4 are referring to different units. One of the important ideas of addition and subtraction is that you can only add or subtract numbers if they are referring to the same unit - we cannot add apples and oranges. To make the paper and pencil addition/subtraction easier, we arrange the numbers vertically, one on top of the other. When we do this, we also line up the place values of each number. By doing so, we know we can add or subtract the numerals in each column because they are referring to the same unit.

Although many textbooks will include the vertical notation even while studying the basic addition and subtraction facts, the importance of the notation is about this idea of lining up the place values so that we can add or subtract like numerals. With the basic facts, such an idea is rather implicit.

If we can help students make this understanding explicit, they can use it when they study addition and subtraction of decimal numbers and fractions. When we "line up the decimal points," what we are really doing is to line up the place values. Thus, we are simply following the same idea. When we add or subtract fractions, we need a common denominator, because the numerators will tell us how many while the denominators tell us what we are counting. Thus, in order to add two fractions, we have to have the two numerators referring to the same unit, or the same denominator. And, when we add, what is being counted do not change, thus the denominator stays the same. Thus, teaching of addition/subtraction algorithms is when we lay this important foundation - not just teaching them an efficient and reliable computational strategies.

Sunday, February 14, 2010

M1N2 - Understanding "place values"

M1N2. Students will understand place value notation for the numbers 1 to 99. (Discussions may allude to 3-digit numbers to assist in understanding place value.)

I have written about this standard previously. In that entry, I discussed different rules of our numeration system. In this post, I want to discuss a bit about what it means to understand "place value."

When you ask young children problems like 24 + 32 before they learn addition of two-digit numbers formally, they would often say something like this: "I know 20 and 30 is 50 and 4 and 2 is 6. So, the answer is 56." So, does this child understand "place value"? It is difficult to say. English number words beyond 20 has a very distinct and easily recognizable pattern. 21 is read "twenty one," 22 "twenty two," etc.. Young children easily notice that "twenty" and "one." Thus, they can easily "decompose" the number words into "twenty" and "two," but that is not enough to say they understand our number system. Understanding of our number system requires not only recognizing 21 is made up of 20 and 1, but also 21 is made up of "2 tens and 1." Because children are often familiar with the decade number words, "ten, twenty, thirty, forty, fifty, sixty, ..." they can determine that "twenty and thirty is fifty." Children who understand "place value" can say that 20+30 is the same things as 2 tens plus 3 tens, thus 2+3=5 tens.

Clearly understanding of "place value" is important for children's understanding of computational algorithms starting in Grade 2. However, this understanding is one of the important goals when we have children think about how to solve problems like 20+30 in Grade 1 (M1N3g). The focus of M1N3g is not to develop computational strategies but really to deepen their understanding of our number system.

Saturday, February 6, 2010

M5M1g - Developing Area Formulas (9)

M5M1. Students will extend their understanding of area of geometric plane figures.
g. Derive the formula for the area of a circle.

In the last post, we established the relationship between the diameter and the circumference of a circle, i.e. the circumference of a circle is always π times as long as the diameter of the circle. Unlike the area formula for polygons we have looked at previously, to establish the formula for the area of circles, we need to know something about the circumference as it will become clear a little later.

But, before we get to the formula, let's investigate the area of a circle a bit more intuitively. The picture below shows a circle with an inscribed square (black) and a circumscribed square (red):

If you compare the area of these two squares to the square that has the radius of the circle as a side (shaded), you see that the inscribed circle has the area twice of the shaded square, and the area of the circumscribed square is 4 times of the shaded square.

It should be clear that the area of the circle is greater than the area of the inscribed square but less than the area of the circumscribed square. Since the area of the shaded square is (radius)^2 (I can't figure out how to make "2" into a superscript...), we can say the following about the area of circle:

[Note: the exponentiation notation isn't discussed in elementary schools, so it is probably better to write it (radius x radius).] If you draw a circle on a grid paper (let's say with the radius of 10 cm), you can refine this approximation even further. The area of circle turns out to be a little more than 3 times of (radius)^2.

In everyday situation, knowing that the area of a circle is a little more than 3 times of (radius)^2 may be good enough. However, sometimes you may want to know more exact value. So, how can we derive the formula for the area of circles? How can we change a circle into a familiar shape? There are a few different possible routes, but I will discuss the most common (I think) approach.

Suppose you have some pizzas left over. Your refrigerator is too full to put the whole box in. What would you do? One thing you might do is to re-arrange the slices like the picture below shows:

When we rearrange, we know that we still have just as much pizza as we did before. But the resulting shape looks more like familiar shapes we have seen before, in this case, a trapezoid. So, we try to use this idea as we derive the formula for the area of a circle.

Suppose we subdivide a circle into 6 equal sectors, then re-arrange those sectors. It will look like this:

If we cut the same circle into 8 sectors then re-arrange the sectors, it will look like this:

If we keep increasing the number of sectors, each sector will get thinner and thinner. Here are the pictures showing what happens when the same circle is cut into 12 sectors and then 24 sectors:

As the number of sectors increases, the shape we get after we re-arrange them will get closer and closer to a rectangle:

Because the re-arranging of sectors do not change the area, the area of this rectangle is equal to the area of the original circle. So, we just need to calculate the area of this rectangle. The area of rectangle can be calculated by multiplying its length and width. By paying attention to where these parts came from the original circle, we see that the length (vertical dimension in this picture) is equal to the radius of the circle, while the width is a half of the circumference of the original circle. Therefore,

But, we already know that,

By substituting the second one in our formula for the area of a circle (because we only want a half of circumference), we end up with the formula for the area of a circle,

The process of changing the given shape into a familiar one used here is a bit different from what we did with polygons. In fact, it will probably be very difficult for children to actually cut circles and re-arrange the sectors - it can be done by providing circles with the sectors already drawn. However, even when we provide pre-drawn sectors, realistically, we can cut the circle into 12 or so sectors at most. At that point, the result may not look like a rectangle - it may look much more like a parallelogram. Since we already know the formula for the area of a parallelogram, we can also use that idea, too.

Deriving the area formula for circles is definitely more complicated than deriving other formulas. Unlike other formulas, there may have to be more "demonstration" than actual hands-on activities. Some may question whether or not it is an appropriate learning goal for Grade 5. However, it is a Grade 5 standard in the GPS, and we need to think about how we can make the formula more meaningful to students. What we discussed here is just one approach to deriving the formula. I encourage readers to investigate other ways of approaching this topic.

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