Tuesday, November 2, 2010

3.OA.1 and M2N3a - Writing multiplication equations

3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

M2N3. Students will understand multiplication, multiply numbers, and verify results.
a. Understand multiplication as repeated addition.
Now that Georgia has adopted the mathematics standards developed by the Common Core State Standards Initiative, I will incorporate the CCSS in my discussion of Georgia standards. I have previously discussed M2N3a in a June 2007 post. In that post, I raised an issue of treating multiplication as repeated addition.

In the CCSS, multiplication is introduced in Grade 3 in the domain of "Operations and Algebraic Thinking." The first standard in the cluster related to multiplication, the CCSS states the following:
1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
In the public draft released in spring, 2010, there was a statement about multiplication as repeated addition, just like M2N3a. However, that statement has been removed. Instead, the CCSS focuses on the meaning of multiplication as an operation to find the total amount when objects are arranged in equal groups. Then, in Grade 5, the CCSS states that the meaning of multiplication to be expanded and consider multiplication as scaling (or re-sizing). I think the approach the CCSS has taken is much more appropriate than what the current GPS states. For those of you interested in the discussion of whether or not multiplication is repeated addition, I encourage you to read a series of columns written by a Stanford University mathematician, Keith Devlin (June 2008, July-August 2008, and September 2008).

Today, however, I would like to focus on the implicit idea in the CCSS - and the current GPS does not even touch upon this idea. Toward the end of my previous blog, I discussed the order in which you write multiplication sentences. In it, I made it clear that my preference is to write the multiplicand, i.e., the number of objects in a group, first then the multiplier - I might argue that is THE correct way mathematically. However, the CCSS actually suggests we write multiplication sentences in the opposite order. Thus, 5x7 is interpreted as "the total number of objects in 5 groups of 7 objects each." Although I have stated in the past that what is important is we have an agreement on the order, I have run into several situations recently that revealed writing the multiplicand first is the way to go.

But, let's first start with how we state/write/read multiplication sentence. A common way teachers and students read multiplication sentence is "5 times 7 is 35." However, if the sentence is representing the situation with 5 groups of 7 in each, a mathematical way of reading the sentence is "7 multiplied by 5 is 35." When we use the phrasing, "N is multiplied by M," it is clear that M is the number of groups - that is, N is taken M times. Thus, one surface level issue is that the order in which we read multiplication sentences and how they are written may not align. Some might argue that this is a non-issue. After all, the same thing happens with division, too. We say "35 divided by 7," but we also say, "how many times does 7 go into 35?" When we write division problem on paper, the divisor may follow the division symbol or it may be outside of the long division symbol (thus to the left of the dividend).

To me, however, the issue is fundamental, and writing the multiplier first creates some difficulties in mathematical discourses. Let me share some examples. In the 4th grade CCSS standard, student are expected to understand multiplication of fractions by whole numbers. The CCSS document is very careful to remain consistent with the order, so in the examples they include always have the multiplier in front, such as 3 x (2/5). Once we agree that we write the multiplier first, problems such as (2/5) x 3 are treated in Grade 5. [There is actually a similar distinction with respect to multiplication and division of decimal numbers in the current GPS. Students learn about multiplying and dividing decimal numbers by whole numbers in Grade 4, and multiplication and division by decimal numbers are discussed in Grade 5.]

Then, in Grade 5, the CCSS treats multiplication by fractions: "Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b." So, (2/3) x (4/5) is interpreted as taking 2/3 of 4/5. Thus, we first divided 4/5 by 3 (thus students must have learned how to divide a fraction by a whole number before this topic), then we take 2 groups of it. Thus (2/3) x (4/5) = 2 x (4/5 ÷ 3). Thus, the multiplier, 2/3, gets split around the multiplicand, 4/5. If you write the multiplicand first, taking 2/3 of 4/5 will be written as (4/5) x (2/3) = (4/5) ÷ 3 x 2 = (4x2)/(5x3). This seems to be much easier to connect to the formula (a/b)(c/d) = ac/bd. [Another issue here is why there is no parentheses around q ÷ b in the CCSS. It seems like you must first find the "partition of q into b equal parts," but the order of operations says we go from left to right. Without parentheses, the statement "a x q ÷ b" means multiply q by a, then partition the result into b equal parts.]

Another example is when discussing how to create equivalent fractions. Again, the CCSS is very careful about the order in which multiplication is written. Thus, they say, "Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size." Typically, this idea is discussed as "if you multiply both the numerator and the denominator of a fraction by the same number, the value of fraction remains the same." More often than not, this relationship is written in textbooks as a/b=an/bn - the multiplier written after the multiplicand. But, this notation would not match our agreement on how to write multiplication.

One final example is not explicitly mentioned in the CCSS nor the GPS. However, once we learn division by fractions, we often makes the statement, "division is the same as multiplication by the reciprocal of the divisor." When we use mathematical notations, we will typically write (a/b) ÷ (c/d) = (a/b) x (d/c). However, if division is the same as multiplication "by the reciprocal," that is, the reciprocal is the multiplier, it should really be written as (a/b) ÷ (c/d) = (d/c) x (a/b).

In general, many familiar ways we discuss/write multiplication assumes that the multiplier comes after the multiplicand. Thus, "5x7" should not be "5 times" of 7, but rather 5 "7 times." Perhaps because I am not a native English speaker, I also get a different sense when I hear "5 times 7" in one breath and "5" (pause) "times 7." The former gives me the sense of 5 groups of 7 but the latter makes me think of 7 sets of 5. Anyway, I wish we can eventually agree to write the multiplicand first - perhaps in the next revision of the CCSS.

By the way, some people might wonder about how this idea works in algebra. After all, when we think of simplifying "5x + 3x," it is much easier to think of this as having 5 x's and 3 x's altogether, thus 8 x's. On the other hand, we want students to understand the slope of a linear function, like "3" in y=3x+2, as the "rate of change." A rate, however, is typically amount per unit. Thus, "3x" in this context suggests we have x units of 3. In algebra (and other higher level mathematics), I believe there are actually two competing conventions. One is the order in which we write multiplication and the other is the convention of writing numbers (and constants) before variables in a term. In algebra, I believe, the latter convention wins perhaps because it makes manipulation of algebraic expressions simpler. But, I think it is still very important that students pay close attention to how we write multiplication when they first learn it in elementary grades.

Thursday, October 21, 2010

M6M4 ab: Surface area formulae

M6M4. Students will determine the surface area of solid figures (right rectangular prisms and cylinders).
a. Find the surface area of right rectangular prisms and cylinders using manipulatives and constructing nets.
b. Compute the surface area of right rectangular prisms and cylinders using formulae.
Surface area is simply the sum of the area of all the faces of a solid. Thus, as long as we can calculate the area of each face, there is nothing really new involved in this standard. The only tricky part here is to figure out the shape of the lateral face of a cylinder. But, students learn about the nets of various solids including cylinders in Grade 4. Thus, the surface area of a cylinder is the sum of the area of the two bases (circles) of the cylinder, and the area of the lateral face which is the rectangle with the dimensions equal to the height of the cylinder and the circumference of the base. We can summarize it in a formula like this:

Surface Area = 2 x (Area of base) + (Circumference of the base) x Height

Although I don't think it is that critical that students know this formula or the formula for prisms, it may be useful to have students explore the surface area of (rectangular) prisms not just as the sum of the areas of the faces. In fact, the first indicator discusses the use of nets in determining the surface area. If the surface area is simply the sum of the areas of the faces, there is really no need to use a net. So, what might be the reason for using nets to calculate the surface area of (rectangular) prisms?

We know that there are many different nets for a prism. However, a common net of a prism has all the lateral faces forming a "train" of rectangles and the two bases on the opposite sides of this "train" like this one.

Instead of calculating the area of each of the faces, you can consider the "train" of the lateral faces as one big rectangle, like this:

The length (vertical side in the drawing above) is equal to the height of the prism. The width (horizontal side) is actually the perimeter of the base. Thus, we can calculate the sum of the areas of the lateral faces as (Perimeter of the base) x height, too. But, then, the calculation of the surface area of a prism can be summarized in this formula:

Surface Area = 2 x (Area of base) + (Perimeter of base) x Height.

Perhaps investigating the surface area of prisms from this perspective allows us to use the same formula for all prisms and cylinders. However, I still don't think it is that important for students to know the formula...

Tuesday, September 21, 2010

M6M3. Students will determine the volume of fundamental solid figures (right rectangular prisms, cylinders, pyramids and cones).
a. Determine the formula for finding the volume of fundamental solid figures.
b. Compute the volumes of fundamental solid figures, using appropriate units of measure.

There is actually a standard in Grade 5 that discusses the volume:
M5M4. Students will understand and compute the volume of a simple geometric solid.
c. Derive the formula for finding the volume of a cube and a rectangular prism using manipulatives.
d. Compute the volume of a cube and a rectangular prism using formulae.
So, what are the difference between these two standards? There are two obvious differences in these two standards. First, the Grade 5 standard involves the volume of "simple geometric solids," while the Grade 6 standard deals with "fundamental solid figures." Specifically, in Grade 6, students are expected to determine the volume of cylinders, pyramids, and cones in addition to cubes and rectangular prisms learned in Grade 5. So, the Grade 6 standard deals with a wider range of solids than the Grade 5 standard does.

Another difference is that, in Grade 5, students are to derive the formula using manipulatives while the Grade 6 standard does not mention the use of manipulatives. So, how do we expect Grade 6 students to derive the formula?

In Grade 5, students may determine the volume of cubes and rectangular prisms by filling them with unit cubes. Those experiences parallel what students might have done as they determine the area of squares and rectangles using unit squares. From these experiences, students learn that the dimensions of cubes and rectangular prisms can tell us the number of unit cubes that fit in each dimension. Thus, they can conclude that the volume of a rectangular prism can be calculated by multiplying its length, width and height.

The solids students explore in Grade 6 cannot be filled with unit cubes because of their shapes. So, how can students determine the formula for those solids? One important step is to re-visit the formula for the volume of cubes and rectangular prisms. When we determine the number of unit cubes inside a rectangular prism, we typically figure out the number of unit cubes in one layer, then multiply the result with the height, which signifies the number of layers. However, the first product, the number of unit squares in a single layer is equal to the area of the rectangular base. Thus, we can express the formula for calculating the volume of a rectangular prism as (Area of Base) x height, instead of length x width x height.

When we consider the volume formula for a rectangular prism as (Area of Base) x height, a natural question is whether or not this formula can be applied to prisms whose bases are something other than rectangles. Students can explore this question with triangular prisms and other prisms. Through such an exploration, they will find that the formula applies to any prism - and cylinders.

The volume formula for pyramids (and cone) is slightly different. It may be difficult to derive the volume formula for pyramids/cones directly. In fact, what we need to do is to relate the volume of a pyramid/cone to the related prism/cylinder, which has the congruent base and the same height as the pyramid/cone. A common way to establish this relationship is to have students actually fill up both a pyramid and the related prism (there are commercially made sets available for this purpose) with water or rice grains. Through such experimentations, students can establish the relationship that the volume of a pyramid/cone is a third of the volume of the related prism/cylinder. Thus, the volume formula for a pyramid is simply (Area of the base) x height ÷ 3 - if students have already learned multiplication of fractions before this unit, the formula can be written as (1/3) x (Area of the base) x height.

It may be useful to have students actually cut out (or the teacher demonstrate cutting) a cube into 3 congruent square pyramids like this - I apologize the poor quality of my 3-D drawing, and I hope you get the idea from this picture.

Note that these pyramids are different from most pyramids students seen in K-8 curriculum. Pyramids students study typically has the vertex that is not on the base to be directly above the center of the base. These pyramids, in contrast, has the vertex directly above one of the vertices of the base.

Clearly, such a demonstration does not establish the 1:3 relationship of the volume of any pyramid to the volume of the related prism. However, it may still be a worthwhile experience for students to have. There is, I believe, a commercially made puzzle that asks you to make a cube out of 3 congruent pyramids.

Saturday, September 4, 2010

M7G3 - Proportional Relationships (7)

M7G3. Students will use the properties of similarity and apply these concepts to geometric figures.
b. Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.

This standard is another example of how proportional relationships play an important role in the middle school mathematics curriculum. We say two figures are similar if one can be made to overlap the other exactly through a combination of translation (slide), rotation (turn), reflection (flip), and dilation (magnification). The parts of two similar figures that match up are called corresponding angles, sides, etc. In a pair of similar figures, we know that corresponding angles are congruent and the ratios of corresponding segments are constant - and the value of this ratio is the scale factor. For example, the two quadrilaterals shown below are similar.

Therefore, angles A and E, B and F, C and G and D and H are congruent, respectively. Moreover, the ratios of the lengths of sides, AB: EF, BC:FG, CD:GH, and DA:HE, are constant, and in this case the ratio is 1:2. The scale factor depends on which of the two figure we consider as the base of the comparison. So, if we consider quadrilateral ABCD as the base, the scale factor, in this case, is 2. On the other hand, if we consider quadrilateral EFGH as the base, the scale factor is 1/2.

Suppose AB = 4 cm, BC = 2 cm, CD = 5 cm, and DA = 6 cm. Then, EF = 8 cm, FG = 4cm, GH = 10 cm, and HE = 12 cm. Let's organize these lengths in a table.

Now, you see that as the length in ABCD doubles and triples (from 2 cm to 4 cm or 6 cm), the length of EFGH also doubles and triples. Even when the length becomes 2.5 times as long, from 2 cm to 5 cm, in ABCD, the corresponding length also becomes 2.5 times as long, 4 cm to 10 cm. Thus, the lengths in these two figures are in a proportional relationship. In general, if two figures, X and Y, are similar, the lengths in these two figures are in a proportional relationship. Thus, we can apply all the tools we discussed previously in representing this relationship. So, if we use a double number line, the relationship can be represented something like this:

Thus, if we know a side in Figure X is 15 cm and the corresponding side in Figure Y is 6 cm, we can use that relationship to determine the length of any side can be determined if we know the length of the corresponding sides. Suppose, we know another side in Figure X is 20 cm, the relationship can be represented in a double number line like this:

On the other hand, if you know the length of a side in Figure Y is 4 cm, the relationship will be represented like this:

Another feature of a proportional relationship is that the quotients of corresponding quantities are constant. So, if we divide the lengths in Figure X by the corresponding lengths in Figure Y, the quotients are constant. We can also use this relationship to represent the two situations above like this:

These tables basically show the four values (including the missing value represented by a ?) from the double number line representations above. Note that in the second table, the columns are in the reverse order. A number line has a particular direction, i.e., as you move to the right, the numbers become larger, However, a table does not have such an inherent directionality. So, for students, it might be more natural if we place the relationship as they are presented.

In any event, since 15 x 0.4 = 6, ? = 20 x 0.4 -- 0.4 is the scale factor if we consider Figure X as the base. For the second problem, we can say that since 6 x 2.5 = 15, ? = 4 x 2.5 -- 2.5 is the scale factor if we consider Figure Y as the base.

As is the case with the conversion of measurements from one unit to another, what is important is to help students develop an understanding that mathematics is a web of relationships. The focus of this standard is not just for students to find the missing lengths in similar figures. We also want them to understand that what they have learned previously, namely proportional relationships, can be used to represent, interpret, and investigate new situations.

Saturday, August 14, 2010

M6M1 - Proportional Relationships (6)

M6M1. Students will convert from one unit to another within one system of measurement (customary or metric) by using proportional relationships.

Some people consider the idea of proportional relationship as the culmination of the elementary school mathematics and the cornerstone of the middle school mathematics. This standards is one example of how proportional relationships play a role in different parts of the middle school mathematics.

Let's think about a situation of converting linear measurements between inches and feet.

You can easily see that as the numbers for inches become 2, 3, 4,... times as much, the numbers for inches also become 2, 3, 4, ... times as much. Therefore, these numbers are in a proportional relationship. Thus, we can use all the tools we have discussed previously to convert from one unit to another.

Suppose you want to find out how many inches 34 feet may be, you can set up the double number line representations in this way.

This representation shows that we know the per-one (or per-unit) quantity and you want to know the number corresponding to 34 units. So, you can use multiplication to find the missing number: ? = 12 x 34.

Going the other direction, for example, converting 104 inches to feet, can be represented in the same way.

Again, we know the per-unit quantity, and you want to know how many units would correspond to 104. Thus, this is a quotitive (measurement) division situation. So, you can find the missing number by division: ? = 104 ÷ 12.

To solve all these unit conversion problems, students do need to know (or be able to look up) one relationship between the two units - and it doesn't have to be 1 to something else. If you know that 2 feet = 24 inches, that's good enough to set up a double number line representation. You can solve it like you do with other proportion problems.

In principle, the situation remains the same whether you are converting within or across different measurement systems. If you know that 1 inch is approximately 2.5 cm, that is enough information for students to convert between inches and centimeters - approximately, but all measurements are approximation, anyway. Although students in earlier grades should be able to convert measurements from one unit to another in some simple cases, once students learn about proportional relationships, they no longer have to think of it in isolation. The idea of proportional relationships, thus, unifies many of the ideas students have learned previously. And, helping students to revisit some of those ideas and look at them from a new perspective is something we need to emphasize, not just the procedure of solving proportional problems.

Wednesday, July 28, 2010

Proportional Relationships (5)

M6A2. Students will consider relationships between varying quantities.
c. Use proportions (a/b=c/d) to describe relationships and solve problems, including percent problems.

While discussing "segment/tape diagrams," I discussed how those diagrams can be used to solve problems involving percents. In the last post, while discussing models for proportional problems, I discussed how double number line may be used to represent problems involving proportional relationships.

Percents describes the relative size of quantities compared to the base quantity. It turns out that "percents" and the actual quantities are in a proportional relationship. For example, suppose the base quantity is 80. The table below summarizes the relationship between the size of quantities being compared and corresponding percentages.

You can see that they are in a proportional relationship because as the quantity becomes 2, 3, 4, ... times as much, the percentages also become 2, 3, 4, ... times as much.

So, if quantities and percentages are in a proportional relationship, then we can also use double number lines to represent problems involving percents, too. So, here are 3 examples.

The first double number line representation may be for a problem like the following:
At Jackson Elementary School, there were 80 fifth grade students this year. Next year, they anticipate that the fifth grade class to be 115% of this year's fifth grade class. How many fifth graders will there be?
The second represents a problem like this:
At Jackson Elementary School, there were 80 fifth grade students this year. Next year, they are expecting 92 fifth grade students. What percents of this year's fifth grade class will the next year's class be?
Finally, the third one represents a problem like this one:
At Jackson Elementary School, they are expecting 92 fifth graders next school year. This is 115% of this year's fifth grade class. How many fifth graders are there this year?
While discussing Process Standards 5, I shared how a segment/tape diagram to represent and solve problems involving percents. The double number line is a different representation. Double number lines representing multiplication or division problems always included a "1" on one of the number lines. In these situations, there is no "1," but by placing a "1" on either number line, a solution approach that combines division and multiplication - the approach discussed in the previous entry - may become apparent.

Saturday, June 12, 2010

M6A2b - Proportional Relationships (4)

M6A2. Students will consider relationships between varying quantities.
b. Use manipulatives or draw pictures to solve problems involving proportional relationships.

In the last three posts, we considered two proportional situations. They are,
c) the distance traveled and the time of travel (at a constant speed), and
f) the amount of meat and the price of meat
The tables below show the values of these quantities:

"Problems involving proportional relationships" from these contexts might be something like the following:
Jim can walk 9 miles in 3 hours. If he maintained the same speed, how far can he walk in 6 hours?

4 pounds of meat cost $18. How much will 10 pounds of the same meat cost?
So, what kinds of pictures might we draw to solve these problems? Actually, you may find it rather difficult to draw pictures for these problems. We can draw pictures that might represent the contexts of the problems, but those pictures may not be too helpful in actually solving the problems. What about manipulatives? What manipulatives would you use to solve these problems? I am not sure what I would use.

If it is difficult to use a picture or manipulative to solve these problems, what is this standard talking about? Perhaps "pictures" here are really referring to diagrams. One particular form of diagrams is double number lines. I used double number lines extensively to talk about multiplication and division of decimal numbers (November 2008). But they can be useful to represent problems involving proportional relationships. Here are the double number line representations of the two problems above.

When students are familiar with double number lines with multiplication and division, they will notice the difference between these double number line representations and those of typical multiplication and division problems. Here are examples of multiplication and division double number line representations:

Do you notice the difference? In the two representations of the problems involving proportional relationship, there is no "1" in the representation. If we put a "1" in the representation, then we can see a solution path. For example, let's use the second problem. If we put a "1" on the top number line, it will look like this:

Now, the left side part of the representation,

is really a partitive division situation. Thus, by dividing 18 by 4, we can find that # = 4.5. Now, double number line representation looks like this:

Now, if we can ignore the middle part of this representation, it will look like this:

and this is a multiplication situation, isn't it. So, multiplying 4.5 by 10, we can obtain the missing quantity.

So, double number line representations can not only represent problems involving proportional relationships, they can also suggest ways to solve the problems, too. Of course, if we want students to be able to use double number lines as their thinking tool at this stage, they do need to be familiar with double number lines. Thus, it is important for teachers of different grade levels to discuss what representations they want to emphasize. It is important for students to be able to use multiple representations. But, if there is any representation, like double number line, that may be used across grades, then that representation should be consistently introduced/developed/used across grades.

Sunday, May 23, 2010

M6A2ae: Proportional Relationships (3)

M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
e. Graph proportional relationships in the form y = kx and describe characteristics of the graphs.

In the previous post, we analyzed the ways two quantities that are in a proportional relationship using a table. Today, we want to look at the idea of analyzing proportional relationships - and identifying features that are unique to proportional relationships - using graphs. So, let's consider one of the relationship we talked about last time:

When you graph this set of data, it will look like this:

Since these quantities are both continuous quantities, we can actually connect the data points using a line (actually a ray):

Let's compare this graph to graphs of three other situations. The first one is the siblings' ages.

The second situation is the candle situation: the length of candles burned and the length of the remaining candle.

The last case is the length and the width of rectangles with a fixed area measurement.

Since the first situation involves the discrete quantities - and since I don't know how to graph the curve for the last one, I am just going to plot these data points.

When you compare these graphs to the graphs of the proportional relationship from earlier, you immediately notice that the graph of the inverse proportional situation isn't a straight line. However, the other three situations seem to result in straight lines. Although it is not really appropriate to use a line to represent the siblings' ages data with a straight line, I'm going to do so to illustrate the similarities and differences - and I'm showing all three lines on the same coordinates.

From these graphs, we noticed that one difference seems to be that the graph of the proportional situation goes through the origin, but not the other two. As it turns out this is indeed unique to proportional situations. The other two cases, constant sum and constant difference situations, result in a straight line. One commonality among the three situations is that the rate of change is constant. Thus, the characteristic of the data sets that are represented in straight lines. I think this might be an idea that is worth discussing explicitly in Grades 7 and 8 when linear equations/functions are studied more formally.

By the way, the fact that the graphs of proportional relationships go through the origin relates to the fact that double number lines we use to represent multiplication and division situations are "hinged" at 0 - in other words, both quantities will go to 0 at the same time. In fact, proportional relationships are assumed in all multiplication and division situations. In middle grades, that fact should become explicit instead of being an implicit assumption.

Saturday, May 8, 2010

6A2a: Proportional Relationships (2)

M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.

In the previous post, I discussed how we can analyze situations where two quantities are changing simultaneously. From that analysis, we defined what a proportional relationship was - two quantities are in a proportional relationship if the quantities change in such a way that their quotient stays constant. This relationship may be represented as y÷x = k, or y = kx, where k is the constant.

Let's think about how else this relationship may be seen by looking at a couple of specific instances. The two proportional situations we discussed last time were:
c) the distance traveled and the time of travel (at a constant speed), and
f) the amount of meat and the price of meat
The tables below show the values of these quantities:

So, what commonalities do you notice about the way quantities are changing in these tables? One thing students might see quickly is that, in both situations, the quantity are changing by the same amount. In this case, both time and amount are increasing by 1 unit as you go from left to right. The distance is increasing by 3 miles while the price is increasing by $4.50. Of course, this observation is really the function of the way we listed these quantities. We could have skipped some instances like this:

Or, we could have listed these pairs unordered:

So, one thing student can learn, more generally about collecting and displaying data, is that when we organize them systematically, we might be able to observe patterns more easily. But, is there any relationship we can observe in these tables even if the data are not organized as neatly?

Let's look at the way the distance changes as the time goes from 4 hrs to 8 hours, 5 hours to 10 hours, 15 hours and 30 hours. In other words, what happens to the distance as the time doubles? What about the price as the amount of meat doubles? What if we the time changed from 10 hours to 30 hours, or 30 hours to 90 hours - i.e., if the time becomes 3 times as long? What if the amount of meat changes from 1 pound to 4 pounds, 2 pounds to 8 pounds, 3 pounds to 12 pounds - i.e., if the amount of meat becomes 4 times as much?

In these proportional situations, when one quantity becomes 2, 3, 4,... times as much, the other quantity is also becoming 2, 3, 4,... times as much. Let's see if that is also the case in other situations. Since the constant quotient relationships is an increase-increase situation, we really don't have to consider an increase-decrease situation. So, the only other increase-increase situation was the constant difference situation. So, let's look at the ages of two siblings shown in the table below:

So, when Ariel becomes twice as old, will Desmond also becomes twice as old? For example, if Ariel's age goes from 10 years old to 20 years old, what happens to Desmond's age. When Ariel is 10 years old, Desmond is 13 years old. That tells us that Desmond is 3 years older than Ariel. So, when Ariel is 20 years old, Desmond will be 23 years old. Clearly 23 is not the double of 13. So, what we noticed about the proportional relationships above is indeed unique. In fact, in most, if not all, Japanese textbooks, proportional relationship is defined using this characteristic: Two quantities are in a proportional relationship if as one quantity becomes 2, 3, 4, ... times as much, the other quantity also becomes 2, 3, 4, ... times as much.

In the same way, Japanese textbooks define inverse proportional relationships this way: Two quantities are in an inversely proportional relationship if as one quantity becomes 2, 3, 4, ... times as much, the other quantity becomes 1/2, 1/3, 1/4, ... times as much. As I stated last time, it is important that students compare and contrast these various situations from the same angle so that they can identify what characteristics are unique to proportional relationships. So, I think it would be useful for students to analyze a variety of situations from this particular perspective, i.e., when one quantity becomes 2, 3, 4, ... times as much, what happens to the other quantity.

Sunday, April 25, 2010

M6A2 Proportional Relationship (1)

M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
d. Describe proportional relationships mathematically using y = kx, where k is the constant of proportionality.

There are many quantities around us that vary in relationship to each other. For example, here are some examples of pairs of quantities that vary simultaneously:
a) ages of two siblings on January 1 each year
b) the number of pages of a book that have been read and the number of pages to be read
c) the distance traveled and the time of travel (at a constant speed)
d) the speed and the time it takes to travel a fixed distance
e) the length of a candle that has been burned and the remaining length
f) the amount of meat and the price of meat
g) the length and the width of a rectangle with a fixed area
h) time of the day in Atlanta and Los Angels

Let's look at these situations a little more carefully. How are the ways the quantities change similar or different? One thing you notice is that in situations a, c, f, and h, as one quantity increases the other also increases. We can cal these increase-increase situations. In contrast, in situations b, d, e, and g, as one quantity increases, the other decreases. So at one level, we can sort these situations into increase-increase and increase-decrease situations.

But, let's dig a little deeper. Let's look at each group more carefully. How are the ways quantities changing different from each other? Let's look at the increase-decrease situations (b, d, e, and g) first. As the two quantities in each of these situations change, is there anything that is not changing - mathematically, the idea of "invariance" is a very important one. You notice that in situations b and e, the sum of the two quantities remain the same. For example, the total number of pages in a book is the sum of the number of pages already read and the number of pages to be read. In contrast, in situations d and g, what stays constant is the product of the two quantities, the distance traveled in d and the area in g.

Now, let's look at a, c, f, and h. As the two quantities in each situation change, is there anything that is staying the same. In situations a and h, what stays the same is the difference between the two quantities. For example, the difference between the ages of two siblings on January 1 will always be the same no matter how old they become. In contrast, in situations c and f, what stays the same is the quotient of the quantities.

So, these situations can be sorted into four categories based on what stays constant in each situation: constant sum, constant difference, constant product, and constant quotient. Based on this way of sorting, we can also express the relationship between the two quantities using mathematical equations in the following ways (k is a constant):
constant sum: x + y = k
constant difference: x - y = k
constant product: x*y = k
constant quotient: y÷x=k

Of these four ways two quantities change simultaneously, we call the last situation, i.e., constant quotient, a proportional relationship. This relationship can be written in mathematical equation as y÷x = k, or y = kx (M6A2 d & e). Moreover, the constant product relationship, xy = k, or y = k÷x, is called an inverse proportional relationship.

When we want students to understand a new concept, it is very important and useful if we provide situations to compare and contrast several cases - examples and non-examples. Clearly there are many other quantities that change in relationship to each other that do not necessarily fit into these four categories - for example, the amount of time you study for a test and your score on a test. Thus, restricting the situations to examine to these four types may be a bit arbitrary. However, sometimes we may want to investigate only those situations that will allow us to analyze them in a particular way. It does not mean that we should investigate other, more messy situations. However, we may not need non-examples that are too complicated.

Tuesday, April 13, 2010

M7N1c - Integers

M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
c. Add, subtract, multiply, and divide positive and negative rational numbers.

I usually don't venture into the 6-8 standards. But, since we discussed the compensation strategies recently, I thought I would discuss how two of those strategies can be used to derive the methods of calculations with integers.

Recall that the equal addition principle of subtraction states that if we add (or subtract) the same number to both the minuend and the subtrahend, the difference stays the same. Thus, 93 - 18 = (93 + 2) - (18 + 2) = 95 - 20. Another property of subtraction students encounter early on is that subtracting 0 will not change the number, that is A - 0 = A. By combining these two properties of subtraction, we can think about a problem like 8 - (-3) this way:
"We know subtracting 0 does not change the number. So, what can I do to change the subtrahend (-3) to 0? Add 3. But, the equal addition principle of subtraction says I have to add the same number to the minuend to keep the difference the same. So,
8 - (-3) = (8 + 3) - (-3 + 3) = (8 + 3) - 0 = 8 + 3."

Thus, you can see that subtracting a negative number is the same as adding the opposite.

We noted that there is a parallel between the compensation strategies for subtraction and division. We can actually use the equal multiplication principle of division to think about division of fraction problems, by combining it with another parallel property, dividing by 1 does not change the number. So, if you are given 3/5 ÷ 2/3, we can think like this:
"We know dividing by 1 does not change the number. So, how can we change (2/3), the divisor, into 1? Multiply by its reciprocal, of course. But the equal multiplication principle of division says I will have to multiply the dividend by the same number, too. So,3/5 ÷ 2/3 = (3/5 x 3/2) ÷ (2/3 x 3/2) = (3/5 x 3/2) ÷ 1 = 3/5 x 3/2."
Thus, we see that the division of fractions is the same as the multiplication by the reciprocal of the divisor.

Of course, strictly speaking, there is a minor glitch in both of these arguments. We established the four compensation strategies with whole numbers. But, we don't know if they still hold if we expand the range of numbers to integers/rational numbers. So, there is a circularity in these arguments. So, I'm not advocating these strategies to establish the computation algorithms, specially since there are other ways where students can meaningfully develop algorithms. However, I think these mathematical relationships are still interesting.

Thursday, April 1, 2010

M*P5: Tape diagrams

M*P5. Students will represent mathematics in multiple ways.

In the recently released draft of the Common Core Standards, there is a noticeable emphasis on linear models such as number lines. I have discussed how many Japanese textbooks use double number lines to discuss multiplication and division of fractions, as well as some proportional problems. However, the Common Core Standards also include a model that is called "tape diagram." In their glossary, "tape diagrams" is defined as follows:
Drawings that look like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar models or graphs, fraction strips, or length models."
In an earlier post, I discussed how a tape diagram may help children represent addition and subtraction situations. The primary purpose of such diagrams is to help students decide the appropriate operation, that is addition or subtraction. However, Japanese textbooks also use tape diagrams, or segment diagrams, to deal with problems in upper grades, too.

Consider a problem like this one:
A fifth grade class counted the number of cars that went by the front entrance of the school between 9 o'clock and 10 o'clock. The total number of cars counted were 156. There were 3 times as many passenger cars as trucks. How many passenger cars and how many trucks were counted?
For this problem, you can use a diagram like the following:

From this diagram, we can see that the total number of cars are made up of 4 equal segments, one of which is equal to the number of trucks and the other three are equal to the number of cars. Since the four segments are equal, we can divide 156 by 4 to find out how many cars each segment represent.

Here is another problem:
There are 3.5 times as many fifth graders at School A as School B. There are 115 more students at School A than at School B. How many students are there at School A and at School B?
This problem can be represented as follows:

From this diagram, students can determine that 115 is made up of 5 equal segments since the last short segment is a half of the other segments, each of which is equal to the number of students at School B. So, 115÷5=23 represents a half of School B. Thus, the number of students at School B is 46 students. The number of students at School A is 23x7=161.

You might notice that these problems can be easily solved if we use algebra, but having diagrams such as tape/segment diagrams, students can develop the foundation for solving these problems algebraically.

There are other types of problems for which tape/segment diagram may be useful. Consider this problem:
At Jimmy's school, there were 475 students last year. This year, there are 24% fewer students. How many students are at Jimmy's school this year?
You can represent this problem using a tape/segment diagram like this:

From this diagram, we can tell that the number of students this year should be 100-24=76% of last year's student population, 475. Thus, we can find the answer by multiplying 475 by 0.76. Alternately, we can subtract 475x0.24 from 475, too. [Click here for a discussion on how double number lines may be used with problems involving percents.]

What we need to keep in mind about these representations is that they are supposed to be students' thinking tools, not just teachers' explanation tools. In order to help make these representations as their own thinking tools, these representations have to be carefully taught. In the Japanese textbooks, they start building these linear models starting in Grade 2 and help students experience increasingly more complicated representations gradually and systematically. I believe the emphasis on linear models in the Common Core Standards is important, but just showing these models to students will not automatically produce positive results.

Tuesday, March 23, 2010

M2N2e: ways to compensate

M2N2. Students will build fluency with multi-digit addition and subtraction.
e) Use basic properties of addition (commutative, associative, and identity) to simplify problems (e.g. 98 + 17 by taking two from 17 and adding it to the 98 to make 100 and replacing the original problem by the sum 100 + 15).

Students, and adults, often use different mental computation strategies. The one that is discussed in this standard is often explained by using the associative property of addition: 98 + 17 = 98 + (2 + 15) = (98 + 2) + 15

However, we can also explain it slightly differently. "98 + 17" means we are putting together 98 and 17. If we pretended 98 were 100, that means we actually have 2 more than we are supposed to. So, if we don't want to change the final answer, we have to make 17 smaller by 2. In other words, 98 + 7 = (98 + 2) + (17 - 2). In general, if we added a number to one of the addends, we have to subtract the same number from the other addend to compensate.

What about subtraction? Let's think about 83 - 18. Subtracting 20 mentally is much easier. But if we subtract 20 instead of 18, we will be taking away 2 more than we are supposed to. So, to compensate for that, we must make the starting number bigger by 2, too. That is, 83 - 18 = (83 + 2) - (18 + 2). Alternately, you might think if we make 83 into 89, then there will be no re-grouping needed. But, in that case, you are starting with 6 more. So, if we want to keep the answer the same, we must take away 6 more than 18 as well. Thus, 83 - 18 = (83 + 6) - (18 + 6). As it turns out, for subtraction, if we add (or subtract) the same number to both the minuend and the subtrahend, the difference stays the same. This idea is sometimes called the equal addition principle of subtraction.

What about multiplication? How do we compensate? Let's think about 35 x 16. If we had 70, it might be easier to multiply mentally. But if we realize that 35 x 16 means 16 groups of 35 [I'm using the Japanese convention of writing the number in a group first]. So, if we make 35 into 70, you are actually putting 2 of those 35's together, and there will be only 8 groups. Or 70 x 8. Thus, we see that 35 x 16 = (35 x 2) x (16 ÷ 2). In general, if we multiply a factor by a number, then we must divide the other factor by the same number to keep the product the same.

For division, let's think about 112 ÷ 14. One way to interpret 112 ÷ 14 is to figure out how many in each group if we split 112 into 14 equal groups. The answer should be the same if we only consider 7 groups with a half as many total. So, 112 ÷ 14 = 56 ÷ 7. In general, if we multiply (or divide) both the dividend and the divisor by the same number, the quotient does not change. In the GPS, this particular idea is actually explicitly mentioned in M4N3(d). I sometime call this relationship the equal multiplication principle of division. Probably the most common place where we see the use of this principle is with problems like 2400 ÷ 400.

When you look at these four ways of making compensations, you notice that there are parallels between addition/multiplication and subtraction/division. With addition and multiplication, we do "opposite" to the two numbers to keep the result the same. However, with subtraction and division, we do the same to both numbers. Although only the division situation is mentioned explicitly in the GPS, looking at these compensation strategies may be useful in helping students develop a deeper understanding of the four arithmetic operations and how they may relate to each other.

Sunday, March 14, 2010

M1N3 f - Mastering the basic addition and subtraction

M1N3. Students will add and subtract numbers less than 100, as well as understand and use the inverse relationship between addition and subtraction.
f. Know the single-digit addition facts to 18 and corresponding subtraction facts with understanding and fluency. (Use strategies such as relating to facts already known, applying the commutative property, and grouping facts into families.)

Many of today's elementary school mathematics textbooks discuss a variety of thinking strategies children can use to figure out the basic addition facts. Some textbooks even organize their addition units according to those strategies: add 1/2, doubles, doubles plus/minus 1, make 10, etc.. Young children often "invent" these strategies. In fact, these strategies are the results of children's developing number sense. Enriching children's number sense, for example, composing and decomposing numbers (MKN2b, M1N3c), is a major emphasis in primary grades. Thus, this particular standards has two purposes: helping children master basic facts and helping them further their number sense.

Clearly, we want children to be able to recall the basic addition facts quickly, and some people may wonder why we need to bother with these different strategies. There are many reasons to include students' invented strategies in primary grades mathematics instruction, but to me the following three are the major reasons. First, these strategies are natural for children. If we take the idea of "starting with where children are," then we should think about how to take advantage of children's natural thinking processes. Another reason is that these invented strategies are the results of and promote further development of children's number and operation senses. I believe that the ability to see numbers and calculations flexibly is a powerful mathematical tool. If that is the case, it seems to make little sense to squash children's natural ability to think and force them to memorize the basic facts first then try to teach these flexible ways of thinking later. Such an approach seems to be rather inefficient. Finally, I believe that a major reason we teach mathematics in elementary schools is to help students become better thinkers. Thus, we should be always encouraging students to think. Quick recall is a goal, but if we want students to continue developing their thinking ability, we must dedicate some time in mathematics classrooms that focuses on children's thinking.

Anyway, although these strategies should be discussed as children naturally "invent" them, there is one particular strategy that should be treated intentionally. That strategy is the make-10 strategy. For example, 9 + 6 can be thought of as (9 + 1) + 5 = 10 + 5 = 15. For subtraction, like 13 - 8, children can think 13 - 8 = (10 - 8) + 3 = 2 + 3 = 5, or 13 - 8 = (13 - 3) - 5 = 10 - 5 = 5. 10 is such an important number in our numeration system. Thus, developing the ability to think with 10 systematically must be a major goal of mathematics teaching. For some of the invented strategies, I don't think it is necessary for all children to be able to use them. However, the make-10 strategy is mathematically so significant that all children should understand and be able to use it effectively. This way of thinking also helps students to go beyond the counting-by-one approach. If we consider older students counting on their fingers a problem, we have to offer them an alternative that can be just as effective and perhaps more efficient. The make-10 strategy is one such strategy.

Monday, March 8, 2010

M3N5 - What are decimal fractions?

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

When the GPS was first released, some people wondered what the phrase found in this standard, "decimal fractions," meant. If you research the Internet, you will find that "decimal fractions" are fractions with powers of 10 as denominators. This interpretation was emphasized in the 2008 revision of the GPS. Thus, M3N5(b) states, "Understand that a decimal fraction (i.e. 3/10) can be written as a decimal (i.e. 0.3)." The corresponding standards in the original GPS, M3N5(c), stated, Understand a one place decimal fraction represents tenths, i.e., 0.3 = 3/10."

However, I believe this was an unnecessary change which actually made the revised GPS a bit incoherent. It seems clear that the phrase, "decimal fractions," in the original GPS was used to mean "decimal numbers." Although the phrase "decimal fractions" isn't commonly used in the existing literature, when it is used, it typically means decimal numbers - or fractional quantities expressed in decimal format. Clearly, it is important for students to understand the equivalence of 3/10 and 0.3, but separating out fractions with powers of 10 as denominator seems to make a little sense mathematically. Furthermore, there are other statements in the GPS where this interpretation of "decimal fractions" creates some problems.

For example, the first sentence describing Grade 3 Number and Operations states, "Students will use decimal fractions and common fractions to represent parts of a whole." By examining the actual standards, we notice that students are also introduced to decimal numbers in Grade 3, but if we interpret "decimal fractions" as fractions with powers of 10 as denominators, then there is no reference to decimal numbers in the description of the standard. Similarly, the description of Grade 4 Number and Operations states, " Students will further develop their understanding of addition and subtraction of decimal fractions and common fractions with like denominators." However, students are to learn addition and subtraction of decimal numbers, M4N5.

In fact, everywhere except in M3N5, the GPS makes much better sense if we interpret "decimal fractions" to mean "decimal numbers." This is a great example how a simple phrase plays an important role in interpreting the standards. I really wish the state DOE will actually publish a document that will further elaborate what they meant.

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.