M5M1. Students will extend their understanding of area of geometric plane figures.

M5A1. Students will represent and interpret the relationships between quantities algebraically.

The topic of deriving the area formulas lie in the intersection of the four of the five content strands in the GPS: Number & Operations, Geometry, Measurement, and Algebra. Obviously, since we are determining the area, we are measuring the size of various shapes. Although we can determine the area by covering the given shape with unit squares or drawing it on a grid paper, we are going beyond measurement based on counting. Rather, we are calculating the area. Therefore, students will have to utilize their understanding of number and operations - and students can practice their skills as they study this topic as well. As we try to calculate the area of new shapes, students will have to somehow make familiar shapes for which they know how to calculate the area. In that process, they will have to utilize their knowledge and skills of geometry. Finally, we are not just interested in calculating the area. We want to generalize the process and derive formulas that can be applied to other shapes of the same type. Thus, students are utilizing and developing their understanding and skills of algebra as they generalize and express the relationship among length measurements of different parts of the given shape and its area.

Although the GPS does not expect students to derive area formulas for other polygons, deriving formulas for other polygons may be helpful for students to deepen not only their understanding of area formulas but also advancing their understanding of algebraic thinking and representations. Two possible polygons to consider are trapezoids and kites. The formula for calculating the area of trapezoids may be familiar to many people, so I will focus on the formula for the area of kites today.

Students can derive the formulas for these shapes using what they already know. The general steps are:

(1) Find the area of various trapezoids (or kites) by making a familiar shape (or a collection of familiar shapes) using the strategies discussed before.

(2) Determine what measurements (or the given trapezoid/kite) will be needed to calculate the area.

(3) Determine which strategy of calculating area might be useful for generalization.

(4) Derive the formula - i.e., summarize the process of calculation in an equation using variables (words or letters).

Note that a kite is a quadrilateral with two distinct pairs of equal adjacent sides. Squares and rhombuses are special types of kites because they have four equal sides. One of the properties of rhombuses is that their diagonals are perpendicular to each other and they intersect at their mid-points. The diagonals of kites are also perpendicular, but they intersect at the mid-point of one of the diagonals but not necessarily the other. In the figure below, the diagonals are all perpendicular. However, in (a), they are intersecting at the mid-point of both diagonals while in (b) ~ (e), they are intersecting at a mid-point of one of the diagonals but not the other. However, since they all have two distinct pairs of equal adjacent sides, they are all kites. Since all four sides are equal, (a) is also a rhombus.

So, how can children find the area of a kite? One of the big idea from earlier is that, when we are given an unfamiliar shape, we may be able to find the area by making a familiar shape (or a collection of familiar shapes). We have also seen that there are three general strategies that can be used to make a familiar shape: (1) sub-dividing, (2) making-it-bigger, and (3) cutting-and-moving. For example, here are three different ways the area of a kite can be calculated:

In (1), we are using the sub-dividing method. Since we already know how to calculate the area of triangles, we can divide the kite into two triangles. The area of the kite is the sum of the area of the two triangles. In (2), we surround the kite by a rectangle. We can see that the length and the width of this rectangle are the length of the two diagonals of the kite. In (3), we cut and moved the parts of the kite to form a rectangle. One of the dimensions of the rectangle is the same as the length of one of the diagonals while the other dimension is a half of the other diagonal.

With each of these methods, we need to know the lengths of the diagonals. Thus, in order to calculate the area of kites, the length of diagonals are needed. However, in method (1), if we use the diagonal whose mid-point is the point of intersection of diagonals (as this example above shows), we will need to know exactly how the other diagonal is split to calculate the area of the two triangles.

For method (3), we can make a new rectangle differently, yet still with the dimension of one diagonal of the kite by a half of the other diagonal of kite as shown below.

However, it may be difficult for some children to understand what the vertical dimension (in this example) should be a half of the diagonal.

Therefore, although each of these methods may be generalized to derive a formula, method (2) may be simpler one to derive a formula. With method (2), you simply enclose the kite with a large rectangle, and the new rectangle you create has the area that is twice the area of the given kite:

Therefore, the area of the kite can be calculated by dividing the area of the rectangle. Thus, the formula for the area of kites might look like this:

Area of Kites = Diagonal 1 x Diagonal 2 ÷ 2.

What's important is not the actual formula but understanding the general process of generalizing and deriving the formula. This can be said of the formulas for triangles and parallelograms, too. We should emphasize the reasoning involved in the process of deriving the formulas. If students understand the reasoning, they will be able to re-derive the necessary formula even if they can't simply recall it.

## Saturday, December 19, 2009

## Sunday, December 6, 2009

### M5M1 c - Developing Area Formulas (5)

M5M1. Students will extend their understanding of area of geometric plane figures.

c. Derive the formula for the area of a triangle.

In some textbooks, the area of triangles is studied before the area of parallelograms. In those cases, students typically examine how to calculate the area of right triangles. They notice that the area of a right triangle is a half of the rectangle you can make by using two copies of the right triangle.

They will then investigate how the area of more general triangles may be calculated, for example a triangle like this one:

Typically, students will split the triangle into two right triangles and apply the same method as before:

From here, those textbooks often generalize the way to calculate the area of triangles as:

However, sometimes students develop a misconception that the base must be the side of triangle that can be split to form 2 right triangles. In particular, they may not be able to calculate the area of the following triangle:

They want to split the triangle into two right triangles as shown, but they cannot determine the length of "base" and "height" in this case:

What they could do is to use the make-it-bigger approach and form a larger right triangle by adding on a smaller right triangle as shown below:

You can not only calculate the area of the given triangle by using this method, this method can be generalized to support the derivation of the formula. However, the challenge for students to derive (or verify the previously developed) formula is that they have to apply the distributive property as shown below:

Students may be used to applying the distributive property over addition, but they may not have had much experiences with the distributive property of multiplication over subtraction. Moreover, they have to treat the expression 4 ÷ 2 as a quantity.

Although it is possible to discuss the area of triangles before the area of parallelograms, my preference is to discuss parallelograms first. If parallelograms are "familiar" shapes, students can use a variety of methods to find the area of triangles. Here are just a couple students have come up on their own:

From the method on the left, we can see that the area of the triangle is the half of the area of the parallelogram, and the area of the parallelogram may be calculated by multiplying the "base," which is one side of the triangle, and the "height," which is the distance between the base and the parallel line containing the third vertex. From the method on the right, we can see that the area of the triangle is equal to the area of the new parallelogram. The area of the new parallelogram may be calculated by multiplying the "base," which is a side of the given triangle, and a half of the "height" of the triangle.

Either way, we can generate the formula, Area = Base x Height ÷ 2. Just as was the case with parallelograms, it is important that students understand that any of the three sides of the triangle may serve as the base and for each base, there is a corresponding height. The height is the distance between the base and the third vertex, which is the same thing as the length of perpendicular segment from the third vertex to the base.

By the way, it is probably important to write the formula with "÷ 2" since students only learn to model fraction multiplication in Grade 5 according to the GPS.

In some textbooks, the area of triangles is studied before the area of parallelograms. In those cases, students typically examine how to calculate the area of right triangles. They notice that the area of a right triangle is a half of the rectangle you can make by using two copies of the right triangle.

They will then investigate how the area of more general triangles may be calculated, for example a triangle like this one:

Typically, students will split the triangle into two right triangles and apply the same method as before:

From here, those textbooks often generalize the way to calculate the area of triangles as:

However, sometimes students develop a misconception that the base must be the side of triangle that can be split to form 2 right triangles. In particular, they may not be able to calculate the area of the following triangle:

They want to split the triangle into two right triangles as shown, but they cannot determine the length of "base" and "height" in this case:

What they could do is to use the make-it-bigger approach and form a larger right triangle by adding on a smaller right triangle as shown below:

You can not only calculate the area of the given triangle by using this method, this method can be generalized to support the derivation of the formula. However, the challenge for students to derive (or verify the previously developed) formula is that they have to apply the distributive property as shown below:

Students may be used to applying the distributive property over addition, but they may not have had much experiences with the distributive property of multiplication over subtraction. Moreover, they have to treat the expression 4 ÷ 2 as a quantity.

Although it is possible to discuss the area of triangles before the area of parallelograms, my preference is to discuss parallelograms first. If parallelograms are "familiar" shapes, students can use a variety of methods to find the area of triangles. Here are just a couple students have come up on their own:

From the method on the left, we can see that the area of the triangle is the half of the area of the parallelogram, and the area of the parallelogram may be calculated by multiplying the "base," which is one side of the triangle, and the "height," which is the distance between the base and the parallel line containing the third vertex. From the method on the right, we can see that the area of the triangle is equal to the area of the new parallelogram. The area of the new parallelogram may be calculated by multiplying the "base," which is a side of the given triangle, and a half of the "height" of the triangle.

Either way, we can generate the formula, Area = Base x Height ÷ 2. Just as was the case with parallelograms, it is important that students understand that any of the three sides of the triangle may serve as the base and for each base, there is a corresponding height. The height is the distance between the base and the third vertex, which is the same thing as the length of perpendicular segment from the third vertex to the base.

By the way, it is probably important to write the formula with "÷ 2" since students only learn to model fraction multiplication in Grade 5 according to the GPS.

## Saturday, November 14, 2009

### M5M1 b - Developing Area Formulas (4)

M5M1. Students will extend their understanding of area of geometric plane figures.

b. Derive the formula for the area of a parallelogram.

As discussed in the previous post, through activities like finding the area of L-shaped region, students can develop the understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)." Moreover, students can develop the following strategies to make a familiar shapes:

* divide the given shape up into several familiar shapes

* cut and re-arrange to make a familiar shape

* make-it-bigger

Now they are ready to tackle this standard.

In many textbooks, students are asked to find the area of parallelograms like the one shown below using what they already know:

Some students will count the number of unit squares, making appropriate adjustments when only a part of a unit square is inside the parallelogram. Other students will try to change the parallelogram to a rectangle, a familiar shape they already know how to calculate the area of. The typical way that this is accomplished by cutting a triangular segment from one end of the parallelogram and moving it to the other side, as shown below:

Since this rectangle is 6 cm wide and 4 cm long, area can be calculated by 6 x 4, or 24 cm2.

In most textbooks, this method is then generalized to derive the formula for calculating the area of parallelograms: Area of Parallelograms = Base x Height. So, is this the end? Have we successfully addressed this particular standard? I argue that the formula at this stage is an overgeneralization. Students at this point may have difficulty calculating the area of parallelograms like the following:

Some students will try to create a new rectangle like before and notice that "the height (in red) stops here!"

Others might try to turn the figure and make a rectangle like this:

Unfortunately, they can't determine the length and the width of this new rectangle other than actually measuring them, which isn't possible if the figure isn't drawn to scale. Even if the figure is drawn to scale, actually measuring the length and the width will introduce measurement errors. So, what can students do? Actually, there are a lot of things they can do using the understanding they developed through the L-shape lesson. Here are some possibilities:

Note that (a), (c) and (d) use the "cut and re-arrange" strategy, (b) uses the "divide up" strategy, and (e) uses the "make-it-bigger" strategy. In (b), (c) and (d), the "familiar" shape students created are parallelograms that can be changed to rectangles by cutting and re-arranging right triangles.

Some of you may be wondering about (e) since students have not learned how to calculate the area of triangles. In this case, instead of calculating the area of each triangle, this student actually pushed together the two triangles that were used to make a bigger rectangle. The two triangles will make a rectangle whose dimensions are 5 cm by 6 cm.

Actually, some students may use this make-it-bigger strategy with the first parallelograms. If they did, then, this "slanted" parallelograms do not pose any challenge to them since they can use exactly the same strategy to this one as well. This strategy could have been used to derive the formula for calculating the area of parallelograms, too. Look at the figure below:

The area of the original parallelogram (un-shaded part in the figure on the left) can be calculated by subtracting the area of shaded rectangle (in the middle figure) from the large rectangle. However, this difference is really the area of the yellow rectangle in the figure on the right. That means that the area of the parallelogram is the same as the area of rectangle you can build on the base whose length is the distance between the base and its opposite side, or more accurately, the distance between the parallel lines containing the base and its opposite side. If we consider this distance between the base and its opposite side as height, we still have the same formula, Area of Parallelogram = Base x Height.

The important idea here, though, is what constitute as the height. The height of a parallelogram is the distance between the base and its opposite side, and the distance between two parallel lines is the length of a perpendicular segment connecting them. It is not the length of the adjacent side to the base. In case of a rectangle, which is a special type of parallelograms, the adjacent side may be used as the height because it is perpendicular to the base. However, that is not generally the case in parallelograms. Thus, understanding what the height of a parallelogram is may be the most important aspect of deriving the formula. Unfortunately, students don't understand this idea because they aren't asked to grapple with parallelograms like the second one we saw above, or derive the formula through the make-it-bigger strategy. I hope you will seriously consider giving your students this challenge as they try to derive the formula for calculating the area of parallelograms.

b. Derive the formula for the area of a parallelogram.

As discussed in the previous post, through activities like finding the area of L-shaped region, students can develop the understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)." Moreover, students can develop the following strategies to make a familiar shapes:

* divide the given shape up into several familiar shapes

* cut and re-arrange to make a familiar shape

* make-it-bigger

Now they are ready to tackle this standard.

In many textbooks, students are asked to find the area of parallelograms like the one shown below using what they already know:

Some students will count the number of unit squares, making appropriate adjustments when only a part of a unit square is inside the parallelogram. Other students will try to change the parallelogram to a rectangle, a familiar shape they already know how to calculate the area of. The typical way that this is accomplished by cutting a triangular segment from one end of the parallelogram and moving it to the other side, as shown below:

Since this rectangle is 6 cm wide and 4 cm long, area can be calculated by 6 x 4, or 24 cm2.

In most textbooks, this method is then generalized to derive the formula for calculating the area of parallelograms: Area of Parallelograms = Base x Height. So, is this the end? Have we successfully addressed this particular standard? I argue that the formula at this stage is an overgeneralization. Students at this point may have difficulty calculating the area of parallelograms like the following:

Some students will try to create a new rectangle like before and notice that "the height (in red) stops here!"

Others might try to turn the figure and make a rectangle like this:

Unfortunately, they can't determine the length and the width of this new rectangle other than actually measuring them, which isn't possible if the figure isn't drawn to scale. Even if the figure is drawn to scale, actually measuring the length and the width will introduce measurement errors. So, what can students do? Actually, there are a lot of things they can do using the understanding they developed through the L-shape lesson. Here are some possibilities:

Note that (a), (c) and (d) use the "cut and re-arrange" strategy, (b) uses the "divide up" strategy, and (e) uses the "make-it-bigger" strategy. In (b), (c) and (d), the "familiar" shape students created are parallelograms that can be changed to rectangles by cutting and re-arranging right triangles.

Some of you may be wondering about (e) since students have not learned how to calculate the area of triangles. In this case, instead of calculating the area of each triangle, this student actually pushed together the two triangles that were used to make a bigger rectangle. The two triangles will make a rectangle whose dimensions are 5 cm by 6 cm.

Actually, some students may use this make-it-bigger strategy with the first parallelograms. If they did, then, this "slanted" parallelograms do not pose any challenge to them since they can use exactly the same strategy to this one as well. This strategy could have been used to derive the formula for calculating the area of parallelograms, too. Look at the figure below:

The area of the original parallelogram (un-shaded part in the figure on the left) can be calculated by subtracting the area of shaded rectangle (in the middle figure) from the large rectangle. However, this difference is really the area of the yellow rectangle in the figure on the right. That means that the area of the parallelogram is the same as the area of rectangle you can build on the base whose length is the distance between the base and its opposite side, or more accurately, the distance between the parallel lines containing the base and its opposite side. If we consider this distance between the base and its opposite side as height, we still have the same formula, Area of Parallelogram = Base x Height.

The important idea here, though, is what constitute as the height. The height of a parallelogram is the distance between the base and its opposite side, and the distance between two parallel lines is the length of a perpendicular segment connecting them. It is not the length of the adjacent side to the base. In case of a rectangle, which is a special type of parallelograms, the adjacent side may be used as the height because it is perpendicular to the base. However, that is not generally the case in parallelograms. Thus, understanding what the height of a parallelogram is may be the most important aspect of deriving the formula. Unfortunately, students don't understand this idea because they aren't asked to grapple with parallelograms like the second one we saw above, or derive the formula through the make-it-bigger strategy. I hope you will seriously consider giving your students this challenge as they try to derive the formula for calculating the area of parallelograms.

## Saturday, November 7, 2009

### M5M1 - Developing Area Formulas (3)

M5M1. Students will extend their understanding of area of geometric plane figures.

As we discussed in the previous post, the GPS expects students to determine the area of rectangles and squares by counting or calculation. Then, in Grade 5, students are expected to derive and use formulas to determine the area of parallelograms, triangles, and circles. Interestingly, there is nothing about area mentioned in Grade 4. It is listed as one of the "Concepts/Skills to Maintain," but there is no specific standard about the area measurement in Grade 4. Many people might wonder about the feasibility of fifth graders actually deriving the area formulas of parallelograms and triangles on their own. Do they have enough background knowledge? What background knowledge do they need to increase the likelihood of their deriving those formulas?

In a previous post on the idea of teaching through problem solving (April, 2009), how children can learn through problem solving new mathematical ideas. Those mathematical ideas are the ones that will serve as the bridge between M3M4 (area of rectangles and squares) and M5M1 (area of parallelograms, triangles, and circles). As we will see shortly, those specific understandings will be used over and over to derive the formulas. So, in Grade 3, finding the area of L-shapes may be simply a complex application of what they learned, but, in Grade 5, the focus should be on ways of thinking involved in calculating the area. If those understandings are made explicit, students are much more likely to be successful in deriving the area formulas. So, I encourage you to read that post again (or for the first time, if you have not read it before).

By the way, element (f) of this standard says, "Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles, and/or triangles and find the sum of the areas of those shapes." Actually, this element is simply one of the strategies developed in the L-shape lesson, that is, sub-dividing the given unfamiliar shape into a collection of familiar shapes. The only difference is what shapes are available to students as familiar shapes. When students work on the L-shape problem, they only knew how to calculate the area of rectangles and squares. However, after students have learned the formulas for the area of parallelograms and triangles, students can also use those figures. So, in case you are wondering if you can afford to spend an extra time to discuss something that is not explicitly mentioned in the GPS, the L-shape lesson does address the GPS directly, too.

As we discussed in the previous post, the GPS expects students to determine the area of rectangles and squares by counting or calculation. Then, in Grade 5, students are expected to derive and use formulas to determine the area of parallelograms, triangles, and circles. Interestingly, there is nothing about area mentioned in Grade 4. It is listed as one of the "Concepts/Skills to Maintain," but there is no specific standard about the area measurement in Grade 4. Many people might wonder about the feasibility of fifth graders actually deriving the area formulas of parallelograms and triangles on their own. Do they have enough background knowledge? What background knowledge do they need to increase the likelihood of their deriving those formulas?

In a previous post on the idea of teaching through problem solving (April, 2009), how children can learn through problem solving new mathematical ideas. Those mathematical ideas are the ones that will serve as the bridge between M3M4 (area of rectangles and squares) and M5M1 (area of parallelograms, triangles, and circles). As we will see shortly, those specific understandings will be used over and over to derive the formulas. So, in Grade 3, finding the area of L-shapes may be simply a complex application of what they learned, but, in Grade 5, the focus should be on ways of thinking involved in calculating the area. If those understandings are made explicit, students are much more likely to be successful in deriving the area formulas. So, I encourage you to read that post again (or for the first time, if you have not read it before).

By the way, element (f) of this standard says, "Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles, and/or triangles and find the sum of the areas of those shapes." Actually, this element is simply one of the strategies developed in the L-shape lesson, that is, sub-dividing the given unfamiliar shape into a collection of familiar shapes. The only difference is what shapes are available to students as familiar shapes. When students work on the L-shape problem, they only knew how to calculate the area of rectangles and squares. However, after students have learned the formulas for the area of parallelograms and triangles, students can also use those figures. So, in case you are wondering if you can afford to spend an extra time to discuss something that is not explicitly mentioned in the GPS, the L-shape lesson does address the GPS directly, too.

## Tuesday, November 3, 2009

### M3M4 - Developing Area Formulas (2)

M3M4. Students will understand and measure the area of simple geometric figures (squares and rectangles).

a. Understand the meaning of the square unit and measurement in area.

b. Model (by tiling) the area of a simple geometric figure using square units (square inch, square foot, etc.).

c. Determine the area of squares and rectangles by counting, addition, and multiplication with models.

Once students understand that area is the amount of space inside any geometric figures, we are ready to start thinking about ways to measure the area of various shapes. The next step is to pick a unit and actually "cover" shapes to see how many units will be needed. So, what should we use as a unit? Although we will eventually use squares as units, we may want to think about using anything that can cover the plane without a hole or an overlap. Also, using a familiar objects might be helpful to focus students' attention on the process of area measurement. One such familiar object might be index cards. Students can measure the area of the surface of desks or any other large rectangular regions.

If students have many index cards available to them, they will cover the rectangular region in many different ways. Here are three possibilities.

In this particular example, no matter how you cover the rectangle, it takes 24 small rectangles. So, we can say that the area of the rectangle is 24 units.

After measuring the area by actually covering rectangles with units, many students will realize that some ways of covering the given shape is easier to count than others. For example, the arrangements like the one on the left requires us to actually count all of the units to determine how many units were used. On the other hand, since the other two arrangements will result in equal groups (either rows or columns), we can use multiplication to find the area (either 4x6 or 8x3).

At this point, you might want to give students only 3 or 4 unit pieces to see if they can think about ways of calculating the area. A common error at this stage is to do the following:

and .

So, the area is 4x3=12 units. It is important for students to understand here why they cannot rotate the unit as they measure how many units will fit in each dimension of the rectangle. What we are trying to do when we measure the second dimension is how many rows (in this example) of 4 units there are. If we turn the unit as shown on the right, we are no longer counting the number of rows of 4 units.

You may want to ask students what we can do to avoid this type of confusion. Some students will realize that if we use a square as a unit, then it doesn't matter whether we rotate it since squares have 4 equal sides. You can then introduce that the standard units of area measurement are squares with unit length on each side, e.g., 1 cm, 1 in, 1 ft, etc.. Each unit square is said to have the area of 1 cm2, 1 in2, 1 ft2, etc., respectively. Actually, I am not sure exactly how the GPS wants these standards units of area to be handled. Unlike the units for volume, these area units are not mentioned in the GPS. However, it seems strange not to talk about the units when we are talking about the area of rectangles.

By using unit squares, we can also make it easier to determine the number of units that fit along each dimension of a rectangle by simply measuring their lengths. So, if a rectangle is 5 inches wide and 8 inches long, that means we can fit 5 1-inch squares along one row and there will be 8 rows. Therefore, we can multiply 5 and 8 to get 40 cm2. It is important that students understand that when 2 lengths are multiplied together, the product mysteriously becomes the area measurement. The two lengths we are measuring are simply telling us how many unit squares will fit along each side of the given rectangle.

Also note that students are not introduced to letters as variables until Grade 5, the formula should be written (if it is to be written at all) as, Area of Rectangle = Length x Width. Again, it is important to emphasize that this formula is to calculate the area of rectangles. Some students (and adults, unfortunately) will say that area is "length x width," but it is only a formula for a specific shape. Area is the amount of space inside a shape, no matter what the shape is.

a. Understand the meaning of the square unit and measurement in area.

b. Model (by tiling) the area of a simple geometric figure using square units (square inch, square foot, etc.).

c. Determine the area of squares and rectangles by counting, addition, and multiplication with models.

Once students understand that area is the amount of space inside any geometric figures, we are ready to start thinking about ways to measure the area of various shapes. The next step is to pick a unit and actually "cover" shapes to see how many units will be needed. So, what should we use as a unit? Although we will eventually use squares as units, we may want to think about using anything that can cover the plane without a hole or an overlap. Also, using a familiar objects might be helpful to focus students' attention on the process of area measurement. One such familiar object might be index cards. Students can measure the area of the surface of desks or any other large rectangular regions.

If students have many index cards available to them, they will cover the rectangular region in many different ways. Here are three possibilities.

In this particular example, no matter how you cover the rectangle, it takes 24 small rectangles. So, we can say that the area of the rectangle is 24 units.

After measuring the area by actually covering rectangles with units, many students will realize that some ways of covering the given shape is easier to count than others. For example, the arrangements like the one on the left requires us to actually count all of the units to determine how many units were used. On the other hand, since the other two arrangements will result in equal groups (either rows or columns), we can use multiplication to find the area (either 4x6 or 8x3).

At this point, you might want to give students only 3 or 4 unit pieces to see if they can think about ways of calculating the area. A common error at this stage is to do the following:

and .

So, the area is 4x3=12 units. It is important for students to understand here why they cannot rotate the unit as they measure how many units will fit in each dimension of the rectangle. What we are trying to do when we measure the second dimension is how many rows (in this example) of 4 units there are. If we turn the unit as shown on the right, we are no longer counting the number of rows of 4 units.

You may want to ask students what we can do to avoid this type of confusion. Some students will realize that if we use a square as a unit, then it doesn't matter whether we rotate it since squares have 4 equal sides. You can then introduce that the standard units of area measurement are squares with unit length on each side, e.g., 1 cm, 1 in, 1 ft, etc.. Each unit square is said to have the area of 1 cm2, 1 in2, 1 ft2, etc., respectively. Actually, I am not sure exactly how the GPS wants these standards units of area to be handled. Unlike the units for volume, these area units are not mentioned in the GPS. However, it seems strange not to talk about the units when we are talking about the area of rectangles.

By using unit squares, we can also make it easier to determine the number of units that fit along each dimension of a rectangle by simply measuring their lengths. So, if a rectangle is 5 inches wide and 8 inches long, that means we can fit 5 1-inch squares along one row and there will be 8 rows. Therefore, we can multiply 5 and 8 to get 40 cm2. It is important that students understand that when 2 lengths are multiplied together, the product mysteriously becomes the area measurement. The two lengths we are measuring are simply telling us how many unit squares will fit along each side of the given rectangle.

Also note that students are not introduced to letters as variables until Grade 5, the formula should be written (if it is to be written at all) as, Area of Rectangle = Length x Width. Again, it is important to emphasize that this formula is to calculate the area of rectangles. Some students (and adults, unfortunately) will say that area is "length x width," but it is only a formula for a specific shape. Area is the amount of space inside a shape, no matter what the shape is.

## Saturday, October 24, 2009

### M3M4 & M5M1 - Developing Area Formulas (1)

M3M4. Students will understand and measure the area of simple geometric figures (squares and rectangles).

M5M1. Students will extend their understanding of area of geometric plane figures.

In the past several posts, I discussed important ideas involved in helping students develop multiplication and division algorithms. With today's post, I want to start a new series on how to help students develop various area formulas (M3M1 & M5M1). In today's post, however, I want to focus my attention on the basic ideas about teaching and learning of area measurement.

I have discussed previously (December, 2008) some basic ideas of teaching measurement. As we teach measuring of any attribute (e.g., length, weight, area, angle, etc.), we must first help students understand the particular attribute we are trying to measure. Without understanding the attribute, measuring it will not make any sense. For that purpose, comparison activities are very useful. Typically, we start with direct comparisons, then move on to indirect comparisons. After students understand the concept, we can start thinking about quantifying the amount of the attribute, i.e., measuring the object. Many people suggest that we start with non-standards units first. One reason for this suggestion is that if we start with standard units, students will have to learn the idea of using units to quantify the attribute and the idea of standard units. Moreover, when we try to measure with standard units, we typically use measurement instruments, such as rulers and protractors. So, students will also have to learn how to use measurement instruments. If we start with non-standard units, students can focus on the notion of quantifying, or measuring, first. Once students understand how a particular attribute can be measured using a non-standard unit, they can use that understanding to both measuring with standard units and learning how to use measurement instruments. After all, the notion of "standard" units probably does not make sense unless you have some experiences with "non-standard" units.

So, what does this all mean when we are teaching the area measurement? Obviously, the first focus should be on helping students understand what area is about. To do so, it seems like we should have students engage in some comparison activities. So, for example, let's ask students to compare the following two rectangles (one is actually a square, but we know that squares are rectangles, don't we?).

Note that I am just showing you the drawing of rectangles, but you should give children cut out pieces to compare. Moreover, I included grid lines to show the dimensions of the rectangles, but you may not want to do so when you give these shapes to students.

Anyway, when students are given these two shapes and asked "Which is bigger?" you see generally two different ways students will compare these shapes. Some children will compare the shapes by overlapping them (on the left below). Others will put the shapes next to each other (on the right).

We can ask students if these two ways of comparing are actually comparing the same attribute (we may not want to use this particular word with 3rd graders). Students may not know, but they can understand that if the conclusions we get from these two ways are different, then, they couldn't be comparing the same attribute. So, have them try comparing these two shapes using both ways. For example, if we put the shapes next to each other and rotate one of them around the other shapes, you see that the two shapes are the same size:

When you overlap shapes, we see that the square is actually "bigger" than the rectangle:

Since the results are different, these two ways of comparison are indeed comparing two different attributes. We know that the first comparison was comparing the length around these shapes, or perimeter, while the second comparison is about area, i.e., the amount of space inside the shapes.

I am not saying that children will understand the difference between the perimeter and the area by doing this one activity. However, it is important for children to have a number of comparison activities to compare the length around and comparing the amount of space inside. When two same objects give different results like the one above several times, students might develop a better sense of the difference between these two attributes.

Once students understand what the area as an attribute is about, we can now move into the discussion of measuring it. So, in the next post, we will discuss M3M1, in which students think about the area of rectangles and squares.

M5M1. Students will extend their understanding of area of geometric plane figures.

In the past several posts, I discussed important ideas involved in helping students develop multiplication and division algorithms. With today's post, I want to start a new series on how to help students develop various area formulas (M3M1 & M5M1). In today's post, however, I want to focus my attention on the basic ideas about teaching and learning of area measurement.

I have discussed previously (December, 2008) some basic ideas of teaching measurement. As we teach measuring of any attribute (e.g., length, weight, area, angle, etc.), we must first help students understand the particular attribute we are trying to measure. Without understanding the attribute, measuring it will not make any sense. For that purpose, comparison activities are very useful. Typically, we start with direct comparisons, then move on to indirect comparisons. After students understand the concept, we can start thinking about quantifying the amount of the attribute, i.e., measuring the object. Many people suggest that we start with non-standards units first. One reason for this suggestion is that if we start with standard units, students will have to learn the idea of using units to quantify the attribute and the idea of standard units. Moreover, when we try to measure with standard units, we typically use measurement instruments, such as rulers and protractors. So, students will also have to learn how to use measurement instruments. If we start with non-standard units, students can focus on the notion of quantifying, or measuring, first. Once students understand how a particular attribute can be measured using a non-standard unit, they can use that understanding to both measuring with standard units and learning how to use measurement instruments. After all, the notion of "standard" units probably does not make sense unless you have some experiences with "non-standard" units.

So, what does this all mean when we are teaching the area measurement? Obviously, the first focus should be on helping students understand what area is about. To do so, it seems like we should have students engage in some comparison activities. So, for example, let's ask students to compare the following two rectangles (one is actually a square, but we know that squares are rectangles, don't we?).

Note that I am just showing you the drawing of rectangles, but you should give children cut out pieces to compare. Moreover, I included grid lines to show the dimensions of the rectangles, but you may not want to do so when you give these shapes to students.

Anyway, when students are given these two shapes and asked "Which is bigger?" you see generally two different ways students will compare these shapes. Some children will compare the shapes by overlapping them (on the left below). Others will put the shapes next to each other (on the right).

We can ask students if these two ways of comparing are actually comparing the same attribute (we may not want to use this particular word with 3rd graders). Students may not know, but they can understand that if the conclusions we get from these two ways are different, then, they couldn't be comparing the same attribute. So, have them try comparing these two shapes using both ways. For example, if we put the shapes next to each other and rotate one of them around the other shapes, you see that the two shapes are the same size:

When you overlap shapes, we see that the square is actually "bigger" than the rectangle:

Since the results are different, these two ways of comparison are indeed comparing two different attributes. We know that the first comparison was comparing the length around these shapes, or perimeter, while the second comparison is about area, i.e., the amount of space inside the shapes.

I am not saying that children will understand the difference between the perimeter and the area by doing this one activity. However, it is important for children to have a number of comparison activities to compare the length around and comparing the amount of space inside. When two same objects give different results like the one above several times, students might develop a better sense of the difference between these two attributes.

Once students understand what the area as an attribute is about, we can now move into the discussion of measuring it. So, in the next post, we will discuss M3M1, in which students think about the area of rectangles and squares.

## Saturday, October 17, 2009

### M4N3 - Developing multiplication algorithms (7)

M4N3. Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.

So far, we have discussed the following:

(1) extending the multiplication table to 10x10

(2) multiplying multiples of 10 and 100 by 1-digit numbers

(3) multiplying 2- and 3-digit number by 1-digit numbers

(4) multiplying by multiples of 10.

Now, we are ready to tackle multiplication of 2- and 3-digit numbers by 2-digit numbers. Before we get started, I wanted to say that, to me, teaching of an algorithm means helping students make their own strategies into written procedures instead of imposing a specific algorithm upon students. Of course, that doesn't mean "anything goes." Rather, teachers must think carefully about how to influence students' thinking naturally. Moreover, it may be possible for teachers to sequence students' experiences in such a way that the algorithm students develop "naturally" is something very similar to, or exactly the same as, the conventional algorithm. For that purpose, the area model of multiplication can play a very important role. Therefore, the use of the model along with base-10 blocks before reaching this point is an integral part of the process. So, how do we help students expand their written methods into multiplication of 2- and 3-digit numbers by 2-digit numbers?

Let's think about 12x23 first. How can students use what they have learned so far to think about ways to calculate this problem? There are at least three possible ways. At the most abstract level, students might be able to think of 12x23 as 12x20+12x3 - i.e., 23 groups of 12 can be split into 20 groups of 12 and 3 groups of 12. Then, each of 12x20 and 12x3 are already discussed. If students can think about this way, they can record the process using the vertical notation,

or .

The notation on the right is basically the standard algorithm for multiplication.

Another possibility is for students to go back to the area representation of multiplication. 12x23 means that we are making a rectangle with the dimension of 12 units by 23 units. The product is represented by the area of this rectangle. So, if you construct this rectangle using base-10 blocks, and using the fewest number of blocks (i.e., use large blocks whenever possible), you can make a rectangle like this one:

By examining the arrangement, we see that there are 1 by 2 rectangles made of flats (200), 2 by 2 rectangles of longs (40), 1 by 3 rectangles made of longs (30) and 2 by 3 rectangles of units (6). So, the product is 200+40+30+6=276. After students have become comfortable with the area model representation with base-10 blocks, you may want to encourage students to move toward drawing instead of using actual base-10 blocks. Sometimes you can make this transition simply by giving students multiplication problems with larger factors. Students will realize that actually making rectangles using base-10 blocks is too tedious.

When students become comfortable with drawing rectangles, they might realize that it is still rather tedious. This is when you may be able to suggest if they could use an adaptation of a notation that we used when we were multiplying 2- or 3-digit number by 1-digit number. Some students may be able to start at this point, without going all the way back to using base-10 blocks. That judgment must be made by teachers, using their knowledge of students. Anyway, the notation might look like this for 12x23:

Again, after students have become fluent with this notation, you might want to bring their attention to the four products (in the example here, 200, 40, 30, and 6). Noticing that these are the products of the two tens digits, the tens digit and the ones digit (in both direction) and the two ones digits. So, you can introduce a new notation that records the same information as this diagram does:

You can then negotiate with your students a consistent order in which you calculate these four products (typically called "partial products) so that we can make sure that we have accounted for all of them. If you really want students to understand the conventional multiplication algorithm, you will start with the ones digit of the multiplier (the bottom number) and multiply the ones and then the tens digits of the multiplicand (the top number). You will then multiply the tens digit of the multiplier with the ones and then the tens digits of the multiplicand. So, this problem would look like this:

If you combine the first two partial products and the last two partial products, you will have:

Note that the example we used, 12x23, did not involve any re-grouping. In a way, this is the most "basic" situation. As students move from one notation to another, you may want to consider moving back to a basic situation. Once students become comfortable with the notation (area model, symbolic notation, or whatever), then you want to look at other situations such as those involving re-grouping and a 0 in the factor/product.

When extending the multiplicand to 3-digit numbers, for example, 587x34, you may want to go back to the diagram notation - it will be rather difficult to actually model these multiplication with base-10 blocks. From the diagram, you can move to the notation that will explicitly record all partial products, then eventually to the conventional algorithm.

As usual, you do want to pay close attention to the numbers (factors) you use. Some students have difficulty with 0's - either in the factors or in the product/partial products, so you want to pay particular attention to those situations.

So far, we have discussed the following:

(1) extending the multiplication table to 10x10

(2) multiplying multiples of 10 and 100 by 1-digit numbers

(3) multiplying 2- and 3-digit number by 1-digit numbers

(4) multiplying by multiples of 10.

Now, we are ready to tackle multiplication of 2- and 3-digit numbers by 2-digit numbers. Before we get started, I wanted to say that, to me, teaching of an algorithm means helping students make their own strategies into written procedures instead of imposing a specific algorithm upon students. Of course, that doesn't mean "anything goes." Rather, teachers must think carefully about how to influence students' thinking naturally. Moreover, it may be possible for teachers to sequence students' experiences in such a way that the algorithm students develop "naturally" is something very similar to, or exactly the same as, the conventional algorithm. For that purpose, the area model of multiplication can play a very important role. Therefore, the use of the model along with base-10 blocks before reaching this point is an integral part of the process. So, how do we help students expand their written methods into multiplication of 2- and 3-digit numbers by 2-digit numbers?

Let's think about 12x23 first. How can students use what they have learned so far to think about ways to calculate this problem? There are at least three possible ways. At the most abstract level, students might be able to think of 12x23 as 12x20+12x3 - i.e., 23 groups of 12 can be split into 20 groups of 12 and 3 groups of 12. Then, each of 12x20 and 12x3 are already discussed. If students can think about this way, they can record the process using the vertical notation,

or .

The notation on the right is basically the standard algorithm for multiplication.

Another possibility is for students to go back to the area representation of multiplication. 12x23 means that we are making a rectangle with the dimension of 12 units by 23 units. The product is represented by the area of this rectangle. So, if you construct this rectangle using base-10 blocks, and using the fewest number of blocks (i.e., use large blocks whenever possible), you can make a rectangle like this one:

By examining the arrangement, we see that there are 1 by 2 rectangles made of flats (200), 2 by 2 rectangles of longs (40), 1 by 3 rectangles made of longs (30) and 2 by 3 rectangles of units (6). So, the product is 200+40+30+6=276. After students have become comfortable with the area model representation with base-10 blocks, you may want to encourage students to move toward drawing instead of using actual base-10 blocks. Sometimes you can make this transition simply by giving students multiplication problems with larger factors. Students will realize that actually making rectangles using base-10 blocks is too tedious.

When students become comfortable with drawing rectangles, they might realize that it is still rather tedious. This is when you may be able to suggest if they could use an adaptation of a notation that we used when we were multiplying 2- or 3-digit number by 1-digit number. Some students may be able to start at this point, without going all the way back to using base-10 blocks. That judgment must be made by teachers, using their knowledge of students. Anyway, the notation might look like this for 12x23:

Again, after students have become fluent with this notation, you might want to bring their attention to the four products (in the example here, 200, 40, 30, and 6). Noticing that these are the products of the two tens digits, the tens digit and the ones digit (in both direction) and the two ones digits. So, you can introduce a new notation that records the same information as this diagram does:

You can then negotiate with your students a consistent order in which you calculate these four products (typically called "partial products) so that we can make sure that we have accounted for all of them. If you really want students to understand the conventional multiplication algorithm, you will start with the ones digit of the multiplier (the bottom number) and multiply the ones and then the tens digits of the multiplicand (the top number). You will then multiply the tens digit of the multiplier with the ones and then the tens digits of the multiplicand. So, this problem would look like this:

If you combine the first two partial products and the last two partial products, you will have:

Note that the example we used, 12x23, did not involve any re-grouping. In a way, this is the most "basic" situation. As students move from one notation to another, you may want to consider moving back to a basic situation. Once students become comfortable with the notation (area model, symbolic notation, or whatever), then you want to look at other situations such as those involving re-grouping and a 0 in the factor/product.

When extending the multiplicand to 3-digit numbers, for example, 587x34, you may want to go back to the diagram notation - it will be rather difficult to actually model these multiplication with base-10 blocks. From the diagram, you can move to the notation that will explicitly record all partial products, then eventually to the conventional algorithm.

As usual, you do want to pay close attention to the numbers (factors) you use. Some students have difficulty with 0's - either in the factors or in the product/partial products, so you want to pay particular attention to those situations.

## Friday, October 9, 2009

### M3N3d - Developing multiplication algorithms (6)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.

d.Understand the effect on the product when multiplying by multiples of 10.

This standard talks about multiplying by multiples of 10, for example 37x30. This situation is different from multiplying multiples of 10, 100, etc. (which we have discussed in a previous post) because we now have 30 groups of 37. Now, if we study this idea after students have already developed a paper-and-pencil algorithm, these problems can be considered as a special case where there will be a 0 in the product. So, procedurally, there are different ways to deal with these problems. Some will carry out the calculation exactly in the same manner as they do with other multipliers:

After students get used to this calculation, they might try to combine the steps to make it more efficient:

From this perspective, this multiplication isn't much different from something like 35x18. The important idea is that we have to write a 0 in the ones place as a place holder.

However, M3N3d states that students must understand "the effect on the product when multiplying by multiples of 10." Moreover, according to the GPS, students do not study how to multiply by 2-digit number until Grade 4 (next post). So, it seems rather odd to talk about multiplying by multiples of 10, which are 2-digit number, at this point. If students' don't know how to multiply by 2-digit number, then we can't focus on the procedural aspect discussed above. Rather, we want students to understand what is going on when we multiply by multiples of 10. Although we cannot use the idea of 10 as a unit in the same way as we did when we were multiplying multiples of 10, we can still use the idea of 10 as a unit when the multipliers are multiples of 10. For example, you can think of 37x30 as 37x3x10. Alternately, you can think of 37x30 as 37x10x3. Either way, multiplying a 2- or 3-digit number by 3 is something students have already learned. What students may not have studied is multiplying 2- (or 3-) digit number by 10. So, that seems to be the primary focus of this standard.

As we explore multiplying 2- and 3-digit numbers by 10, we may again want to go back to the area model of multiplication. For example, if students are to model 17x10 using base-10 blocks, they might at first construct something like this by simply extending what they have done previously:

At this point, some might notice that we can actually use a flat on the left side since there are 10 longs. Moreover, on the right side, since there are 10 rows of units, we can replace each column by a long, resulting in an arrangement like this:

Students can also record the process more abstractly like this, too:

They can also consider cases like 40x10 by extending their thinking of 40 as four 10's. If you have 10 groups of four 10's, you can think of that as 4 groups of ten 10's as well, or 100x4.

From these exploration, students may notice that when you multiply 2- and 3-digit numbers by 10, the product will contain the same set of numerals in the same order but every numeral is moved one place to the left - and there is a 0 in the ones place as a place holder.

Although we may be able to consider multiplying by multiples of 10 as a special case of multiplying by 2-digit numbers, students still need to learn the effect of multiplying by 10 before they can explore multiplying by 2-digit numbers. Moreover, once you study the effect of multiplying by 10, extending it to multiplication by multiples of 10 may be useful to help students deepen their understanding of multiplication operation. Although the formal study of properties of multiplication is done in Grade 4, Grade 3 students can use the associative property to reason about the effect of multiplying by multiples of 10. I believe that's the point behind this standard, not just the procedural fluency.

This standard talks about multiplying by multiples of 10, for example 37x30. This situation is different from multiplying multiples of 10, 100, etc. (which we have discussed in a previous post) because we now have 30 groups of 37. Now, if we study this idea after students have already developed a paper-and-pencil algorithm, these problems can be considered as a special case where there will be a 0 in the product. So, procedurally, there are different ways to deal with these problems. Some will carry out the calculation exactly in the same manner as they do with other multipliers:

After students get used to this calculation, they might try to combine the steps to make it more efficient:

From this perspective, this multiplication isn't much different from something like 35x18. The important idea is that we have to write a 0 in the ones place as a place holder.

However, M3N3d states that students must understand "the effect on the product when multiplying by multiples of 10." Moreover, according to the GPS, students do not study how to multiply by 2-digit number until Grade 4 (next post). So, it seems rather odd to talk about multiplying by multiples of 10, which are 2-digit number, at this point. If students' don't know how to multiply by 2-digit number, then we can't focus on the procedural aspect discussed above. Rather, we want students to understand what is going on when we multiply by multiples of 10. Although we cannot use the idea of 10 as a unit in the same way as we did when we were multiplying multiples of 10, we can still use the idea of 10 as a unit when the multipliers are multiples of 10. For example, you can think of 37x30 as 37x3x10. Alternately, you can think of 37x30 as 37x10x3. Either way, multiplying a 2- or 3-digit number by 3 is something students have already learned. What students may not have studied is multiplying 2- (or 3-) digit number by 10. So, that seems to be the primary focus of this standard.

As we explore multiplying 2- and 3-digit numbers by 10, we may again want to go back to the area model of multiplication. For example, if students are to model 17x10 using base-10 blocks, they might at first construct something like this by simply extending what they have done previously:

At this point, some might notice that we can actually use a flat on the left side since there are 10 longs. Moreover, on the right side, since there are 10 rows of units, we can replace each column by a long, resulting in an arrangement like this:

Students can also record the process more abstractly like this, too:

They can also consider cases like 40x10 by extending their thinking of 40 as four 10's. If you have 10 groups of four 10's, you can think of that as 4 groups of ten 10's as well, or 100x4.

From these exploration, students may notice that when you multiply 2- and 3-digit numbers by 10, the product will contain the same set of numerals in the same order but every numeral is moved one place to the left - and there is a 0 in the ones place as a place holder.

Although we may be able to consider multiplying by multiples of 10 as a special case of multiplying by 2-digit numbers, students still need to learn the effect of multiplying by 10 before they can explore multiplying by 2-digit numbers. Moreover, once you study the effect of multiplying by 10, extending it to multiplication by multiples of 10 may be useful to help students deepen their understanding of multiplication operation. Although the formal study of properties of multiplication is done in Grade 4, Grade 3 students can use the associative property to reason about the effect of multiplying by multiples of 10. I believe that's the point behind this standard, not just the procedural fluency.

## Wednesday, September 30, 2009

### M3N3c - Developing multiplication algorithms (5)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.

In the previous post, I discussed how students can develop a paper-and-pencil algorithm for multiplying 2-digit numbers by 1-digit numbers. Let's consider how we can help students extend the procedure to multiplication of 3-digit numbers by 1-digit number.

How can we multiply 312 x 3? How can students use what they have learned so far to calculate this? One possibility is to think of 312 as 300+12. Then, we can multiply 300x3 and 12x3. Both of these are already learned ideas. If students have already understood how to multiply a 2-digit number by a 1-digit number using a paper-and-pencil method, they can then combine their learning and record this multiplication something like this:

When extending the multiplicand from 2-digit to 3-digit, therefore, there isn't really any new concept involved. Even the idea of looking at 312x3 as 300x3+12x3 is really the same idea as looking at 12x3 as 10x3+2x3, i.e., the distributive property of multiplication, which will be formally studied in Grade 4.

One important thing to think about when we study multiplying 3-digit numbers by 1-digit numbers is different situations where re-grouping must take place, or when there is a 0 (or more) in either the multiplicand or the product. The example we just saw, 312x3, does not involve re-grouping and there is no 0 in the multiplicand nor the product. So, in a way, it is a "general" case of multiplying 3-digit numbers by 1-digit numbers. But, here are some of other cases:

Re-grouping is involved

• 227x3

• 227x5

• 162x3

• etc.

0 is involved

• 406x7

• 365x4

• 527x4

• etc.

I encourage you to think about other cases. As teachers, we must also think about how we want to deal with them. We can carefully sequence those cases and have students think about how they can adapt the written procedure they developed those situations. As you do, it will be helpful if you explicitly ask students what is different about each case compared to the most general one that we start with.

As we look at those special cases, it is important that students understand what is actually happening when we are multiplying 3-digit numbers by 1-digit numbers. For that, it might be useful to go back to the notation system that we used when we developed when we were multiplying 2-digit numbers by 1-digit numbers. For example, let's think about 427x4. Since we can think of 427x4 as 400x4+27x4, and we can use a pictorial notation like this:

Or, we can use more symbolic notation like this (with the previous agreement that we start recording with the partial product of the ones digits first):

We can combine some of the steps involved in this notation and develop a notation like this:

No matter how you approach this topic, what we cannot do is to start with the standard algorithm, which is the most sophisticated way of recording the processes. Help students extend what they have previously learned, which may be the standard algorithm for multiplying 2-digit numbers by 1-digit number by thinking about the structure of numbers and the meaning of operations. If necessary, go back to the intermediate notations that were used while developing the algorithm for multiplying 2-digit number by 1-digit numbers. By experiencing this extension, students can then think about how they can extend the algorithm for multiplying 3-digit numbers by 1-digit numbers to multiplying 4-digit (or even longer) numbers by 1-digit numbers. They have not only the experiences of multiplying two numbers but also the experience of "extending" their procedure from one case to another. So they can ask themselves not just "How did I multiply 2- or 3-digit numbers by 1-digit numbers?" but also "How did I extend the algorithm for multiplying 2-digit numbers to 3-digit numbers?" Therefore, when teaching multiplication of 3-digit numbers by 1-digit numbers, what is important is not the procedure but the idea of how to extend the previously learned procedure (2-digit multiplicands) to a new situation (3-digit multiplicands).

In the previous post, I discussed how students can develop a paper-and-pencil algorithm for multiplying 2-digit numbers by 1-digit numbers. Let's consider how we can help students extend the procedure to multiplication of 3-digit numbers by 1-digit number.

How can we multiply 312 x 3? How can students use what they have learned so far to calculate this? One possibility is to think of 312 as 300+12. Then, we can multiply 300x3 and 12x3. Both of these are already learned ideas. If students have already understood how to multiply a 2-digit number by a 1-digit number using a paper-and-pencil method, they can then combine their learning and record this multiplication something like this:

When extending the multiplicand from 2-digit to 3-digit, therefore, there isn't really any new concept involved. Even the idea of looking at 312x3 as 300x3+12x3 is really the same idea as looking at 12x3 as 10x3+2x3, i.e., the distributive property of multiplication, which will be formally studied in Grade 4.

One important thing to think about when we study multiplying 3-digit numbers by 1-digit numbers is different situations where re-grouping must take place, or when there is a 0 (or more) in either the multiplicand or the product. The example we just saw, 312x3, does not involve re-grouping and there is no 0 in the multiplicand nor the product. So, in a way, it is a "general" case of multiplying 3-digit numbers by 1-digit numbers. But, here are some of other cases:

Re-grouping is involved

• 227x3

• 227x5

• 162x3

• etc.

0 is involved

• 406x7

• 365x4

• 527x4

• etc.

I encourage you to think about other cases. As teachers, we must also think about how we want to deal with them. We can carefully sequence those cases and have students think about how they can adapt the written procedure they developed those situations. As you do, it will be helpful if you explicitly ask students what is different about each case compared to the most general one that we start with.

As we look at those special cases, it is important that students understand what is actually happening when we are multiplying 3-digit numbers by 1-digit numbers. For that, it might be useful to go back to the notation system that we used when we developed when we were multiplying 2-digit numbers by 1-digit numbers. For example, let's think about 427x4. Since we can think of 427x4 as 400x4+27x4, and we can use a pictorial notation like this:

Or, we can use more symbolic notation like this (with the previous agreement that we start recording with the partial product of the ones digits first):

We can combine some of the steps involved in this notation and develop a notation like this:

No matter how you approach this topic, what we cannot do is to start with the standard algorithm, which is the most sophisticated way of recording the processes. Help students extend what they have previously learned, which may be the standard algorithm for multiplying 2-digit numbers by 1-digit number by thinking about the structure of numbers and the meaning of operations. If necessary, go back to the intermediate notations that were used while developing the algorithm for multiplying 2-digit number by 1-digit numbers. By experiencing this extension, students can then think about how they can extend the algorithm for multiplying 3-digit numbers by 1-digit numbers to multiplying 4-digit (or even longer) numbers by 1-digit numbers. They have not only the experiences of multiplying two numbers but also the experience of "extending" their procedure from one case to another. So they can ask themselves not just "How did I multiply 2- or 3-digit numbers by 1-digit numbers?" but also "How did I extend the algorithm for multiplying 2-digit numbers to 3-digit numbers?" Therefore, when teaching multiplication of 3-digit numbers by 1-digit numbers, what is important is not the procedure but the idea of how to extend the previously learned procedure (2-digit multiplicands) to a new situation (3-digit multiplicands).

## Friday, September 25, 2009

### M3N3c - Developing multiplication algorithms (4)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.

This is the fourth in a series of posts in which I am discussing the development of multiplication algorithms. Up to this point, students were calculating mentally. The focus has been more on consolidating students' understanding of our number system and the meaning of multiplication by using those understanding to figure out multiplication beyond the basic facts. Today's standard is the first step toward developing paper-and-pencil algorithms. As I begin my post, let me emphasize that teaching of an algorithm for any operation should focus on helping students develop the algorithm on their own. In other words, we need to move away from the show-and-tell approach where teachers show students how to multiply using the multiplication algorithm and then have them practice over and over. Practice is important, but students should first develop the algorithm themselves. Of course, that does NOT mean that we just leave students on their own. Rather, teachers must plan carefully to guide students' thinking.

One useful idea in developing a multiplication algorithm is the area model of multiplication. In Grade 3, students learn about area of rectangles and squares. When students cover a rectangle with unit squares, they notice that they are arranged rows and columns of equal sizes. Because all rows (or columns) are equal, we can use multiplication to efficiently determine the area. This idea can be used to model multiplication where the two factors are represented by the two dimensions of a rectangle and the product is represented by the area. So, for example, 4x6=24 can be modeled as shown below.

Notice that since you can turn the rectangles around without changing the area, this is also a useful model to show why the commutative property of multiplication is true. It is also useful to model the distributive property of multiplication.

When you model multiplication problems like 14x7 using base-10 blocks, you can certainly try to make 7 groups of 14 (1 long and 4 units). However, we want to encourage students to organize the model more systematically using the area model. The area model representation will make it much easier to determine the product by observation if the same type of blocks are grouped together.

Eventually, we want to help students move beyond modeling with actual base-10 blocks. One useful approach to do so is to have students draw what they would have done with base-10 blocks. Thus, drawing the picture like the one above. Grid papers can be very helpful in that process. However, as they become comfortable with drawing pictures, they realize that drawing can be rather tedious. Given our goal is to determine the product, what we want to know is how many longs and how many units we have. Thus, we can model the multiplication explicitly showing only the information we need. Here is an example for 14x7.

Once students become comfortable with modeling multiplying 2-digit number by 1-digit number this way, we can ask if they can think of a way to represent this model using a vertical notation like we did with addition and subtraction. Here are two possibilities:

Students can see that 70+28 and 28+70 are the same. Thus, we can write it either way. At this point, it is ok to suggest that we agree to write the product of the ones digit first. Again, after students practice this notation, they might notice that the ones digit for the partial product of the tens digit on the multiplicand and the multiplier is always 0. Therefore, the ones' digit of the product is always the ones digit of the partial product of the ones digit of the multiplicand and the multiplier, 8 in the example above. Then, we have to add the tens digits of the partial products to find the product. This process can be combined if you use a notation like this:

This may be a slightly different notation than some of us are used to, where the tens digit of the partial product above the tens digit of the multiplicand. That notation sometimes causes students to add the re-grouped digit and the tens digit of the multiplicand before multiplying by the multiplier - that is students end up doing (2+1)x7 instead of 1x7+2. Writing the re-grouped digit below the horizontal bar (the equal sign) might minimize that error.

In the next post, I will discuss how this procedure may be extended to multiplying 3-digit numbers

This is the fourth in a series of posts in which I am discussing the development of multiplication algorithms. Up to this point, students were calculating mentally. The focus has been more on consolidating students' understanding of our number system and the meaning of multiplication by using those understanding to figure out multiplication beyond the basic facts. Today's standard is the first step toward developing paper-and-pencil algorithms. As I begin my post, let me emphasize that teaching of an algorithm for any operation should focus on helping students develop the algorithm on their own. In other words, we need to move away from the show-and-tell approach where teachers show students how to multiply using the multiplication algorithm and then have them practice over and over. Practice is important, but students should first develop the algorithm themselves. Of course, that does NOT mean that we just leave students on their own. Rather, teachers must plan carefully to guide students' thinking.

One useful idea in developing a multiplication algorithm is the area model of multiplication. In Grade 3, students learn about area of rectangles and squares. When students cover a rectangle with unit squares, they notice that they are arranged rows and columns of equal sizes. Because all rows (or columns) are equal, we can use multiplication to efficiently determine the area. This idea can be used to model multiplication where the two factors are represented by the two dimensions of a rectangle and the product is represented by the area. So, for example, 4x6=24 can be modeled as shown below.

Notice that since you can turn the rectangles around without changing the area, this is also a useful model to show why the commutative property of multiplication is true. It is also useful to model the distributive property of multiplication.

When you model multiplication problems like 14x7 using base-10 blocks, you can certainly try to make 7 groups of 14 (1 long and 4 units). However, we want to encourage students to organize the model more systematically using the area model. The area model representation will make it much easier to determine the product by observation if the same type of blocks are grouped together.

Eventually, we want to help students move beyond modeling with actual base-10 blocks. One useful approach to do so is to have students draw what they would have done with base-10 blocks. Thus, drawing the picture like the one above. Grid papers can be very helpful in that process. However, as they become comfortable with drawing pictures, they realize that drawing can be rather tedious. Given our goal is to determine the product, what we want to know is how many longs and how many units we have. Thus, we can model the multiplication explicitly showing only the information we need. Here is an example for 14x7.

Once students become comfortable with modeling multiplying 2-digit number by 1-digit number this way, we can ask if they can think of a way to represent this model using a vertical notation like we did with addition and subtraction. Here are two possibilities:

Students can see that 70+28 and 28+70 are the same. Thus, we can write it either way. At this point, it is ok to suggest that we agree to write the product of the ones digit first. Again, after students practice this notation, they might notice that the ones digit for the partial product of the tens digit on the multiplicand and the multiplier is always 0. Therefore, the ones' digit of the product is always the ones digit of the partial product of the ones digit of the multiplicand and the multiplier, 8 in the example above. Then, we have to add the tens digits of the partial products to find the product. This process can be combined if you use a notation like this:

This may be a slightly different notation than some of us are used to, where the tens digit of the partial product above the tens digit of the multiplicand. That notation sometimes causes students to add the re-grouped digit and the tens digit of the multiplicand before multiplying by the multiplier - that is students end up doing (2+1)x7 instead of 1x7+2. Writing the re-grouped digit below the horizontal bar (the equal sign) might minimize that error.

In the next post, I will discuss how this procedure may be extended to multiplying 3-digit numbers

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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.