# Learning Math With Manipulatives - Base Ten Blocks (Part III)

In the first two parts, representing, adding, and subtracting numbers using base ten blocks were explained. The use of base ten blocks gives students an effective tool that they can touch and manipulate to solve math questions. Not only are base ten blocks effective at solving math questions, they teach students important steps and skills that translate directly into paper and pencil methods of solving math questions. Students who first use base ten blocks develop a stronger conceptual understanding of place value, addition, subtraction, and other math skills. Because of their benefit to the math development of young people, educators have looked for other applications involving base ten blocks. In this article, a variety of other applications will be explained.

One common way of teaching multiplication is to create a rectangle where the two factors become the two dimensions of a rectangle. This is easily accomplished using graph paper. Imagine the question 7 x 6. Students colour or shade a rectangle seven squares wide and six squares long; then they count the number of squares in their rectangle to find the product of 7 x 6. With base ten blocks, the process is essentially the same except students are able to touch and manipulate real objects which many educators say has a greater effect on a student's ability to understand the concept. In the example, 5 x 8, students create a rectangle 5 cubes wide by 8 cubes long, and they count the number of cubes in the rectangle to find the product.

Multiplying two-digit numbers is slightly more complicated, but it can be learned fairly quickly. If both factors in the multiplication question are two-digit numbers, the flats, the rods, and the cubes might all be used. In the case of two-digit multiplication, the flats and the rods just quicken the procedure; the multiplication could be accomplished with just cubes. The procedure is the same as for one-digit multiplication - the student creates a rectangle using the two factors as the dimensions of the rectangle. Once they have built the rectangle, they count the number of units in the rectangle to find the product. Consider the multiplication, 54 x 25. The student needs to create a rectangle 54 cubes wide by 25 cubes long. Since that might take a while, the student can use a shortcut. A flat is simply 100 cubes, and a rod is simply 10 cubes, so the student builds the rectangle filling in the large areas with flats and rods. In its most efficient form, the rectangle for 54 x 25 is 5 flats and four rods in width (the rods are arranged vertically), and 2 flats and five rods in length (with the rods arranged horizontally). The rectangle is filled in with flats, rods, and cubes. In the whole rectangle, there are 10 flats, 33 rods, and 20 cubes. Using the values for each base ten block, there is a total of (10 x 100) + (33 x 10) + (20 x 1) = 1350 cubes in the rectangle. Students can count each type of base ten block separately and add them up.

Base ten blocks are so flexible, they can even be used to divide! There are three methods for division that I will describe: grouping, distributing, and modified multiplying.

To divide by grouping, first represent the dividend (the number you are dividing) with base ten blocks. Arrange the base ten blocks into groups the size of the divisor. Count the number of groups to find the quotient. For example, 348 divided by 58 is represented by 3 flats, 4 rods, and 8 cubes. To arrange 348 into groups of 58, trade the flats for rods, and some of the rods for cubes. The result is six piles of 58, so the quotient is six.

Dividing by distributing is the old "one for you and one for me" trick. Distribute the dividend into the same number of piles as the divisor. At the end, count how many piles are left. Students will probably pick up the analogy of sharing quite easily - i.e. We need to give everyone an equal number of base ten blocks. To illustrate, consider 192 divided by 8. Students represent 192 with one flat, 9 rods and 2 cubes. They can distribute the rods into eight groups easily, but the flat has to be traded for rods, and some rods for cubes to accomplish the distribution. In the end, they should find that there are 24 units in each pile, so the quotient is 24.

To multiply, students create a rectangle using the two factors as the length and width. In division, the size of the rectangle and one of the factors is known. Students begin by building one dimension of the rectangle using the divisor. They continue to build the rectangle until they reach the desired dividend. The resulting length (the other dimension) is the quotient. If a student is asked to solve 1369 divided by 37, they begin by laying down three rods and seven cubes to create one dimension of the rectangle. Next, they lay down another 37, continuing the rectangle, and check to see if they have the required 1369 yet. Students who have experience with estimating might begin by laying down three flats and seven rods in a row (rods vertically arranged) since they know that the quotient is going to be larger than ten. As students continue, they may recognize that they can replace groups of ten rods with a flat to make counting easier. They continue until the desired dividend is reached. In this example, students find the quotient is 37.

Up until now, the value of the cube has been one unit. For older students, there is no reason why the cube couldn't represent one tenth, one hundredth, or one million. If the value of the cube is redefined, the other base ten blocks, of course, have to follow. For example, redefining the cube as one tenth means the rod represents one, the flat represents ten, and the block represents one hundred. This redefinition is useful for a decimal question such as 54.2 + 27.6. A common way to redefine base ten blocks is to make the cube one thousandth. This makes the rod one hundredth, the flat one tenth, and the block one whole. Besides the traditional definition, this one makes the most sense, since a block can be divided into 1000 cubes, so it follows logically that one cube is one thousandth of the cube.

Numbers don't stop at 9,999 which is the maximum you can represent with a traditional set of base ten blocks. Fortunately, base ten blocks come in a variety of colors. In math, the ones, tens, and hundreds are called a period. The thousands, ten thousands, and hundred thousands are another period. The millions, ten millions and hundred millions are the third period. This continues where every three place values is called a period. You may have figured out by now that each period can be represented by a different colour of place value block. If you do this, you eliminate the large blocks and just use the cubes, rods, and flats. Let us say that we have three sets of base ten blocks in yellow, green, and blue. We'll call the yellow base ten blocks the first period (ones, tens, hundreds), the green blocks the second period, and the blue blocks the third period. To represent the number, 56,784,325, use 5 blue rods, 6 blue cubes, 7 green flats, 8 green rods, 4 green cubes, 3 yellow flats, 2 yellow rods, and 5 yellow cubes. When adding and subtracting, trading is accomplished by recognizing that 10 yellow flats can be traded for one green cube, 10 green flats can be traded for one blue cube, and vice-versa.

Base ten blocks can be used to add and subtract integers. To accomplish this, two colours of base ten blocks are required - one colour for negative numbers and one colour for positive numbers. The zero principle states that an equal number of negatives and an equal number of positives add up to zero. To add using base ten blocks, represent both numbers using base ten blocks, apply the zero principle and read the result. For example (-51) + (+42) could be represented with 5 red rods, 1 red cube, 4 blue rods, and 2 blue cubes. Immediately, the student applies the zero principle to four red and four blue rods and one red and one blue cube. To finish the problem, they trade the remaining red rod for 10 red cubes and apply the zero principle to the remaining blue cube and one of the red cubes. The end result is (-9).

Subtracting means taking away. For instance, (-5) - (-2) is represented by taking two red cubes from a pile of five red cubes. If you can't take away, the zero principle can be applied in reverse. You can't take away six blue cubes in (-7) - (+6) because there aren't six blue cubes. Since a blue cube and a red cube is just zero, and adding zero to a number doesn't change it, simply include six blue cubes and six red cubes with the pile of seven red cubes. When six blue cubes are taken from the pile, 13 red cubes remain, so the answer to (-7) - (+6) is (-13). This procedure can, of course, be applied to larger numbers, and the process might involve trading.

By no means have I explained all of the uses of base ten blocks, but I have covered most of the major uses. The rest is up to your imagination. Can you think of a use for base ten blocks when teaching powers of ten? How about using base ten blocks for fractions? So many math skills can be learned using base ten blocks simply because they represent our numbering system - the base ten system. Base ten blocks are just one of many excellent manipulatives available to teachers and parents that give students a strong conceptual background in math.

The base ten blocks skills described above can be applied using worksheets from http://www.math-drills.com. The worksheets come with answer keys, so students can get feedback on their ability to correctly use base ten blocks.

**Multiplying One- and Two-Digit Numbers**One common way of teaching multiplication is to create a rectangle where the two factors become the two dimensions of a rectangle. This is easily accomplished using graph paper. Imagine the question 7 x 6. Students colour or shade a rectangle seven squares wide and six squares long; then they count the number of squares in their rectangle to find the product of 7 x 6. With base ten blocks, the process is essentially the same except students are able to touch and manipulate real objects which many educators say has a greater effect on a student's ability to understand the concept. In the example, 5 x 8, students create a rectangle 5 cubes wide by 8 cubes long, and they count the number of cubes in the rectangle to find the product.

Multiplying two-digit numbers is slightly more complicated, but it can be learned fairly quickly. If both factors in the multiplication question are two-digit numbers, the flats, the rods, and the cubes might all be used. In the case of two-digit multiplication, the flats and the rods just quicken the procedure; the multiplication could be accomplished with just cubes. The procedure is the same as for one-digit multiplication - the student creates a rectangle using the two factors as the dimensions of the rectangle. Once they have built the rectangle, they count the number of units in the rectangle to find the product. Consider the multiplication, 54 x 25. The student needs to create a rectangle 54 cubes wide by 25 cubes long. Since that might take a while, the student can use a shortcut. A flat is simply 100 cubes, and a rod is simply 10 cubes, so the student builds the rectangle filling in the large areas with flats and rods. In its most efficient form, the rectangle for 54 x 25 is 5 flats and four rods in width (the rods are arranged vertically), and 2 flats and five rods in length (with the rods arranged horizontally). The rectangle is filled in with flats, rods, and cubes. In the whole rectangle, there are 10 flats, 33 rods, and 20 cubes. Using the values for each base ten block, there is a total of (10 x 100) + (33 x 10) + (20 x 1) = 1350 cubes in the rectangle. Students can count each type of base ten block separately and add them up.

**Division**Base ten blocks are so flexible, they can even be used to divide! There are three methods for division that I will describe: grouping, distributing, and modified multiplying.

To divide by grouping, first represent the dividend (the number you are dividing) with base ten blocks. Arrange the base ten blocks into groups the size of the divisor. Count the number of groups to find the quotient. For example, 348 divided by 58 is represented by 3 flats, 4 rods, and 8 cubes. To arrange 348 into groups of 58, trade the flats for rods, and some of the rods for cubes. The result is six piles of 58, so the quotient is six.

Dividing by distributing is the old "one for you and one for me" trick. Distribute the dividend into the same number of piles as the divisor. At the end, count how many piles are left. Students will probably pick up the analogy of sharing quite easily - i.e. We need to give everyone an equal number of base ten blocks. To illustrate, consider 192 divided by 8. Students represent 192 with one flat, 9 rods and 2 cubes. They can distribute the rods into eight groups easily, but the flat has to be traded for rods, and some rods for cubes to accomplish the distribution. In the end, they should find that there are 24 units in each pile, so the quotient is 24.

To multiply, students create a rectangle using the two factors as the length and width. In division, the size of the rectangle and one of the factors is known. Students begin by building one dimension of the rectangle using the divisor. They continue to build the rectangle until they reach the desired dividend. The resulting length (the other dimension) is the quotient. If a student is asked to solve 1369 divided by 37, they begin by laying down three rods and seven cubes to create one dimension of the rectangle. Next, they lay down another 37, continuing the rectangle, and check to see if they have the required 1369 yet. Students who have experience with estimating might begin by laying down three flats and seven rods in a row (rods vertically arranged) since they know that the quotient is going to be larger than ten. As students continue, they may recognize that they can replace groups of ten rods with a flat to make counting easier. They continue until the desired dividend is reached. In this example, students find the quotient is 37.

**Changing the Values of Base Ten Blocks**Up until now, the value of the cube has been one unit. For older students, there is no reason why the cube couldn't represent one tenth, one hundredth, or one million. If the value of the cube is redefined, the other base ten blocks, of course, have to follow. For example, redefining the cube as one tenth means the rod represents one, the flat represents ten, and the block represents one hundred. This redefinition is useful for a decimal question such as 54.2 + 27.6. A common way to redefine base ten blocks is to make the cube one thousandth. This makes the rod one hundredth, the flat one tenth, and the block one whole. Besides the traditional definition, this one makes the most sense, since a block can be divided into 1000 cubes, so it follows logically that one cube is one thousandth of the cube.

**Representing and Working With Large Numbers**Numbers don't stop at 9,999 which is the maximum you can represent with a traditional set of base ten blocks. Fortunately, base ten blocks come in a variety of colors. In math, the ones, tens, and hundreds are called a period. The thousands, ten thousands, and hundred thousands are another period. The millions, ten millions and hundred millions are the third period. This continues where every three place values is called a period. You may have figured out by now that each period can be represented by a different colour of place value block. If you do this, you eliminate the large blocks and just use the cubes, rods, and flats. Let us say that we have three sets of base ten blocks in yellow, green, and blue. We'll call the yellow base ten blocks the first period (ones, tens, hundreds), the green blocks the second period, and the blue blocks the third period. To represent the number, 56,784,325, use 5 blue rods, 6 blue cubes, 7 green flats, 8 green rods, 4 green cubes, 3 yellow flats, 2 yellow rods, and 5 yellow cubes. When adding and subtracting, trading is accomplished by recognizing that 10 yellow flats can be traded for one green cube, 10 green flats can be traded for one blue cube, and vice-versa.

**Integers**Base ten blocks can be used to add and subtract integers. To accomplish this, two colours of base ten blocks are required - one colour for negative numbers and one colour for positive numbers. The zero principle states that an equal number of negatives and an equal number of positives add up to zero. To add using base ten blocks, represent both numbers using base ten blocks, apply the zero principle and read the result. For example (-51) + (+42) could be represented with 5 red rods, 1 red cube, 4 blue rods, and 2 blue cubes. Immediately, the student applies the zero principle to four red and four blue rods and one red and one blue cube. To finish the problem, they trade the remaining red rod for 10 red cubes and apply the zero principle to the remaining blue cube and one of the red cubes. The end result is (-9).

Subtracting means taking away. For instance, (-5) - (-2) is represented by taking two red cubes from a pile of five red cubes. If you can't take away, the zero principle can be applied in reverse. You can't take away six blue cubes in (-7) - (+6) because there aren't six blue cubes. Since a blue cube and a red cube is just zero, and adding zero to a number doesn't change it, simply include six blue cubes and six red cubes with the pile of seven red cubes. When six blue cubes are taken from the pile, 13 red cubes remain, so the answer to (-7) - (+6) is (-13). This procedure can, of course, be applied to larger numbers, and the process might involve trading.

**Other Uses**By no means have I explained all of the uses of base ten blocks, but I have covered most of the major uses. The rest is up to your imagination. Can you think of a use for base ten blocks when teaching powers of ten? How about using base ten blocks for fractions? So many math skills can be learned using base ten blocks simply because they represent our numbering system - the base ten system. Base ten blocks are just one of many excellent manipulatives available to teachers and parents that give students a strong conceptual background in math.

The base ten blocks skills described above can be applied using worksheets from http://www.math-drills.com. The worksheets come with answer keys, so students can get feedback on their ability to correctly use base ten blocks.

Learning Math With Manipulatives - Base Ten Blocks (Part III)
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