| In the first two parts, representing, adding, | | | | down another 37, continuing the rectangle, |
| and subtracting numbers using base ten blocks | | | | and check to see if they have the required |
| were explained. The use of base ten blocks | | | | 1369 yet. Students who have experience with |
| gives students an effective tool that they | | | | estimating might begin by laying down three |
| can touch and manipulate to solve math | | | | flats and seven rods in a row (rods |
| questions. Not only are base ten blocks | | | | vertically arranged) since they know that the |
| effective at solving math questions, they | | | | quotient is going to be larger than ten. As |
| teach students important steps and skills | | | | students continue, they may recognize that |
| that translate directly into paper and pencil | | | | they can replace groups of ten rods with a |
| methods of solving math questions. Students | | | | flat to make counting easier. They continue |
| who first use base ten blocks develop a | | | | until the desired dividend is reached. In |
| stronger conceptual understanding of place | | | | this example, students find the quotient is |
| value, addition, subtraction, and other math | | | | 37.Changing the Values of Base Ten BlocksUp |
| skills. Because of their benefit to the math | | | | until now, the value of the cube has been one |
| development of young people, educators have | | | | unit. For older students, there is no reason |
| looked for other applications involving base | | | | why the cube couldn't represent one tenth, |
| ten blocks. In this article, a variety of | | | | one hundredth, or one million. If the value |
| other applications will be | | | | of the cube is redefined, the other base ten |
| explained.Multiplying One- and Two-Digit | | | | blocks, of course, have to follow. For |
| NumbersOne common way of teaching | | | | example, redefining the cube as one tenth |
| multiplication is to create a rectangle where | | | | means the rod represents one, the flat |
| the two factors become the two dimensions of | | | | represents ten, and the block represents one |
| a rectangle. This is easily accomplished | | | | hundred. This redefinition is useful for a |
| using graph paper. Imagine the question 7 x | | | | decimal question such as 54.2 + 27.6. A |
| 6. Students colour or shade a rectangle seven | | | | common way to redefine base ten blocks is to |
| squares wide and six squares long; then they | | | | make the cube one thousandth. This makes the |
| count the number of squares in their | | | | rod one hundredth, the flat one tenth, and |
| rectangle to find the product of 7 x 6. With | | | | the block one whole. Besides the traditional |
| base ten blocks, the process is essentially | | | | definition, this one makes the most sense, |
| the same except students are able to touch | | | | since a block can be divided into 1000 cubes, |
| and manipulate real objects which many | | | | so it follows logically that one cube is one |
| educators say has a greater effect on a | | | | thousandth of the cube.Representing and |
| student's ability to understand the concept. | | | | Working With Large NumbersNumbers don't stop |
| In the example, 5 x 8, students create a | | | | at 9,999 which is the maximum you can |
| rectangle 5 cubes wide by 8 cubes long, and | | | | represent with a traditional set of base ten |
| they count the number of cubes in the | | | | blocks. Fortunately, base ten blocks come in |
| rectangle to find the product.Multiplying | | | | a variety of colors. In math, the ones, tens, |
| two-digit numbers is slightly more | | | | and hundreds are called a period. The |
| complicated, but it can be learned fairly | | | | thousands, ten thousands, and hundred |
| quickly. If both factors in the | | | | thousands are another period. The millions, |
| multiplication question are two-digit | | | | ten millions and hundred millions are the |
| numbers, the flats, the rods, and the cubes | | | | third period. This continues where every |
| might all be used. In the case of two-digit | | | | three place values is called a period. You |
| multiplication, the flats and the rods just | | | | may have figured out by now that each period |
| quicken the procedure; the multiplication | | | | can be represented by a different colour of |
| could be accomplished with just cubes. The | | | | place value block. If you do this, you |
| procedure is the same as for one-digit | | | | eliminate the large blocks and just use the |
| multiplication - the student creates a | | | | cubes, rods, and flats. Let us say that we |
| rectangle using the two factors as the | | | | have three sets of base ten blocks in yellow, |
| dimensions of the rectangle. Once they have | | | | green, and blue. We'll call the yellow base |
| built the rectangle, they count the number of | | | | ten blocks the first period (ones, tens, |
| units in the rectangle to find the product. | | | | hundreds), the green blocks the second |
| Consider the multiplication, 54 x 25. The | | | | period, and the blue blocks the third period. |
| student needs to create a rectangle 54 cubes | | | | To represent the number, 56,784,325, use 5 |
| wide by 25 cubes long. Since that might take | | | | blue rods, 6 blue cubes, 7 green flats, 8 |
| a while, the student can use a shortcut. A | | | | green rods, 4 green cubes, 3 yellow flats, 2 |
| flat is simply 100 cubes, and a rod is simply | | | | yellow rods, and 5 yellow cubes. When adding |
| 10 cubes, so the student builds the rectangle | | | | and subtracting, trading is accomplished by |
| filling in the large areas with flats and | | | | recognizing that 10 yellow flats can be |
| rods. In its most efficient form, the | | | | traded for one green cube, 10 green flats can |
| rectangle for 54 x 25 is 5 flats and four | | | | be traded for one blue cube, and |
| rods in width (the rods are arranged | | | | vice-versa.IntegersBase ten blocks can be |
| vertically), and 2 flats and five rods in | | | | used to add and subtract integers. To |
| length (with the rods arranged horizontally). | | | | accomplish this, two colours of base ten |
| The rectangle is filled in with flats, rods, | | | | blocks are required - one colour for negative |
| and cubes. In the whole rectangle, there are | | | | numbers and one colour for positive numbers. |
| 10 flats, 33 rods, and 20 cubes. Using the | | | | The zero principle states that an equal |
| values for each base ten block, there is a | | | | number of negatives and an equal number of |
| total of (10 x 100) + (33 x 10) + (20 x 1) = | | | | positives add up to zero. To add using base |
| 1350 cubes in the rectangle. Students can | | | | ten blocks, represent both numbers using base |
| count each type of base ten block separately | | | | ten blocks, apply the zero principle and read |
| and add them up.DivisionBase ten blocks are | | | | the result. For example (-51) + (+42) could |
| so flexible, they can even be used to divide! | | | | be represented with 5 red rods, 1 red cube, 4 |
| There are three methods for division that I | | | | blue rods, and 2 blue cubes. Immediately, the |
| will describe: grouping, distributing, and | | | | student applies the zero principle to four |
| modified multiplying.To divide by grouping, | | | | red and four blue rods and one red and one |
| first represent the dividend (the number you | | | | blue cube. To finish the problem, they trade |
| are dividing) with base ten blocks. Arrange | | | | the remaining red rod for 10 red cubes and |
| the base ten blocks into groups the size of | | | | apply the zero principle to the remaining |
| the divisor. Count the number of groups to | | | | blue cube and one of the red cubes. The end |
| find the quotient. For example, 348 divided | | | | result is (-9).Subtracting means taking away. |
| by 58 is represented by 3 flats, 4 rods, and | | | | For instance, (-5) - (-2) is represented by |
| 8 cubes. To arrange 348 into groups of 58, | | | | taking two red cubes from a pile of five red |
| trade the flats for rods, and some of the | | | | cubes. If you can't take away, the zero |
| rods for cubes. The result is six piles of | | | | principle can be applied in reverse. You |
| 58, so the quotient is six.Dividing by | | | | can't take away six blue cubes in (-7) - (+6) |
| distributing is the old "one for you and one | | | | because there aren't six blue cubes. Since a |
| for me" trick. Distribute the dividend into | | | | blue cube and a red cube is just zero, and |
| the same number of piles as the divisor. At | | | | adding zero to a number doesn't change it, |
| the end, count how many piles are left. | | | | simply include six blue cubes and six red |
| Students will probably pick up the analogy of | | | | cubes with the pile of seven red cubes. When |
| sharing quite easily - i.e. We need to give | | | | six blue cubes are taken from the pile, 13 |
| everyone an equal number of base ten blocks. | | | | red cubes remain, so the answer to (-7) - |
| To illustrate, consider 192 divided by 8. | | | | (+6) is (-13). This procedure can, of course, |
| Students represent 192 with one flat, 9 rods | | | | be applied to larger numbers, and the process |
| and 2 cubes. They can distribute the rods | | | | might involve trading.Other UsesBy no means |
| into eight groups easily, but the flat has to | | | | have I explained all of the uses of base ten |
| be traded for rods, and some rods for cubes | | | | blocks, but I have covered most of the major |
| to accomplish the distribution. In the end, | | | | uses. The rest is up to your imagination. Can |
| they should find that there are 24 units in | | | | you think of a use for base ten blocks when |
| each pile, so the quotient is 24.To multiply, | | | | teaching powers of ten? How about using base |
| students create a rectangle using the two | | | | ten blocks for fractions? So many math skills |
| factors as the length and width. In division, | | | | can be learned using base ten blocks simply |
| the size of the rectangle and one of the | | | | because they represent our numbering system - |
| factors is known. Students begin by building | | | | the base ten system. Base ten blocks are just |
| one dimension of the rectangle using the | | | | one of many excellent manipulatives available |
| divisor. They continue to build the rectangle | | | | to teachers and parents that give students a |
| until they reach the desired dividend. The | | | | strong conceptual background in math.The base |
| resulting length (the other dimension) is the | | | | ten blocks skills described above can be |
| quotient. If a student is asked to solve 1369 | | | | applied using worksheets from The worksheets |
| divided by 37, they begin by laying down | | | | come with answer keys, so students can get |
| three rods and seven cubes to create one | | | | feedback on their ability to correctly use |
| dimension of the rectangle. Next, they lay | | | | base ten blocks. |