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

Photographer: Annie Spratt | Source: Unsplash

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 color 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.


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.

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 color 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 colors of base ten blocks are required — one color for negative numbers and one color 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).

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.



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Dr. D. M. Hardy

Dr. D. M. Hardy

I have a M.Ed. in Instructional Technology, and an Ed.D. in Adult & Career Education. I enjoy spreading knowledge, because we all need to be life-long learners.