Base Ten Blocks provide a spatial model of our base ten number system. Base Ten Blocks typically consist of four different concrete representations that are introduced in elementary math and utilized well into middle school.
Units = Ones; Measure 1 cm x 1 cm x 1 cm
Rods = Tens; Measure 1 cm x 1 cm x 10 cm
Flats = Hundreds; Measure 1 cm x 10 cm x 10 cm
Cubes = Thousands; Measure 10 cm x 10 cm x 10 cm
Providing names based on the shape rather than on the value allows for the pieces to be renamed when necessary. For example, when studying decimals, a class can use the flat to represent a unit and establish the value of the other pieces from there.
The size relationships among the blocks make them ideal for the investigation of number concepts. Initially students should explore independently with Base Ten Blocks before engaging in structured activities. As they move the blocks around to create designs and build structures, they may be able to discover on their own that it takes 10 smaller blocks to make one of the next larger blocks. Students' designs and structures also lead them to employ spatial visualization and to work intuitively with the geometric concepts of shape, perimeter, area, and volume.
Base Ten Blocks are especially useful in providing students with ways to physically represent the concepts of place value and addition, subtraction, multiplication, and division of whole numbers. By building number combinations with Base Ten Blocks, students ease into the concept of regrouping, or trading, and can see the logical development of each operation. The blocks provide a visual foundation and understanding of the algorithms students use when doing paper-and-pencil computation. Older students can transfer their understanding of whole numbers and whole-number operations to an understanding of decimals and decimal operations.
Working with Base Ten Blocks
Place-value mats, available in pads of 50, provide a means for students to organize their work as they explore the relationships among Base Ten Blocks and determine how groups of blocks can be used to represent numbers. Students may begin by placing unit blocks, one at a time, in the ones column on a mat. For each unit they place, they record the number corresponding to the total number of units placed (1, 2, 3...). They continue this process until they have accumulated 10 units, at which point they match their 10 units to 1 rod and trade those units for the rod, which they place in the tens column. Students continue in the same way, adding units one at a time to the ones column and recording the totals (11, 12, 13...) until it is time to trade for a second rod, which they place in the tens column (20). When they finally come to 99, there are 9 units and 9 rods on the mat. Adding one more unit forces two trades: first 10 units for another rod and then 10 rods for a flat (100). Then it is time to continue adding and recording units and making trades as needed as students work their way through the hundreds and up to the thousands. Combining the placing and trading of rods with the act of recording the corresponding numbers provides students with a connection between concrete and symbolic representations of numbers.
Base Ten Blocks can be used to develop an understanding of the meaning of addition, subtraction, multiplication, and division. Modeling addition on a place-value mat provides students with a visual basis for the concept of regrouping.
Subtraction with regrouping involves trading some of the blocks for smaller blocks of equal value so that the "taking away" can be accomplished. For example, to subtract 15 from 32, a student would trade one of the rods that represent 32 (3 rods and 2 units) for 10 units to form an equivalent representation of 32 (2 rods and 12 units). Then the student would take away 15 (1 rod and 5 units) and be left with a difference of 17 (1 rod and 7 units).
Multiplication can be modeled as repeated addition or with rectangular arrays. Using rectangular arrays can help in understanding the derivation of the partial products, the sum of which is the total product.
Division can be done as repeated subtraction or through building and analyzing rectangular arrays.
By letting the cube, flat, rod, and unit represent 1, 0.1, 0.01, and 0.001, respectively, older students can explore and develop decimal concepts, compare decimals, and perform basic operations with decimal numbers.
The squares along each face make the blocks excellent tools for visualizing and internalizing the concepts of perimeter and surface area of structures. Counting unit blocks in a structure can form the basis for understanding and finding volume.
Assessing Students’ Understanding
Base Ten Blocks provide a perfect opportunity for authentic assessment. Watching students work with the blocks gives you a sense of how they approach a mathematical problem. Their thinking can be "seen" through their positioning of the blocks. When a class breaks up into small working groups, you can circulate, listen, and raise questions, all the while focusing on how individuals are thinking.
The challenges that students encounter when working with Base Ten Blocks often elicit unexpected abilities from those whose performance in more symbolic, number-oriented tasks may be weak. On the other hand, some students with good memories for numerical relationships have difficulty with spatial challenges and can more readily learn from freely exploring with Base Ten Blocks. Thus, by observing students’ free exploration, you can get a sense of individual styles and intellectual strengths.
Having students describe their creations and share their strategies and thinking with the whole class gives you another opportunity for observational assessment. Furthermore, you may want to gather student's recorded work or invite them to choose pieces to add to their math portfolios.