Base Ten Blocks

Exploring Base Ten Blocks

The Base Ten Blocks provide a spatial model of our base ten number system. The smallest blocks–cubes that measure 1 cm on a side–are called units. The long, narrow blocks that measure 10 cm by 1 cm by 1 cm are called rods. The flat, square blocks that measure 10 cm by 10 cm by 1 cm are called flats. The largest blocks available, cubes that measure 10 cm on a side, are called cubes. When working with base ten place value experiences, we commonly use the unit to represent ones, the rod to represent tens, the flat to represent hundreds, and the cube to represent thousands. Providing names based on the shape rather than 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.

Base Ten units, longs, flats, cube

The size relationships among the blocks make them ideal for the investigation of number concepts. Initially, however, 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 of a smaller block to make one of the next larger block. 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 are able to see the logical development of each operation. The blocks provide a visual foundation and understanding of the algorithms children use when doing paper-and-pencil computation. Older children 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 children to organize their work as they explore the relationships among the blocks and determine how groups of blocks can be used to represent numbers. Student 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 units 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 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 understanding of the meanings of addition, subtraction, multiplication, and division. Modeling addition on a place-value mat provides students with a visual basis for the concept of regrouping.

Base Ten addition sample

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 child 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.

Base Ten multiplication sample

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.

Base Ten understanding and finding volume sample

Assessing Students’ Understanding

The use of Base Ten Blocks provides 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 are able to 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 children's free exploration, you can get a sense of individual styles and intellectual strengths.

Having children 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 children's recorded work or invite them to choose pieces to add to their math portfolios.

Base Ten portfolio sample

Skills Taught

  • Area
  • Average
  • Comparing
  • Counting
  • Counting On
  • Decimals
  • Deductive Reasoning
  • Estimation
  • Following Directions
  • Game Strategies
  • Interpreting Data
  • Logic
  • Making Patterns
  • Measuring
  • Mental Math
  • Multiples of 10
  • Number
  • Number Relationships
  • Number Sense
  • Organizing Data
  • Pattern Recognition
  • Patterns
  • Patterns & Functions
  • Place Value
  • Predicting
  • Probability
  • Properties of Cubes
  • Properties of Geometric Shapes
  • Properties of Rectangles
  • Properties of Squares
  • Shape Recognition
  • Sorting and Classifying
  • Spatial Visualization
  • Square Roots
  • Surface Area
  • Volume
  • Weight

Subjects Taught

  • Addition
  • Decimals
  • Division
  • Geometry
  • Measurement
  • Money
  • Multiplication
  • Percents
  • Probability & Statistics
  • Subtraction

Related Manipulatives

Color Tiles
Base Ten Place Value Mat

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Primary Grade Activities (PreK to 2nd)

Base Ten Blocks provide children with a physical model of our place value system of numeration. By using Base Ten Blocks, children develop understanding of the processes of addition and subtraction. Their understanding can then be connected to the abstract symbolism.

The names of unit, rod, and flat are used to describe the blocks interchangeably with one, ten, and hundred. This is to keep the emphasis on the concrete blocks. While the word ten is likely to trigger the symbolic image of 10, the 1 and the 0, the word "rod" describes the materials without the interference of the symbols.

As with other materials, allow students the chance to build with the blocks before requiring their attention to a specific activity. Children may or may not discover the relationship that 10 of one block makes 10 of the next larger block. It is necessary to establish this relationship before other instruction is initiated.

Here are two activities for getting to know Base Ten Blocks and using them to learn and understand the math that they represent.

Activity Curriculum Strands Topics
Race for a Flat
In this game for two pairs of children, players take turns rolling number cubes and finding the sums of the numbers rolled. They take Base Ten Blocks to represent the sums in an effort to be the first to accumulate blocks with a total value of 100.
Place value
Sum It Up!
Children use Base Ten Blocks to model a number as the sum of two addends. Then they find ways to model the same number with different pairs of addends.
Number Counting
Place value

Intermediate Grade Activities (3rd to 5th)

Working with concrete materials, even after students have learned the abstract procedures, provides them the opportunity to experience a concept in a new way. Though students may be able to solve problems with paper and pencil, perhaps even automatically, being asked to use the Base Ten Blocks requires them to reason in a different way. This is beneficial for deepening their understanding and developing their flexibility.

Using Base Ten Blocks during instruction is based on the approach that students’ understanding of a concept is best developed concretely. Once a student demonstrates concrete understanding, the teacher connects that concrete experience to the standard symbolism. Students then use the abstract symbolism in conjunction with the manipulatives.

It is important when approaching instruction in this way that enough time is provided to develop the concept concretely without the interference of the symbolism. Time used at this level to develop a solid base of understanding is well spent. Allow students to use the materials as long as they need to. When you encourage them to try the written algorithm without the material, also encourage them to visualize the blocks. It is helpful for them to connect what they write to what they actually do with the blocks.

Here are two activities using Base Ten Blocks in different contexts.

Activity Curriculum Strands Topics
Modeling Multiplication
Students use Base Ten Blocks to build rectangular arrays that model the multiplication of two 2-digit numbers.
Spatial visualization
Properties of rectangles
Race for a Whole
In this game for two to four players, students roll number cubes to indicate numbers of tenths and hundredths. They collect Base Ten rods and units to represent the tenths and hundredths, respectively, in an effort to be the first to accumulate blocks with a total value of 1.
Probability & Statistics
Place value
Game strategies

Middle Grade Activities (6th to Algebra)

Base Ten blocks help students make connections to decimal operations and operations with algebraic expressions in the middle grades. The proportional relationships among the pieces can be used to develop understanding of the number system and ratio/proportion. In addition, the pieces can be used to model solid figures and develop understanding of volume and surface area.

Here are two activities for using Base Ten Blocks with middle grades students.

Activity Curriculum Strands Topics
Building Boxes
Students use Base Ten Blocks to build as many different rectangular prisms as they can from eight rods.
Spatial visualization
Double the Dimensions
Students use Base Ten Blocks to design and build structures. They determine the volume and surface area of their structures and then predict how these will change when they "double" their structures.
Surface area
Spatial visualization
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