Concrete Models

Students need to approach decimals using concrete materials, before moving to a pictorial representation and finally to abstract numbers and symbols.

There are many models that have been used to explore decimals.  One main criteria is that students can actually make a whole with ten tenths and a tenth with ten hundredths – the model is proportional.


Money is not actually suitable as a tool for introducing decimals for this reason.  Ten dimes can’t be physically joined to construct a dollar – this relationship relies on an abstract understanding.


Many teachers and programs use commercial base 10 materials for exploring decimals. Usually the flat Basetenblocksis considered the whole, and then the rod is a tenth of the whole, with the unit cube the hundredth. This model has the advantage of being widely available, fairly cheap, and proportional.  It is also easily used for tenths and hundredths. However, you need to explicitly remove the student’s earlier conceptions of using these materials for whole numbers.


thousandthTo use commercial base 10 for thousandths in grade 6, the kits are easily augmented with 1000th chips – 10 of these small chips stack to make a cube.


Fraction circles are actually a fairly unsophisticated model for tenths – we usually want students to start using rectangular models for fractions as they are more versatile and can be accurately subdivided easier. They are also fairly expensive, which limits the amount of tenths and wholes students can have to represent decimals.

The decimal strip model, downloadable below, is one I made up for grade 4. It appears to have had good success, and can be used for rounding decimals, adding and subtracting using the place value mat included, and even multiplication using an array model. One drawback is in constructing hundredths for grade 5; they will be fairly small. If you give it a try, please send me some feedback.

Decimal Strip ModelInstructions and blackline masters for using the decimal strip model.


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