### What are Decimal Fractions?

Decimal fractions are a notational system for recording certain types of fraction. They are not an idea that is separate from other numbers, but are linked to both the whole-number place-value system, and common fraction ideas. The Venn diagram below shows this relationship:

It is important that students learn to see decimal fractions as merely a certain group of fractions, represented symbolically in a particular format. Teachers can foster this understanding with questions like these:

• How can you write the fraction six hundredths another way?
• Who can show me a picture that represents fifty-seven hundredths?
• What is one tenth of the number 46?
• Shade these rectangles to match the fractions:

What are Percents?

It is important that students learn that percents are a certain type of fraction, expressed in parts out of one hundred (per cent). As explained in the Introductory page to these notes, students often seem to develop fractured, isolated ideas about rational numbers. They seem to believe that percents are different to fractions, and that concepts needed for common fractions, decimals, and percents are all separate and have to be learned one at a time. The graphic below shows two pictorial models, labeled as “0.75”, “75 hundredths”, and “75%”. The models show that these labels are equivalent, and refer to the same numerical amount:

On the contrary, students should learn to perceive all rational numbers as being very similar, and as expressing the same ideas in somewhat different forms. Students will need a great deal of practice in naming fractions in different ways, converting one to another, and showing them using pictorial or concrete models. For example, students could be asked to fill in the following table:

### Modeling Decimal Fractions & Percents

As with most mathematics concepts, decimal fractions and percents should be modeled for students, to help them see the relationship between the number, the symbol and the name:

The diagram above is based on a strategy suggested by Payne & Rathmell in 1975 for teaching numeration concepts. Students are asked questions relating to each of the arrows in the diagram, such as: “Write the symbol for forty-seven hundredths”, or “Shade this hundredths grid to show the fraction 0.47”. In so doing students should learn to associate the various representations for decimal numbers. The same models and strategy can be used to discuss percents, since they are also based on 100 parts of a whole.

### Square Region Models

The model used for the decimal fraction above is a square grid divided into 100 smaller squares, each representing one hundredth. This is an effective model for hundredths, provided that the idea of the large square representing a whole is firmly established. This model can be used as a follow-up to a similar model for tenths:

By using the same-sized squares divided into ten strips and one hundred squares, students can see the equivalence of various decimal fractions. They can also learn to link decimals and percents, and to make comparisons between them, and to convert from one form to the other.

### Place-value Block Models for Decimals

Another model for decimal fractions is based on an extension of the familiar place-value blocks. Place-value blocks generally model thousands, hundreds, tens and ones only. By purchasing or making smaller blocks by cutting up a one block, tiny place-value blocks may be used to represent tenths, hundredths and thousandths. The graphic below shows the appearance of these decimal place-value blocks, taken from a software title called Hi-Flyer Decimals. This software was demonstrated at the NCTM 1999 Annual Meeting in San Francisco.

The mini-blocks are based on the size of a “one” block, which is a 1 cm cube. The other places are represented by blocks that are progressively one tenth of the size of the previous block, making the “thousandth” block a cube that is a mere 1 mm (1/25″) in size.

### Summary

1. Decimal fraction notation is used to represent fractions with denominators that are powers of ten (tenths, hundredths, thousandths, …).
2. Decimals are a subset of all rational numbers; students need to see decimals in the context of other fractions.
3. Percents are a fraction concept based on a denominator of 100.
4. Decimals and percents may be modeled using hundred grids or cut-up place-value blocks.

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<< Part III:   Understanding Place Value

#### Reference

Payne, J & Rathmell, E 1975, ‘Number and Numeration’, In JN Payne (ed.), Mathematics Learning in Early Childhood, pp. 125-160, nctm, Reston, VA.

Graphics credit: Peter Price

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