- Part I: Using the Power of Computers to Teach Decimal Fraction and Percent Concepts
- Part II: What Are Rational Numbers?
- Part III: Understanding Place Value
**Part IV: Decimal Fractions and Percents**- Part V: Using Software to Teach Decimal Fractions

### 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:

### Teaching About Decimals

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

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

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<< Part III: Understanding Place Value

>> Part V: Using Software to Teach Decimal Fractions

#### 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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[related-posts]

PatriciaNovember 16, 2011 at 4:43 PMThank you very helpful when checking my sons homework.

Peter PriceNovember 16, 2011 at 9:49 PMPatricia,

Thanks for the comment. I am very glad that this was helpful. Lots of children find decimal fractions difficult, so teachers and parents need to teach the concepts very carefully.

Peter

MaggiNovember 29, 2011 at 8:54 AMI am in 5th grade, and not to offend the maker of this, it is still challenging me!

Peter PriceDecember 1, 2011 at 8:22 PMMaggi, thanks for the comment. I am not sure what part of this page is challenging to you. Perhaps that it a good thing! Do please let me know if you have any questions. Thanks for visiting!

Ram Chandra PoudelDecember 28, 2011 at 4:00 AMI am a primary teacher in Nepal. I found your article is very useful in my class room. I hope you will post more teaching strategy in future. thank you verrrrrrrrrrr……….y much.

Peter PriceJanuary 5, 2012 at 9:00 AMThanks for the comment, Ram Chandra! I’m happy to hear that our content is being used in Nepal to help students to learn mathematics. Don’t hesitate to contact me again if you have a question, suggestion or comment.

NEENAJanuary 4, 2012 at 1:39 AMTHANX A LOT THESE TIPS R VERY HELPFUL IM GRADE THREE TEACHER AND LAST YEAR I FACED A LOT DIFFICULTY IN TEACHING DECIMAL NFRACTIONS THANX A LOT ONCE AGAIN ,NEENA

Peter PriceJanuary 5, 2012 at 8:58 AMThanks, Neena

I’m glad the site is useful to you.

Peter

PaulaJanuary 21, 2012 at 12:53 AMHome school mom here. I have taught this concept to 2 other children and found it hard to explain clearly. Wow, this makes it so much easier. Thanks! I can finally teach it in a very understandable way!! Wish I would have had you as a math teacher. 🙂

Thank you,

Paula

Peter PriceJanuary 22, 2012 at 2:26 PMPaula, thanks for this comment. I’m really glad to hear that this article has helped you.

Home school moms rule! You are all amazing to teach your own children across all subjects.

All the best explaining decimal fractions to your child!

martarMarch 24, 2012 at 1:54 AMI like your website, very helpful. However I got stumped the other day. I was showing a student how to model decimals. The student wanted to know how to model 8.312. I was not sure how to show the student how to model this number. Can you explain??

Peter PriceMarch 26, 2012 at 12:34 PMHi Martar, thanks for the question.

Modeling decimals is really tricky, mostly because the pieces are so darned small (especially thousandths), and seeing the pieces in relation to the “one” is difficult.

The only models I’ve come up with are a square region model, which can be divided into ten even strips and than 100 little squares. Thousandths can be shown this way, via little tiny strips inside the 100 squares.

The other model is a “cut up base ten block”. Take a wooden “one” block and carefully cut it with a razor blade into 10 slices (tenths), then 100 chips (hundredths), then up to 1000 little 1 mm cubes (thousandths).

See the next post in this series for software I designed to illustrate this model on a computer. We have had great success using this for students to learn the sizes of decimal fractions, all the way down to thousandths.

sophiaNovember 22, 2013 at 7:23 AMstupid

Peter PriceNovember 22, 2013 at 9:30 AMSophia, I am sorry to hear that you did not like this page. I am trying to help teachers with their teaching of math; perhaps it didn’t help you as a student. I hope you find the help you are looking for. If you want to contact me directly, let me know here and I’ll gladly have an email conversation with you.