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Connecting Fraction and Decimal Representations
Fractions and decimals are two different ways to represent the same quantities. A fraction shows a relationship between a numerator and a denominator, while a decimal uses place value to show parts of a whole. In this topic, we focus on how these representations are connected and how students can move between them meaningfully.
We will look at different types of decimal representations. Some fractions have terminating decimals, which end after a certain number of decimal places, such as 0.4 or 0.125. Other fractions have repeating decimals, where a digit or group of digits repeats forever, such as 0.3333… or 0.16666…. We will also distinguish these from irrational decimals, which go on forever without repeating, such as π or √2.
One way to connect fractions and decimals is by using equivalent fractions. For example,
2/5 = 4/10 = 0.4
This works because tenths, hundredths, and thousandths are directly connected to decimal place value. We will also use prime factorization to predict whether a fraction will have a terminating or repeating decimal. When a fraction is in simplest form, its decimal will terminate if the denominator has only factors of 2 and/or 5. If the denominator has any other prime factors, the decimal will repeat.
As future teachers, it is important to help students see fractions and decimals as connected representations, not separate topics. This understanding supports later work with percents, ratios, proportions, measurement, and algebra.
Student Learning Goals
By the end of this topic, students should be able to:
Explain how fractions and decimals can represent the same quantity.
Identify terminating, repeating, and irrational decimals.
Use equivalent fractions to write fractions as decimals.
Connect tenths, hundredths, and thousandths to decimal notation.
Use prime factorization of the denominator to predict whether a fraction will terminate or repeat.
Explain why denominators with only factors of 2 and/or 5 produce terminating decimals.
Distinguish between repeating decimals and irrational decimals.
Key Vocabulary
Fraction - A number written as a ratio of two integers, such as 3/4.
Decimal - A number written using place value to show parts of a whole.
Terminating Decimal - A decimal that ends, such as 0.75.
Repeating Decimal - A decimal with a digit or pattern that repeats forever, such as 0.666666…..
Irrational Decimal - A decimal that goes on forever without repeating.
Equivalent Fractions - Fractions that look different but represent the same amount.
Prime Factorization - Writing a number as a product of prime numbers.
Simplest Form - A fraction form where the numerator and denominator have no common factor other than 1.
Example Strategies
2/5 = 4/10 - Terminating - 0.4
3/4 = 75/100 - Terminating - 0.75
7/8 = 875/1000 - Terminating - 0.875
1/3 - Denominator has factor 3 - Repeating - 0.333…
5/6 - 6 = 2 × 3, includes factor 3 - Repeating - 0.8333…
1/7 - Denominator has factor 7 - Repeating0.142857142857…
Prime Factorization Test
To predict whether a fraction has a terminating or repeating decimal:
Write the fraction in simplest form.
Find the prime factorization of the denominator.
Look at the prime factors.
Only factors of 2 - Terminates
Only factors of 5 - Terminates
Only factors of 2 and 5 - Terminates
Any other prime factor - Repeats
Examples:
3/20 - Since 20 = 2 × 2 × 5, the decimal terminates.
4/15 - Since 15 = 3 × 5, and 3 is not 2 or 5, the decimal repeats.
6/12 - Even though 12 has a factor of 3, the fraction simplifies to 1/2. Since the simplified denominator is 2, the decimal terminates.
Teacher Connection
Students often experience fractions and decimals as separate units in school, but mathematically they are deeply connected. A decimal is another way to describe fractional parts based on powers of ten. This is why equivalent fractions with denominators of 10, 100, or 1000 are so useful.
It is also helpful for future teachers to make the difference between repeating and irrational decimals very clear. Repeating decimals may go on forever, but they have a predictable pattern and can be written as fractions. Irrational decimals also go on forever, but they do not repeat and cannot be written as fractions.
Helpful teacher questions include:
What fraction does this decimal represent?
Can this fraction be renamed with tenths, hundredths, or thousandths?
Is the fraction in simplest form?
What are the prime factors of the denominator?
Does the decimal end, repeat, or continue without a pattern?
How do you know this decimal is exact and not just an approximation?
Quick Reflection Question
Why is it important to simplify a fraction before deciding whether its decimal representation terminates or repeats? Use an example such as 6/12 or 10/40 in your explanation.