## Terminating and Repeating Decimals

Decimal numbers are numbers written in our base ten system: the value of a digit in each place is 10 times the value the same digit would have one place to its right. Students have been studying the base ten system for years, but spend a little time reviewing place value and multiplying by powers of 10. Then, students can tie place value and mental math concepts into their study of the connections between fractions and decimals.

Materials: Chart paper

Preparation: None

Prerequisite Skills and Concepts: Students should be quite familiar with the base ten system, including the value of a digit in any given place. Assure yourself that students are quite comfortable with how multiplying by powers of ten moves the decimal point in the product. The concept of equivalent fractions is also important for this lesson.

Approach this topic through the need to write fractions as decimals and decimals as fractions.

• Ask: What is the decimal notation for the fraction ?
Most students will know, without computing, that this is 0.25. Point out that the proper way to say 0.25 is twenty-five hundredths and that this automatically gives an equivalent fraction for .
• Ask: How can you show that = ?
Students should describe finding an equivalent fraction by multiplying numerator and denominator by the same number, in this case, 25. Reinforce the fact that multiplying or dividing the numerator and denominator by the same number has the same effect as multiplying or dividing by one: it does not change the value of the fraction, just its appearance. Be sure students conclude, after trying a number of examples, that if a fraction has an equivalent with a denominator that's a power of 10, then the fraction can be written as a terminating decimal.
• Ask: How can you writeas a decimal?
It's unlikely that students have memorized the decimal equivalent for , so you should discuss dividing the numerator by the denominator to find a decimal equivalent for any fraction. Have a student show the work on the chalkboard and show that can also be written as a terminating decimal (0.3125).
• Say: Now, show how to writeas a decimal.
Have a student show the work on the chalkboard. When the student realizes that the same remainder keeps recurring, stop and discuss the definition of repeating decimal: a decimal with a digit or digits that repeat in an identical pattern indefinitely.
• Say: Let's make a table of terminating and repeating decimals that are equivalent to fractions with denominators 2 through 9.

You can complete this table as a class project, or split up the task among several groups. Be sure to spend plenty of time looking for patterns:

-all thirds, sevenths, ninths are repeating decimals.
-the ninths have a really easy pattern to remember. = 0.111..., = 0.222..., = 0.333..., etc.
-in the sevenths, the same digits repeat in a different order for each numerator.
Terminating and Repeating
Decimals
Operations with Integers