Terminating and Repeating Decimals
After students have a good grasp of terminating and repeating decimals, you want to be sure that they can write any rational number in either its fractional or its decimal form and that they can compare rational numbers no matter what form they take.
Preparation: Write on the board this list of terminating decimals: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Use a 3-column format with the columns labeled Decimal, Fraction, Simplified Fraction.
Wrap-Up and Assessment Hints
- Say: Writing a terminating decimal as a fraction is simple. You're almost there when you read the decimal. Read the decimal numbers I've written on the board. Tell me whether the fraction form is in simplest form.
Students should have no trouble reading these decimals and telling you which have a fraction form with a lesser denominator. Have a student record the fractional forms of these decimals.
- Ask: Think about the table we've just made. Is there something special about the decimals that have a simpler fractional form?
Students should notice that what look like even numbers all simplify, while the odd numbers don't. This isn't exactly what's going on here, so add 0.25 and 0.75 to the list and lead a discussion about what's really happening: When you look at the numerals that make up a decimal number, you're looking at the numerator of a fraction. The denominator is deduced by the placement of the decimal point. If the numerator is a factor of the denominator (as in the case of 0.2, 0.25, 0.5) or if the numerator has a common factor with the denominator (as in the case of 0.4, 0.6, 0.75, 0.8), then there is a simplified fractional form of the decimal.
- Now erase the table of terminating decimals and replace it with a 2-column table of repeating decimals
( , , , ). Label the columns Decimal, Fraction.
- Ask: Can anyone fill in the second column of this table?
Most of your students should recognize as and as . Some may recognize as , but it's unlikely that anyone will recognize as .
- Say: You can use algebra to help you write any repeating decimal as a fraction.
- Now help students identify the repeating decimal as x. Discuss what happens if you subtract x from 10x, if you subtract 10x from 100x, and so forth. Start with one of the fraction-equivalents they know and demonstrate multiplying the repeating decimal by a power of 10 that lets you eliminate the repeating part using subtraction:
You now have a simple algebraic equation to solve. .
- Ask: Can you think of several ways to compute the fraction for ?
Encourage students to try subtracting 10x -x and 100x 10x to see how their results compare. This exercise leads nicely into figuring out how to find the fractional equivalent for .
- A fun exercise is to try writing as a fraction. This method will tell you that = 1. You and the students can have a lively mathematical discussion about why this might be so.
- Now that students can write rational numbers in either their fractional or their decimal form, help them use this information to compare rational numbers that are very close to each other on the number line and look quite similar in decimal form: 0.6 and ; 0.1 and ; etc.
- The main points you want students to get from work with terminating and repeating decimals are that they are all rational and that sometimes the fraction form of a number is a little easier to work with than the decimal form (and vice versa).
Terminating and Repeating
Operations with Integers