Am I Terminating or Repeating?
Lesson 7 of 12
Objective: Objective: Students will use the calculators to determine which decimal is terminating and which is repeating.
Teacher Guided Notes:
- Use a calculator or computer to demonstrate division of 2 integers which always results in repetition or termination of the decimal and discuss why calculators sometimes give a different display. Apply this to converting fractions to decimals, for example: 1/8 (terminating), 1/7 (repeating).
- For those students who are ready for an extension, here is a simple way to use a calculator to convert a decimal back into a fraction. Show students how the decimal remainder of the quotient of two whole numbers can be converted into a whole number remainder on a calculator. Start with an improper fraction, divide to display the quotient, subtract the whole number part of the decimal, and multiply the decimal remainder by the divisor. Students should round up the result to the next whole number. For example, 45/7 on a TI-30x calculator displays as 6.4285714285. Multiply 0.4285714285 by 7 to determine that the quotient with a remainder is 6 R 3. Check the result by converting the fraction 3/7 to decimal form. Explain that some calculators may truncate a decimal (simply dropping undisplayed digits) while other will round it
For those students who are ready for an extension, this strategy makes connections to finding the nth term of a pattern and it reinforces that a repeating decimal is rational. After demonstrating long division that results in a repeating decimal, ask students to find the 100th digit in the decimal fraction. For example, 1/7 in decimal form has the repeated 142857, which has 6 digits. After students have tried some strategies, guide them to consider how many 6’s are in 100. The remainder of 100÷6 is 4. The 4th digit in the repeated is 8, which will also be the 100th digit in the decimal fraction. Challenge students to find the 50th digit of the decimal fraction for 2/7.
Using a calculator, discover which of the fractions are terminating decimals and which repeating decimals are.
-Arrange students into groups. Each group will take part of a set of fractions and convert them in to decimals. The fraction set for the whole class will be all the proper fractions having 2 through 11 as denominators, not including fractions with a numerator of one. Students should examine their decimal equivalents and compare them to the set where the numerators were one. Are their new results the same as when the numerator was one?
-Ask them to generalize which denominators result in a terminating decimal and which result in a repeating decimal. (Teacher note: students may conclude that terminating denominators are factors of powers of 10; for example, 10 is divisible by 2, 5, and 10, 100 is divisible by 4, etc. Students should see that these powers of 10 define the place values in our decimal system.)
-Ask students to repeat the first set of divisions using pencil, paper, and long division. ½ and 1/11
Note: For 1/11 Students may notice that in this particular problem, the “bring down” step occurs twice in order to bring down enough zeros to create 100, which is a dividend greater than 11.
-Homework: Have students pick 5 fractions in which the denominators are between 3 and 11 (like above), but have numerators that are not equal to 1. Then, have them do long division problems to show the decimal equivalent of the fractions.
Activity: Students will complete the class Activity lab sheet. Use the teacher guided notes to take you through the instruction of the lab sheet. You may opt to use groups for this activity, each student may complete on their own, or you may do as a whole group.
Homework: Have students use their Interactive Math Journals, the title for today’s lesson activity should be Terminating decimals/Repeating Decimals. Have students discover which fractions out of ½, 1/3, ¼, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, and 1/11 are terminating and repeating.
Students may use the worksheet available in order to organize and list the fractions and their titles. This will become a resource that students may use in their interactive notebooks. This lesson is intended to be completed before moving into long division. This will lend itself to discussion of the meaning of rational numbers, and an introduction to non-rational numbers which will be further discussed in 8th grade.
Once students understand how to determine if a fraction terminates or repeats challenge them to discover more terminating and repeating decimals on their own. Have students log their discoveries in their notebooks.