Infinite Decimal Expansions

5 min

Narrative

In this activity, students practice using long division to calculate the decimal form of a rational number. Students are expected to notice a repeating pattern (MP8).

Launch

Arrange students in groups of 2. Since the goal is for students to notice a repeating pattern, do not provide access to calculators. If needed, remind them of the previous activity where the decimal expansion of 211\frac2{11} was shown to be 0.18‾0.\overline{18} using long division and repeated reasoning.

Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion. 

Student Task

The first 3 digits after the decimal for the decimal expansion of 37\frac37 have been calculated. Find the next 4 digits.

Long division calculations for decimal expansion, showing place value.
Long division calculations for decimal expansion, showing place value. The first line indicates the given place values of the quotient, 0 point 4 2 8. The second line indicates the division sentence 3 divided by 7; the number 7 is the left most number, followed by the long division symbol, and the number 3 inside; the 3 lines up vertically with the 0 above. On the third line reads as "minus twenty eight," with the 2 directly below the 3 from above. The fourth line reads "twenty," with the 2 directly below the 8 in 28. The fifth line reads "minus fourteen," with the 1 directly below the 2 in 20 and the 4 directly below the 0 in 20. The sixth line reads "sixty," with the 6 directly below the 4 in 14. The seventh line reads "minus fifty six" with the 5 directly below the 6 in 60, and the 6 directly below the 0 in 60. The eight line reads "4" with the 4 directly below the 6 in 56. A vertical line is drawn through all of the lines, falling between the 0 and the 4 in "0 point four two eight" and the 2 and 8 in twenty eight.

 

Sample Response

5714

Synthesis

The goal of this discussion is to make sure students understand that all rational numbers have a decimal expansion that eventually repeats. Ask students to share the next 4 digits and record them on the long division calculation for all to see. Discuss:

  • “Without calculating, what number do you think will be next? Why?” (I think 2 will be the next digit because I can see the starting pattern has begun again.)

Continue the calculation and verify that 2 comes next and continue until reaching 4 again. Point out that this cycle will continue indefinitely—we can predict what will happen at each step because it is exactly like what happened 6 steps ago.

Tell students that all rational numbers have a decimal expansion that eventually repeats. Sometimes they eventually repeat 0s, like in 38=0.3750000...\frac38=0.3750000 . . . . Sometimes they repeat several digits like in 37= 0.428571‾\frac37= 0.\overline{428571}. If necessary, remind students that in overline notation, the line goes over the digits that repeat.

Be careful in the use of the word “pattern,” as it can be ambiguous. For example, there is a pattern to the digits of the number 0.12112111211112 . . . , but the number is not rational.

Standards
Building On
  • 7.NS.2.d·Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
  • 7.NS.A.2.d·Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Addressing
  • 8.NS.1·Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
  • 8.NS.A.1·Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

15 min

15 min