Comparing Fraction Claims on a Number Line
Three students compare 3/4 and 5/8. Student A says 5/8 is larger because 8 is bigger than 4. Student B says 3/4 is larger because 4 is smaller, so the pieces are bigger. Student C says they are equal because both are "missing one piece." The teacher draws a number line from 0 to 1 and asks the class to place both fractions. A fourth student points out that 3/4 = 6/8, which settles part of the argument. The class must decide who is right, explain the flaw in each wrong claim, and then compare 3/4 with 7/10 using a valid strategy.
1. Evaluate each of the three claims and state which student is correct, with justification.
Read the worked answer
Student B is correct: 3/4 > 5/8.
- Student A's flaw: a larger denominator means smaller pieces, not a larger value. Eighths are smaller than fourths, so comparing denominators alone is invalid.
- Student C's flaw: "missing one piece" ignores piece size. 3/4 is missing 1/4 (a large piece); 5/8 is missing 3/8 (three small pieces). The missing amounts differ.
- Student B's reasoning is sound: fourths are larger pieces, and 3 of them exceed 5 eighths.
The cleanest justification uses a common denominator: 3/4 = 6/8, and 6/8 > 5/8. On a number line, 3/4 sits at the same point as 6/8, to the right of 5/8.
2. Compare 3/4 and 7/10 using two different valid strategies and state which is greater.
Read the worked answer
Strategy 1 - common denominator: The least common denominator of 4 and 10 is 20. 3/4 = 15/20 and 7/10 = 14/20. Since 15/20 > 14/20, 3/4 > 7/10.
Strategy 2 - benchmark to 1: 3/4 is 1/4 away from 1, and 7/10 is 3/10 away from 1. Compare the gaps: 1/4 = 5/20 and 3/10 = 6/20. The smaller gap belongs to 3/4, so 3/4 is closer to 1 and therefore larger.
Both strategies agree: 3/4 is greater than 7/10. A number-line sketch confirms 3/4 = 0.75 and 7/10 = 0.70, so 3/4 lies to the right.