Comparing Book Fractions
Two third graders, Maya and Leo, are reading copies of the same 80-page book in the same edition. Maya has read 3/4 of her book. Leo has read 5/8 of his book. During conference time, Maya says, “I read more because fourths are bigger than eighths.” Leo says, “I read more because 8 is greater than 4.” Their teacher asks them to compare the fractions using the same whole, an 80-page book, and a number line from 0 to 1 marked in eighths. The students must decide who has read more pages, explain the mistake in each statement, and show how many pages each student has left. The teacher also wants them to name a benchmark fraction that helps compare 3/4 and 5/8 without multiplying.
1. Who has read more pages? Use the 80-page whole to show your comparison.
Read the worked answer
Maya has read more pages. For Maya: 80 ÷ 4 = 20 pages in each fourth, and 3 × 20 = 60 pages. For Leo: 80 ÷ 8 = 10 pages in each eighth, and 5 × 10 = 50 pages. Since 60 > 50, Maya has read more. On a number line in eighths, 3/4 = 6/8, and 6/8 > 5/8, which agrees with 60 > 50.
2. Maya says fourths are bigger than eighths, so she read more. Is her conclusion right for the wrong reason, or wrong? Explain.
Read the worked answer
Her conclusion is right, but her reasoning is not enough. It is true that 1/4 is larger than 1/8, but the number of parts matters too. Maya has 3 fourths, which is 6 eighths. Leo has 5 eighths. Compare 6/8 and 5/8: 6/8 is greater, so Maya read more. The correct comparison is 3/4 = 6/8 > 5/8, not simply “fourths are bigger.”
3. Name a benchmark fraction that shows both 3/4 and 5/8 are greater than 1/2. Why is that benchmark alone not enough to decide who read more? Then give the pages each student has left.
Read the worked answer
Both fractions are greater than 1/2: 3/4 > 1/2 and 5/8 > 1/2. But the benchmark 1/2 only tells us both students passed half the book; it does not tell which fraction is larger. To decide, compare in eighths: 3/4 = 6/8, and 6/8 > 5/8. Maya has 80 − 60 = 20 pages left. Leo has 80 − 50 = 30 pages left.