Fraction comparison practice

Yes. These can be compared because eighths are smaller than sevenths, so \(\frac{3}{8} \) has 3 small parts and \(\frac{3}{7} \) has the same number of larger parts, so \(\frac{3}{7} \) is larger.

No. Twelfths are smaller than thirds, but \(\frac{5}{12} \) has more small parts, and \( \frac{2}{3} \) has fewer larger parts. That's not enough information to know which is bigger.

Yes. Both fractions are made of fifteenth pieces. \(\frac{4}{15} \) has only 4 pieces, and \( \frac{6}{15} \) has more of the same sized pieces, so \( \frac{6}{15} \) is larger.

Yes. Eighths are larger than fifteenths. There are more larger pieces (3) in \(\frac{3}{8} \), and fewer (only 2) smaller pieces in \( \frac{2}{15} \). More larger pieces is bigger, so \(\frac{3}{8} \) is larger.

No, Fourths are bigger than sixths, but there are more sixths in \( \frac{5}{6} \) and fewer fourths in \(\frac{3}{4} \). We're comparing more small pieces to fewer large pieces, and that's not enough information to compare.

Yes, \(\frac{3}{4} \) is \(\frac{1}{4} \) less than 1. \( \frac{5}{6} \) is \( \frac{1}{6} \) less than 1.
We can compare and say \(\frac{1}{4} \) is greater than \( \frac{1}{6} \).
That means that \(\frac{3}{4} \) is farther from 1 than \( \frac{5}{6} \), so it's smaller than \( \frac{5}{6} \).

No. \(\frac{7}{8} \) is \(\frac{1}{8} \) less than 1 and \( \frac{19}{21} \) is \( \frac{2}{21} \) less than 1
We know \(\frac{1}{8} \) is bigger than \( \frac{1}{21} \), but there is only one eighth (fewer) and there are two twenty-firsts, which is more small pieces, so we don't know which is bigger: \(\frac{1}{8} \) or \(\frac{1}{21} \).
That means we can't use this reasoning to figure out which fraction is bigger.

No, for the same reason as the previous example.
\(\frac{7}{8} \) is \(\frac{1}{8} \) less than 1 and \( \frac{10}{12} \) is \( \frac{2}{12} \) less than 1
We know \(\frac{1}{8} \) is bigger than \( \frac{1}{12} \), but there is only one eighth (fewer) and there are two twenty-firsts, which is more small pieces, so we don't know which is bigger: \(\frac{1}{8} \) or \(\frac{1}{12} \).
That means we can't use this reasoning to figure out which fraction is bigger.

Yes, \(\frac{7}{9} \) is \(\frac{2}{9} \) less than 1. \( \frac{10}{12} \) is \( \frac{2}{12} \) less than 1.
We know ninths are bigger than twelfths, so since there are the same number of parts, we can and say \(\frac{2}{9} \) is greater than \( \frac{2}{12} \).
That means that \(\frac{7}{9} \) is farther from 1 than \( \frac{10}{12} \), so it's smaller than \( \frac{10}{12} \).

Thirds are larger than fourths, but there are more fourths and fewer thirds, so we don't have enough information to compare the fractions this way.

Fifths are bigger than eighths, but you also need to check (and explain) that there are the same number (or more) of the large pieces as compared to the small pieces to know that the total fraction is bigger.

\(\frac{7}{8} \) is only one part away from the whole, and \(\frac{10}{12} \) is two parts away from the whole, but those parts aren't the same size, If we can't be sure which is bigger \(\frac{7}{8} \) or \(\frac{2}{12} \), then we can't use the distance from the whole to explain which fraction is bigger

This explanation's biggest problem is that it's vague.
It might be saying that 7>3 (or 6>3 or 5>3)so we have more parts taken in the first fraction than the second. If that's what it's saying, then there's a problem because it's not explaining what happens with the size of the parts. In fact, this start isn't going to help much with these fractions because the fraction with more parts also has smaller parts, so that doesn't give us enough information to compare the two.

Alternately, it might be saying that the first fraction is bigger because 8>4. That's probably a bigger misconception because eighths are smaller than fourths, so knowing that 8>4 isn't a reason to thing the first fraction is larger.

This is another vague explanation
It seems to be saying that it takes more eighths to make a whole than fourths, so eighths are smaller. It doesn't include an explanation of the total amount because it doesn't use the information about how many parts are taken.

This explanation is incomplete because it doesn't tell how we know that \(\frac{7}{8} \) is closer to a whole than \(\frac{3}{4} \).

This seems to be an explanation, but if you look closer, all this says is that \(\frac{7}{8} \) is bigger because it's bigger.

This explanation is missing an explanation of how you know that \(\frac{2}{5} \) is smaller than \(\frac{1}{2} \) and \(\frac{5}{8} \) is larger than \(\frac{1}{2} \)

This explanation is missing the detail that both of the fractions are eighths, so the pieces are the same size for both fractions.