# Fraction comparison practice

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## Which of these pairs can be compared directly (without using by considering residual amounts) by arguing from the
number and size of pieces? (Same numerator, Same denominator and similar arguments)

\(\frac{3}{8} \) and \(\frac{3}{7} \)

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.

\(\frac{5}{12} \) and \( \frac{2}{3} \)

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.

\(\frac{4}{15} \) and \( \frac{6}{15} \)

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.

\(\frac{3}{8} \) and \( \frac{2}{15} \)

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.

\(\frac{3}{4} \) and \( \frac{5}{6} \)
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.

## Which of these pairs of fractions can be compared using the residual strategy?

\(\frac{3}{4} \) and \( \frac{5}{6} \)
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 \( \frac{5}{6} \).

\(\frac{7}{8} \) and \( \frac{19}{21} \)
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.

\(\frac{7}{8} \) and \( \frac{10}{12} \)
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.

\(\frac{7}{9} \) and \( \frac{10}{12} \)
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 \( \frac{10}{12} \).

**Find the error or omission in each of these explanations**

A.
\(\frac{2}{3} \) is bigger than \(\frac{3}{4} \) because thirds are larger than fourths.
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.

B.
\(\frac{3}{5} \) is bigger than \(\frac{3}{8} \) because fifths are bigger than eighths.
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.

C.
\(\frac{7}{8} \) is bigger than \(\frac{10}{12} \) because \(\frac{7}{8} \) is fewer parts away from the whole.
\(\frac{7}{8} \) 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

D.
\(\frac{7}{8} \) is bigger than \(\frac{3}{4} \) because it has more parts.
OR
\(\frac{6}{8} \) is bigger than \(\frac{3}{4} \) because it has more parts.
OR
\(\frac{5}{8} \) is bigger than \(\frac{3}{4} \) because it has more parts.

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.

E.
\(\frac{7}{8} \) is smaller than \(\frac{3}{4} \) because it has more parts in the whole.
OR
\(\frac{6}{8} \) is smaller than \(\frac{3}{4} \) because it has more parts in the whole.
OR
\(\frac{5}{8} \) is smaller than \(\frac{3}{4} \) because it has more parts in the whole.

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.

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

G.
\(\frac{7}{8} \) is bigger than \(\frac{3}{4} \) because 7 is a bigger part of 8 than 3 is of 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.

H.
\(\frac{2}{5} \) is smaller than \(\frac{5}{8} \) because \(\frac{2}{5} \) is smaller than \(\frac{1}{2} \)
and \(\frac{5}{8} \) is larger than \(\frac{1}{2} \) .
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 \(\frac{1}{2} \)

I.
\(\frac{2}{8} \) is smaller than \(\frac{3}{8} \) because 2 is less than 3
This explanation is missing the detail that both of the fractions are eighths, so the pieces are the same size for both fractions.