Number and Operations—Fractions: Comparing Fractions with Different Numerators and Denominators (CCSS.4.NF.2)
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Common Core 4th Grade Math › Number and Operations—Fractions: Comparing Fractions with Different Numerators and Denominators (CCSS.4.NF.2)
Lena drank $\frac{2}{3}$ of a small bottle of juice. Max drank $\frac{3}{4}$ of a large bottle of juice. Without knowing if the bottles are the same size, which symbol compares $\frac{2}{3}$ and $\frac{3}{4}$ correctly for the amounts they drank?
$\frac{2}{3} > \frac{3}{4}$
$\frac{2}{3} < \frac{3}{4}$
$\frac{2}{3} = \frac{3}{4}$
Cannot be compared because they refer to different wholes
Explanation
Comparisons are valid only when fractions refer to the same whole. Here the bottles are different sizes (different wholes), so even though both fractions are greater than $\tfrac{1}{2}$, we cannot decide who drank more without equal wholes. A model with two bars of different lengths (one showing $\tfrac{2}{3}$ and one showing $\tfrac{3}{4}$) illustrates that the shaded amounts are not directly comparable.
Which symbol compares $\frac{3}{8}$ and $\frac{2}{5}$ correctly?
$\frac{3}{8} > \frac{2}{5}$
$\frac{3}{8} = \frac{2}{5}$
$\frac{3}{8} < \frac{2}{5}$
Cannot be compared because they have different denominators
Explanation
Benchmark first: both are less than $\tfrac{1}{2}$ because $\tfrac{1}{2}=\tfrac{4}{8}$ and $\tfrac{1}{2}=\tfrac{5}{10}$ while $\tfrac{2}{5}=\tfrac{4}{10}$. The gaps from $\tfrac{1}{2}$ are $\tfrac{1}{8}$ for $\tfrac{3}{8}$ and $\tfrac{1}{10}$ for $\tfrac{2}{5}$. Since $\tfrac{1}{10}<\tfrac{1}{8}$, $\tfrac{2}{5}$ is closer to $\tfrac{1}{2}$ and is larger, so $\tfrac{3}{8}<\tfrac{2}{5}$. Model check: on fraction bars (same-size wholes), the $\tfrac{2}{5}$ bar shades slightly more than the $\tfrac{3}{8}$ bar. Confirm: with a common denominator, $\tfrac{3}{8}=\tfrac{15}{40}$ and $\tfrac{2}{5}=\tfrac{16}{40}$, so $15<16$.
Which symbol compares $\frac{4}{9}$ and $\frac{5}{12}$ correctly?
$\frac{4}{9} < \frac{5}{12}$
$\frac{4}{9} > \frac{5}{12}$
$\frac{4}{9} = \frac{5}{12}$
Cannot be compared because they have different denominators
Explanation
Benchmark first: both are less than $\tfrac{1}{2}$. Compare closeness to $\tfrac{1}{2}$ using thirty-sixths: $\tfrac{1}{2}=\tfrac{18}{36}$, $\tfrac{4}{9}=\tfrac{16}{36}$ (2 thirty-sixths below), and $\tfrac{5}{12}=\tfrac{15}{36}$ (3 thirty-sixths below). $\tfrac{4}{9}$ is closer to $\tfrac{1}{2}$, so it is larger than $\tfrac{5}{12}$. Model check: area models with the same-size wholes show slightly more shading for $\tfrac{4}{9}$. Confirm: with a common denominator $36$, $16>15$ so $\tfrac{4}{9}>\tfrac{5}{12}$.
Which symbol compares $\frac{5}{6}$ and $\frac{7}{12}$ correctly?
$\frac{5}{6} > \frac{7}{12}$
$\frac{5}{6} < \frac{7}{12}$
$\frac{5}{6} = \frac{7}{12}$
Cannot be compared because they have different denominators
Explanation
Benchmark first: both are less than $1$, but $\tfrac{5}{6}$ is $\tfrac{1}{6}$ below $1$ while $\tfrac{7}{12}$ is $\tfrac{5}{12}$ below $1$. Since $\tfrac{1}{6}=\tfrac{2}{12}<\tfrac{5}{12}$, $\tfrac{5}{6}$ is closer to $1$ and is larger. Model check: on a number line from $0$ to $1$, the point for $\tfrac{5}{6}$ is to the right of $\tfrac{7}{12}$. Confirm: $\tfrac{5}{6}=\tfrac{10}{12}$ and $\tfrac{7}{12}$ stays $\tfrac{7}{12}$, and $10>7$.