Number and Operations—Fractions: Adding and Subtracting Mixed Numbers with Like Denominators (CCSS.4.NF.4)

Think of four groups of one-third: $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=\frac{4}{3}$. A fraction bar picture would show 4 equal pieces the size of $\frac{1}{3}$. $\frac{1}{12}$ multiplies only the denominator, $\frac{3}{4}$ flips the fraction, and $\frac{2}{3}$ shows only two groups instead of four.

Six groups of one-fifth make six fifths: $\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{6}{5}$. A fraction bar model would show 6 equal $\frac{1}{5}$ pieces. $\frac{1}{30}$ multiplies only the denominator, $\frac{5}{6}$ flips the numbers, and $\frac{2}{5}$ counts only two groups.

Three groups of two-fifths is $\frac{2}{5}+\frac{2}{5}+\frac{2}{5}=\frac{6}{5}$. This is also $3\times\frac{2}{5}=\frac{3\times 2}{5}=\frac{6}{5}$. A fraction bar model shows 6 pieces the size of $\frac{1}{5}$. $\frac{6}{15}$ multiplies numerator and denominator, $\frac{2}{15}$ multiplies only the denominator, and $\frac{3}{5}$ forgets to make 3 groups of $\frac{2}{5}$.

Compute $5\times\frac{3}{8}=\frac{15}{8}=1\frac{7}{8}$. Think of a fraction bar: five groups of $\frac{3}{8}$ make 15 pieces the size of $\frac{1}{8}$. The answer lies between 1 and 2 whole pounds. $\frac{15}{40}$ multiplies both numerator and denominator, $\frac{3}{40}$ multiplies only the denominator, and $\frac{5}{8}$ counts just one group.