Multiplying mixed numbers in 5th grade

Answer. 1

The answer is $1\frac{1}{2}$ because $3$ groups of $\frac{1}{2}$ cup make $\frac{3}{2}$ cup, which is the same as $1\frac{1}{2}$ cups.

1. 1There are $3$ equal glasses, and each holds $\frac{1}{2}$ cup. That means multiply $3$ by $\frac{1}{2}$.
2. 2Find the total number of cups.
3. 3Change the improper fraction to a mixed number. So Leo pours $1\frac{1}{2}$ cups in all. An equivalent answer such as $\frac{3}{2}$ is also correct.

Common mistake. A common mistake is adding $3$ and $\frac{1}{2}$ to get $3\frac{1}{2}$. The story describes $3$ equal groups of $\frac{1}{2}$ cup, so it calls for multiplication, not addition.

The idea underneath. A whole number times a fraction means repeated equal groups, so $3 \times \frac{1}{2}$ is the same as adding $\frac{1}{2}$ three times.

The story stating that Mia uses $\frac{3}{4}$ of a $24$-foot ribbon is correct because it matches both the shaded fraction, $\frac{3}{4}$, and the whole amount, $24$ feet.

1. 1The brace over the entire model is labeled 24, so the whole amount is 24.
2. 2Count the equal parts in the whole model. There are 4 equal parts.
3. 3Count the shaded parts. 3 of the 4 equal parts are shaded.
4. 4The matching story must say $\frac{3}{4}$ of $24$. The story stating that Mia uses $\frac{3}{4}$ of a $24$-foot ribbon matches both the fraction and the whole.

Common mistake. A common mistake is counting the 1 unshaded part instead of the 3 shaded parts, producing $\frac{1}{4}$ instead of $\frac{3}{4}$. The correct story must match both the shaded count and the brace label.

The idea underneath. A model-to-story match must preserve the structure of the model: the whole, the equal parts, and which parts are selected.

The best statement is 'The whole is divided into the wrong number of equal parts.' because the story needs 4 equal rows for $\frac{3}{4}$, but the model shows 6 equal rows.

1. 1First, decide whether the model family fits the story. A rectangle with a length and a width is best shown with an area model, so the model type does fit.
2. 2Next, check the denominators. Each denominator tells how many equal parts the whole should be divided into.
3. 3The model shows 5 equal columns, which is correct. However, it shows 6 equal rows instead of 4, so the row factor in the model is $\frac{3}{6}$ instead of $\frac{3}{4}$.
4. 4So the best statement is that the whole is divided into the wrong number of equal parts.

Common mistake. A common mistake is choosing 'The model matches the story' because an area model fits a rectangle story. You also have to check that each denominator gives the correct number of equal parts.

The idea underneath. Matching a fraction model to a story is a two-step check: choose the right model family, then use the denominators to verify the equal parts.

The correct statement says Leo correctly found $\frac{8}{9}$, but in Step 2 he changed $\frac{1}{3}$ to $\frac{4}{9}$ instead of $\frac{3}{9}$.

1. 1Leo needs two steps: first find $\frac{2}{3}$ of $1\frac{1}{3}$ miles, then add that result to $1\frac{1}{3}$ miles.
2. 2Step 1 is correct.
3. 3The mistake is in Step 2. When $\frac{1}{3}$ is renamed with denominator $9$, it should become $\frac{3}{9}$, not $\frac{4}{9}$.
4. 4Using the correct ninths gives the right total of $2\frac{2}{9}$ miles.
5. 5So the statement that says Leo correctly found $\frac{8}{9}$ but renamed $\frac{1}{3}$ as $\frac{4}{9}$ instead of $\frac{3}{9}$ is the best description of his work.

Common mistake. A common mistake is checking only the multiplication line. Here the first line is correct, but the addition step uses $\frac{4}{9}$ for $\frac{1}{3}$ instead of $\frac{3}{9}$.

The idea underneath. In a two-step fraction story, the operations can be right and the work can still be wrong if an equivalent fraction is rewritten incorrectly.