Unit fractions in 3rd grade

The fraction $\frac{1}{2}$ shows $1$ of $2$ equal parts. The correct model is the rectangle divided into $2$ equal vertical strips with $1$ strip shaded.

1. 1Look at the fraction $\frac{1}{2}$. The bottom number is the denominator — it tells you the whole must be split into $2$ equal parts.
2. 2The top number is the numerator. It is $1$, so exactly $1$ of those $2$ equal parts should be shaded.
3. 3Find the model with $2$ equal parts AND exactly $1$ part shaded. The correct model is the rectangle divided into $2$ equal vertical strips with $1$ strip shaded.

Common mistake. A common mistake is choosing a model that has $2$ parts when the parts are not all the same size. A unit fraction needs the whole to be split into equal parts.

The idea underneath. A unit fraction $\frac{1}{2}$ has three things to check: the whole is split into $2$ parts, all $2$ parts are the same size, and exactly $1$ part is shaded.

Answer. 2

Shade exactly $2$ of the $6$ equal parts to show $\frac{2}{6}$ of the circle.

1. 1Look at the fraction shown above. The denominator is $6$, so the circle is split into $6$ equal parts. The numerator is $2$, so you need to shade $2$ of those parts.
2. 2Click $2$ sectors of the circle to shade them. The order in which you click does not matter — what matters is that exactly $2$ of the $6$ parts end up shaded.
3. 3When $2$ of the $6$ equal parts are shaded, the picture shows $\frac{2}{6}$ of the whole circle.

Common mistake. A common mistake is shading $6$ parts (mixing up the numerator and denominator) or shading $4$ parts (the parts that would be left unshaded). The numerator $2$ tells you how many parts to shade; the denominator $6$ tells you how many equal parts make up the whole circle.

The idea underneath. $\frac{2}{6}$ means $2$ copies of the unit fraction $\frac{1}{6}$. Each sector you shade is one $\frac{1}{6}$, so shading $2$ sectors gives you $\frac{2}{6}$ of the whole circle.

Sara identified the numerator and denominator correctly.

1. 1The circle has 8 equal parts, and 2 of them are shaded. The fraction $\frac{2}{8}$ shows the shaded part.
2. 2Sara said the 2 tells how many pieces are shaded. In the circle, 2 pieces are shaded. That part is right.
3. 3Sara said the 8 tells how many equal parts make the whole. The circle has 8 equal parts. Each part is $\frac{1}{8}$ of the whole. That part is right too.
4. 4So Sara was right about both numbers.
5. 5The 2 is the numerator and the 8 is the denominator, so Sara was right about the numerator and the denominator. The correct answer is "Sara identified the numerator and denominator correctly."

Common mistake. Check each part of the sentence against the circle before you look for a mistake. Here both parts are right.

The idea underneath. In this picture, the bottom number counts all the equal parts in the whole. It also tells us each part is $\frac{1}{8}$ of the whole. The top number counts the parts included in the fraction; here those are the shaded parts.

$\frac{5}{6}$ is $5$ copies of the unit fraction $\frac{1}{6}$, so $\frac{5}{6} = \frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$.

1. 1Look at the fraction $\frac{5}{6}$ below. The denominator $6$ means the whole is split into $6$ equal parts. Each part is the unit fraction $\frac{1}{6}$. The numerator $5$ tells us we have $5$ of those parts.
2. 2Add $\frac{1}{6}$ $5$ times — once for each part:
3. 3Pick the choice that shows $\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$.

Common mistake. A common mistake is swapping the roles of the numerator and denominator — for example, writing $6$ copies of $\frac{1}{5}$ instead of $5$ copies of $\frac{1}{6}$. The numerator (top) counts the copies; the denominator (bottom) names each piece's size.

The idea underneath. Any fraction is built from unit fractions. The fraction $\frac{5}{6}$ is $5$ copies of $\frac{1}{6}$ — the bottom number names the unit, the top number counts the copies.