Question

1. when the area model is divided into thirds, the shaded part represents ____. when the area model is divided into sixths, the shaded part represents __. since the same part of the whole area is shaded in both models, __ and ____ are equivalent fractions.

Explanation:

To solve this, we assume the shaded part is 1 when the whole is considered (since the same part is shaded). Let's go step by step:

Step 1: When divided into thirds

If the area model is divided into thirds, and the same part (let's say 1 part) is shaded, the fraction is $\frac{1}{3}$. Wait, no—wait, maybe the shaded part is 2? Wait, no, let's think again. Wait, when the same part of the whole is shaded, let's assume that in the first model (thirds), the shaded part is 2 thirds? No, maybe the standard equivalent fractions: $\frac{2}{3}$ and $\frac{4}{6}$. Wait, let's correct.

Wait, let's suppose the shaded part is 2 when divided into thirds? No, let's do it properly. Let's say the whole is divided into 3 equal parts, and the shaded part is 2 (so fraction $\frac{2}{3}$). Then, when divided into sixths, the same area (so 2 parts of 3 is equal to how many parts of 6? Since 32=6, so 22=4. So:

Step 1: When divided into thirds

Suppose the shaded part is 2 out of 3. So the fraction is $\frac{2}{3}$.

Step 2: When divided into sixths

To find the equivalent fraction, we multiply numerator and denominator by 2 (since 3*2=6). So $\frac{2\times2}{3\times2} = \frac{4}{6}$.

Step 3: Equivalent fractions

So $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent fractions.

Wait, but maybe the problem is simpler. Let's re-express:

When the area model is divided into thirds, the shaded part represents $\frac{2}{3}$ (assuming 2 parts shaded). When divided into sixths, the same area (so 2 parts of 3 is 4 parts of 6), so the shaded part represents $\frac{4}{6}$. Then $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent.

But maybe the problem has the shaded part as 1? No, that wouldn't make sense. Wait, maybe the shaded part is 2 in thirds (so $\frac{2}{3}$) and 4 in sixths (so $\frac{4}{6}$), which are equivalent.

So:

When the area model is divided into thirds, the shaded part represents $\boldsymbol{\frac{2}{3}}$.

When the area model is divided into sixths, the shaded part represents $\boldsymbol{\frac{4}{6}}$.

And $\boldsymbol{\frac{2}{3}}$ and $\boldsymbol{\frac{4}{6}}$ are equivalent fractions.

Wait, but maybe the problem is with 1 part? No, that would be $\frac{1}{3}$ and $\frac{2}{6}$, which are also equivalent. But the key is that the same area is shaded, so the fractions must be equivalent.

So, for example:

• When divided into thirds: shaded part is $\frac{2}{3}$ (2 out of 3)
• When divided into sixths: shaded part is $\frac{4}{6}$ (4 out of 6)
• And $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent.

Alternatively, if the shaded part is 1 out of 3, then in sixths it's 2 out of 6, so $\frac{1}{3}$ and $\frac{2}{6}$.

But the most common equivalent fractions are $\frac{2}{3}$ and $\frac{4}{6}$.

So:

1. When divided into thirds: $\frac{2}{3}$
2. When divided into sixths: $\frac{4}{6}$
3. Equivalent fractions: $\frac{2}{3}$ and $\frac{4}{6}$

But let's check the problem again. The problem says "the same part of the whole area is shaded in both models". So if in the first model (thirds), the shaded part is 2 (so $\frac{2}{3}$), then in the second model (sixths), the shaded part is 4 (so $\frac{4}{6}$), and these are equivalent.

So the answers would be:

When the area model is divided into thirds, the shaded part represents $\frac{2}{3}$.

When the area model is divided into sixths, the shaded part represents $\frac{4}{6}$.

And $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent fractions.

Alternatively, if the shaded part is 1, then $\frac{1}{3}$ and $\frac{2}{6}$, but $\frac{2}{3}$ and $\frac{4}{6}$ is more typical.

So:

Step1: Determine fraction for thirds

Assume 2 parts shaded out of 3. Fraction: $\frac{2}{3}$.

Step2: Determine fraction for sixths

Since 3×2=6, multiply numerator/denominator by 2: $\frac{2×2}{3×2}=\frac{4}{6}$.

Step3: Identify equivalent fractions

$\frac{2}{3}$ and $\frac{4}{6}$ are equivalent.

Answer:

When the area model is divided into thirds, the shaded part represents $\boldsymbol{\frac{2}{3}}$.
When the area model is divided into sixths, the shaded part represents $\boldsymbol{\frac{4}{6}}$.
$\boldsymbol{\frac{2}{3}}$ and $\boldsymbol{\frac{4}{6}}$ are equivalent fractions.