Scaffolded Math and Science: Dividing Fractions by Fractions using ... - Free Printable

It looks like you've uploaded an image of a "Dividing Fractions Using Models" Task Card Activity, which is a teaching tool designed to help students understand how to divide fractions using visual models. The image shows two example task cards:

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🔹 Task Card #1:

Problem:
$$
\frac{1}{2} \div \frac{1}{3} = ?
$$

Visual Model:
- A rectangle is divided into 2 equal parts (representing $\frac{1}{2}$), with one part shaded red.
- Another rectangle is divided into 3 equal parts (representing $\frac{1}{3}$), with one part shaded blue.

The goal is to determine how many thirds fit into one-half.

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🔹 Task Card #2:

Problem:
$$
\frac{2}{5} \div \frac{2}{3} = ?
$$

This card includes a blank grid model where students are expected to draw or shade the fractions and then divide them visually.

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Let’s solve Task Card #1 step by step using both the model and math.

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## ✔ Step-by-Step Solution for Task Card #1:
$$
\frac{1}{2} \div \frac{1}{3}
$$

🧠 What does this mean?

We want to know: How many $\frac{1}{3}$s are in $\frac{1}{2}$?

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🔍 Visual Model Explanation:

1. Draw a whole rectangle and divide it into 2 equal parts → each is $\frac{1}{2}$.
2. Shade one of those halves (red).
3. Now, think about $\frac{1}{3}$: divide the same whole into 3 equal parts.
4. How much of the shaded half ($\frac{1}{2}$) can be covered by $\frac{1}{3}$?

But to make it easier, let's use a common denominator approach.

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✔ Method: Use Common Denominator

Convert both fractions to have the same denominator.

- $\frac{1}{2} = \frac{3}{6}$
- $\frac{1}{3} = \frac{2}{6}$

Now, the problem becomes:
$$
\frac{3}{6} \div \frac{2}{6}
$$

Since the denominators are the same, we can divide the numerators:
$$
\frac{3}{6} \div \frac{2}{6} = \frac{3}{2} = 1\frac{1}{2}
$$

✔ So, $\frac{1}{2} \div \frac{1}{3} = \frac{3}{2}$ or $1\frac{1}{2}$

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📊 Visual Interpretation:

Imagine a bar representing 1 whole.

- Divide it into 6 equal parts (LCM of 2 and 3).
- $\frac{1}{2} = 3$ out of 6 parts.
- $\frac{1}{3} = 2$ out of 6 parts.

Now ask: How many groups of 2 sixths fit into 3 sixths?

→ One full group (2/6), and a half group left over → total: $1\frac{1}{2}$

So, $\frac{1}{2} \div \frac{1}{3} = 1\frac{1}{2}$

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🔄 General Rule Reminder:

To divide fractions:
$$
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
$$

Apply it here:
$$
\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} = 1\frac{1}{2}
$$

✔ Matches our model!

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## 🧩 Task Card #2: $\frac{2}{5} \div \frac{2}{3}$

Let’s solve this too.

🔍 Step-by-step:

$$
\frac{2}{5} \div \frac{2}{3}
$$

Use the rule:
$$
= \frac{2}{5} \times \frac{3}{2} = \frac{6}{10} = \frac{3}{5}
$$

✔ Answer: $\frac{3}{5}$

🧱 Visual Model (for students):

- Draw a rectangle and divide it into 5 parts; shade 2 → $\frac{2}{5}$
- Now, divide the same whole into 3 parts → $\frac{2}{3}$ is 2 of those parts
- But since we’re dividing, we need to see how many times $\frac{2}{3}$ fits into $\frac{2}{5}$

Wait — actually, $\frac{2}{3}$ is larger than $\frac{2}{5}$, so it doesn’t fit fully. That’s why the answer is less than 1.

So, $\frac{2}{5} \div \frac{2}{3} = \frac{3}{5}$ means: $\frac{2}{5}$ is $\frac{3}{5}$ of $\frac{2}{3}$.

You can show this by:
- Drawing a bar for $\frac{2}{5}$ and another for $\frac{2}{3}$
- Seeing what fraction of the second fits into the first

But the key takeaway is that visual models help students see division as "how many times does one fraction fit into another?"

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## ✔ Summary:

| Problem | Answer | Explanation |
|--------|--------|-------------|
| $\frac{1}{2} \div \frac{1}{3}$ | $1\frac{1}{2}$ | One-half contains one and a half thirds |
| $\frac{2}{5} \div \frac{2}{3}$ | $\frac{3}{5}$ | Two-fifths is three-fifths of two-thirds |

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## 💡 Teaching Tip:
Encourage students to:
1. Use area models or bar models.
2. Find a common denominator to compare sizes.
3. Use the "invert and multiply" rule after understanding with models.

This activity helps build conceptual understanding before moving to abstract rules.

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Let me know if you'd like printable versions of these task cards or more examples!