Dina calculated 49*51+49*49=4900 98*37+2*37=3700 99*37+37=3700 Explain how she performed the calculations and why her method works. Then give three similar calculations based on the same approach and find the answers mentally.

Dina is using clever simplifications to make her calculations easier. Let’s break down how she performed each one.

### 1. \(49 \times 51 + 49 \times 49 = 4900\)

This expression can be simplified using the **distributive property**:
\[
49 \times 51 + 49 \times 49 = 49 \times (51 + 49) = 49 \times 100 = 4900
\]
Dina grouped the two terms with the common factor \(49\), simplifying it to \(49 \times 100 = 4900\).

### 2. \(98 \times 37 + 2 \times 37 = 3700\)

Here, Dina is using the fact that \(98 + 2 = 100\), which makes the multiplication easier:
\[
98 \times 37 + 2 \times 37 = (98 + 2) \times 37 = 100 \times 37 = 3700
\]
She grouped the two terms into a single multiplication of \(100 \times 37 = 3700\).

### 3. \(99 \times 37 + 37 = 3700\)

This is similar to the second example:
\[
99 \times 37 + 37 = (99 + 1) \times 37 = 100 \times 37 = 3700
\]
Again, Dina is using a combination to make the multiplication by 100.

### Why Her Method Works

Dina is using the **distributive property of multiplication** to combine terms that allow for easier arithmetic. By factoring common terms or strategically adding to form multiples of 100, the calculations become simple mental math, particularly when multiplying by 100.

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### Similar Calculations Based on This Approach

1. \( 48 \times 52 + 48 \times 48 \)
   \[
   = 48 \times (52 + 48) = 48 \times 100 = 4800
   \]

2. \( 97 \times 41 + 3 \times 41 \)
   \[
   = (97 + 3) \times 41 = 100 \times 41 = 4100
   \]

3. \( 98 \times 25 + 2 \times 25 \)
   \[
   = (98 + 2) \times 25 = 100 \times 25 = 2500
   \]

Each of these calculations follows Dina's method of simplifying the expressions by factoring or combining terms to create round numbers (such as multiples of 100), which are easier to multiply mentally.

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Would you like more examples or further clarification?

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Here are 5 relative questions:
1. Can Dina's method be applied to all numbers?
2. How can the distributive property simplify other arithmetic problems?
3. What mental math strategies could complement this method?
4. Can you prove that Dina's method works for other two-term factorizations?
5. How could you apply similar simplifications to division problems?

**Tip:** The distributive property is a powerful tool for simplifying arithmetic, especially when dealing with round numbers or common factors.

Dina calculated 49*51+49*49=4900 98*37+2*37=3700 99*37+37=3700. Explain how she performed the calculations and why her method works. Then give three similar calculations based on the same approach and find the answers mentally.

Explore how Dina simplifies complex multiplication problems using the distributive property of multiplication. Learn the steps behind her mental math approach with examples like 49 * 51 + 49 * 49 = 4900 and similar calculations. Suitable for Grades 6-8 students looking to master multiplication techniques.

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