Question

1. solve 52 × 26 using 2 partial products and an area model.

1. solve the following using 2 partial products. visualize

a.
68
× 23
______
__ × __
______
__ × __
______

c.
16
× 25
______

Explanation:

3. Solve \( 52 \times 26 \) using partial products and an area model.

Step 1: Decompose the numbers

Decompose \( 52 \) into \( 50 + 2 \) and \( 26 \) into \( 20 + 6 \).

Step 2: Find partial products (area model)

• Multiply \( 50 \times 20 = 1000 \) (area of the large rectangle)
• Multiply \( 50 \times 6 = 300 \) (area of one smaller rectangle)
• Multiply \( 2 \times 20 = 40 \) (area of another smaller rectangle)
• Multiply \( 2 \times 6 = 12 \) (area of the smallest rectangle)

Step 3: Sum the partial products

Add all the partial products: \( 1000 + 300 + 40 + 12 = 1352 \)

Step 1: Decompose the numbers

Decompose \( 68 \) into \( 60 + 8 \) and \( 23 \) into \( 20 + 3 \). We will use two partial products (multiplying by the tens and then the ones, or vice versa). Let's use multiplying \( 68 \) by \( 20 \) and \( 68 \) by \( 3 \).

Step 2: Calculate first partial product

Multiply \( 68 \times 20 \). \( 68 \times 20 = 1360 \) (this can also be thought of as \( 60 \times 20 + 8 \times 20 = 1200 + 160 = 1360 \))

Step 3: Calculate second partial product

Multiply \( 68 \times 3 \). \( 68 \times 3 = 204 \) (this can also be thought of as \( 60 \times 3 + 8 \times 3 = 180 + 24 = 204 \))

Step 4: Sum the partial products

Add the two partial products: \( 1360 + 204 = 1564 \)
For the blanks (if we decompose \( 23 \) into \( 20 + 3 \) and multiply each by \( 68 \)):

• First partial product: \( 68 \times 20 \)
• Second partial product: \( 68 \times 3 \)

Or if we decompose \( 68 \) into \( 60 + 8 \) and \( 23 \) into \( 20 + 3 \), and take two partial products (e.g., \( 60 \times 23 \) and \( 8 \times 23 \)):

• First partial product: \( 60 \times 23 = 1380 \)
• Second partial product: \( 8 \times 23 = 184 \)

Then \( 1380 + 184 = 1564 \)

Step 1: Decompose the numbers

Decompose \( 16 \) into \( 10 + 6 \) and \( 25 \) is already a whole number, or decompose \( 25 \) into \( 20 + 5 \) and multiply by \( 16 \). Let's use decomposing \( 25 \) into \( 20 + 5 \).

Step 2: Find partial products

• Multiply \( 16 \times 20 = 320 \)
• Multiply \( 16 \times 5 = 80 \)

Step 3: Sum the partial products

Add the partial products: \( 320 + 80 = 400 \)

Answer:

\( 52 \times 26 = 1352 \)

4a. Solve \( 68 \times 23 \) using partial products.