Questions: 3-4 Multiplication of Whole Numbers Question 15, 3.4.B-21 Part 1 of 4 The accompanying model illustrates 11 * 12. a. Explain how the partial products are shown in the figure. b. Draw a similar model for 12 * 23. c. Draw a similar base-five model for the product 14(five) * 22(five). Explain how the model can be used to find the answer in base five, a. Choose the correct answer below and fill in the answer box to complete your choice. A. The partial products are 100, 10, 20, and 2, which gives the product . B. The partial products are 1000, 100, 200, and 20, which gives the product . C. The partial products are 100, 30, and 2, which gives the product .

3-4 Multiplication of Whole Numbers
Question 15, 3.4.B-21
Part 1 of 4
The accompanying model illustrates 11 * 12.
a. Explain how the partial products are shown in the figure.
b. Draw a similar model for 12 * 23.
c. Draw a similar base-five model for the product 14(five) * 22(five). Explain how the model can be used to find the answer in base five,
a. Choose the correct answer below and fill in the answer box to complete your choice.
A. The partial products are 100, 10, 20, and 2, which gives the product .
B. The partial products are 1000, 100, 200, and 20, which gives the product .
C. The partial products are 100, 30, and 2, which gives the product .

Transcript text: 3-4 Multiplication of Whole Numbers Question 15, 3.4.B-21 Part 1 of 4 The accompanying model illustrates $11 \cdot 12$. a. Explain how the partial products are shown in the figure. b. Draw a similar model for $12 \cdot 23$. c. Draw a similar base-five model for the product $14_{\text {five }} \cdot 22_{\text {five }}$. Explain how the model can be used to find the answer in base five, a. Choose the correct answer below and fill in the answer box to complete your choice. A. The partial products are $100,10,20$, and 2 , which gives the product $\square$ ]. B. The partial products are $1000,100,200$, and 20 , which gives the product $\square$ . C. The partial products are 100,30 , and 2 , which gives the product $\square$ .

Solution

Solution Steps

Solution Approach

a. To explain how the partial products are shown in the figure, we need to break down the multiplication of 11 and 12 into simpler components. This involves using the distributive property of multiplication over addition.

b. To draw a similar model for $12 \cdot 23$, we can use the area model or grid method, breaking down each number into tens and units, and then multiplying each part.

c. For the base-five model of $14_{\text{five}} \cdot 22_{\text{five}}$, we need to convert the numbers to base ten, perform the multiplication, and then convert the result back to base five. The model can be used to visualize the multiplication process in base five.

Step 1: Calculate Partial Products for $11 \cdot 12$

To find the partial products for $11 \cdot 12$, we break down the numbers as follows: \[ 11 = 10 + 1 \quad \text{and} \quad 12 = 10 + 2 \] Using the distributive property: \[ 11 \cdot 12 = (10 + 1) \cdot (10 + 2) = 10 \cdot 10 + 10 \cdot 2 + 1 \cdot 10 + 1 \cdot 2 \] The partial products are: \[ 100, 20, 10, \text{ and } 2 \] Summing these partial products: \[ 100 + 20 + 10 + 2 = 132 \]

Step 2: Calculate Partial Products for $12 \cdot 23$

To find the partial products for $12 \cdot 23$, we break down the numbers as follows: \[ 12 = 10 + 2 \quad \text{and} \quad 23 = 20 + 3 \] Using the distributive property: \[ 12 \cdot 23 = (10 + 2) \cdot (20 + 3) = 10 \cdot 20 + 10 \cdot 3 + 2 \cdot 20 + 2 \cdot 3 \] The partial products are: \[ 200, 30, 40, \text{ and } 6 \] Summing these partial products: \[ 200 + 30 + 40 + 6 = 276 \]

Step 3: Calculate the Product of $14_{\text{five}} \cdot 22_{\text{five}}$ in Base Five

First, convert the base-five numbers to base-ten: \[ 14_{\text{five}} = 1 \cdot 5^1 + 4 \cdot 5^0 = 5 + 4 = 9_{\text{ten}} \] \[ 22_{\text{five}} = 2 \cdot 5^1 + 2 \cdot 5^0 = 10 + 2 = 12_{\text{ten}} \] Next, multiply the base-ten equivalents: \[ 9_{\text{ten}} \cdot 12_{\text{ten}} = 108_{\text{ten}} \] Finally, convert the result back to base-five: \[ 108_{\text{ten}} = 4 \cdot 5^2 + 1 \cdot 5^1 + 3 \cdot 5^0 = 413_{\text{five}} \]