Explain how modeling partial products can be used to find the products of greater numbers
Modeling partial products is a method used to find the product of two or more larger numbers by breaking them down into smaller, more manageable parts. It involves multiplying individual digits and summing up the products.
Here's a step-by-step explanation of how modeling partial products can be used to find the products of greater numbers:
Step 1: Break down the numbers into their place values.
For example, consider multiplying 123 by 456. Break down 123 into 100, 20, and 3, and break down 456 into 400, 50, and 6.
Step 2: Multiply each digit of the first number with each digit of the second number.
Start with the rightmost digit of the second number (6 in this case) and multiply it by each digit of the first number (3, 2, and 1). Write down the partial products under each digit:
6 x 3 = 18 (write down 18 under the first digit)
6 x 2 = 12 (write down 12 under the second digit)
6 x 1 = 6 (write down 6 under the third digit)
Step 3: Shift the position of the partial products.
Move the partial products one position to the left for each new digit of the second number. In this case, shift the partial products once to the left for the tens position and twice to the left for the hundreds position.
Step 4: Sum up the shifted partial products.
Add up all the shifted partial products to get the final product. In this case, add up 18000, 12000, and 600, which gives the final product of 56088.
Using the modeling partial products method breaks down the multiplication problem into smaller, more manageable steps. By breaking the numbers into their place values and multiplying each digit separately, you can then sum up the partial products to get the final product of larger numbers.
Modeling partial products is a useful method for finding the product of greater numbers because it breaks down the multiplication process into smaller, more manageable steps. This method involves breaking down each number into its place value components and then multiplying these components before adding them together to find the final product.
To model partial products, follow these steps:
1. Decompose the numbers: Write each number being multiplied vertically, with each digit in its proper place value position. For example, if you are multiplying 1234 by 5678, you would write it as:
1234
× 5678
2. Multiply the ones digits: Start with the rightmost digit on the second number (in this case, 8), and multiply it by each digit in the first number, one at a time. Write the product under the line, aligned with the ones place. In this example, it would look like:
1234
× 5678
______
9872
3. Multiply the tens digits: Move to the next digit on the second number (in this case, 7), and repeat the process. Multiply it by each digit in the first number, keeping in mind the place value. Write the products under the line, aligned with the tens place. In this example, it would look like:
1234
× 5678
______
8648
7406
4. Continue with the rest of the digits: Repeat the process for each subsequent digit in the second number, multiplying it by each digit in the first number and writing the products in their proper place value positions.
5. Add the partial products: Once you have multiplied all the digits, add up the partial products to find the final product. In this example, it would look like:
1234
× 5678
______
7006652
So, by modeling partial products, we have found that the product of 1234 and 5678 is 7006652. This method allows us to break down the multiplication into smaller steps, making it easier to compute the product of larger numbers.