Understanding Area Models
The area model is a pedagogical tool for teaching multiplication. The area model is a great visual way to break down a multiplication problem into smaller, more manageable parts, making it easier to see how multiplication works step by step.
For starters, the area model helps students see how numbers can be broken down into tens and ones (place value), making the multiplication easier by handling smaller numbers first.
The area model also helps students understand the distributive property. In this case, it shows that26 times 18 is the same as open parenthesis 20 plus 6 close parenthesis open parenthesis 10 plus 8, close parenthesis.
$$ 26 \times 18 \text{ is the same as } (20 + 6)(10 + 8) $$
This represents an application of the distributive property. Here's how:open parenthesis 20 plus 6 close parenthesis, open parenthesis 10 plus 8 close parenthesis, equals, open parenthesis 20 times 10 close parenthesis, plus, open parenthesis 20 times 8 close parenthesis, plus, open parenthesis 6 times 10 close parenthesis, plus, open parenthesis 6 times 8 close parenthesis.
$$ (20 + 6)(10 + 8) = (20 \times 10) + (20 \times 8) + (6 \times 10) + (6 \times 8) $$
This is exactly what the area model shows by breaking the problem into four partial products. By placing partial products in a grid, students can see how each part of the number contributes to the final product. This visual aid makes it easier to understand how multiplication works and why the partial products must be added together.
In conclusion, the area model is an excellent way to teach students how to handle larger multiplication problems. It makes the process clear, visual, and more manageable by breaking it down into smaller, simpler steps, while also reinforcing the distributive property.
Explanation of This Problem
Let's break down the problem 26 times 18 \( 26 \times 18 \) using the area model.
First, break the numbers into place values: The number 26 is broken into 20 plus 6 \( 20 + 6 \) (tens and ones). The number 18 is broken into 10 plus 8 \( 10 + 8 \) (tens and ones).
Then, set up the area model as shown in the problem, divided into four smaller sections. Label the top of the rectangle with 20 plus 6 \( 20 + 6 \) and the side with 10 plus 8. \( 10 + 8. \)
Now, fill in each section of the rectangle by multiplying the numbers that meet at each section:
Top left: 20 times 10 equals 200. \( 20 \times 10 = 200 \)
Top right: 6 times 10 equals 60. \( 6 \times 10 = 60 \)
Bottom left: 20 times 8 equals 160. \( 20 \times 8 = 160 \)
Bottom right: 6 times 8 equals 48 \( 6 \times 8 = 48 \)
To find the total product, add up all the partial products:200 plus 60 plus 160 plus 48 equals 468
$$ 200 + 60 + 160 + 48 = 468 $$
So, 26 times 18 equals 468. \( 26 \times 18 = 468 \)
