You broke apart facts like $7 \times 6 = 5 \times 6 + 2 \times 6$, and you can find the area of a rectangle. Here the two ideas meet: cut a rectangle into two smaller rectangles.
Why it works
Cutting a rectangle does not change how much space it covers. The two pieces together cover exactly the whole. Both pieces are 6 tall, so their areas are $6 \times 5$ and $6 \times 3$. Adding them gives the whole area: $6 \times 8 = 6 \times 5 + 6 \times 3$. This is the same break-apart idea, drawn as area.
Key idea
$a \times (b + c) = a \times b + a \times c$. To find a big area, cut the rectangle into two easier rectangles and add their areas.
Worked example: Cut the long side
Find the area of a 6 by 8 rectangle by cutting 8 into $5 + 3$.
- Left part: $6 \times 5 = 30$.
- Right part: $6 \times 3 = 18$.
- Whole: $30 + 18 = 48$.
Answer: 48 square units
Worked example: Bigger numbers
Find $7 \times 13$ using area.
- Cut 13 into $10 + 3$.
- $7 \times 10 = 70$ and $7 \times 3 = 21$.
- $70 + 21 = 91$.
Answer: 91
Common mistake
Adding a side instead of an area. $6 \times 5 + 3$ leaves out most of the right part: that part is 6 rows of 3, not just 3. Each part is a rectangle, so find each area by multiplying.