Lesson 3
Reasoning to Find Area
Let’s decompose and rearrange shapes to find their areas.
Problem 1
Find the area of each shaded region. Show your reasoning.
![3 figures on a grid labeled A, B, C.](https://cms-im.s3.amazonaws.com/FpCukLdcaxLBKn6HZRsxPfvU?response-content-disposition=inline%3B%20filename%3D%226-6.1.A.PP.Image.14a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A.PP.Image.14a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T003803Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1f21152589239f87d63e61081c77c6f1282afe02d816812c1f079dc08b535259)
Problem 2
Find the area of each shaded region. Show or explain your reasoning.
![4 figures labeled A, B, C, D.](https://cms-im.s3.amazonaws.com/Bb6GhX6Ua9ZT8XAPsQL4JV31?response-content-disposition=inline%3B%20filename%3D%226-6.1.A.PP_Image_16.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A.PP_Image_16.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T003803Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=15c83dee644d41c9950915206f070bc60169b537a00438c9d078b344cd1574bf)
Problem 3
Two plots of land have very different shapes. Noah said that both plots of land have the same area.
![Two shapes labeled “plot A” and “plot B”.](https://cms-im.s3.amazonaws.com/sdzePpjwxeJzFKUdQ2MEpKKq?response-content-disposition=inline%3B%20filename%3D%226-6.1.A.PP.New.Image.12.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A.PP.New.Image.12.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T003803Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=42b56d8e42671c700dda1928dcfa12539903820c0aa9bdd1befd48ac94041226)
Do you agree with Noah? Explain your reasoning.
Problem 4
A homeowner is deciding on the size of tiles to use to fully tile a rectangular wall in her bathroom that is 80 inches by 40 inches. The tiles are squares and come in three side lengths: 8 inches, 4 inches, and 2 inches. State if you agree with each statement about the tiles. Explain your reasoning.
- Regardless of the size she chooses, she will need the same number of tiles.
- Regardless of the size she chooses, the area of the wall that is being tiled is the same.
- She will need two 2-inch tiles to cover the same area as one 4-inch tile.
- She will need four 4-inch tiles to cover the same area as one 8-inch tile.
- If she chooses the 8-inch tiles, she will need a quarter as many tiles as she would with 2-inch tiles.