## Solving for Area (Rectangles)

It’s a peanut butter and jelly sandwich.

It’s also the set up for a math problem I presented my fourth grader, Simon, with.

Story problem goes like this: Dad makes Si a PBJ using bread that measures 4 inches by 5 inches. But before Si can eat it, Mom cuts a perfect 1 inch by 2 inch rectangle out of the soft yummy middle for herself (*yes, sometimes I do mess with the boy-os like this*). What is the remaining surface area of that top slice of bread?

Remember solving for these “find the area of the shaded region” kind of problems?

**Recognize how to break a problem into manageable pieces.**Here, the thought would be: There is no specific formula to solve for area of a “square doughnut.” But I do know the formula for a rectangle… Oh! Can I solve this problem by manipulating the area of two rectangles?**Identify the applicable formula(s).**The area of a rectangle is length times width.- If necessary,
**solve for missing values given initial set of information.**Not needed in this example. **Perform the actual algorithms.**4 x 5 = 20. 1 x 2 = 2.- Know how to
**manipulate calculated numbers to arrive at final answer.**It is not enough to just crunch out “length times width.” In this problem, I need to subtract area of smaller rectangle from area of larger rectangle. **Clearly state the complete final answer**(including units of measurement). The answer is not just “18.” At the very least, I need to write “18 square inches.” Or better yet, I could write something like “the surface area of the remaining sandwich face is 18 square inches.

This is not easy stuff. Think of multiple part math problems like patting your head, rubbing your tummy, marching in place while singing a tune all at the same time. Each individual piece may not be that big of a deal, but pulling it altogether can be tricky.

Here’s another area problem. I worked on something similar with a bunch of fourth graders the other day (I volunteer at school once a week – math workshop with a 4th grade class is one of my stops).

Want to give it a try?

The kids at school had a terrible time with this problem…. which then really piqued my interest on how to help them (or help their parents help them) understand how to attack it.

Some comments about this problem & some pitfalls you might want to keep in mind:

- There is some missing information that will have to be solved for. I.e., I did not label every length in the original figure. Assume all corners are right angles.

- There is more than one way to solve this problem. In the above picture, I’ve drawn out the four ways the original shape could be broken up. While I think it most natural to sum the area of two small rectangles (Method #1 or Method #2), it seemed that many of the children gravitated towards finding the area of the large rectangle first and then subtracting out the area of the corner rectangle (Method #4).

- Kids seem quick to dismiss their own approach to a problem in favor of following the other children’s calculations. I encourage them to try their own way first and then compare final answers with their neighbors.

- I ask students if their answers agree with each other. If not, I ask them to figure out what they did differently from each other. If yes, I confirm if the answer is correct or not. Always possible that the children all made the same mistake.

- Solutions can go awry anywhere along the 6 steps specified above. Patience, organization and ability to compute those basic algorithms super important.

After working with the other fourth graders, I was curious to see how my own boy would approach this type of math problem. Hence my quizzing him.

We worked out the second problem independently of each other. Si & I each arrived at the answer, 203 square centimeters. Does your answer agree with ours?