#SOL21 – Day 23 – Math Problem

Stephan has a piece of wood that measures 18 inches long and 12 inches wide. He wants to make a tray with sides 2 inches high. Maurice says that he can make a tray that has a base measuring 10-1/2 inches by 10-1/2 inches. Is he correct? Explain.

The last problem on today’s 6th grade math test was a doozie! At least half of the kids called me over just to make sense of it. “Is the tray supposed to have two sides or four?” “How thick is the wood?” “Do the sides go on top of the base or on the edges of it?” One kid stared at the paper and finally wrote, “I have no clue!”

According to the answer key, students were supposed to compare the area of the board (18 x 12″ = 216 in²) to the area of the base (10.5″ x 10.5″ = 110.25 in²) plus the area of the four sides (4″ x 2″ x 10.5″ = 84 in²). Since 216 in² > 194.25 in², Maurice is correct.

Well, I’ve followed enough sewing patterns in my life to know that you can’t just calculate the surface area and call it good, so I made a sketch of the piece of wood. I found that after you cut out the 10.5″ x 10.5″ square, you’re left with an L-shaped piece. From that, you can cut three 2-inch pieces, but the scraps that are left are 1.5″ wide. That means that there’s not a section wide enough to actually cut out the last 2-inch side!

I decided to take this home to my family and see how each of them would solve it. It was hilarious to compare the different approaches! The 7th grader had an answer within seconds. She didn’t even need a piece of paper. “No, he can’t do it,” she said.

“I’m going to need more proof than that.”

“Fine. The base is 10-1/2 inches, so you can’t cut any 2″ pieces off the 12″ side, but there’s 7-1/2″ on the other side, so you can only cut three pieces from there.” Boom.

The 11th grader who’s on the fabrication subgroup for the high school robotics teams went off to work in the other room with a piece of paper. We kept overhearing mutterings of, “This is hard!” and “How thick is the wood?” and “Is this supposed to be shaped like a plus?”

In the meantime, the 9th grader disagreed with his younger sister. “Yah, he can do it. Look, you cut the base, and then you cut three sides. You make the last side by cutting out these two pieces and sticking them together.” With that, both he and his sister left to go hang out in their bedrooms.

Finally, the 11th grader came back down with the answer that it couldn’t be done. When I asked how he figured it out he said, “I made a CAD drawing. There isn’t a wide enough strip left to cut out the final side.”

“Well, what if you joined two smaller pieces together to make the last side?” I asked.

He wrinkled up his face like I was asking him to put pickle juice on his ice cream. “I guess you could…if you want your tray to fall apart.”

My husband, the same one who created the cake pop stand out of Lego a couple of days ago, then showed his drawing. “You can actually cut out these two staircase-like shapes. When you shift them and join them, you’ll have your 10-1/2 inch square base and still have the right-sized scraps to cut out the four 2″ sides.”

My son and I exchanged a look silently acknowledging that that was a clever solution. “By the way, this was on the 6th grade test my students took today,” I said.

“What?!?” replied the 11th grader.

I can’t wait to discuss this problem in class tomorrow!