The Playground Problem
My students love to work together to solve mathematical problems. Cesar and Jorge are trying to make sense of the problem and share their ideas with the others.
Cristal shows her friends how she made three-fourths and then found two-thirds of that."
Delighted, he said, "Yes!" At last the problem started making sense to him.
Jessi made 4 equal parts and colored 3 of these 4 parts to represent the playground area. "Two-thirds of this is the blacktop," he said. "That's half of the park area." He also colored 1/4 of the playground area brown to represent the sand area for the swings and a slide.
Next he had to figure out the second part of the problem.
Jorge and Alexis made thirds, but then were not sure how to find 3/4 of that.
I went over to see what he was talking about and he showed me that 3/6 and 2/4 were both equal to 1/2 so the blacktop areas of both parks were equal. I laughed and said, "You are geniuses!"
Eligio looked surprised as I took the picture. "We're geniuses?" he remarked. I reassured him that I was convinced of it.
Even though Patricio was not exact in forming his "equal" sixths, it still worked because the 1/2 benchmark on each paper was accurate.
Alejandra shows her 4 equal parts.
Rigoberto created 12 equal parts, but then he was not sure what to do next.
Leslie explains to her friends how it is possible to have 3/4 of the park be playground with 2/3 of that covered with blacktop.
Esmeralda, Kimberly, and Galilea had fun illustrating the play equipment on their playgrounds.
Esmeralda proves that 2/4 = 3/6, so her blacktop areas are equal.
Although still timid about speaking in front of the class, Galilea proudly explains her thinking to us, confident that her solutions are correct.
We spent 4 days doing this problem, but it is worth the effort if you can encourage the rich discussion with the children. This is giving them a good foundation for 5th grade when they will actually learn to solve problems like this using multiplication to prove it as well.
Cristal shows how she solved 3/4 of 2/3. She divided the 2/3 into 8 equal parts to find 3 of 4 equal parts of the 2/3. Others later showed her that these were actually twelfths of the whole playground--as you can see in Alejandra's illustration (6/12 = 1/2).
I'm really proud of all of their efforts as they begin to think like mathematicians and use the Math Practice Standards! Great job at making sense of the problem and persevering in solving it.