## 4.MD.A.2

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals....

“Proportional reasoning is one of the most fundamental topics in middle grades mathematics. Students’ ability to reason proportionally affects their understanding of fractions and measurement in elementary school, and it supports their understanding of functions and algebra in middle school and beyond. Given the importance of ratio and proportion, it is typical to see extensive class time devoted to the topic in upper elementary and middle school grades.” –NCTM, posted 3-11-2013

Alejandra tried to solve the pizza problem with 28 cubes. She was able to successfully divide it into two halves, but then was not able to cut the remaining 14 cubes into thirds for Cristal to get a 1/3 of the remaining pizza.

Jason helped her discover that the problem could easily be solved with 24 cubes. “First, just divide it in half,” Jason said. “That’s 12 and 12. Pablo eats half and then gives 1/3 of the remaining pizza to Cristal. That’s 1/3 of 12, so it’s 4. And 3 times 4 is 12, so it works.”

I let them present their solution to the class and then asked the children, “What other numbers work for this pizza problem? And what would be a reasonable size for the pizza that Pablo and Alexis bought?”

Galilea tried 36 cubes. She said that Pablo ate 18/36 of the pizza and gave 6/36 to Cristal, leaving 12/36 for Alexis. “The whole pizza has 36 pieces. That’s 36/36 or one whole,” she smiled.

Alejandra, Patricio,and Rigoberto also tried 6 pieces of pizza and 24 pieces to represent the whole pizzas. They decided that 6 pieces was too small to share with 3 people--especially since Pablo ate half!

Cesar made two pizzas. The first one had 6 pieces. “Pablo ate ½ and gave one piece to Cristal,” he said. “That leaves 2/6 for Alexis and
2/6 = 1/3.” I showed him that 1/3 of ½ = 1/6 and he added that to his paper.

Cesar’s second pizza had 24 pieces. After dividing the cubes in half and then cutting the remainder into 3 equal parts, he saw that Pablo must have given Cristal 1/6 or 4 pieces of pizza. At this point, several of the children realized that the total amount of pizza pieces had to be an even number and a multiple of 3.

As a result, Eligio tried 66 cubes. He said that Pablo ate half and wrote ½ = 33/66. He divided up the remaining pizza into 3 equal parts and wrote that Cristal ate 11/66 or 1/6 of the pizza. I was fascinated with the numbers Eligio chose, because it was easy to see the same pattern that kept appearing around the classroom for several others:  ½ + 1/6 + 1/3 = 1 entire pizza. In Eligio’s case, though, you didn’t even need to factor out the elevens to see the same pattern of 3/6 + 1/6 + 2/6 = 6/6 = 1.

We made a table to summarize our data.

Doing all of this math made Pablo really hungry!  It was almost lunchtime at school, so he decided that the pizza in his story problem must have had at least 12 pieces—maybe even 18, since he had to give ½ of it away.  Alexis said that 4 pieces of pizza were plenty for him, so he decided that 12 pieces was a reasonable size of pizza to buy. “Besides,” he laughed, “it won’t cost so much.”