A Rich Problem For Third Graders

8 pieces

8 pieces

4 pieces

Last week I wrote a story problem on the board for morning work. I am always looking for ways to stretch my students' thinking, to develop their conceptual knowledge and to get them actively problem solving as much as possible.

The problem read like this: Henry had 4 pizzas. They were all the same size. He cut each one into eight pieces. He ate 20 pieces. How much of the four pizzas did he eat?

Students dutifully drew the pizzas out and divided them into eighths. They carefully shaded in 2 pizzas (16 pieces) and 4 pieces of the next pizza.

Here is where is got interesting. When asked how much of the four pizzas Henry ate, a student answered: 2 1/2. Another student answered 20/32.

Students began to look at me with the "O.K. so what is the right answer?" look. This was tricky, since the question wasn't how much pizza did Henry eat (2 1/2) but how much of the total pizza did he eat?

I wanted students to do the math thinking and to really use everything they knew about fractions to help them understand the different answers and the difference between the two questions.

We made some assumptions "Yes, there are 2 1/2 pizzas shaded in. Yes, Henry started with 4 pizzas, or 32 pieces. Yes, the question is asking how much of the all four pizzas did he eat."

I began asking questions. When we want to show the fraction of something shaded in, what does the denominator represent?  Lets do that. I also stopped at this point and drew 4 rectangles side by side, divided into eighths. I shaded in 20 parts. We formulated a fraction for this picture. We went back to the pizza problem.
Suddenly a light bulb went on: A student raised her hand. "OHHHHHHH, when we said 2 1/2, we were only looking at the pizzas shaded in. We didn't count the unshaded pizza.

We went on to have a great discussion about what the question was really asking. These opportunities for rich discussions don't come up every day, but I love it when they happen!