Not sure which one I saw first, but I got the idea for this lesson hook from at least two teachers: Jensilvermath, and Pam Wilson. Both are creative educators, and generous online colleagues, who share their ideas, resources, and materials.

One of Aesop’s Fables tells the story of a crow who comes across a half-full pitcher of water in the desert. He cannot reach the water until he figures out that by dropping pebbles into the vessel, the displacement causes the water level to rise until he can quench his thirst. Using this narrative as our lesson hook, students were given a cup full of marbles, and a graduated cylinder partially filled with water. They were asked to predict how many marbles they would need to reach 2000 mL, and then how many more until the water overflowed.

I have found that giving too much structure can take some of the life out of a task, but not enough structure, and students flounder. In this case, I asked them a direct question, but did not suggest any methods at first. As we were right in the midst of linear equations, my assumption was that they would jump right to dropping their marbles into the cylinder, creating a scatterplot, find an average rate of change and line of best fit. But students always surprise me. They asked for an extra graduated cylinder to do some experimentation, and pulled out the scale to start weighing marbles. They traced the cylinder base to see how many marbles fit in that circle. As we had more than one color, it was important to them to see if the lighter blue marbles were consistent with the dark blue – something I hadn’t even considered. One group even qualified their prediction with the caveat, “…if the ratio of light blue to dark blue marbles is consistent with our sample, then

this prediction should hold.” What a nice expression of understanding. Reminder to self: always give students as much freedom as possible. Let them run until they really need help.

The students who dropped their marbles into the cylinder one at a time collected data points as the water level rose. They created scatterplots of this data, and calculated an average rate of change. Next, they used this information to find an equation for a line of best fit, which helped them to make a confident prediction about how many marbles they would need to bring the water all the way to the top. We took out enough marbles to test their predictions, and added them to the cylinder until the water level reached 2000 mL and then until it overflowed. Cheers and groans for the accuracy of their predictions.

Creating ways for students to create mathematical models and make predictions is one of the most important opportunities that I can set up for them. These types of tasks help students to connect the math from their classroom to questions that they will come across in the real world. Even if they will not need to calculate the number of marbles to overflow a cylinder, they will almost certainly need to use similar problem-solving skills, and equally importantly, they will have to decide what math skills they need to apply to novel situations. Students react very strongly when they see the “answer” to this type of task – very different from how most students react when looking up the answer in the back of the math textbook. Even reluctant mathematicians couldn’t help but look closely as we counted the last few marbles out!

I used this video to introduce the *Crow and the Pitcher*. It’s short but gets the point across. I shot a video of our cylinder, and edited it into a 3-Act format while I had the supplies out. I think that if you can get your hands on some marbles and a vessel, you may as well do this hands-on, but in case you don’t have a bunch of marbles handy, or if video is your preferred medium, I’ve published the materials below for you to use. Did I give enough information in act 2 or did I forget something? Please do let me know if you use any of this, and how it goes …and don’t forget to check out the Action Version…