During a recent Wednesday, the Herons explored the idea of a "Deca Tree," a tree that has ten trunks, each with ten branches, each with ten twigs, each with ten leaves. After we thought about what that would mean a little together, I asked the students to figure out how many leaves the tree would have if a woodcutter cut off one trunk, then one branch, then one twig, then one leaf.
Students could work individually or in pairs. Many took advantage of our plaza and some sidewalk chalk to show their work.
This type of problem is called a "Low threshold, high ceiling" problem because it can be approached in many different ways and all students can successfully work on it. There are profound concepts of number that can be explored by students who are ready for that sophistication...but there is just as valuable work to be done by students who are exploring trying to conceptualize (or even draw) such big numbers.
Many students began by a quick sketch to get the idea of the tree. One student said confidently, "It's ten to the fourth power." Then he paused and asked, much more hesitantly, "Right?" Another student ran over and agreed. "I did ten times ten times ten times ten so that's ten to the fourth," she agreed. These kinds of problems encourage conversation and collaboration.
In fact, quite a debate broke out as answers began to trickle in. The ten to the fourth group decided to multiply 9x9x9x9 to get their final answer because "you take one off each time." The student who had originally multiplied ten four times to get the total number of leaves had gone about it differently, using subtraction. They got different answers. Who was right?
I never answer that question - instead, I ask them to convince the other group of the accuracy of their answer. Often it doesn't take long for one group to exclaim "OH! I see now." And indeed, that's just what happened. When students are asked to defend their math thinking to each other, they see the value of carefully writing out what they've done and being able to use a common math language to communicate. The work they do in problem of the week helps them to get ready to communicate mathematically in situations like this. They also realize that getting the right answer is only a tiny portion of the work we do as mathematicians.
In this case, students were really interested in trying to figure out why 9 to the fourth power didn't work. Math is a creative endeavor and can be full of wonder...you just have to find the right problems.