For this project, we are creating a book of strings. The students can use their strings from this exponents unit, but they have to write a report or record a video explaining the way in which the expressions of each string are "tied" together. They especially have to explain the step involving the zero power and the meaning of the negative exponents and how they connect to the positive exponents.
My goal is that students will explain how these string patterns help them think of/about algorithms for simplifying expressions involving exponents. For example, they might explain that "2-4 can be thought of as 24 and then one can take the reciprocal. This means that 2-4 = 1/16."
I start this session off by reviewing some strings:
I review this string with the class and look at many different aspect of the patterns we observe. As a class, we review the different ways that each pattern leads to the meaning of 5^0 and 5^-1. This string was constructed to demonstrate a simple way to understand zero and negative exponents. A students solving it can start with what they know (the positive exponents) and work their way toward the unknown (zero and negative exponents)
3 x 3^3 =
3 x 3^2 =
3 x 3^1 =
3 x 3^0 =
3 x 3^-1 =
3 x 3^-2 =
3 x 3^-3 =
3 x 3^-4 =
3 ^-4 =
Again, students can start with what they know 3^4 and by solving the second expression, they might realize that somehow 3 x 3^3 = 3^4. This gives them an opportunity to explain why this might make sense. Even though we have't formally reviewed any laws of exponents, we would talk about the intuitive idea that 3^3 x 3 = 3^4. This string develops that idea and helps students understand positive and negative exponents. More interesting and complex strings will touch upon many ideas in exponents.
Students are encouraged to work alone on this project, but to make their string "publishable." Since we are assembling a book of their strings, their work needs to be authentic and eye catching. People should want to read their book.
At first, many students may not like the idea of producing publishable work, since they think that math is "only about the numbers," and "has nothing to do with how good it looks." However, I am teaching them about more than just math (which is elegant by definition) and I want them to function in the world in which we live. In this world, you need to have a great idea and know how present it well in order to to put it into action. Having the great idea is only a good start.
Many of my students will want to type their work. I use this as an opportunity for them to learn how to make equations look great in a word processing document.
Here are the Microsoft guildelines for entering an equation:
Here are the Apple Pages guidelines:
Here are the Google Doc guidelines (although they are very tough to use):
Students are encouraged to write and explain their thinking, but they can also use other formats. Anything that can be scanned or linked to in a digital pdf document works for this project.
This is the guide I would give to them: The String Project Guide.docx
This is the rubric I would give: Math String Rubric.doc
This project is started in class, but I have them finish it over the week at home. They meet me during lunch or after school if they need help. After the strings are reviewed and graded, I assemble all the strings and create our first volume of “Law and Order.”