To get this class rolling, I give students a few Startup Strings and ask them to analyze each set of expressions. I give them the task, "After you compete each string, find a way of describing the pattern you see."
String 1 is meant to illustrate how we add exponents when multiplying to equal bases. It starts off with combinations they will recognize and then transitions into negative combinations.
String 2 is mean to help students identify how they can deal with a power of a power by pairing equal expressions and then contrasting them with an unequal expression. This will help students recognize that expressions like (3^3)^4 = 3^12, not 3^7
String 3 is presented as an opposite from string 1. The emphasis can be put on the idea that multiplication and division are opposite operations and thus imply opposite algorithms. When we multiply equal bases we add exponents, when we divide we subtract
String 4 is meant to help students process the idea that (ab)^x = a^x * b^x and that (a+b)^x does not necessarily equal a^x + b^x
As I stated in the Start Up, String 1 is meant to illustrate how we add exponents when multiplying to equal bases. It starts off with combinations they will recognize and then transitions into negative combinations. I have a student share their results and explain. I ask direct questions, like, "What pattern do you see?" And I ask lots of what if questions like, "What if the bases are different? What if the base is negative? Will this law still apply?"
2^2 x 2^0 = 2^2
2^2 x 2^1 =
2^2 x 2^3 =
2^3 x 2^4 =
2^4 x 2^5 =
2^5 x 2^4 =
2^4 x 2^3 =
2^3 x 2^2 =
2^2 x 2^1
2^1 x 2^0 =
2^1 x 2^-1 =
2^-1 x 2^-2 =
After discussing this string as a class, I give the class a few problems to work on:
The amount of problems I ask students to complete depends on the time available, but I like to get a few questions out and see how students are doing. This work time is meant for individual work, but I am flexible if students need more support.
String 2 helps students identify how they can deal with a power of a power by pairing equal expressions and then contrasting them with an unequal expression. Again we review the string and I ask them what they notice about each expression. I check for reasoning by asking, "will this only work with bases of 2 and 3 and with exponents of 2 and 3?"
2^2 x 2^2 =
2^2 x 2^2 x 2^2 =
2^2 x 2^3 =
3^2 x 3^2 x 3^2 =
I like to introduce algebraic reasoning in this discussion by asking "How can we show that this will always work?" I want to help my students think algebraically, in terms of variables. It helps to also discuss why this law builds on the previous rule. For example, (3^2) ^3 = 3^2 x 3^2 x 3^2, so this law is simply an extension of our previous law.
Then, I ask students to complete a set of practice for this string: (ab)c.docx
An important pair of examples to discuss with students is that (x+y)^2 does not equal (xy)^2. I like to ask that one at the end as a challenge. However, most students struggle to simplify these expressions correctly. The need to deal with them directly can not be forgotten.
I consider String 3 to be in opposition to String 1. In this discussion I like to place emphasis on the idea that multiplication and division are inverse operations and thus imply opposite algorithms. When we multiply equal bases we add exponents, when we divide, therefore, we subtract.
4^3 x 4^3 =
2^3 x 2^2 =
2^3 / 2^2 =
My goal for this string is to really tap into student understanding of adding exponents as the appropriate strategy when equal bases are being multiplied. I support this way of thinking about the string by showing my students why this makes sense. I show how the pairs of factors form terms and can be cancelled as "1's" as you divide them. It is critical to be careful, to make this process meaningful, to discuss it in relation to multiplication.
Like String 2's reasoning built on String 1's conclusions, this law also builds on the others. My experience is that if I do not help students to make the connections, they consider each law as a separate rule to memorize (instead of developing the mindset that they are all different perspectives of the way we work with exponents). Again, I provide a set of practice problems: ab divided by ac.
String 4 helps students process the idea that (ab)^x = a^x * b^x and that (a+b)^x does not necessarily equal a^x + b^x. This law is fundamental to the way we deal with radicals in high school and is something I spend a lot of time revisiting.
(2 x 3)^2 =
2^2 x 3^2 =
(3 x 4)^2 =
3^2 x 4^2 =
(3 x 5)^2 =
3^2 x 5^2 =
(1 + 2)^2 =
1^2 + 2^2 =
(4 + 5)^2 =
4^2 + 5^2 =
Students frequently rush and "distribute" the exponent over addition and subtraction. They apply a misinterpreted perception of the distributive property. Over the years this seems to compound issues they have with the laws of exponents. As students work on (ab)^x.docx, most of my questions here get them on track to understanding the law and applying it situations where we also have (a/b)^x. I stress the importance of the law and explain the frequent misinterpretation in the summary.
The summary conversation is about the major misconception I try and identify for them during this lesson:
(a + b)^x does not always equal (ab)^x
I have found that a fun way to approach this is statement is to ask, "When is this true?"
I think that it helps students to realize that they can test different cases to find out when it is and isn't true. The types of cases they test vary in each situation, but I encourage them to try simple positive and negative values and try special values like 1 and 0. They can test fractions and extremes as well, but they need to start with their comfort zone. The goal here is to get students thinking, "I can't remember the exact law, but let me test it with some easy numbers. That should give me a clue."
We finish by stressing that since (a + b)^x does not always equal (ab)^x, we don't treat it as a law. We are studying properties which are always true. There might be properties that are sometimes or rarely true, but those are dealt with on a case by case basis.