Remember...the Properties of Powers

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Objective

SWBAT recall power properties by testing special cases, following patterns, and writing the expressions in different ways.

Big Idea

Students often see properties of powers as independent rules to be memorized.This lesson is to dissuade students from memorizing without understanding.

LAUNCH

10 minutes

To begin the lesson, project the LAUNCH on the board. Ask students to answer these questions at their desks. Call on volunteers to go to the board and show their work.

It is a good idea to "milk" this exercise a bit and ask students to find several different values that make each of these equations true. A few students will quickly see that both 0 and 1 make the equations true. Ask them to go up to the board and demonstrate this. Once all students see that the equations are equal with these values ask:

"So if you are to check answers to problems involving powers, is it a good idea to use 0 or 1?"   Students should answer, "NO!"


APPLICATION

25 minutes

Hand each student a Remember The Properties Application Handout

On this worksheet

  • Some problems will ask students to use a "special case" to verify their reasoning. By this I mean that students should check to see if an equation is true by substituting values for the variable(s). A calculator should be allowed.
  • Some problems will ask students to use repeated multiplication to prove that an equation is true. By this point in the unit, this idea should be familiar to students. 
  • In some cases students will have to demonstrate that a pattern is not true by providing a "counterexample". Remind students that a counterexample is simply a "special case" for which the answer is false. Ask students how many counterexamples are needed to disprove a problem? Make sure that they understand that only 1 counterexample is necessary. 

The reason for this activity (i.e., testing with special cases) is to help students realize the notion of consistency, which allows them to prove or disprove a problem with assurance. 

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Question 1: "Milk" this question by asking students to provide a counterexample to disprove the other other two options, x30,  2x11

Question 2: Ask students to try using 1 and 0 to re-confirm that these are not good testing options. 

Question 3: Students here will have to figure, they have to first think of a wrong way of writing a property, then try substituting 2 and see if 2 makes it a true statement.  An example of a correct answer is:

(bm)n = bm+n  

Using 2 gives: (32)2 = 32+2 hence, (32)2 = 34

Question 4: This is another example confirming that 2 is not a good checking option. 

Question 5: Students should show using power of a quotient first, then using quotient of powers inside the parenthesis first. I'd tell them that I will not be looking for one particular route, as long as their route is correctly done and leads to the correct answer. 

CLOSURE

10 minutes

To close the lesson, write the six properties that we have seen in this unit so far on the board. Then tell students that they show that none of these properties are true if they make the following changes to an equation:

  • each multiplication symbol is replaced by a plus sign
  • each division is replaced by a minus sign

If time permits, I plan to go through a couple of examples from students to ensure that they realize that the properties are related to multiplication and division, not to addition and subtraction.

Extension

In this lesson I have not emphasized stating the names of each of the properties. This extension exercise was chosen to provide a chance to review the names of the properties at the tail end of the lesson.