Teaching Phase: How will the teacher present the concept or skill to students?
Pick three students and have them stand in the front of the room with the cards for 2 cubed, 2 to the fourth and 2 to the fifth power with the "2" sides showing.
The teacher could use the students' names, for example -
Charlie has 2 cubed, Frank has 2 to the fourth and Fionna has 2 to the fifth.
What is the difference between Charlie's 2 cubed and Frank's 2 to the fourth? The value of 2 to the fourth is 2 times (twice) the value of 2 cubed. You multiplied by 2.
What about the difference between Frank's 2 to the fourth and Fionna's 2 to the fifth? Once again you multiplied by 2, since the base is 2, and the exponent increased by 1, from 4 to 5.
What would the next number be? 2 to the sixth or 64
Have another student (Sean) come up and take the 2 to the sixth card.
Following this pattern, what would be the next number? 2 to the seventh or 128
Have another student (Heather) come up and hold the 2 to the seventh card.
What's being done to Sean's 2 to the sixth to get Heather's 2 to the seventh? Multiplying by 2 again
Have the students flip their cards to reveal the sides that have x instead of 2.
What is the difference between Charlie's x cubed and Frank's x to the fourth now? x to the fourth is 2 times x cubed. Remember how we multiplied by 2 before? Why did we multiply by 2? Because it is the base value
So what is the difference between each of these students' expressions? The previous one is being multiplied by x, the base value, when the exponent increases by 1.
Say "Oops, I made a mistake. I forgot to include the lower exponent values. Johnny, can you come up here please?" Give him the x squared card.
How might I find the value of Charlie's x cubed after finding the value of Johnny's x squared. Multiply by 2. Charlie's x cubed is 2 times more than Johnny's x squared. What if I were trying to determine the value of Johnny's x squared after finding the value of Charlie's x cubed? Divide by 2, the base.
So, what are we doing to find the next value, when the exponent is larger by 1 and the base is the same? Multiplying by the base again.
If the exponent is smaller by 1 and the base is the same, how do we find the next value? Divide by the base.
Have the students flip their cards back over to the sides with the 2s.
What would come before Johnny's 2 squared? 2
What would the exponent be on that? 1
Have a student (Olivia) come up and hold the power of 1 card with the 2 facing forward.
What value would come before Olivia's 2 to the 1? Zero will be a possible answer.Zero is the exponent, but not the value. Ask what operation we used to get from 2 squared to 2 to the first? Divide by 2. Is 2 divided by 2 equal to 0?
What if we had 3 instead of 2 as the base? 4? 5? They would all be divided by themselves and the answer would be 1.
So what can we conclude about anything to the power of zero? It will always equal 1.
Have a student come up and hold the zero power card with the 2 showing, then have everyone flip over to show the x-sides of the cards.
So, x to the zero power is equal to 1.
Then have them flip back to the 2. I want to go one more step lower. If I go one more step lower (have another student come up), what will the exponent be? What will the value be? Remember, we divide by the base, 2, to get the next step down, so the previous value of 1 is going to be divided by 2 to get 2 to the negative 1.
Leave this as a fraction.
Have another student come up and hold the negative one exponent card. Write the fraction ½ on the board. I want to go another step lower. What do I do?
Show the divide by 2 on the board "Can you divide a fraction by a whole number?"Multiply by the inverse
Change the divide by 2 on the board to multiply by ½. "What does this equal?" ¼
Before I have another student come up and hold a card, let's look at this in terms of x.
We had 1/x for x to the negative 1 power. What do we need to do to get the 1/x to the next lower power?
(divide by x by multiplying by the reciprocal, 1/x)
When I multiply these fractions, what do I get? ()
Have a student come up and get the negative 2 power card.
What is the next card going to look like?
The next one?
What can we conclude about negative exponents from this?
Have the students sit back down.
Write on the board, . What does this equal in fraction form? 1/25
Write on the board . What does this equal to? 1/27
Write on the board . What does this equal? someone is bound to say Point out how the base of the exponent will not include the coefficient, and that the correct answer is
Write on the board . What does this equal to? 1
Write on the board . What does this equal to?
We know that , right? So, let's replace that in our expression's denominator.
We get This is the same as saying .
What is an efficient way to divide by a fraction? What do you do? Multiply by the reciprocal.
So, this becomes
So, when we have a negative exponent in the denominator, how can we simplify the expression? Express the denominator (a power with the negative exponent) as a fraction and multiply the numerator by the reciprocal of the denominator.
Let's look at this with a variable. What does
What about ? ?
Now let's take what we have just learned and apply it to some of the multiplying and dividing we have previously done. Before this our answers always had positive exponents. This is no longer the case.
Use what you have already learned to multiply 5 squared times 5 to the negative third power? = = 1/5 Explain how you simplified the expression. Students may subtract exponents or cancel the two 5/5s.