August 30, 2012
EQ: How do I identify exponential growth?
Bellwork:
1. Bethany's grandmother has been sending her money for her birthday
every year since she turned 1. When she was one, her grandmother
sent her $5. Every year she sends Bethany twice as much money as she
did the previous year. How much money will Bethany receive for her 5th
birthday? Explain how you came up with your answer.
August 30, 2012
Investigation
You have been assigned the task of making ballots for the upcoming
student elections. Because the school is trying to save paper, you want to
see how many ballots can be cut out of a single piece of paper. Using the
scissors and paper provided make a table that displays many ballots you
can make with and number of cuts (n). Start by recording the number of
ballots created for 1, 2, 3, 4, and 5 cuts. Then try to create a rule that
shows how many ballots will be created with n cuts. To simplify the
process, each cut should cut the piece(s) of paper exactly in half.
How many ballots could you make if you could make 20 cuts? 30?
How many tomes would you have to cut a piece of paper in order to
create enough ballots for all 800 students at our school?
You have 15 minutes.
August 30, 2012
So what did your data look like? Is the relationship between cuts and
total ballots linear? How do you know?
Number of Ballots
Cuts
Ballots
1
2
2
4
3
8
4
16
5
32
n
???
If the relationship isn't linear, what is it? Add row or column to your table
that shows how you got from the number of cuts to the number of ballots.
That is to say, what mathematical operation did you perform?
Cuts
Ballots
My
Thinking
4
2
2
4
3
8
16
1x2
1x2x2
1x2x2x2
1x2x2x2x2
1
5
32
n
???
1x2x2x2x2 1 n number
of 2's
x2
August 30, 2012
The relationship between the number of cuts and the total number of
ballots made is known as an exponential relationship. In an exponential
relationship, a fixed change in the independent variable results in the
dependent variable being multiplied by a fixed amount. In our ballot
example, every time the cuts (I.V.) increased by 1, the number of ballots
(D.V.) was multiplied by 2.
Exponential relationships are easy to recognize in tabular form. In an
exponential relationship, there is no constant rate of change, and the
dependent variable starts out by growing or shrinking at a slow rate, but
then increases rapidly.
x
y
1
3
2
9
3
27
4
81
5
243
6
729
The table above shows the equation 3x. Notice how the relationship
starts out growing slowly, but then shoots up quickly.
August 30, 2012
Exponential relationships are pretty easy to identify in equations as well,
but there are 3 different ways to write exponential equations:
Expanded Form: When we write a number as a product of multiple
factors.
6x6x6x6x6x6x6
Exponential Form: When we write a number using a base raised to an
exponent or "power".
67
The number above is read as "six to the seventh power." the number six
is the base and the number 7 is the exponent or "power."
Standard Form: Standard form is what we get when we evaluate an
expression in expanded or exponential form.
67 = 279,936 so 279,936 would be considered standard form.
Work with your partner and think of one situation in which we might wnat
to use each of the 3 different forms.
August 30, 2012
EQ: How do I operate with exponents?
Bellwork:
Write each number in exponential notation:
1. 5 x 5 x 5 x 5 x 5 x 5 x 5
2. 8 x 8 x 8 x 8 x 8
3. 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
Write each number in expanded form:
4. 46
5. 78
6. 212
Write each number in standard form:
7. 25
8. 54
August 30, 2012
What do exponential relationships look like in graphic form?
Total Ballots Produced
Sketch a graph of your data from the ballot investigation. What does the
graph look like?
Ballot Making Investigation
Number of Cuts
August 30, 2012
But that graph really only gives us a small picture of an exponential
relationship. Lets go back to our table of data:
Cuts
Ballots
1
2
2
4
3
8
4
16
My
Thinking
1x2
1x2x2
1x2x2x2
1x2x2x2x2
5
32
n
???
1x2x2x2x2 1 n number
of 2's
x2
Was anyone able to write a rule that defines the relationship?
B = 2n
Total ballots is equal to 2 to the power of n, or 2 to the nth power.
If we were to write this using our standard variables x and y, it would
be: y = 2x
Using a graphing calculator, sketch a graph of y = 2x
Zoom out to see what the graph looks like for many values. Try
graphing y = 3x or y = 4x
August 30, 2012
Operating with Exponents
Now that we know what exponents look like, its important for us to know
what to do when we see them in equations and expressions. When we
operate with exponents, we perform the four basic mathematical
operations (+, -, ×, ÷).
Consider the following expression:
23 × 22
Find the value of the expression in standard notation.
8 × 4 = 32
That wasn't so hard was it? But how would we express the product in
exponential notation? First let's look at how we would write the
expression in expanded form. Write each of the factors from the
expression above in expanded form.
(2 × 2 × 2) × (2 × 2)
Do we really need the parentheses in the expression above? What
happens if you remove them?
2×2×2×2×2
Now write the expression again in exponential form:
25
August 30, 2012
So we've decided that:
23 × 22 = 25
What relationship do you notice between the exponents?
Rule #1
ax × ay = a(x+y)
Practice: Simplify the following expressions and leave your answer in
exponential form.
1. 53 × 55
2. 48 × 46
3. 74 × 712
4. 212 × 215
5. 234536 × 234154
August 30, 2012
So what about division? consider the following expression:
47 ÷ 43
Again, we can start by writing the expression in expanded form:
(4 × 4 × 4 × 4 × 4 × 4 × 4) ÷ (4 × 4 × 4)
What is another way of writing a division problem?
That's right! We can write it as a fraction.
(4 × 4 × 4 × 4 × 4 × 4 × 4)
(4 × 4 × 4)
Then we can use what we know about the multiplicative identity.
Remember, the multiplicative identity is the value which we can multiply
by any other value without changing its identity. In this case (and most
others) the multiplicative identity is 1. So in our fraction above, we can
cancel 3 pairs of 4's because 4/4 is 1. This leaves us with:
4 × 4 × 4 × 4 = 44
So,
47 ÷ 43 = 44
What is the relationship between the exponents in the equation above?
August 30, 2012
Rule #2
Practice:
1. 79 ÷ 74
2. 912 ÷ 95
3. 415 ÷ 45
4. 1232 ÷ 122
5. 196354 ÷ 196213
ax ÷ ay = a(x-y)
August 30, 2012
So now your asking, we've done multiplication and division, what could
possibly be left? Well, there is addition and subtraction, but unfortunately
there's not much we can do to add or subtract exponential relationships.
But what if we took one exponential relationship and took it to a higher
power?
Consider the following expression:
(33)4
Just like before, we can write the expression in expanded form. You try it
this time. See of you can write a rule that tells us what to do when we
raise one exponent to another power.
(3 × 3 × 3) × (3 × 3 × 3) × (3 × 3 × 3) × (3 × 3 × 3)
3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 312
(33)4 = 312
August 30, 2012
Rule # 3
(ax)y = a(x × y)
Practice:
1. (52)5
2. (36)2
3. (27)8
4. (14)6
5. (234120)3
August 30, 2012
There are also a few special cases. Consider the following expression:
50
This one's a little harder to figure out. I find it easiest to do this one in
table form.
53 = 5 x 5 x 5 x 1
2
5 =5x5x1
51 = 5 x 1
50 = 1
Rule #4
a0 = 1
Practice: I'm not going to insult your intelligence by making you practice
this one.
August 30, 2012
Another special case: What happens when I have a negative exponent?
Consider the following expression:
5-3
Lets go back to our table from the last rule and continue it.
53 = 5 x 5 x 5 x 1
52 = 5 x 5 x 1
51 = 5 x 1
50 = 1
-1
5 =1÷5
-2
5 =1÷5÷5
-3
5 =1÷5÷5÷5
5-1 =
1
5
If we rewrite the division as fractions, we get:
1
1
-2
5 =
5-3 =
53
52
August 30, 2012
Rule #5
a-x = 1
ax
Practice
1. 3-2
2. 2-5
3. 6-3
4. 9-4
5. 8-8
August 30, 2012
*****CAUTION*****
Negative bases can be tricky. It's important to follow the order of
operations to the letter. Consider the following expression:
-32
It's tempting to evaluate the expression like this:
-32 = (-3) × (-3) = 9
But the order of operations says differently. PEMDAS tells us to do
exponents before we affix negative signs. Why? Well, a negative sign is
like multiplying a value by -1. Multiplication comes after exponents in the
order of operations. So...
-32 = (-1) ×(3) × (3) = -9
However, if we want to include the negative sign with the number 3, all
we have to do is use some cleverly placed parentheses.
(-3)2 = 9
August 30, 2012
August 30, 2012