Breakout Session #4Properties of

Exponents and

Rates of Change

Presented byDr. Del Ferster

What’s in store for today? We’ll spend a bit of time looking at some

“test-type” problems. We’ll refresh some basic properties of

exponents◦Fractional powers and radicals◦Negative powers

We’ll spend a bit of time on rates of change◦For linear functions, this is SLOPE

Then, we’ll look at some problems that deal with rates of change.

Practice Problems Properties of

Exponentsand

Rate of Change

Let’s do some problems!

Answers to practice problems

1. , so C

2. , so A

3. , so B

12x

2x

12x

Practice Problem Answers (continued)

4. need 3 gallons, so B

5. 200 minutes, so A

6. 15 points per hour, so C

7. B

Practice Problem Answers (continued)

8. , so A

9. 16 feet per second, so D

10. 48 feet per second, so C

1 mile per minute15

Practice Problem Answers (continued)

Free Response Questions

1.

2A. $10,800 profit per summer

2B. 3 summers

5 322 2x y z

Properties of Exponents

Negative powers, and

fractional powers

How do we handle negative exponents?

Is there a way to write radical expressions as exponential forms, and vice versa?

A couple of questions to get us started

Negative ExponentsIf a ≠ 0 is any real number and n is a positive integer, then

In effect, any base that has a negative power, gets relocated. If it was originally on top of the fraction, it moves to the bottom.

After the move, the power goes positive.

1nnaa

Examples: Negative Exponents

4

3 7

4

3

1.

2. 5

43.10

x

r x

xy

4

1x

Simplify each expression. Your final answer should have no negative powers.

3

7

5rx

3

4

25yx

Fractional Exponents

For example,

and,

For example,

1/ n na a

31/ 3x x

/ /( ) or equivalentlym n m m n mnna a a a

23 32 / 3 2x x or x

Examples: Fractional Exponents

1/ 2

2 / 3

1/ 3

1. 4

2. 8

3. 125

First, rewrite as a radical, then simplify each expression.

2 14 2

23 2 238 8 2 4

1 33

1 1 15125125

RatesOf

Change

What do we mean when we give a rate of change?

What kinds of things have a CONSTANT rate of change?

What are some examples of rate of change, that we use in “everyday life”?

A couple of questions to get us started

When you have two variables changing values at the same time, the rate of change describes the rate at which the first variable is changing compared to the rate at which the 2nd variable is changing.

Rate of Change

Change in the dependent variableChange in the independent variable

Vertical change Horizontal change

Rate of change can be expressed in many different ways:

xy

xxyy

12

12

A line has a positive rate of change if the line is increasing from left to right.A line has a negative rate of change if the line is decreasing from left to right.

Types of rate of change

A line has a zero (0, or no) rate of change if the line is horizontal.

A line has an undefined rate of change if the line is vertical.

The words GRADIENT and RATE and SLOPE all mean exactly the same thing.

If you can solve for one of these, you can solve for any, because they’re all the same.

Gradient, Rate, and Slope

Here are the basics:1. There will always be 2 variables (numbers)2. Gradient means “How does a change in one variable affect the other?”3. To get the answer you will usually have to subtract (to get the change) and 4. you will almost always have to divide

(to fine the affect).

Just the FACTS!

IF you can subtract and divide then you can solve all gradient/rate/slope (or more impressively, rate of change) problems!

What was that?

We begin with word problems. The trick here is to read carefully and to figure out what to do with the information you are given.

Remember: You may have to SUBTRACT. You will almost always have to DIVIDE.

An balloon rises 1000 meters in 5 minutes. At what rate is it rising?

Express your answer in meters per minute.

How about an example?

200 meters per minute

Five seconds into flight, a rocket is 2,000 feet above the earth. Twenty five seconds into flight, the rocket is 42,000 feet above the earth. How fast is the rocket rising (in feet per second)?

Let’s jazz up that last problem

42000 200025 5

40000

20 2000 feet per second

A balloon is released at an altitude of 500 meters at 2:15. At 2:45 the balloon had an altitude of 3500 meters. Find the rate at which the balloon is rising.

Balloons are fun, let’s do another problem!

3500 500 3000 100 meters per minute30 30

Now, let’s explore the concept, when the data is provided by a graph.

Rate of change problems and graphs

A student released a balloon and recorded data on the graph at the left. According to the graph at what rate did the balloon rise from time 2 minutes to time 7 minutes?

Remember:◦There are always 2 variables (numbers).◦One is called the DEPENDENT VARIABLE

◦This is usually y When dividing, this one goes on TOP.

◦The other is called the INDEPENDENT VARIABLE.

◦This is usually x or t It goes on the bottom.

A bit more of the “math stuff”

A bit of a trick!Do rates of change remain

constant?

Example 1—Changes in rate of change

Consider the graph to the left. Let’s explore this question:

Does the rate change over time?

To find out let’s calculate the rate of change for several different time periods and see if it changes.

First what is the rate from time 0 to time 4?

How about the rate from time 3 to time 6?

Ok, one more, how about from time 2 to time 9?

Example 1--Continued

Did the rate change over time?

Finally…

How can the rate be the same when that line is OBVIOUSLY going UP?◦Remember, that RATE is SLOPE!◦If you were a skier going down that black

line, would you think that the slope was changing? In other words, is the line “STEEPER” in

some places than others?

Hold on there…wait a minute..

Tying it all together!The slope of the line stays the same, and since slope and rate are really the same thing, the rate must stay the same as well.

If the slope doesn’t change, then the rate doesn’t change.

Can you draw a graph, that would indicate a change in the rate? What could it look like?

Comparing rates of change on a graph.

Graph A Graph B

Consider these two graphs.1. Is the rate in each graph changing?2. Something is different though, what?

It appears that the slope of the line in graph B is steeper than the slope of the line in graph A.

The STEEPER the slope, the GREATER (FASTER) the rate of change.

Continuation of the comparison

Now for a wicked twist!The “line” on this graph isn’t straight, what does this mean?

Is the slope increasing or decreasing?

Imagine that you were riding the bike up this slope, would you say that it is getting steeper or less steep?

What’s the difference between these 2 graphs?

The scales are the same, the labels are the same, so what’s the difference?

The graph on the left represents the RATE of a car going 50 miles in 1 hour (50 mi/hr)

The graph on the right represents the RATE of a car going 20 miles in 1 hour (20 mi/hr).

Since the slope of the graph on the right is less steep than the slope of the graph on the left, the car’s RATE is less◦ In summary: Constant slope = constant rate;

Steep = Fast; less steep = slower.

THE ANSWER!!!

What’s the difference between these 2 graphs?

The scales are the same, the labels are the same, so what’s the difference?

The graph on the left represents a car moving at a constant speed. ◦We know this because slope is rate and

slope is c_____ In the graph on the right the slope does

change.◦It starts off fairly flat, then the slope gets

steeper and steeper.◦What does this mean?

THE ANSWER!!!

1. Find the rate of change for the data in the table below.

Skills check on rate of change

Tickets Cost5 $756 $907 $1058 $120

Skills Check (Continued)

2. Find the rate of change for the data in the graph.

Skills Check (Continued)

3. Find the slope of the line. risesloperun

1.

2.

3.

Answers to the skills check$15 per ticket

400 calories per hour

43

I’ve compiled a packet of problems that allow us refine our skills related to rates of change.

Some problems are good for middle school grades, some for high school grades.◦With some editing on your part, the

middle school ones can be used with higher elementary grades, I think.

Frankly, I think this is what we should spend time on with our students.

Time for some problems

Wrapping UpThanks for your attention and participation.◦You have my utmost respect for working hard all day with your kids, and still hanging in there for what we’ve done!

If I can help in any way, don’t hesitate to shoot me an email, or give me a call.