Suggested Activities

Unit 2: Algebra Essentials

Rate of Change

An Introduction to Slope

Schema Activator A rate of change is a ratio that shows a

change in one quantity with respect to a change in another quantity. Examples:

hours worked vs. dollars earned miles ran vs. calories burned miles traveled vs. fuel in tank weight in pounds vs. price of bananas

List 2 examples of your own.

Rate of Change Independent variables are quantities that are

manipulated or changed.

hours worked miles ran miles traveled weight in pounds

Dependent variables are quantities that are changed as a result of manipulating the independent variable.

dollars earned calories burned fuel in tank price of bananas

Rate of Change: WRITE THIS DOWN!

Rate of Change = Change in Dependent Variable

Change in Independent Variable

RATIO!

Independent variable? Dependent? The rate of change is constant in the table.

Find the rate of change.

What does this tell us?

Rate of Change: Example 1

Time(hours)

Temperature

(F)

1 -2

4 7

7 16

10 25

13 34

Independent variable? Dependent? The rate of change is constant in the table.

Find the rate of change.

What does this tell us?

Rate of Change: Example 2

People Cost (dollars)

2 7.90

3 11.85

4 15.80

5 19.75

6 23.70

Rate of Change: You Try It!

Number of Days

Rental Charge

1 $60

2 $75

3 $90

4 $105

5 $120

The rate of change is constant in each table. Find the rate of change.

Miles Ran Calories Burned

1 50

2 100

3 150

4 200

5 250

Facebook Challenge: The Solution

Schema Activator

The graph below represents the value of an iPad based on the number of years that have passed since it was

originally purchased.

1.Calculate the rate of changeas shown in the graph.

2.What is the y-intercept?

3.What does it mean in the contextof this problem?

Resale Value of a Refrigerator

Years after original purchase

Am

ount

s ($

)

3 6 9 12 15 18

100

200

300

400

500

600

It’s Slinky Time!Materials your team needs:

1. A Slinky2. A foam cup3. Pennies

Your task:1. Attach the foam cup to one end of the Slinky.2. Hang the Slinky from a doorway, ceiling, corner of

the room, etc.3. Measure the length of the Slinky from one end to

the other (where the Slinky meets the cup).4. Add one penny to the cup.5. Measure the length of the Slinky again – record!6. Repeat steps 4-5 to fill in the data table.

Tonight’s homework:1. Graph your data. Include labeled axes and a title!2. Remember: (# of pennies is the independent

variable, so it should be on the x-axis!)

The Slinky Activity: Follow-Up

Return to your Slinky team. Compare each other’s graphs. Do they look the same? If not, what could be

a cause for the variation between graphs? Do you see any pattern(s) when comparing

the number of pennies to the length of the Slinky?

Can you identify the independent/dependent variables?

Select one graph to represent your team’s findings.

Be prepared to share.

How much do you owe?

Fill in your traffic ticket. In the “SPEED LIMIT” box list 65.

Make up your own “ALLEGED SPEED” … how fast were you actually driving?

What type of car were you driving? Be creative!

The fine for speeding in your state is $85 plus $5 for each mile above the speed limit you were driving.

Calculate how much you need to pay.

Let’s compare our answers.

Can we write an equation that can be used to calculate any amount due based on the driver’s speed?

What’s the equation? Hint: Notice that you must pay $85 regardless

of your speed, in addition to $5 for every mile over the speed limit. Use a variable to represent the speed at which you’re traveling.

Amount Owed = 85 + (Your Speed – 65) • 5 Or, we can write it like this:▪ y = 85 + 5(s – 65)

Now, let’s graph that equation. Set your window to:

Xmin=0 Xmax=120 Xscl=20 Ymin=0 Ymax=200 Yscl=20 Xres=1

Using the Table feature on your calculator, about how much would somebody owe if they were driving: 101 mph? 90 mph? 85 mph?

Check your predictions by substituting for the variable s in our equation:

y = 85 + 5(s – 65)

ExponentialExponentialGrowth & DecayGrowth & Decay

Real-World…(practical) Applications!

Schema ActivatorSchema Activator

Do the following exponential functions represent growth or decay? (Think about the car and bank account examples we’ve worked on.)

1. y = (1.08)x

2. y = (½)x

3. y = (0.25)x

4. y = ½(8)x

5. y = (¼)x

Team 1Team 1

Population:  The population of the popular town of Jersey City in 2008 was estimated to be 250,000 people with an annual rate of increase (growth) of about 2.4%. 

1. Write an equation that models the population P based on the number of years x following 2008.

2. How many people would you expect to be living in Jersey City in 2012?

3. High-Five Challenge: How long will it take for Jersey City’s population to double?

Team 2Team 2

Money:  Jeiny invests $300 at a bank that offers 5% interest compounded annually.

1. Write an equation to model the growth of the investment.

2. What would be the value of Angelis’ account after 8 years?

3. High-Five Challenge: After how many years will Angelis’ account be worth four times her initial investment?

Team 3Team 3

Cars: Matt bought a new car at a cost of $25,000.  The car depreciates approximately 15% of its value each year.

1. Write an equation that models the decay of the car’s value.

2. What will be the value of the car in 4.5 years?3. How much money did Matt lose?4. High-Five Challenge: When will the value of

Matt’s car first fall below $3,500?

Team 4Team 4

More Money:  Jordi invests $1,285 at a bank that offers 4.25% interest compounded annually.

1. Write an equation to model the growth of the investment.

2. What would be the value of Will’s account after 2 years?

3. High-Five Challenge: What is the minimum interest rate at which Will should invest $1,285 to have at least $1,500 after 3 years?

Exploring Quadratics

1. Today you will explore the graph of a quadratic function.

2. In groups of 6, you will complete the Kitchen Paraboloids handout and answer the questions that follow.

3. Group member roles:1. Water pourer2. Measurement master3. Data recorder4. Results reporter