# Unit 2: Algebra Essentials

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Unit 2: Algebra Essentials
Suggested Activities Unit 2: Algebra Essentials

An Introduction to Slope
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!

Rate of Change: Example 1
Independent variable? Dependent? The rate of change is constant in the table. Find the rate of change What does this tell us? Time (hours) Temperature (F) 1 -2 4 7 16 10 25 13 34

Rate of Change: Example 2
Independent variable? Dependent? The rate of change is constant in the table. Find the rate of change What does this tell us? People Cost (dollars) 2 7.90 3 11.85 4 15.80 5 19.75 6 23.70

Rate of Change: You Try It!
The rate of change is constant in each table. Find the rate of change. Number of Days Rental Charge 1 \$60 2 \$75 3 \$90 4 \$105 5 \$120 Miles Ran Calories Burned 1 50 2 100 3 150 4 200 5 250

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. Calculate the rate of change as shown in the graph. What is the y-intercept? What does it mean in the context of this problem?

A Slinky A foam cup Pennies Your task: Attach the foam cup to one end of the Slinky. Hang the Slinky from a doorway, ceiling, corner of the room, etc. Measure the length of the Slinky from one end to the other (where the Slinky meets the cup). Add one penny to the cup. Measure the length of the Slinky again – record! Repeat steps 4-5 to fill in the data table. Tonight’s homework: Graph your data. Include labeled axes and a title! Remember: (# of pennies is the independent variable, so it should be on the x-axis!)

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.

The Traffic Ticket Problem
How much do you owe? The Traffic Ticket Problem

You were caught speeding.
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!

How much do you owe? 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.

What’s the equation? 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.

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

Make predictions. 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 = (s – 65)

Exponential Growth & Decay
Real-World…(practical) Applications!

Do the following exponential functions represent growth or decay
Do the following exponential functions represent growth or decay? (Think about the car and bank account examples we’ve worked on.) y = (1.08)x y = (½)x y = (0.25)x y = ½(8)x y = (¼)x Schema Activator

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%.  Write an equation that models the population P based on the number of years x following 2008. How many people would you expect to be living in Jersey City in 2012? High-Five Challenge: How long will it take for Jersey City’s population to double? Team 1

Money:  Jeiny invests \$300 at a bank that offers 5% interest compounded annually.
Write an equation to model the growth of the investment. What would be the value of Angelis’ account after 8 years? High-Five Challenge: After how many years will Angelis’ account be worth four times her initial investment? Team 2

Cars: Matt bought a new car at a cost of \$25,000
Cars: Matt bought a new car at a cost of \$25,000.  The car depreciates approximately 15% of its value each year. Write an equation that models the decay of the car’s value. What will be the value of the car in 4.5 years? How much money did Matt lose? High-Five Challenge: When will the value of Matt’s car first fall below \$3,500? Team 3

More Money: Jordi invests \$1,285 at a bank that offers 4
More Money:  Jordi invests \$1,285 at a bank that offers 4.25% interest compounded annually. Write an equation to model the growth of the investment. What would be the value of Will’s account after 2 years? 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? Team 4

Exploring Quadratics Today you will explore the graph of a quadratic function. In groups of 6, you will complete the Kitchen Paraboloids handout and answer the questions that follow. Group member roles: Water pourer Measurement master Data recorder Results reporter