2An Introduction to Slope Rate of ChangeAn Introduction to Slope
3Schema ActivatorA 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 earnedmiles ran vs. calories burnedmiles traveled vs. fuel in tankweight in pounds vs. price of bananasList 2 examples of your own.
4Rate of ChangeIndependent variables are quantities that are manipulated or changed.hours workedmiles ranmiles traveledweight in poundsDependent variables are quantities that are changed as a result of manipulating the independent variable.dollars earnedcalories burnedfuel in tankprice of bananas
5Rate of Change: WRITE THIS DOWN! Rate of Change = Change in Dependent VariableChange in Independent VariableRATIO!
6Rate 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)1-2471610251334
7Rate 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?PeopleCost (dollars)27.90311.85415.80519.75623.70
8Rate of Change: You Try It! The rate of change is constant in each table. Find the rate of change.Number of DaysRental Charge1$602$753$904$1055$120Miles RanCalories Burned1502100315042005250
10Schema ActivatorThe 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?
11It’s Slinky Time! Materials your team needs: Your task: A SlinkyA foam cupPenniesYour 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!)
12The 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.
13The Traffic Ticket Problem How much do you owe?The Traffic Ticket Problem
14You 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!
15How 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.
16What’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.
17Here’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=0Xmax=120Xscl=20Ymin=0Ymax=200Yscl=20Xres=1
18Make 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)
20Do 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)xy = (½)xy = (0.25)xy = ½(8)xy = (¼)xSchema Activator
21Population: 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
22Money: 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
23Cars: 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
24More 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
25Exploring QuadraticsToday 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 pourerMeasurement masterData recorderResults reporter