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1 Activity A: Sidewalk Replacement. The following activity is from Introductory Algebra and focuses and synthesizes reading a graph with writing an equation. The piecewise function is typical in real world situations but adds a level of complexity to a standard type question. Sidewalk Replacement. A root from a large tree has lifted the sidewalk in front of your home. The city requires that you get it fixed. To do it yourself, you would have to rent a jackhammer to remove the old concrete, pay to have the old concrete hauled away, and buy and finish the new concrete. You should also consider the value of your time. The total cost of renting a jackhammer is given by the graph in the figure. 1. How much is a 2-hour rental? 2. How much is an 8-hour rental? 3. If your budget is $125, for how many complete hours can you rent the jackhammer? 4. If your budget is $90, for how many complete hours can you rent the jackhammer? 5. For each hour after the first 4 hours, what is the cost to rent per hour? 6. Write an equation relating the number of hours and the total cost. Define your variables. 7. Unlimited Concrete Services (UCS) offers sidewalk removal and replacement at $6.50 per square foot. The sidewalk is 5 feet wide and 25 feet long. The UCS bid does not include charges for root removal. Compare the cost of a 10-hour rental of the jackhammer with that of hiring UCS.

2 Activity B: Gas Mileage The following activity comes from Introductory Algebra and focuses on slope and rate of change of linear functions. The synthesis occurs in part 4 where students relate steepness of a line with gas mileage and in part 5 where students use slope to calculate future odometer readings without first finding an equation of a line. Gas Mileage. Roberta s odometer read 12,538 miles when she filled her gas tank on Monday. Later that week, she stopped and filled her tank again, adding 4 gallons of gas. At that time, her odometer read 12,662 miles. Ralph s odometer read 13,431 when he filled his tank on Tuesday. He also filled up later that week, adding 7 gallons of gas. When he filled his tank, his odometer read 13,536 miles. 1. Write two ordered pairs that correspond to Roberta s gas record and two ordered pairs that correspond to Ralph s record. Use gallons for the independent variable and miles for the dependent variable. 2. Calculate the slope of Roberta s gas record using the two ordered pairs you found in part 1. Calculate the slope of Ralph s gas record using the two pairs you found in part 1. What are the units for the slope? Does Roberta or Ralph get better gas mileage? 3. Draw a graph using the ordered pairs you found in part 1. Plot both sets of ordered pairs on the same graph. Draw a line connecting Roberta s ordered pairs. Draw a second line connecting Ralph s ordered pairs. 4. Use the graph to find which driver gets better gas mileage. Why? 5. Assuming Roberta s and Ralph s gas mileages remain the same, what will their odometers read after they use 6 more gallons?

3 Activity C: Winterizing Sprinkler Systems The following activity is from Introductory Algebra and focuses on writing and solving an inequality in two variables. A standard word problem is extended by asking students to come up with solutions that do not lie on the boundary line; interpreting the intercepts of a real world situation and by introducing assumptions into the scenario that make a standard problem richer. Winterizing Sprinkler Systems. Walter wants a flat screen television that costs $800. In order to earn money, he plans to rent an air compressor and hire himself out to blow water out of sprinkler systems for the winter (this is known as winterizing sprinkler systems). Walter can charge $45 for each job. The air compressor costs $150 per day to rent. What combinations of number of jobs and days of rental will allow him to make enough money to buy the television? 1. List five possible combinations of jobs and days that will allow him to make at least $ Write an expression for the money earned by winterizing x sprinkler systems. 3. Write an expression for the cost of renting a compressor for y days. 4. Write an inequality that describes the money earned less the rental cost. Show that your solutions from part 1 satisfy the inequality. 5. Find the intercepts. What is the meaning of the x-intercept? Why is the y-intercept negative? 6. Graph the inequality. 7. Suppose it takes Walter 45 minutes to winterize each system, including driving time between jobs. Walter is willing to work 9 hours a day. How does this information affect your graph of the solutions? Are there any other unstated but likely restrictions to the solutions?

4 Activity D: Hot Water Heater The following activity is from Introductory Algebra and is from a section on solving systems of equations with tables and graphs. Complexity is added to a standard problem by the addition of proportional rates, percentage discounts and rebates. Hot Water Heater. A homeowner in Jacksonville, Florida, is replacing her electric hot water heater. She is considering electric, hybrid heat pump, and solar models and is wondering if and when an initial investment in alternatives to electricity will pay off in savings from using less electricity in the future. Maintenance costs on all models are roughly the same, and the expected lifetimes of the three models are similar. Electric model: A 50-gallon electric water heater costs $309 plus $300 to install. Hybrid model: A hybrid heat pump water heater costs $1498 plus $350 to install. Solar model: A solar water heater costs $4970 installed; however, you can get an $800 local rebate and a 30% federal rebate (after the local rebate is taken off). The Energy Use Guide on the electric water heater estimates the annual cost of operation at $514. The Energy Use Guide on the hybrid heat pump estimates a $234 annual cost. However, the Energy Use Guide is based on a U.S. average of cents per kilowatt-hour (kwh). In Jacksonville, the rate is cents per kwh. A solar water heater in Jacksonville can be expected to provide 85% of the homeowner s hot water needs. The other 15% is provided by a small back-up electric hot water heater that comes with the installed system. The cost for the remaining 15% would be proportional to the cost of running the electric water heater. 1. How much per year (to the nearest dollar) does the electric hot water heater cost to run? How much per year does the hybrid heat pump cost to run? 2. How much per year (to the nearest dollar) in electric would the solar system cost? 3. Write an equation for each of the three models. Let the total cost in dollars, C, depend on the number of years of operation, t. 4. Without doing any calculations or considering finance costs, which model, over many years, would be the cheapest? Why? 5. How many years (to the nearest month) would it take before the hybrid heat pump would be cheaper than the electric model? 6. How many years (to the nearest month) would it take before the solar model would be cheaper than the electric model? 7. How many years (to the nearest month) would it take before the solar model would be cheaper than the heat pump? 8. Draw a graph showing the total costs of the three models. Use your graph to confirm your

5 Activity E: Comparing Quadratic Solution Methods The following activity is form Intermediate Algebra. In this activity students must analyze a quadratic and justify their choice of solution technique. Choosing a solution method. Listed below are six methods for solving quadratic equations. Draw a graph and find x-intercepts Make a table and find x where ( ) or y =0 Factor and use the zero product rule Use the square root Complete the square and take the square root Use the quadratic formula Below are 12 equations. Match each equation with a method form the list above, solve with that method and suggest advantages and disadvantages of the method you choose. Below are Table 5 and Figure 24, which may or may not be useful in solving some equations ( )( ) 5. 6., where a > 0 and b > 0 7. ( )( ) , where a > 0 12.

6 Activity F: Stopping Distance The following activity is from Intermediate Algebra and is from a section on direct variation. Richness is added to a standard problem in this real world situation by the formula stopping distance is the sum of reaction distance plus braking distance. Stopping Distance. Stopping distance is the sum of the reaction distance (traveled during the driver s reaction to a situation) and the braking distance (traveled after brakes are applied and before the car comes to stop). Table 3 and figure 16 give reaction distance, and braking distance (in feet) in terms of speed (mph) TABLE 3: Reaction Distance and Breaking Distance as a Function of Speed Speed (mph) Reaction Distance (feet) Braking Distance (feet) Write an equation for the relationship between speed, s, and reaction distance, R 2. What is the reaction distance of you are travelling 90 mph? 3. Write an equation for the relationship between speed, s, and braking distance B. 4. What is the braking distance if you are travelling 90 mph? 5. Write an equation for the total stopping distance, T. 6. What is the total stopping distance if you are traveling 90 mph? 7. If you triple your speed by what factor does your reaction distance increase? 8. If you triple your speed, by what factor does your braking distance increase? 9. What are the units on the vertical axis? What are the units on the horizontal axis? If you put your equation in part 1 into the form, what units will be on k 10. Write an equation for the relationship between speed and braking distance where speed is measured in km/hour and braking distance is measured in meters.

7 Activity G: Healthy Bacteria The following activity is from Intermediate Algebra and comes from a section introducing exponential functions. This standard problem in biology is enriched by having students examine growth per minute and growth per hour by using the power laws of exponents. Healthy Bacteria. The human gastrointestinal tract contains many healthy bacteria that aid in the digestion of food. The possible benefit of taking supplementary bacteria in pill or diet form is continually being researched. One source of lactobacilli contains between 2 billion and 20 billion live bacteria per capsule. Typical lactobacilli are about in size. The bacteria regenerate (double) every hour under optimal growing conditions. Suppose 2 billion lactobacilli were placed in a Petri dish of size If the dish were placed in an incubator under optimal growing conditions, how long would it take before the bacteria covered the dish? 1. How many bacteria would it take to fill the dish? 2. By what percent do bacteria grow every hour? 3. Write an exponential equation of the form ( ), with t in hours, that models the bacteria growth 4. Write an equation based on part 3 and then use a calculator table to find, to the nearest minute, the time it would take for bacteria to cover the dish. 5. By what percent do bacteria grow every minute (use three decimal places)? 6. Write an exponential equation of the form ( ), with t in minutes, that models the bacteria growth. 7. Suppose the incubator wasn t at optimal temperature and instead of regenerating every hour, the bacteria regenerated every 6 hours. Write an exponential equation of the form ( ) that models the bacteria growth under these conditions. 8. Write an equation based on part 7 and then use a calculator table to find, to the nearest minute, the time it would take for the bacteria to run out of room 9. By what percent do the bacteria grow every hour and by what percent do the bacteria grow every minute under these conditions

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