• Variety items

    These items are short-answer tasks that may be used alone or in combination.  For each item, the student should be encouraged to articulate the answer in the cell phone context, and not to settle for a numerical or other mathematical result devoid of the context.  In some cases, this means the students may have to interpret an answer (e.g., round to the nearest unit) to give the mathematical result meaning in the context.

    V1. Populations

    Here is a map of Dauphin County.  On the map we have marked several places with large numbers of people.  To decide where to place towers, the company divides the county into a north part and a south part, as shown by the thick red line. How many more customers does the company reach by serving the south region rather than the north region?

    Answer: The population in the north is 9,346.  The population in the south is 76.797.  Serving the south region would mean reaching 76,797-9,346=67,451 people.

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    V2. Comparing Populations

    Here is a map of Dauphin County.  On the map we have marked several places with large numbers of people. Of all the marked places on this map, which has the greatest population? Which has the least population?

    Answer: The greatest number is 48,540, the population of Harrisburg.  The least number is 299, which is the population of Pillow.

     

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    V3. Interpreting Grids

    If the lines on the grid are 1-inch apart, what is the area in square miles of the land represented by one square on the grid? 

    Answer: 1 inch x 1 inch represents 3/4 mile x 3/4  mile = (3/4 miles)^2=9/16 square miles, or approximately 0.56 square miles

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    V4. Our County¡¯s Area

    The scale on this map claims 1 inch represents 3/4 miles.  What is the area in square miles of Dauphin County?  Explain how you arrived at your answer.

    Answer: 

    Way : I counted the number of 1-inch squares lying within the county boundary lines and estimated the number of partial squares lying within the county boundary.  I multiplied this answer by (3/4)2, which is the number of square miles represented by 1 square inch on the map.  261 squares lie within the county. They are marked on the map.  Also inside the county are approximately 33 almost-filled squares (green) and 39 half squares (blue).  The area of the county is approximately square inches or 313.5x0.752= 176 square miles.

    Way : Cut out the county and then cut the county along the grid lines.  Tape together the partial squares to get full squares ¨C or as many things close to full squares as possible ¨C and then count the total number of full squares.

    Enactment: Way  and Way seem to be very different, yet they have some things in common.  Tell what mathematical ideas are in both Way  and Way (e.g., the 1-inch square as the standard unit of measure, counting full squares, having to ¡®put together¡¯ partial squares to get full squares through addition in Way  and physically in Way ).

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    V5. Who Has Phone Service

    There were approximately 252,933 people in Dauphin County in 2002.   If the company has a service for people in all of the places shown on map, will more than half of the people in the county have service in their homes? 

    Answer: The total population of the places on the map is 86,143.  Based on the population of these places only, no.  But, if we take into consideration that some people living in the areas between the places are also included in the services to the places. If we try to place the towers to include as many of the people living outside the labeled places, it seems that we might reach more than half of the people.

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    V6. High Populations

    This map shows some of the most populated places in Dauphin County. 

    a.         What is the total number of people who live in these population centers?

    Answer: The sum of the numbers on the map is 86,143

    b.     If the total population of Dauphin County is 252,933, what percent of the population lives in these places?

    Answer: 86,143 ¡Â 252,933 x 100% ¡Ö 34%

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    V7. Calculating Distance

    Here is a map of Dauphin County.  The scale on the map is 1 inch = 3/4 mile.  How far is Harrisburg from Highspire according to this map?

    Answer: 5 1/4x 3/4 = (21/4)(3/4)=63/16¡Ö3.9 miles.

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    V8. Air Miles

    The scale on this map claims 1 inch represents 3/4 miles. How many miles (by air) is Paxtang from Hummelstown? 

    Answer: 5 inches x 3/4 mile per inch = 15/4 miles ¡Ö 3.75 miles.

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    V9. Decreasing Populations

    The population of Steelton Borough was 5796 in 2002.  This population has been decreasing lately at the rate of -1.1% every two years.   Write an equation that predicts the population of Steelton n years after 2002.

    Answer: P=5796x(1-1.1/100)n, where P is the population n years after 2002.

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    V10. Rectangles

    Imagine a rectangle drawn around Dauphin County so that the sides of the rectangle are horizontal and vertical.  Sketch the rectangle on the map below.  What would be the area of that rectangle?

    Answer:

    Way : Locate a rectangle with the greatest horizontal distance and the greatest vertical distance as marked on the map below.  Count the blocks to get the length and width of the rectangle.  The rectangle shown is 19 blocks by 28 blocks.  The area of the rectangle would be the product of these two lengths, converted from inches into miles: .

    Way : Determine how to locate a rectangle without a horizontal side. Use the Pythagorean Theorem to compute the length and width, and then the area, of this rectangle. To find the shorter side length, count the number of squares along the legs of DABC.  AC=12 squares.  CB=18 squares.    To find the longer side length, counting gives us AD=23 and DE=14.  Then, .  The area of the rectangle would be  or .  Converting the area to square miles, we have  or .

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    V11. Estimating with Rectangles

    It is possible to use this map to estimate the area of Dauphin County.

    a.     At most, how many miles is it from the north of the county to the south?  How many miles is it from the east to the west? 

    Answer: Count 28 blocks across the ¡°tallest¡± part of the county.  Converting this into miles gives  miles north to south. Similarly, count the blocks across the widest part of the county to get 19 blocks.  So, it is  miles east to west.

    b.         Imagine a rectangle with the distances from part a as its length and width.  What would be the area of that rectangle?

    Answer: Using the widest and highest parts of the county from part a, we know the area of the rectangle would be .

    c.     The area that you calculated in part b is an estimate of the area of Dauphin County.  Do you expect the actual area of the county is more or less than the area of that rectangle? Explain you thinking.

    Answer: Less.  The rectangle includes the shaded that lie outside the boundary of the county.

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    V12. Cell Phone Coverage

    If a cell phone tower covers a two-mile radius, would it be possible to place one tower so that tower would cover both Steelton and Highspire?

    Answer:  Yes.  According to the map, the towns are 3 1/2 inches, or 3 1/2x3/4=7/2x3/4=21/8=2 5/8¡Ö2.625 miles apart.  A circle with 2-mile radius and center at the midpoint of the segment connecting Steelton and Highspire would cover both towns since 2>(1/2)(2 5/8)=21/16=1 5/16¡Ö1/31.

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    V13.  Tower Placement

    Suppose a cell tower serves everyone within one mile of the tower.  If you had only one tower to place and you wanted to be sure to have it serve as many people as possible in all of these places, where would you place it? 

    Answer:  Harrisburg is clearly the most populated place on the map; its population is much greater than the population of any other place.  So, we definitely want to include Harrisburg.  To get as many more people as possible, we what to place a circle with a 1-mile radius so that includes Harrisburg and as many other places to get the largest possible additional total population. A tower placed as shown by the large red dot on this map would reach Dauphin and Harrisburg. A tower placed at the small blue dot almost reaches Harrisburg and Steelton.  To be sure to include as many people in these places as possible, put the tower at the large red dot.

    Enactment:

    Answers may vary.  In particular, there are a few slightly different locations that are close to the red dot that also are feasible locations for the tower.

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    V14. Dual Coverage

    A cell tower typically covers a one-mile radius.  If a cell phone tower is placed in Highspire, would it also cover Steelton?

    Answer: No, the distance from Highspire to Steelton is 3 1/2 inches, or 3 1/2x3/4=7/2x3/4=21/8=2 5/8=2.625 miles.  This distance is greater than 2 miles, the maximum diameter of a circle representing a cell tower.

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    V15. County Angles

    The blue circle on the following map shows the region a cell phone tower covers.   The tower is placed exactly on the boarder between Dauphin County and Lebanon County.

    a.     Estimate the measure of the angle. 

    Answer: This looks like about 120¡ã.

    b.     What fraction of a circle is this? 

    Answer: 120/360=1/3.

    c.     What is the area of the sector that is not in Dauphin County

    Answer: Two-thirds of the circle lies in Dauphin County.  The area is approximately 2/3 of 4¦Ð square miles or 8/3¦Ð¡Ö8.37 square miles.

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    V16. Costly Towers

    If each cell phone tower costs approximately $150,000 to buy from the manufacturer, write an expression for the cost of the following:

    a.     22 towers

    Answer: 22x$150,000 or $3,300,000

    b.     43 towers

    Answer: 43x$150,000 or $6,450,000

    c.     n towers

    Answer: $150,000n

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    V17. Operating Costs

    If 210,000+1767n represents the cost of operating one new tower for n months, what would 210000 and 1767 represent in terms of cell phone costs?

    Answer: 1767 represents how much the company pays monthly to operate the tower.  210000 represents the money the company had to spend to set up the tower.

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    V18. Functioning Costs

    A cell phone company pays approximately $150,000 for each cell tower it purchases from a company in Missouri. 

    a.     Fill in the following chart.

    number of towers (n)               0       1          2       3       4

    purchase price of towers (c)    0       ___      ___   ___   ___

    Answer:  The chart should read as follows:

    number of towers (n)                  0      1                  2                  3                  4

    purchase price of towers (c)     0      150,000     300,000     450,000     600,000

    b.     Write an expression for the cost, c, of n towers.  Explain how you arrived at this equation.

    Answer: The pattern in the table is a linear pattern.  The cost per tower, 150,000, would be the slope.  The cost of 0 tower, 0, would be the vertical intercept.  So, the expression is c=150000n.

    c.     A cell phone company can purchase a tower for $150,000.  The company needs to pay approximately $60,000 in zoning costs to install a tower. Write an expression for the zoning costs, z, of installing n towers.

    Answer: The slope would be 60,000.  The horizontal intercept is 0.  The equation would be z=60000n.

    d.     c+z represents the cost the company pays up front to buy and install n towers.  Write an expression using n to complete this equation:  c+z=____________.

    Answer: The sum of c and z is the sum of 150000n and 60000n. 150000n+60000n=210000n.

    e.     Each year, the company pays several other fees for the tower.  The cost to least the land is approximately $60,000.  The annual maintenance charge is approximately $1200.  The yearly inspection is approximately $10,000.  Write an equation for the annual cost, a, of n towers.

    Answer:  a=(1200+10000+60000)n or a=71200n.

    f.      The questions you just answered involved the cost of n towers.  Ankland writes the equation t= 71200y+210000.  His classmates agree his equation makes sense.  What do you think 71200, 210000, y and t represent in Ankland¡¯s equation?

    Answer: 71200 is the cost per year of one tower.  210000 is the initial fixed cost of purchasing a tower.  y is a number of years.  t would be the total cost of operating the tower for y years.

    g.     According to Ankland¡¯s equation, what is the value of t when y=3?  What does this value mean to the cell phone company?

    Answer: If y=3, t=423600.  $423,600 is the amount of money it would cost the company to install and use a tower for 3 years.

    h.     If y=0, what is the value of t in Ankland¡¯s equation?  What doest this value of t mean to the cell phone company?

    Answer:  When y=0, t= 210000.  $210,000 is the amount it would cost the company to run the tower for 0 years.  It is only the amount of money the company would pay to purchase the tower.  I would wonder why the company would spend this much money to buy a tower and not use it!

    i.      One billion dollars is a lot of money.  If t = 1,000,000, what is the value of y?  Write your answer as a sentence that includes the values of t and y.

    Answer: If t=1,000,000,000, y=(1000000-210000)¡Â71200.  In a sentence:  A company would spend $1,000,000,000 to run a tower for approximately 11 years.

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    V19. Set-up Costs

    A cell phone company spends a lot of money to set up and run a cell phone tower.  The company has two big costs when it first sets up a tower.  The tower itself costs $150,000 to purchase one tower.  The company then spends $60,000 in zoning costs to have the land available to them.  What is the total initial cost the company pays?

    Answer: 150000+60000=210000 or $210,000 is the initial cost.

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    V20. Cell Phone Revenue

    A farmer sets up a cell phone tower on her farm. She charges cell phone companies to use her tower.  From each of four companies, she receives $8,560 per month.  How much money does she make each year from her tower?

    Answer: 12x4x8560=410880.  In one year, they make $410,880.

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    V21. Tower Maintenance

    For a city like Harrisburg, a cell phone company has to pay several fees in order to run a cell phone tower.  The following chart shows these charges.  How much would it cost the company to run one tower for one year?

    $150,000   Purchase the tower and have the company deliver and install it

    $60,000     Zoning costs to be able to use the land

    $10,000     Cost per year to lease the land

    $10,000     Cost per year to have the tower inspected

    $1,200       Cost per year for tower maintenance

    Answer:  150000+60000+10000+10000+1200=221200 dollars.

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    V22. Tower Highs and Lows

    A 300-foot high cell phone tower gives good service up to approximately 2 miles from the tower.  Assume the service area depends on the height of the tower.

    a.     Smaller cell phone towers are 200 feet tall.  How far from this shorter tower should it give good service?

    Answer: Using similar triangles as shown here, we have .  The shorter tower would provide good service up to 2x2/3=4/3=1 1/3¡Ö 1.3 miles from the tower.

    b.     The smallest typical tower is 150 feet tall.  For how many miles would this tower provide good service?

    Answer:  Similar triangles this time suggest .  Good service by this tower would be about 2x150/300= 1 mile from the tower.

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    V23. Tower Usage

    A cell phone tower serves cell phones that are at most 2 miles away from the tower.  If this circle represents the distance from the tower in which a cell phone can use the tower, what is the area of this circle?

    Answer: The radius of the circle would be the 2 miles between the tower and the phones at the edge of the service area.  The area of the circle is  where r=2, or 4¦Ð square miles or approximately 4(3.14)=12.56 square miles.

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    V24. Minute Use

    A cell phone calling plan offers 450 minutes a month for $69.99.  A person with this plan used 279 minutes last month.  How many unused minutes did the person pay for?

    Answer: 450-279=171 minutes

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    V25. Rural Towers

    In rural areas cell phone towers may be 20 miles apart.  If you drive 75 miles along a straight highway through rural Pennsylvania, how many towers would you expect to see if towers are spaced 20 miles apart?

    Answer: 75¡Â20=3.75, which means I should expect to see either 3 or 4 towers.

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    V26. Picture Bits

    The quality of picture transmitted from one cell phone to another depends on the resolution of the picture.  An acceptable resolution is 72 pixels per inch.  How many pixels would be needed to fill 8 inches?

    Answer: Using dimensional analysis yields the result as follows:

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    V27. Minute Charges

    The following chart describes several cell phone plans.

    Company

    Monthly Cost

    Anytime Minutes Included

    Night and Weekend Minutes

    Long Distance Charges

    Over Anytime Minute Charges

    Nextel

     

    $39.99

    500

    Unlimited

    $0.20 per minute

    $0.40 per minute

    Cingular

     

    $47.49

    500

    Unlimited

    Included

    $0.40 per minute

    Verizon

    $39.99

    500

    Unlimited

    $0.20 per minute

    $0.40 per minute

    *The above information appeared on the following websites: www.nextel.com, www.cingular.com, www.verizon.com.

    If a customer uses all 500 anytime minutes included in a plan, what is the cost per minute

    a.     Of the Nextel plan?

    Answer: 39.99¡Â500¡Ö0.08 or 8 cents per minute.

    b.     Of the Cingular plan?

    Answer: 47.49¡Â500¡Ö0.095 or 9.5 cents per minute.

    c.     Of the Verizon plan?

    Answer: 39.99¡Â500¡Ö0.08 or 8 cents per minute.

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    V28. What¡¯s Paid For         

    Ms. Sellers has the Cingular plan and used 279 minutes last month.  She pays $47.49 per month for the plan.  How many unused minutes did she pay for?

    Answer: 500-279=221 unused minutes

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    V29. Choosing a Plan

    Jos¨¦ wants to change cell phone plans.  He kept track of his phone calls for one month to help him decide on which of the three cell phone plans to choose.  He used 525 anytime minutes, 63 weekend minutes, and 2 hours of long distance during the month.  Which plan would you advise Jos¨¦ to purchase to get the best deal?  Explain your thinking.

    Company

    Monthly Cost

    Anytime Minutes Included

    Night and Weekend Minutes

    Long Distance Charges

    Over Anytime Minute Charges

    Nextel

     

    $39.99

    500

    Unlimited

    $0.20 per minute

    $0.40 per minute

    Cingular

     

    $47.49

    500

    Unlimited

    Included

    $0.40 per minute

    Verizon

    $39.99

    500

    Unlimited

    $0.20 per minute

    $0.40 per minute

    *The above information appeared on the following websites: www.nextel.com, www.cingular.com, www.verizon.com.

    Answer: On the Nextel or Cingular plan, Jos¨¦ would pay the same amount. The weekend minutes are unlimited on both plans, so the 63 weekend minutes do not matter.  The 525-500=25 over anytime minutes would cost the same on both plans (25x.4=10 dollars).  The long-distance minutes would cost nothing on the Cingular plan but would cost 2x60x.2=24 dollars on the other plans.  So, Jos¨¦ should choose the Cingular plan.

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    V30. General Cell Costs

    If Mr. Okran used 535 anytime minutes, what would his monthly bill be for each plan described in the chart below?  What would be his monthly bill on a plan with a monthly cost of $d and an over anytime minute charge of $c per minute?

    Company

    Monthly Cost

    Anytime Minutes Included

    Night and Weekend Minutes

    Long Distance Charges

    Over Anytime Minute Charges

    Nextel

     

    $39.99

    500

    Unlimited

    $0.20 per minute

    $0.40 per minute

    Cingular

     

    $47.49

    500

    Unlimited

    Included

    $0.40 per minute

    Verizon

    $39.99

    500

    Unlimited

    $0.20 per minute

    $0.40 per minute

    *The above information appeared on the following websites: www.nextel.com, www.cingular.com, www.verizon.com.

    Answer:

    Nextel ¨C 39.99+(535-500)x0.4=53.99 dollars

    Cingular ¨C 47.49+(535-500)x0.4=61.49 dollars

    Verizon ¨C 39.99+(535-500)x0.4=53.99 dollars

    General answer ¨C d+(535-500)c=d+35c dollars