• Particular materials

    Mathematics Tasks

     

    Free-Response Items

    F1. CFL Replacement Questions

    CFLs fit in 29 of the 35 old light fixtures in a house in central Iowa.[1]  Is it correct to say CFLs fit in more than half of the fixtures?  Explain your answer in at least two mathematically different ways.  Tell how the answers are mathematically different.

    F2. CFL Conversion Questions

    One type of CFL is a spiral tube with a base that screws into a typical light fixture, much like an incandescent bulb.  CFLs of this type come in different sizes, as shown in the photo to the right. On each of these packages (or in on-line catalogs), there is a variety of information printed on each size.  Here is a sample of that information for the smallest CFL shown in the photo:

    Uses less energy

    Utiliza menos energia

    7w             25w

    375      è     160

    Lumens      Lumens

    a.   The “7w”, or 7 watts, on the left side is a measure of the size of the CFL.  The “25w” in the column on the right indicates that this CFL corresponds to a 25-watt incandescent bulb, a common size for a small incandescent bulb. Other common sizes for incandescent bulbs are 75 watts and 100 watts.  What size CFL (in watts) would you expect to correspond to a 75-watt incandescent bulb? What size CFL would you expect to correspond to a 100-watt incandescent bulb?

    b.   Another common type of incandescent bulb is a 60-watt incandescent bulb.  What number of watts would be appropriate for the corresponding CFL?

    c.  According to information on the package, a 19-watt CFL corresponds to a 75-watt incandescent bulb and a 23-watt CFL corresponds to a 100-watt incandescent bulb.

    i.    Compare this information with your results in part a.

    ii.    Assume the package information is correct.  What would you expect to see on a package as the CFL wattage that corresponds to a 60-watt incandescent bulb?

    d.   Another way to figure out what number of watts for a CFL would match the 60-watt incandescent bulb involves using a graph.  The following data came from CFL packages like those in the photograph.

    CFL watts

    Incandescent watts

    Lumens

    7

    19

    23

    25

    75

    100

    375

    1200

    1600

    It is possible to use the information to study the relationship between the watts of CFLs and the watts of incandescent bulbs.

    i.    Enter the CFL watt values as L1, the incandescent watt values as L2, and the lumen values as L3.

    ii.    Plot the CFL watts on the horizontal axis and the incandescent bulb watts on the vertical axis.  Be sure to determine a good window to use before creating the graph.

    iii.   Fit a linear function to the data. Record an equation to represent the graph.

    iv.   What is the slope of the line?  What is the intercept of the line?  What do these values mean in terms of the CFLs?

    A second way to use information in the chart and now in your calculator is to find a mathematical way to associate watts and lumens.  This would involve the relationship between the energy needed by the light and the amount of light the bulb produces.

         Create the scatter plot and fit a line to the data for CFL watts (in L1) and lumens (in L3).

    *    Use the scatter plot and graph to determine how many watts correspond to 870 lumens.

    F3. Circular CFLs

    CFLs come in different shapes.  One type of CFL has a circular tube. The greatest distance across the entire CFL is 8 inches.  The greatest distance across the inside of the CFL is  inches.   Estimate the surface area of the CFL.

     

    Variety items

    V1.      Comparing Bulb Life

    A CFL lasts an average of 10,000 hour. A typical incandescent bulb lasts 750 hours.  How many incandescent bulbs would you need to last as long as one CFL?

    V2.      Savings

    Replacing a 100-watt incandescent bulb with a 32-watt CFL can save at least $30 in energy costs over the life of the bulb. [2] If a business in town replaces 17 100-watt incandescent bulbs with 32-watt CFLs, what would be the savings over the life of these bulbs?

    V3. CFL Surface Areas

    For any one CFL, the total surface area of the tube(s) determines how much light the CFL produces.  Some CFLs have 2, 4 or 6 tubes.  Some CFLs have circular or spiral shape tubes.[3] Should a CFL with 4 tubes produce twice as much light as a CFL with 2 tubes?  Explain.

    V4.      Lifetime of CFLs

    A tube in a CFL lasts approximately 10000 hours.[4]  If a CFL is installed in your classroom today, should you expect the CFL to last until the end of the school year?

    V5. Conversions to Find Costs

    The following figure is part of a chart that appears in a web page.[5]

    Determine how they calculated $21.90 for the Annual Energy Cost of a 100-Watt Incandescent bulb.  Assume six hours per day and electric rate of 10 cents per 1-kilowatt hour.

    V6.      Register Tape Questions

    The following portion of a register tape[6] shows the prices of several types of CFLs.  Use this information to answer the following questions:

    a.   The 13-watt mini-twist CFL comes in a package of three bulbs.  What is the customer paying for each 13-watt bulb, excluding sales tax?

    b.   What would it cost to buy three 23-watt CFLs with 6% sales tax?

    c.   What would the cost of one 60-watt bulb be if you purchased them in a package of 4?

    d.   The custodian will be replacing each light fixture in your classroom with one 13W Mini CFL.  This means that one CFL bulb will replace each light fixture in your room.  Estimate the total of this replacement.  Then calculate the actual cost. How did your estimate compare to the actual cost?  Show your work and explain your reasoning.

    e.   The register tape here was from a purchase made through Penn State.  No sales tax was included.  What would be the sales tax on this purchase if the sales tax rate is 6%?

    f.    At the bottom of the register tape receipt, Lowe’s guarantees the lowest prices.  If these prices are not the lowest, Lowe’s will beat any other stores prices by 10%.If a customer found all of the bulbs at a lower price at another store, how much money would Lowe’s need to give back to the customer?

    g. You have $30 to purchase bulbs.  Create a shopping list of the bulbs you would buy to get as close to $30 without overspending. (Note:  There will be no sales tax added t o this purchase.)

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    Mathematics handouts

     

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    Technology guides

    Quick Tricks

    Plotting Data Points and Fitting Curves

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    Visual components

     

    Picture of incandescent light bulb

    Picture of CFL

    Register tape with prices of various CFLs:

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    Web “in-sites”

    CFL introductory video.  To initiate discussion of CFLs, one might use the GE Lighting video available at http://www.gelighting.com/na/newsletter/movies/learnmore.html.  The video has an animated speaker (a male scientist?) and it lasts less than two minutes.  At the end of this video, students may gather information about different types of CFLs.  This information would be very useful to students in designing their plan about what kind of lighting changes should be made in various locations and for various purposes throughout their school or home (as in the major project).  [Caution:  As long as the site is open in the browser window, it continues to make a sound (similar to sandpaper rubbing against wood) every few seconds.  If leaving the site open makes sense, it may be good to turn the volume down or off, particularly if students are seated close to the machine or speakers.]

    What is light? Students may raise this question as they think about CFLs and compare them to other lighting materials.  A very short but accessible description of what is light (first sentence and first long paragraph) appears at http://www.gelighting.com/na/institute/introlight.html [7]:

    Introduction to Lighting

     

    Simply put, light is a form of traveling energy.

     

    Light from the sun brings energy to the earth… energy that can be absorbed by plants for photosynthesis, by the oceans to evaporate water and cause rain, by photocells and solar panels to create electricity, and so on.

     

    When light enters the human eye and falls on the retina, it sets off photochemical and neurological processes that result in seeing. Radio waves, television transmissions, microwaves, ultra violet light, X-rays are all forms of electromagnetic waves just like light, but with a different wavelength. The human eye cannot perceive these wavelengths, but instruments can pick them up.

     

    Thomas Edison invented the first practical electric lamp in 1879. He also invented the phonograph, moving pictures, the mimeograph machine, carbon microphones and so on. Edison’s original lamp used a carbon filament in a vacuum. Today we use tungsten wire in a bulb filled with argon gas.

     

    Edison’s original lamp converted less than 1% of the electricity into light. Today’s household bulbs convert 6% to 7% into light, the rest being wasted as heat. Compact fluorescent lamps today can be 50 times more efficient than Edison’s original lamp and will last for years.

    The  background on Edison and the details about the amount of electricity converted to light may provide additional motivation for why CFLs are an appealing alternative to incandescent bulbs.

    Science connection via on-line quiz. Classroom mathematics conversation around CFLs may connect nicely with science conversations about heat, light, energy, and environment.  A particularly nice way to connect this module to science topics is through the “Fun Lighting Quiz” offered by Energy Star[8].  The quiz is on line at http://www.energystar.gov/index.cfm?c=ilg.pr_lighting_quiz_index .  For example, question number 4 (http://www.energystar.gov/index.cfm?flag=advance&cfid=90679&cftoken=44490458&fuseaction=ilg.execQuiz&x=40&y=14) opens the door to a nice opportunity to challenge a basic mal-conception of heating/cooling.

    Question 5 of the 10-question quiz provides nice motivation to understand why CFLs matter as well as to help students negotiate heat and light as different forms of energy.

     

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    Materials and equipment needed

    Students would need the following materials and equipment to work on the items in this module:

    Item

    Materials and equipment

    All

    Paper and pencil for notes and calculations; Calculators for computations

    ???

    Graphing calculator with ability to plot data points (Plot Data Points Lab Guide)

     

    IS THIS TYPE OF CHART USEFUL?

     

    In addition, it would be helpful for students to have access to actual CFLs for curiosity, for motivation, or to better understand the context for some items  (e.g., F3, V3).  Minimally, pictures of the objects would be useful – See Normal Light Bulb Photo.



     

    [3] http://www.solarseasons.com/cfls.htm, p. 1

     

    [5] http://www.solarseasons.com/cfls.htm, p. 1

     

    [7] No permission yet to use the following clip from the web site – it is here temporarily only to give us an idea of what the site holds.  In web version, excerpts from sites may be replace by a link to the site that then appears as a pos-up box.

    [8] ENERGY STAR® is a collaboration between the U.S. Department of Energy, the U.S. Environmental Protection Agency, and participating companies with the goal to prevent pollution by giving consumers energy-efficient product choices without sacrificing on quality. The ENERGY STAR® label makes it easy to identify products that will save money and help protect the environment. The Energy Star Partner of the Year Award announced February 5, 2004 went to GE.