• 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 CFL 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.            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?[3]

    Answer:

    10000÷750=13.33….  You should expect to use 14, or at least 13, incandescent bulbs to last as long as one CFL.

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    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. [4] 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?

    Answer:

    To estimate the savings, multiply the $30 savings for each replacement by the number of replacements, 17.   The savings would be at least $510.  It is mathematically important to say the savings are at least $510 because the savings on each replacement should be at least $30. However, in terms of the context, the duration of some bulbs may be more or less than that of other bulbs.  The prediction may not be accurate enough to guarantee “less than” is appropriate.

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    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.[5] Should a CFL with 4 tubes produce twice as much light as a CFL with 2 tubes?  Explain.

    Answer:

    At first glance it seems the 4-tube CFL should produce twice as much light at the 2-tube CFL because the surface area of twice as many tubes would be twice the surface area of the 2 tubes. However, the CFL with four tubes may or may not produce twice as much light as a CFL with two tubes.  If we know all of the tubes are congruent and have the same electrical power, the CFL with four tubes should produce twice as much light as the CFL with two tubes.  However, if the tubes are not of the same size with the same power, one CFL with four tubes may not produce as much light as a bulb with two tubes.

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    V4.            Lifetime of CFLs

    A tube in a CFL lasts approximately 10000 hours.[6]  If a CFL is installed in your classroom today, should you expect the CFL to last until the end of the school year?

    Answer path:

    Students likely will use specific numbers in their responses.  The answer here is given as a generic formula although students likely will use particular numbers. Estimate the number of hours, h1, of light per school day (number of school days is sd) and the number of hours, h2, of light per non-school day (number of which is nd).  Estimate how many of each type of day exists between now and the end of the school year.  Calculate the total amount of light time needed using h1*sd+h2*nd.

    Enactment:  The expectation is for students to answer this question with particular numbers based on whatever day the task is given to them.  It would be natural to ask them how the calculations they do would change if they were asked the question one month later or one month earlier.  Thinking about how they would change the numbers but still do the same operations leads them to algebraic thinking about variables.  In this particular problem, the total amount of light time needed would be calculated as h1*sd+h2*nd.  Students who are new to thinking about generalized arithmetic might only be able to express this as number of hours on a school day times number of school days plus number of hours on a non-school day times number of non-school days.  As students begin to use variables, subscripts may be a problem; writing the expression as hsd*sd+hnd*nd might be more accessible to them.

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    V5. Conversions to Find Costs

    The following figure is part of a chart that appears in a web page.[7]

     

    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.

    Answer:  is how they seemingly calculated $21.90.

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    V6.            Register Tape Questions

    The following portion of a register tape[8] shows the prices of several types of CFLs.  Use this information to answer the following questions:

    LOWE'S

    STATE COLLEGE, PA

    (814)237-2100

    -SALE-SALES #; S0273SM1 77947      09-17-04 TE#; 1840  PENN STATE UNIVERSITY

     

     

     

    76223              100W 4-pk.                             1.92

     

    76510              60W 4-pk.                               0.78

     

    51430              60W                                        0.49

     

    51800              100W                                      0.59

     

    156922            13W MINI CFL 3-pk.           11.98

     

    67196              13W MINI CFL                      4.98

     

    44258              23W MINI TWIST CFL         6.97

     

     

    VISA  XXXXXXXXXXXX1234         064231 AMOUNT:         27.71

    0273 TERMINAL;                  07  09/17/04 12:27:34

    THANK YOU PENN STATE UNIVERSITY FOR SHOPPING LOWE'S

    RECEIPT REQUIRED FOR CASH REFUND. CHECK PURCHASE REFUNDS REQUIRE 15 DAY WAIT PERIOD FOR CASH BACK. STORE MGR: RAY FORZIAT

     

    WE HAVE THE LOWEST PRICES, GUARANTEED! IF YOU FIND A LOWER PRICE, WE WILL BEAT IT BY 10%.  SEE STORE FOR DETAILS

    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?

    Answer:

    11.98÷3 yields approximately $3.99 per 13-watt bulb.

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

    Answer:

    3x6.97=20.91, 1.06x20.91≈22.1646. OR 1.06x6.97x3≈22.1646.  It would cost approximately $22.16.

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

    Answer:

    0.78÷4 = 0.195.  One of these bulbs would be $0.20.

    Enactment: Students should discuss the issue of rounding.  This discussion may be useful to address the rounding capabilities of the TI-73 or any other graphing calculator.

    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.

    Answer:

    Estimate will vary depending on the number of light fixtures in the classroom.  A reasonable student response would be rounding $4.98 to $5.00 and multiplying by the number of light fixtures.  Actual cost for a given classroom will be $4.98 multiplied by the number of light fixtures.  In their comparison of estimate and actual, students should realize that the estimate value will be slightly more than the actual cost.

    Enactment: Students should discuss the reasonableness of their estimation of $5.00 for the cost of one 13W Mini CFL.  Students should discuss the relationship between the estimated and actual costs.

    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%?

    Answer:

    $27.71 x 6% = $1.66 and $27.71 + $1.66 = $29.37

    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?

    Answer:

    $27.71 x 10% = $2.77

    Enactment: There is a nice chance here to do a bit of rounding.  The calculation of 10% of $27.71 is easy.  So, attention could be given to how we ‘round’ $2.771 to $2.77 to get the amount of the tax.

    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.)

    Answer:

    There are multiple solutions to this problem.  Following are four possible solutions:

    Way ¬:

    Number

    Description

    Unit Price

    Total

    2

    13W Mini CFL 3-pk.

    $11.98

    $23.96

    1

    13W Mini CFL

    $4.98

    $4.98

    2

    60W

    $0.49

    $0.98

     

     

     

    $29.92

    Way­:

    Number

    Description

    Unit Price

    Total

    6

    13W Mini CFL

    $4.98

    $29.88

     

     

     

    $29.88

    Way ®:

    Number

    Description

    Unit Price

    Total

    4

    23W Mini Twist CFL

    $6.97

    $27,88

    4

    60W

    $0.49

    $1.96

     

     

     

    $29.84

    Way ¯:

    Number

    Description

    Unit Price

    Total

    2

    100W 4-pk.

    $1.92

    $3.84

    1

    60W 4-pk.

    $0.78

    $0.78

    1

    60W

    $0.49

    $0.49

    1

    100W

    $0.59

    $0.59

    1

    13W Mini CFL 3-pk.

    $11.98

    $11.98

    1

    13W Mini CFL

    $4.98

    $4.98

    1

    23W Mini Twist CFL

    $6.97

    $6.97

     

     

     

    $29.63

     

    Enactment:  Students strategies may use different combinations of addition and multiplication.  For example, Way ­ is a multiplicative approach while Way ¯ is an additive approach.  It would be important to draw students’ attention to these differences.  Most students might use an additive approach, given that one “adds up” the prices to find the “total” amount.  This is a time when multiplicative thinking could be encouraged!

     



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

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

    [6] http://www.solarseasons.com/cfls.htm, pp. 1-2

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

    [8] Register tape is based on actual purchase from Lowes’, September 2004.