• Lesson ideas

    There are three types of lesson ideas.  Open-ended projects are major activities that engage students in answering non-trivial questions about the real-world contexts.  These activities could be treated as projects that take several days or class meetings.  Free-response items are individual problems or sets of problems. They require students to make decisions, calculate values, and do other things to construct their own answers.  For all or at least most of these open-ended items, there are specific suggestions about how the items may play out in the classroom.  Variety items are stand-alone problems that could be used in many different ways, including review and practice.  These items may be used to revisit previously learned materials while students are working either on a project or on several open-ended items related to CFLs. The variety items also may be incorporated as applications of mathematics into lessons in a standard mathematics curriculum and used separately from the projects and free-response items.  However, some of the variety items may provide insights into things to consider in the projects or free-response answers.

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    Open-ended projects

    P1. School Lighting Project

    Lighting is important in a school.  Your school likely pays a large amount of money for electricity to light the hallways, classrooms, and other spaces in your school. In addition to these monthly electric bills, your school also paid to purchase the light fixtures and to replace light bulbs in those fixtures.  A current claim is the a relatively new type of light bulb may help your school – as well as other schools, offices, and homes – to save money on lighting costs over time.

    This new type of light bulb is a compact fluorescent light bulb, or CFL for short.  Should your school save money if it replaces its incandescent light bulbs with CFLs?  Explain your recommendation.  Write a report to your principal that informs the school about what to do and why the plan makes sense.  Be sure to explain in detail how your plan will help the school save money over the next several years.

    Answer:  It is very clear that the details of students’ projects will vary from one school to another. It is also clear that the answers may vary from student to student.  The expected conclusion in all typical cases (e.g., not a case where the particular school building is about to be closed, making the replacement task a waste of time and resources) is that changing to CFLs will lead to savings eventually but not immediately.  Minimally, the students’ explanations should include sources of fixed costs (e.g., cost of purchasing the new CFLs, installation costs) and sources of savings (e.g., decreased electricity consumption). Students will need data about the current light use in their school.  Graphs and charts may be used effectively to convey the comparisons.  Geometry may come into play if students need to decide where to locate fixtures or to determine the area lit by a particular light source.

    Enactment:

    ·        Take time to develop the context and to motivate the project.  Perhaps the principal or central office person could visit the class in person or via video to set-up the school’s need and to provide students with a sense of the real dollar values attached to the cost of lightening the school.

    ·        It is tempting to give students answers to all of their questions.  It often would be more beneficial to student to encourage them to seek the information rather than to receive it.  There are several potential resources (web sites, books, etc.).  If students have difficulty locating information, it may help to share the Web “in-sites” with them.

    ·        It is tempting to present the students with a data set rather than having students collect the data.  However, this move defeats the spirit of the intended lesson.  In collecting the data students negotiate several measurement issues (e.g., how to measure parts of the CFLs).  They also have to make decisions that involve various areas of mathematics (e.g., rounding, sampling).  However, it may be reasonable to have information for the students about who to ask about the school and to have these people ready to respond.  For example, the custodian may tell them how frequently light bulbs are changed and an office staff person may know the school’s cost of purchasing replacement bulbs.

    ·        Students may need to develop some clever ways around particular data collection problems.  Here are some issues that might arise and ideas about how to address the issues:

    o       Students likely will ask, How many incandescent light bulbs are in the school? If it is not possible to count all of the lights in the school, students can be asked about the number of light bulbs in some small part of the school (e.g., the auditorium).  It is important that chosen space (auditorium) has a large number of bulbs. Note that a CFL does not provide exactly the same area of light as an incandescent bulb.  Also, CFLs can be purchased for use in standard (incandescent) light fixtures; CFLs that do not fit standard fixtures also exist.  Students may need to consider the relative prices of these different kinds of CFLs.

    o       Students likely can see the light fixtures but they may not be able to see the number of incandescent bulbs in each fixture.  If they can see the number of bulbs, they likely cannot see the size of the bulb (e.g., 60 watt versus 100 watt).  This may be a good time to consult with the building staff responsible for the lighting.  They may be able to tell students, for example, what size bulbs are used where.

    ·        If students have difficultly getting started, it might help to ask them what factors they think they need to include.  If students continue to struggle, or as a way of helping students to think about how these factors relate to their project, it might be useful to ask questions about the existing and proposed lighting. Some questions that students may need to answer in their work on this activity are:

    o       How many incandescent bulbs are in the school?

    o       How much light do the incandescent bulbs provide?

    o       How many CFLs would be needed to provide this amount of light?

    o       What amount of power do the existing bulbs use?

    o       Will this be enough power for the replacement CFLs?

    o       What will be the initial cost of installing the CFLs?

    o       Will any new light fixtures be needed?

    o       How much energy will the new bulbs save in a month?  in two months? In three months? In n months?

    o       And then there is the ultimate question: is it a good idea for the school to switch to CFLs?

    ·        Students may have to deal with the fact that they cannot merely replace one incandescent bulb with one CFL.  So, they will have to determine how they will deal with the temptation to claim one CFL is the equivalent to so many incandescent bulbs.  As a result, students see that they cannot simply divide the number of incandescent bulbs by some number to get the number of CFLs needed – after all, some of the bulbs may be lone bulbs in small rooms while other bulbs may be in large collections (such as several dozen bulbs in the auditorium).]

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    P2. Room Lighting Project

    A smaller version of the School Lighting Project is to have students determine whether and how to change the lighting in their classroom, the library, or some other specific room or portion of the school.

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    P3. Home Lighting Project

    A variation on the School or Room Lighting Projects would have students determine whether and how to change the lighting in their homes.  This option adds the obvious opportunity for parental involvement.

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    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.

    Answer:

    Way :    So, the CFLs fit in more than one-half of the fixtures.

    Way :

    Way ƒ:   So, the CFLs fit in more than half of the fixtures.

    Way :   The CFLs fit in more than half of the fixtures.

    Explanation of differences and similarities:  Ways  and ƒ use fractions to make the comparison. Way  compares the fraction of fixtures to one-half using percents; 50% represents one-half.  Way ƒ compares the fraction of fixtures to one-half expressed as an equivalent fraction with an even denominator.  Ways and use whole number relationships to make the comparisons. Way compares the number of fixtures the CFLs fit to half of the fixtures available.  Way compares the number of fixtures fit to the number not fit with the understanding that the number fit and the number not fit would be equal if exactly CFLs fit exactly half of the fixtures.

     

    Enactment: Students should discuss which ways are easier than other ways for the given problem and for related problems.  In the given problem, for example, comparing 29/35 (Way ƒ) to 1/2 is easy when 29/35 is written as an equivalent fraction with an even denominator, 58/70.  58/70 would also be a useful form of 29/35 if students had to compare 29/35 to a number of tenths since 10 divides 70 but 10 does not divide 35.  For example, given 58/70, it is easy to see that 29/35 is greater than 4/10=28/70 with 58>28. However, 29/35 is already in a handy form if the comparison is to some number of fifths. For example, is 29/35 less than 2/5 can be done by quickly expressing 2/5 as 14/35 and seeing 29 is not less than 14.  This discussion may be a useful reminder to students about how and why common denominators are used to compare two fractions.

    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?

    Answer:

    Way : 21-watt and 28-watt CFLs correspond to 75-watt and 100-watt incandescent bulbs, respectively.  The mathematical reasoning for this answer may be 75 watts would be 3 times 25 watts for the incandescent bulbs and so for the CFLs one would expect 3 times 7 or 21 watts.  Similarly, 100=4x25 implies 4x7=28.

    Way :  and rewriting fractions with common denominators suggest a 75-watt incandescent bulb matches a 7x3=21-watt CFL.  Similarly,  suggests a 7x4=28-watt CFL matches a 100-watt incandescent bulb.

     

    Enactment: Students at this grade level likely can apply multiplicative thinking or constant ratio as suggested in the sample answer.  Students learn in the later parts of this item that this mathematical reasoning may not quite match the real-world context.  Have students present these different methods and compare them, noting that factor-product relationship between 25 and 75=25x3 or between 25 and 100=25x4 makes it easy to arrive at 21=7x3 and 28=7x4.  If students do not suggest one of the methods, it may be useful to introduce that method after students attempt the next part.

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

    Answer:

    Way : 60÷25=2.4 and 2.4 x7=16.8.  A 16.8-watt CFL would correspond to the 60-watt incandescent bulb.

    Way : , 60x7÷25=16.8.  A 16.8-watt CFL matches the 60-watt incandescent bulb.

     

    Enactment: Students should see that this question is answered in a way that is nearly identical to the questions in part a.  Both questions ask for a prediction. The fact that 60 is not a multiple of 25 makes part b more difficult than part a.

    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.

    Answer:

    The package information gives watt values that are less than the values I calculated for the incandescent bulbs.

     

    Enactment: Ensure students know their mathematical thinking in part a could be absolutely perfect and the lack of a match to this answer in part ei has to do with the real-world issues yet to be considered.

    I student know two point – not one point – completely defines a line, now may be a good time to have them think about why only the one data point, (7,25), was not enough information to get an answer that would match reality.  A more general point is that having information about only one thing (i.e., one package) simply is not always enough information on which to base a general conclusion about all such things.

    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?

    Answer:

    Way ¬: For the incandescent bulbs, 60 watts is greater than 25 watts and less than 75 watts, but closer to 75 watts than to 25 watts. So, the CFL should be more than 7 watts and less than 19 watts, and probably closer to 19 watts.

    Way ­:  .  The CFL should be labeled as 19x60÷75=15.2 watts.

    Way ®:  .  The CFL would be marked 23x60÷100=13.8 watts.

    Way ¯:  Using  and , the value for the CFL could be something like 13.8 or 15.2 watts.

    Way °:   The value of the ratio seems to decrease as the number of watts becomes larger.  For the 60-watt incandescent bulb, the value seems to be between 0.25 and 0.28 but closer to 0.25.  Suppose the value is 0.26.  The CFL would be labeled 0.26x60 or 15.6 watts.  However the number of watts in all other cases is a whole number.  So, the CFL may be labeled 16 watts.

     

    Enactment: Students may be flustered when they come up with clearly conflicting answers when using what they should know are mathematically correct calculations.  This item leads students to the data analysis technique in part d, thought it may be possible for students to stop with part e. In preparation for future work, have students present their answers in the order of greater precision.  The order of precision from least to most for the given methods is Way ¬, Way ­ or Way ®, Way ¯, then Way °.  In Way °, there is the essence of simultaneously considering all data points, as students would learn to do in part d. Be sure students justify Ways ­ and ® as mathematically correct despite the fact both of the obviously unequal numerical answers could not be correct in the context.  Creating the scatter plot in the next item may help students understand why these conflicting answers happen.

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    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.

    Answer:

    The values should appear in the lists as indicated.

    Enactment: For instructions on how to enter data and plot points, see Plotting Data Points and Fitting Curves. Students may need substantial guidance in this calculator work.  They will have opportunities to practice these calculator skills.

    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.

    Answer:

    The CFL watt values range from 7 to 23.  One good window would have x-values from 0 to 25 in increments of 1.  The incandescent bulb watt values range from 25 to 100.  One reasonable window would have y-values from 0 to 120 in increments of 10.

     

    Enactment:  One nice aspect of using a the value from 0 to 120 in steps of 10 is that 60 will appear exactly as a y-value when the cursor moves around the window.  This happens because the number 60 is in the sequence {0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120}.  Notice that 60 would also appear for a window with y-values from 0 to 120 in steps of 20, as 60 appears in the sequence {0. 20. 40. 60. 80. 100. 120}.  However, 60 does not show up as a y-value if the window runs from 0 to 120 in steps of 25; 60 does not appear in the sequence {0, 25, 50, 75, 100}. While discussing students’ strategies for this problem set, it would be reasonable to have students talk about how they can predict if 60 would show up as an exact value in the window.

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

    Answer:

    The graph of the fitted function appears below, with the expression in x for y at the top of the screen.

    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?

    The slope is approximately 1.7 and the intercept is approximately 10.  This slope implies that an incandescent light uses 1.7 watts for every 1 watt used by the CFL.  The intercept indicates that a 10-watt incandescent light uses the energy of a 0-watt CFL.  This does not seem to make sense in reality.

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    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.

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

    Answer:

    The scatter plot and its window information appear first, followed by the fitted line’s equation as Y2 and its graph.

     

     

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

    Answer:

    Moving the tracer until 870 (or a value close to 870) appears as the value for y.  The number of watts is approximately 13.8, or close to 14 watts.

     

    Enactment: Students may notice by now that the watts are given in integer values.  So, rounding the result of the calculator work makes sense in the situation.

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    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.

    Answer:

    The CFL does not have a shape that matches the usual formulas for surface area.  The rounded nature of the CFL makes it impossible to calculate the surface area as a sum of the area of flat surfaces.  So, to estimate the surface area, a reasonable method is to use a more common figure as a model of the CFL.

    One way to think about the CFL in a simpler form is to image that the CFL is ‘flat’ on top and on the bottom and has vertical sides.  The visual image is that of a three-dimensional “disk”, as shown here:

    The surface area of this model would be the sum of areas of the top, the bottom, the inside, and the outside.  These areas can be calculated as follows:

    Top:  area of a large circle – area of a smaller circle

                =  =

    Bottom: same as top

    Inside vertical part: height x circumference of smaller circle

                =  =

    Outside vertical part: height x circumference of larger circle

                =  =

    Sum: 92.6+92.6+78.4+106≈370

    An estimate of the surface area is 370 square inches.

    Enactment: Students may have other strategies for computing the surface area.  Many valid methods are possible.  If students have difficulty getting started, it may be useful to have them think first of what two-dimensional object reminds them of the CFL. Circle would be an expected response.  They may then see the region of the top of the figure as the area between two concentric circles. This task may be used either for students to apply their understanding of the areas of familiar shapes or to introduce the need for surface area.  Years later in their mathematics experience, students could return to this task and compute the volume and surface area of this figure (a torus) using calculus techniques.

    F4. Comparison Chart Completion

    The following chart appears on a web site.[2]  All of the numbers in the chart are related to other numbers in the chart.  Explain how the person who made this chart could have computed each of the following numbers.

    a.       $40.60

    Answer:

    ($5.91 x 4.5 years) + $14.00 = $40.60

    Enactment: Students should be able to discuss the importance of including the initial cost of the lamp in their calculation. Note:  Because the Ti-73 and any other graphing calculator will perform order of operations on expressions presented, this may be a place to pursue the topic of order of operations.  

    b.  $103.55

    Answer:

    ($21.90 x 4.5 years) + (10 lamps x $0.50) = $103.55

    Enactment: Students may encounter difficulties in the reasoning necessary for this calculation due to the difference between the lamp lives of the CFL and incandescent lights.

    c.       $62.95

    Answer:

    $103.55 - $40.60 = $62.95

    d.      10

    Answer:

     A 100-Watt bulb lasts for only 167 days.  The 27-watt CFL lasts 1642.5 days.  So, it takes 1642.5÷167 or approximately 10 days.

    Enactment: The value obtained through this calculation is actually 9.8.  This could lead to a valuable discussion on the issues involved with rounding.

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