Writing Word Problems

I find it quite easy to complain about context in real-world scenarios.  However, when it comes to writing word problems which obey all of my rules things become much harder.  We’re coming to the end of the unit on triangle trigonometry and my CT asked me if I could write a set of word problems which required the use of basic trig, the law of sines, and the law of cosines.  My personal goal was to make these problems require a context to make sense, complement students’ intuition about the natural world, and be interesting enough to not make students groan when reading the problem.  In retrospect I realized that I used a lot of diagrams – something which I tend to avoid as it simplifies the problem for the students and removes some of the conceptualization aspect.  I wrote some justification for them after each problem.

1)   (Tweaked from Dan Meyer) A high-rise is on fire. Everyone has evacuated except for a man and his son who are waving and coughing from a window on the fourth floor (thirty-five feet off the ground). The firemen arrive on scene with their forty foot ladder.  In order for the ladder to be climbed safely, it needs to be angled at least 25^o from vertical.  The question running through everyone’s mind is whether or not the latter is long enough.

a)    Draw a diagram of the scene.
b)   Use trigonometry to determine whether or not the ladder will reach the trapped people.
c)    What is the maximum distance the ladder can be placed from the building so that it reaches the people?
d)   If the ladder is placed at the maximum distance, what angle will it make with the building?

2)    When John was building houses in Louisiana part of his job was to cover the roof in shingles.  In order to know how much money he needed to spend on shingles, he needed to figure out the surface area of the roof (note: This is actually a problem I had to solve while was building houses in Louisiana).

a)    The standard angle for a roof in Louisiana is 30^o.  What is the length of one side of the roof?
b)   What is the surface area of the entire roof?
c)    Shingles cost twenty dollars per case.  Any good construction worker knows that three cases of shingles cover one hundred square feet.  How much money does he need to spend on shingles for the house?

Note: I included a simple diagram here because there could be ambiguity in the structure of the roof being lengthwise or widthwise.  In giving that information (and only that information) visually I hope to nip the ambiguity in the bud.  If this was a more open-ended, exploratory activity, a much more interesting question would be “which roof would cost less to shingle – building it lengthwise or widthwise,” but given the setting of this problem (review for a test) I left it this way.

3)    On the hills to the west of Stanford there is a giant satellite dish (you can see it from highway 280).  It is surrounded by a barbed wire fence so you can’t get too close, but we can still calculate how tall it is.  From the fence the angle of elevation is 70⁰.  If you back up 90 feet from the fence, the angle of elevation is 65^o.  How tall is the dish?

Note: I included a diagram here because, quite frankly, I think this problem is really hard.  It requires use of linear pairs, the triangle sum theorem, the law of sines, and right angle trigonometry.  I want all of my students to have access to the complexity of this problem and fear that without a diagram some students would miss out on the wonders of the problem.

4)  Every year there is a paddle board race from the Santa Cruz harbor to Moss Landing.  We can use triangle trigonometry find the total distance contestants swim.  The distance from Santa Cruz to Mt. Madonna County Park is 80 miles.  The distance from Moss Landing to Mt. Madonna County Park is 50 miles.  The angle between these two distances is 70^o.  What is the total distance of the race?

Note: I included a diagram here because I believe it makes the problem real for the students.  In order to solve the problem they don’t need a diagram, but I would just be talking about a race from point A to point B and the extra point C.  The context of this problem and accompanying diagram is closely related to the Wason Selection Task.

Are these problems pseudocontext? are they amorphous?  I don’t think so… but I’m not entirely sure.  Some things are easy to define but hard to judge.  The more I think about a real world context the less I feel like I can judge their quality.