[PS] The Piano Lid

Geometry, Prentice Hall (Pearson), 2007, pg. 151


Kate Nowak gets out the knives and goes in:

I think Pearson knows that angles are hard to motivate. You would hope that the multi-jillion dollar conglomerate we paid multi-thousand dollars for books and associated peripherals would use all that money to help me. But they clearly punted on motivating angle measure. The lessons are dry and contextless and the exercises include this monstrosity.

  1. Yes people have to prop piano lids with sticks, but what could possibly happen that we would give a flip about the measures of those angles?
  2. The photo is not taken head-on. I bet that given angle isn’t even really 57 degrees.
  3. Superimposing a diagram of a triangle on a photo does not make it a real world application.
  4. Do “prop sticks” on all pianos intersect the lid at 90 degrees? Why? Is the prop-stick length and angle determined by the width of the piano? Yes, if the right angle is required, by hypotenuse-leg, and no, if it’s not, because it’s side-side-angle. I still don’t really care, but it’s at least a teeny bit more interesting.


The lid of a grand piano is held open by a prop stick whose length can vary, depending upon the effect desired. The longest prop stick makes angles as shown. What are the values of x and y?

A short prop stick makes the angles shown below. What are the values of a and b?


  1. Scan an example of pseudocontext.
  2. Email it to dan@mrmeyer.com
  3. List the textbook title, edition, and publisher.
  4. Give me your interpretation of the term “pseudocontext.”
  5. Let me know if you’d like credit (name, blog or twitter) or if you’d prefer anonymity.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. My wife is a former piano teacher, so I asked her about this. She said there is actually a short stick and tall stick (or sometimes they take the lid off entirely). It helps to direct the sound towards the audience when at different angles. So, the angles might be somewhat important in that way, but not the angles they are asking about.

  2. Thank you, Kate for being so eloquently angry about this one and thank you, Dan for starting these posts about pseudo-context. I feel like all the quiet rage bubbling up inside me about these kinds of problems and all the money they generate has been validated and I won’t end up in a cabin in the woods writing manifestos now that I know y’all are out there too.

  3. It seems like there could be an interesting question in here though it might be difficult to solve without some physics. I’d like to know: how does the volume of the piano vary with the angle of the lid?

  4. this situation could be made into an entire lesson, especially if there was grand piano in the high school. take a wimpy little stick and demonstrate that it can snap under the weight of piano lid or slide out under the weight. you can have discussions of the different forces involved from gravity and with the stick being at an angle and all, what part is lifting it up from gravity vs the part that is trying to get enough grip not to slip out. How would changing angle change all of this? lots of physics, lots of trig. getting out of the class on a “field trip” because i never get enough chances to do that. thanks folks, i got 1 1/2 months to fiddle around with our dying old piano to see how this stick thing is actually all attached and what math i can do with this and book the back stage for our “field trip”. …and my students wonder why i rarely assign work from the textbook. i am glad others out there have the same opinion and are embarrassed to ask stupid questions to students.

  5. Maybe it’s too physics-y, but it seems like a simple iPod game like Scribattle or Monkey might be an easier way of teaching angles, except the question then becomes “What is the best angle?” instead of “What is this angle?”

  6. I agree that this really reeks of pseudocontext, and has little or no connection with (or interest from) most Geometry students.

    As for trying to salvage the context for some possible lesson, my thoughts went along the same lines as @CalcDave and @Gilbert. That is, why in the first place are there different height/angle settings for the lid? Does it have to do with sound volume, or the relative position of the listeners, or some other factor(s)? Acoustically, you could consider angles of reflection where the sound waves bounce off the lid, but I’d think that’s a lesson that’s a bit further down the road for the students.

  7. Heh. I teach with this textbook and just finished this chapter. Overall opinion: the explanations of the actual concepts are good; the problems and proofs that test the concepts are good; the “real-world connection” problems are horrendously contrived. I skipped a problem this lesson that has our hypothetical hero carefully pacing around an entire lake to form a triangle and doing some calculating to determine how long their swim will be. Really? How ’bout Google Maps?

    Fortunately, I don’t care much. I’ve approached this course as having two meaningful objectives for my students: preparing them for future classes and the SAT, and using geometry as a platform to discuss logic and proof versus intuition. The ideas of postulates versus theorems are great, with ties to so many things, even religion. The “real world” problems might have some fun thinking, but I don’t need them to fake relevance.

  8. Does the angle of the lid have anything to do with the resonance of the sound? If so, what angle would provide the best resonance? This seems to be a better problem, perhaps worthy of WCYDWT. However, would this be a practical problem for the classroom? I don’t know.

    Better yet, why do I care about the angle so much as how long does the prop stick have to be?

    I found a good pseudo context the other day that I have to share. I’ll share it as soon as I get it scanned.

  9. Sound behaves much like light. The angle of reflection of the sound will be congruent about the normal. This could launch into a whole wonderful discussion about acoustics using angles.

  10. I have just recently started reading your blog! I tutor math, and I’ve started removing the psuedocontext for my students struggling with decoding questions. I have found that once they think about, “What do I need to know to figure out the question”, the students immediately know what information to extract from the question! I know it’s not exactly the goal of removing pseudocontext you have, but I have found it’s helpful!

  11. This is my favorite part of these pseudocontext posts, where we try to rehab the pseudo out of the pseudocontext. I’d like to point out (in agreement with several of you) that the person who is most concerned about angles here is the guy making the stick. There are several really good questions to ask there, but unfortunately, I don’t think any of them lead to subtracting 57° from 90°. Those questions all involve the length of the stick and, consequently, some trigonometry. Which makes me wonder if this problem is beyond rehabilitation.

  12. This discussion here made me realize another frustration I have with problems like this: they essentially state something as a fact (“the longest prop stick makes the angles as shown”) without actually backing it up. What could be an interesting research/exploratory project (more so for middle schoolers) is then reduced to a dead end subtraction problem.

    To ponder a little on one of Kate’s questions: I think the general question of whether you need a stick to be perpendicular to a slanted, heavy, hinged surface like this to prop it up is actually a great question to think about and explore and is pretty relevant (like in car hoods, for example).

    Also: this post is now number 2 on a Google search for piano prop sticks.

  13. If this is like the grand pianos I know. The hinge is at the edge. The ray that “makes angle x actually is perpendicular to the two rays making angle y and in line with the hinge. It does not exist in the same plane.

  14. I just wanted to pick a bone with Tim (not really, as I think he makes some great points):

    “I skipped a problem this lesson that has our hypothetical hero carefully pacing around an entire lake to form a triangle and doing some calculating to determine how long their swim will be. Really? How ’bout Google Maps?”

    I actually think indirect measurement problems like this provide real context for right triangle problems ( I realize the piano problem deals with angles, but Tom mentioned a problem computing sides). Indirect measurement is more of a useful and motivating example than most pseudocontext in text problems. Google Maps or not, I think it’s reasonable to think of indirect measurement as motivating context. Heck, if we’re taking into account all the tools/ tech that COULD make math easier, I’d never get off the couch…