Precalculus Quiz on Law of Sines, Law of Cosines • Luke Hodge

Introduction

This week’s assessment (a Precalculus quiz on the Law of Sines and the Law of Cosines) is from Luke Hodge. As you read through the assessment and consider adding your comments to the discussion, keep these questions in mind:

  • What do you like (and why)?
  • What would you do differently (and why)?
  • What questions do you have for Luke?

Submission Type

“Share an assessment you don’t hate…”

Assessment

Behold!

Author’s Commentary

“I was reasonably happy with the quiz because a couple of the problems (#4 & #6) allowed for different levels of insight and I got a wide range of answers. Questions #1 – #3 were things we had looked at a bunch. We had done optimizing problems like #4, but this was a new set up. Question #5 was not new, but I had not put much emphasis on “hill” problems. Question #6 and the bonus were completely new.”

On Deck

Sam Shah’s assessment on rational functions

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8 thoughts on “Precalculus Quiz on Law of Sines, Law of Cosines • Luke Hodge

  1. Luke,

    In no particular order, sorry…

    I love the hill problem!! It is authentic; how often is our street grade at 0? I have never seen a question asked quite that way. Nice.

    I also liked the tree with the dangling rope, though I found the sketch a bit confusing because I interpreted the segment marked 45 as another element … (maybe that’s just me).

    The “Extra Copy” idea is pure genius! How often do we totally obliterate the image with erasures or, heaven forbid, scratched out pen marks? That must make your students feel like they can really take risks in their solution attempts. Definitely gonna steal that one.

    I want to try the sticks and triangles problem when I have more time, so I shall not comment yet. It is intriguing. My first thought is that it’s simple, but…

    Out of curiosity, what do you use to make your sketches?

    Very nice work!

    – Jennifer

  2. First off, love the sketches, how did you do them?

    #1 and # 2 – Nice, clean, and clear. Have your students encountered situations like #2 where they have to deduce whether there is more than one triangle, or are there always told that there could be more than one? I find that my students have an awfully hard time thinking of the ambiguous case without a direct prompt to do so.

    #4 – I quite like this problem. How did they do on this? Would it have been interesting for them to have started by listing the possible triangle side combos here?

    #6 – I am curious as to why you have a part A and a part B here. I would have gone straight to part B myself. I’m interested in your thought process here.

    The second bonus question is really interesting to me. Had the students been exposed to a problem like this one before? I’d love to spring this on my Precalc Honors students in the fall and see how they do with it. LOVE IT.

    A few questions for you – (1) How long were the students expected to work on this? (2) I notice on a few occasions that you have two copies of a drawing. Do your students very actively mark up these drawings? (3) Had they already had any assessments of the boring old ‘solve this triangle’ type of problems already? I guess what I’m getting at is the question of whether you think in terms of the exercise versus problem dichotomy.

    Thanks for sharing!!!

  3. I also like # 4 and 6. Thinking on this last year’s precalculus class, they would have had a hard time on those two problems and I’ll bet half would not even have attempted them. I let them get away with too much of a lack of persistence.

  4. Do we have a secure place to discuss solutions? Part of dreaming up these great assessments is looking at student work and deciding what attributes are indicative of different levels of achievement. I’d like to bat around some approaches to solving the problems I find more interesting. Then, next year, it would be great to look at student work. I know that time will be an issue.

  5. Dan, thanks for your comments. I’m intrigued by the idea of sharing student work and various approaches to solving problems we find interesting. Do you have any suggestions for how this might happen? Maybe on a secure page (password protected) on the blog? Or maybe via Google Docs? Will the “security” squash the conversation, and is there any way around that? I look forward to brainstorming with you a bit on this.

    Feel free to reply in the comments, or to hit me up on Twitter (@mjfenton) or via email (mjfenton at gmail dot com).

  6. Responses to questions:

    They had 50 minutes to complete this. I didn’t give a “boring old solve this triangle” type quiz though they had a few problems like that for homework. Most students didn’t need the extra diagrams, but it helps a few.

    @jen, Some sort of hill problem will show up in most pre-calculus books. I like your imposed image. Maybe even get a google earth image with measurements on it? I did get a couple questions on the #2 sketch, but most of them are accustomed to my limited artistic ability.

    Tools: I used something called “efofx” to make the drawings. It is very nice in that it is a “click & edit” object in word documents. It is not nice in that it cost $. You can use the “pen” tool in geogebra, then use the “graphics tool to clipboard” to copy & past the image into a document, but you won’t be able to edit the image.

    #2: Generally my students can determine IF there is more than one possible triangle if I ask them. This was more or less a freebee in that I told them there is more than one triangle. The possibility just doesn’t occur to most of them without a prompt.

    #4: Most students came up with at least one triangle and found an area. Most did not find the optimal triangle, but many made reasonable decisions in their attempts. My intent was to see if they could use the law of cosines while also providing an opportunity for analytical thinking. We had not listed out the possibilities with this sort of problem – more an adhoc sort of approach – hopefully noticing things as we go, like if the two smaller legs are about the same size as the larger, the area will be small. I was influenced by this
    when creating the problem.

    #6: One of those sides is quite a bit more challenging. I figured most would at least have a shot at getting part a) of the problem & maybe that would push them to try the bonus.

    BONUS: They had not seen anything like the second bonus. Sadly, I think very few had enough time to give it much thought. Sometimes taking the time to think up a good bonus doesn’t really pay off on the test, but gives you some ideas for things to look at next year or leave out there as an ongoing bonus.

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