False Position Word Problem

 I recently painted two twenty-unit battalions of Warhammer 40k miniatures (toy soldiers for even bigger nerds) along with one four-unit squadron. This cost $44 in materials (paints, primer, etc.). If I were to paint one more twenty-unit battalion, how much would this cost me (assuming that prices are the same)?



If I were to solve this using the method of false position, I would begin by setting up an equation based on what I know: namely, that 2 sets of twenty and one fifth of a set of twenty cost $44.

2x+x/5=44

In order to deal nicely with the fraction, I will assume that the cost of a set of twenty (x) is five:

2(5)+5/5=44

10+1=44

11=44

Because I am off by a factor of four, I multiply my initial guess by four to get $20. A quick test demonstrates that this is correct:

2(20)+20/5=44

40+4=44

44=44.

Additionally, as a short reflection, I find this method intriguing. It's something I had never encountered before, despite doing a bit of research into historical math myself for other classes as well as always being on the lookout for new methods of approaching algebra for students. Something I particularly like about the method is how it focuses on choosing a reasonable, appropriate first guess. It's an elementary skill for you or I, but the idea of picking a convenient number (in this case 5) is both highly mathematical in its thinking and also very practical when students arrive at problems that involve making a table of values, for instance.

Comments

  1. Susan GerofskyNovember 9, 2020 at 11:17 AM

    OK, but I'd like to see you work this out here using the method of false position as well! Could you please add this in a revised post? Cheers!

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  2. Susan GerofskyNovember 18, 2020 at 11:23 AM

    Thanks for adding your solution! One thing I would suggest avoiding is the use of non-equations with an equal sign (like 11=44). You will want to discourage your students from using the equal sign in this kind of situation, and it's important to model a better way of notating it. (I sometimes use an equal sign with a question mark above it...) Very interesting reflection on this process of false position!

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    Replies
    1. JacobNovember 18, 2020 at 2:07 PM

      It's funny that you should say that: I also use a question mark above the equal sign when writing out by hand. I wasn't exactly sure how to translate that to typing. I suppose I could draw inspiration from the coding world (!=) and go with ?=.

      Delete

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