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Potty Math

by Robert Kaplinsky

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64
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Potty Math

Act One

  • 1.

    Which toilet uses less water?

  • 2.

    What information do we need to know?

  • 3.

    When would the dual flush toilet obviously use less water?

  • 4.

    When would the single flush toilet obviously use less water?

  • 5.

    How could the two toilets use about the same amount of water?

  • 6.

    What assumptions are we making?

Act Two

  • Teacher note
    I have gone back and forth about whether to include this as a lesson. On the one hand, it is a real world situation where I had to choose the toilet that made the most economic and environmental sense. On the other hand it will make teenage boys and girls talk about going potty in class.
    The information that you need to know (and what may not be easily readable on the image) is the amount of water each toilet uses. The toilets use the unit gpf or gallons (of water used) per flush. The dual flush toilet uses 1.6 gpf for “solids” and 1.0 gpf for “liquids” while the single flush always uses 1.28 gpf.
    What is important to realize is that the answer to the question changes depending on how often the toilet is used for “solids” or “liquids”. How you determine this is an assumption I will address later in this section.
    It will be helpful to begin by having students come up with reasonable guesses to this problem. They may get stuck, so if necessary, try to pull out reasoning similar to what I have listed below.
    Consider these three scenarios:
    You use the dual flush toilet 100% of the time for “liquids”. So, it uses 1.0 gpf which is less water than the 1.28 gpf single flush toilet.
    You use the dual flush toilet 100% of the time for “solids”. So, it uses 1.6 gpf which is more water than the 1.28 gpf single flush toilet.
    You use the dual flush toilet 50% of the time for “liquids” and 50% of the time for “solids”. So, it averages 1.3 gpf which is more than the 1.28 gpf single flush toilet.
    Using this type of reasoning, students should be able to reach the conclusion that the toilets’ water usage would be about the same when the dual flush toilet is used slightly less than 50% of the time for “solids” and slightly more than 50% of the time for “liquids”.
    To solve this problem, we want to set up a system of equations and/or inequalities. If you set it up as an equation, you are finding when the toilets use the same amount of water. If you set it up as an inequality, you are finding when one toilet uses less water than the other. I am choosing to set it up to find out when the dual flush toilet uses less water than the single flush toilet. This gives us:
    dual flush usage < single flush usage
    If x is the percentage of the time the dual flush toilet is used for “liquids” and y is the percentage of the time it is used for “solids”, then you get the inequality:
    x*1.0 + y*1.6 < 1.28
    You also know that combining the percentage of time the toilet is used for “liquids” and “solids” will give you 100% so you have the equation:
    x + y = 1
    Solving for y (the percentage of the time the toilet is flushed for “solids”) you find out that y < .4666. meaning that the dual flush toilet must be used for “solids” less than 46.66% of the time for it to use less water. Similarly, solving for x (the percentage of the time the toilet is flushed for “liquids”) you find out that x > .5333. meaning that the dual flush toilet must be used for “liquids” more than 53.33% of the time for it to use less water.
    Now that we know when each toilet will use less water, it is time to address the part of the lesson that might send it down the drain. I have been avoiding discussing what percentage of the time people use toilets for “liquids” or “solids”. You could play it safe and just say that the class will assume people do each 50% of the time or some other number. Alternatively you could do some sort of anonymous poll and tally it up as a class average. I went as far as trying to find data online but that was not a pleasant Google experience. Whatever assumptions you make, I am sure it will be a memorable lesson for all.
    One last note: for a historical perspective, most toilets used to use 3.4 gpf. Today toilets now use 1.6 gpf or even 1.28 gpf. So, both of these toilets would save water compared to older models.
    I have also left the prices in the pictures in case you want to also find out when one toilet would pay for itself from the water savings.
  • ImagePicture of dual flush toilet
  • ImagePicture of single flush toilet

Sequel

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