The Situation

One type of toilet comes in two models: dual flush or single flush.  The dual flush toilet has two buttons: one flushes “solids” with more water and “liquids” with less water.  The single flush toilet always flushes with the same amount of water.

 

The Challenge(s)

  • When would both toilets use the same amount of water?

 

Question(s) To Ask

These questions may be useful in helping students down the problem solving path:

  • What information do we need to know?
  • When would the dual flush toilet obviously use less water?
  • When would the single flush toilet obviously use less water?
  • How could the two toilets use about the same amount of water?
  • What assumptions are we making?

 

Consider This

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.

From my experiences, people approach this problem from one of two ways:

  • Some attempt this problem as a series of guess and check ratios.  Specifically, they know that the single flush toilet always uses 1.28 gpf.  So, they try to find out what ratio of “liquids” to “solids” on the dual flush toilet also gives an average of 1.28 gpf.  They might start with 1:1 and realize that averages 1.3 gpf.  Then they might try 2:1 and realize that averages 1.2 gpf.  From there they narrow down the ratio until they eventually get to 8:7 which has a ratio of exactly 1.28 gpf.
  • Others attempt this problem using one of these two systems of equations (which is my ultimate goal for this lesson):
    • 1.0*x + 1.6*y = total water usage AND 1.28*x +1.28*y = total water usage
      • Where x is the number of “liquid” flushes and y is the number of “solid” flushes
      • Set then equal to each other and solve
    • x*1.0 + y*1.6 = 1.28 AND x + y = 1
      • Where x is the percentage of “liquid” flushes and y is the percentage of “solid” flushes
      • The first equation describes the percentage of flushes for “liquids” and “solids” that would average out to 1.28 gpf
      • The second equation has those two percentages adding up to 1 (or 100%)


If students get stuck, consider sharing 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”.

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.

 

What You’ll Need

  • Picture of dual flush toilet

  • Picture of single flush toilet

 

Student Work

Below are low, medium, and high work samples for the challenge. Only focus on the “What is your conclusion?” part. Note that high is very relative and the samples below are far from an ideal high:

  • Low

  • Low

  • Medium

  • High

  • High

 

Teacher Work

 

Content Standard(s)

  • CCSS 8.EE.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
  • CCSS A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
  • CCSS A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • CCSS F-BF.1 Write a function that describes a relationship between two quantities.

 

Source(s)

 

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