When Should She Take Her Medicine?

 
Your friend is sick and has to take medicine five times a day for seven days.
 

• How often should your friend take the medicine?

 

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

• What is a guess that is too low?
• What is a guess that is too high?
• What is your best guess?
• What assumptions are we making?

 

This simple scenario packs a surprising amount of math.  There are a few assumptions I intentionally left out of “The Situation” that I hope will come out from the problem’s context.  They include:

• You want to spread out the medicine doses so that there is a reasonably equal amount of time between each dose.  That allows your body to have a consistent level of medicine in it to get better.  So, it is not an option to take all five doses at once.
• I have not defined when the friend is awake.  This allows you to consider all sorts of scenarios including defining the day by picking a wake up time and bedtime or making the medicine so important that she has to be woken up to take it.
• In the context of this problem, the fact that she has to take the medicine for seven days is a real-life distractor.

What is interesting about this problem is how trivial the problem is when taking three or less doses a day.  It quickly ramps up in difficulty beginning with four pills.  So, let’s consider all the medicine dosages up to five:

• One dose a day is easy.  Pick a time and take it at the same time every day.
• Two doses a day is also easy.  Take one dose in the morning and one in the evening.
• Three doses a day is still reasonable.  Take one dose in the morning, one in the middle of the day, and one in the evening.
• Four doses a day starts to get a little challenging.  You can take one dose in the morning, one in the evening, and try to space out the other two before and after lunch.
• Five doses is where it really starts getting challenging.  When do you take it?

There are limitless possibilities, so let’s consider what may seem like the easiest possible situation.  The friend only takes doses of medicine while awake, and wakes up at 7 AM and goes to bed at 10 PM.  So, she is up for 15 hours.  The first dose of the day will be when she wakes up and the last dose of the day will be when she goes to bed.  Since the friend needs to take 5 doses a day, it seems like the the friend should take the medicine every 3 hours (15 hours / 5 doses).  Consider how that works out though:

• 7:00 AM – Dose #1
• 10:00 AM – Dose #2
• 1:00 PM – Dose #3
• 4:00 PM – Dose #4
• 7:00 PM – Dose #5

We were expecting the last dose to be taken at 10 PM, when she goes to bed, however the last dose was taken at 7 PM.  So, what happened?

The problem is that while five doses of medicine were administered, there were only four time periods between the medicine.  The image below illustrates this as each red circle represents a dose of medicine along the black line that represents when she is awake.

So instead, we should have divided the 15 hours by 4 time periods giving us 3.75 hours between doses.  This leads to more math as students then need to determine that 3.75 hours is equivalent to 3 hours and 45 minutes.  They can verify this by carefully adding the times.  So, she should take her medicine at:

• 7:00 AM – Dose #1
• 10:45 AM – Dose #2
• 2:30 PM – Dose #3
• 6:15 PM – Dose #4
• 10:00 PM – Dose #5

This is what we were hoping for!

As an extension, this problem gets harder when you:

• Start with a number of waking hours that is not divisible by 4 without a remainder.  So, you could try this many times with different waking hours.
• Increase the number of doses.  What happens if your friend take the medicine 6 or 7 times a day?

 

Kate Fisher has a wonderful description of how she implemented this problem that is well worth reading before trying it out with your students.
 

• CCSS 4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

 

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