From BluWiki

>Wikiteacher Home

Overview

This topic explores variations, generalizations, and extensions of problems related to measure time using hourglasses (technically minute glasses, I suppose) of particular amounts.

Theorems:

• The greatest number that cannot be exressed by ax+by where x,y are relatively prime and a,b are whole numbers is equal to nm-n-m

• related unsolved (open) problems

Motivating questions and problems

• Given two sand hourglasses that can measure 5 and 7 minutes respectively, how can you measure exactly four minutes?
• Given two sand hourglasses that can measure n and n+2 minutes respectively, how can you measure exactly four minutes? (potential motivator for discussing relatively prime numbers)
• Given two sand hourglasses that can measure 5 and 7 minutes respectively, what is the greatest length of time that cannot be measured from the outset if I can only use one hourglass at a time? (equivalent to the chicken mcnugget problem)
• Given two sand hourglasses that can measure 5 and 7 minutes respectively, what is the greatest length of time that cannot be measured from the outset if I can use both hourglasses at the same time?
• Given two sand hourglasses that can measure n and n+2 minutes respectively, what is the greatest length of time that cannot be measured from the outset if I can use both hourglasses at the same time?
• Given two sand hourglasses that can measure n and m minutes respectively, what is the greatest length of time that cannot be measured from the outset if I can use both hourglasses at the same time?
• Given two sand hourglasses that can measure n and m minutes respectively, what is the greatest length of time that cannot be measured if I can use both hourglasses at the same time?
• Is the hourglass problem equivalent to the water bucket problem that asks "How do you measure exactly 4 gallons of water with a 5 and 7 gallon bucket"?

Lesson plans

• Given two sand hourglasses that can measure 5 and 7 minutes respectively, what is the greatest length of time that cannot be measured from the outset if I can only use one hourglass at a time? (equivalent to the chicken mcnugget problem)

With a little work, students should be able to find a number of different times that are possible, and possibly find solutions for 24, 25, 26, 27, and 28. If students seem to be scattered, you could suggest to look for solutions for times under 30. Using the basic concept of induction, students can then prove that any time over 28 will be possible by getting to the appropriate number between 24 and 28 and then adding some number of fives.

You could extend this to the problem with n and m minutes and attempt to find which problems will not have a largest impossible time, and lead students to the idea of relatively prime numbers.

Once students are comfortable with the kinds of problems that will have largest impossible times, they can attempt to come up with an expression to describe that amount. With a good bit of data, students should be able to conjecture that this will be nm-n-m or, alternately, (n-1)(m-1)+1.

• Given two sand hourglasses that can measure 5 and 7 minutes respectively, what is the greatest length of time that cannot be measured from the outset if I can use both hourglasses at the same time?

Approach this in the same way as the above problem. Initially, students will need to find the difference between this problem and the last. Again, with a bit of dat, students should be able to conjecture that, starting with n and n+2 when n is odd, the largest time that cannot be made this will be 2n-2.

Proof: n minutes is possible using the first hourglass. n+2 minutes is possible using the second hourglass. n+4 minutes is possible by flipping over the first hourglass as soon as it runs out and then flipping it over again after the second runs out. Using this same method, you will be able to add any number of 2 minute intervals by going back and forth in this manner. This implies that every odd number after n will be possible.

n+(n+2) is also possible by flipping the first and then the second. This can be rewritten as 2n+2. You can then do 2n+4 by using the above method and accruing 2 minutes in the first hourglass while the second is going. This can be extended to every subsequent even number.

2n can be done by flipping the first hourglass twice.

This leaves the smallest amount that cannot be done as 2n-2, since 2n-1 will be an odd number greater than n+2 as long as n>2.

2n-2 will not be possible because you cannot do anything other than start them both at the same time. The first time you will be able to determine the amount of time that has passed will be n. This would imply that you would need to measure n-2 more minutes. There will not be a way to do this since your two hourglasses will be at n and 2 minutes respectively.

Exercises

Posted here should be links to lessons and sheets intended to practice and reinforce concepts, ideas, and vocabulary

Problems

Posted here should be links to lessons and sheets intended to explore the topic in a more abstract and deep way

Extensions

Posted here should be ideas for projects or subtopics

Assessments

Posted here should be quizzes, tests, or other assessment tools

Follow up topics, or Where to next?

Primality, relative primes, induction

External links

Links to useful outside sites related to this topic.