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Rationals Between...
Age 14 to 16
Challenge Level 
Why do this problem?
This problem investigates rational and irrational numbers. Along the way, there is the chance for some reasoned arguments and the search for counter-examples, while students are challenged to apply their understanding of fractions and decimals.Possible approach
"Can you find a fraction between $\sqrt{65}$ and $\sqrt{67}$?
Give students a little bit of time to come up with ideas, and write up some of the fractions they've found.
"What values is it possible for the denominator to take? Can you find a denominator where there is no fraction that lies between $\sqrt{65}$ and $\sqrt{67}$?"
Again, give students some time to explore. Then share approaches, and if no-one has come up with something similar, show Charlie's and Alison's approaches from the problem.
"Your challenge is to find all the denominators where there is no fraction between $\sqrt{65}$ and $\sqrt{67}$, and to prove that you have found them all!"
Students may wish to use calculators, or investigate using a spreadsheet. Towards the end of the lesson, bring students together to discuss what they found, in particular, focussing on explanations why they know that after a certain point, every denominator will have a fraction in the specified range.
Give students a little bit of time to come up with ideas, and write up some of the fractions they've found.
"What values is it possible for the denominator to take? Can you find a denominator where there is no fraction that lies between $\sqrt{65}$ and $\sqrt{67}$?"
Again, give students some time to explore. Then share approaches, and if no-one has come up with something similar, show Charlie's and Alison's approaches from the problem.
"Your challenge is to find all the denominators where there is no fraction between $\sqrt{65}$ and $\sqrt{67}$, and to prove that you have found them all!"
Students may wish to use calculators, or investigate using a spreadsheet. Towards the end of the lesson, bring students together to discuss what they found, in particular, focussing on explanations why they know that after a certain point, every denominator will have a fraction in the specified range.
Key questions
What happens to the gap between $\frac{p}{q}$ and $\frac{p+1}{q}$ as $q$ gets larger?
Possible extension
An interesting extension question to ponder is whether there are pairs of distinct irrational numbers without any rationals between them. An Introduction to Irrational Numbers might be of interest to students who enjoy working on this problem.Possible support
Students could develop their understanding of fractions with Keep it Simple.You may also like
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