Which numbers can't be written as the ratio of two integers?

Key questions

  1. 1

    What are our key examples of interesting families of integers?

  2. 2

    What are rational and irrational numbers?

  3. 3

    How can we manipulate numerical expressions involving surds?

  4. 4

    How can we show that a number is rational or irrational?

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Resource type Title
Rich example Divide and conquer
Building blocks Ab-surd!
Package of problems A difference of two fractions
Investigation Irrational constructions
Investigation Staircase sequences


Resource type Title
Many ways problem Prime triangles
Scaffolded task $\sqrt{2}$ is irrational
Scaffolded task Staircase sequences revisited
Food for thought Scary sum
Investigation What's possible?
Bigger picture Counting fractions
Bigger picture Death by number
Bigger picture Fractions everywhere
Bigger picture Irrational vs rational: does it matter?

Review questions

Title Ref
Can we express this unit fraction as the sum of two others? R9583
Can we write $\sqrt{2016}+\sqrt{56}$ as a power of $14$? R7658
Can we write $\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}$ as a power of $2$? R9624
Given five numbers of the form $a-b\sqrt{c}$, which is smallest? R9229
How many grid-points can be inside this circle? R7626
How many integers less than $100$ have digits that add to $8$? R5519
What is this multiple of $13$'s final digit? R6962
What links any three consecutive squares? R9507
What's the smallest number with four different prime factors? R7162
When is $4^n-1$ prime? R5927
Which of these numbers does not have a square root of the form $x + y\sqrt{2}$? R6256
Which values does this flowchart print out? R7228
Why is at most one of these numbers rational? R8276
Will these lockers be open or closed? R9596