## 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?

5. 5

What does it mean to say that something is infinite?

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#### Introducing...

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

#### Developing...

Resource type Title
Many ways problem Prime triangles
Scaffolded task $\sqrt{2}$ is irrational
Food for thought Scary sum
Investigation What's possible?
Bigger picture $\pi$ — what's in a number?
Bigger picture Counting fractions
Bigger picture Death by number
Bigger picture Embracing infinity
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