Level 5: Integers, powers and roots

Numbers · level 5 of 8 · rough age 12-13 (a guide for parents; anyone can start anywhere)

In level 4 you calculated with fractions, decimals and negative numbers. In this level the number toolkit grows up. You find out why a negative times a negative is positive, meet the rational numbers (every fraction, negative or not), and learn to write repeated multiplying as powers, including zero and negative powers. Then you use powers of 10 to handle the sizes of planets and blood cells, undo squaring with square roots, and finish by counting in base 2, the way every computer does.

Lesson	You will be able to
Multiplying and dividing integers	Multiply and divide any integers, and give three reasons why negative times negative is positive
Rational numbers	Place, compare and calculate with negative fractions, and show there is always another rational between two
Powers	Use bases and exponents, and explain the laws of exponents by writing out the factors
Zero and negative exponents	Work out \(a^0\) and \(a^{-n}\), explain why they must mean what they do, and say why \(0^0\) is left undefined
Scientific notation	Write very big and very small numbers as \(a \times 10^n\), and multiply and divide them
Square roots and cube roots	Find exact roots, estimate roots like \(\sqrt{40}\) by trapping and squeezing, and find cube roots of negatives
Other bases: binary, base 5 and hexadecimal	Convert between bases, add in binary, and explain why computers use base 2

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