Week 3 Β· Negative numbers, opposites, comparing, and the families of numbers Β· 3 lessons
3
Integers and rational numbers: how the number families fit together
Discover Mathematics P6: Term 1, Unit 2, Lessons 1β3
GoalSay which number families a number belongs to and write rational numbers as fractions a/b.
New words
integer Β· a whole number or its opposite: β¦ β2, β1, 0, 1, 2 β¦rational number Β· a number that can be written as a fraction a/b with b not 0counting number Β· 1, 2, 3, β¦ the numbers we use to countsubset Β· a smaller family that sits inside a bigger one
Explanation
Numbers belong to families that sit inside each other like boxes. The counting numbers 1, 2, 3 β¦ sit inside the natural numbers, the natural numbers sit inside the integers, and the integers sit inside the rational numbers. Integers are the whole numbers and their opposites: β¦ β2, β1, 0, 1, 2 β¦. A rational number is any number that can be written as a fraction a/b, where a is an integer and b is not 0. So 4 = 4/1, 0.75 = 3/4, β1.5 = β3/2 and β45 = β45/1. Every integer is rational, but a rational number like 0.5 is not an integer. To name a number, pick the smallest family it belongs to. In this course we treat 0 as an integer and we will not ask whether it is a natural number.
Which families? 7 belongs to all four. β3 is an integer and a rational number, but not a natural or counting number. 0.75 and β1.5 are rational only.
Number
As a/b
4
4/1
0.75
75/100 = 3/4
β1.5
β15/10 = β3/2
β45
β45/1
β0.2
β2/10 = β1/5
Writing rational numbers as a/b. Whole numbers get denominator 1. A decimal becomes a fraction with a power of 10 and is then simplified.
Plotting on a number line. A is β1.5, B is β1/2, C is 1/2 and D is 3/2. A and D are opposites, and so are B and C. Each one is the same distance from 0 as its opposite.
Class activity
Number family sort. Write ten numbers on cards, such as 12, β4, 0.5, β2.25, 100, β1/3, 0, 3/4, β9, 7. Make two piles: integers, and rational numbers that are not integers. Then tell a grown-up why each card is in its pile.
Practice
1
Which families does β8 belong to?
The integers and the rational numbers (β8 = β8/1). It is not a counting or natural number.
2
True or false: every integer is a rational number.
True. Any integer a can be written as a/1.
3
True or false: every rational number is an integer. Give an example.
False. 0.5 = 1/2 is rational but it is not an integer.
4
Write 0.6 and β2.5 as fractions a/b in simplest form.
0.6 = 6/10 = 3/5 and β2.5 = β25/10 = β5/2.
5
Point P is exactly halfway between β3 and β2. Which rational number does it show?
