MYP 3 Mathematics 1.1 Understanding Different Types of Numbers

MYP 3 Mathematics 1.1 Understanding Different Types of Numbers - Full notes

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MYP 3 Mathematics | 1.1 Understanding Different Types of Numbers

MYP 3 MATHEMATICS · CHAPTER 1: NUMBER CONCEPTS & SYSTEMS

1.1

Understanding Different Types of Numbers

Full notes for MYP Year 3 · IB-style

Contents

1	Counting numbers
2	Integers: numbers below zero
3	Rational numbers: fractions and decimals
4	Irrational numbers
5	Special whole numbers
6	Classifying numbers
7	Data-based questions and further practice
8	Common misconceptions and links
9	Exam-style question with markscheme
10	Check your understanding
11	Summary and topic map

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and do every Quick check before moving to the next section.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
describe natural numbers, whole numbers and integers
use and order negative numbers
write fractions as decimals and recognise recurring decimals
recognise irrational numbers such as √2 and π
recognise square, cube and prime numbers
classify any number into all the sets it belongs to

Guiding questions

Why do we need different types of numbers? How do the number families fit together? How can we tell if a number is rational or irrational?

How this subtopic is assessed

Criterion A questions ask you to classify, order and convert numbers. Criterion B questions look for patterns (e.g. odd and even sums, terminating decimals). Criterion C rewards correct notation and clear reasoning. Criterion D uses temperatures, heights and money.

Always show working and use correct symbols.

Criterion	Name	What it rewards
A	Knowing and understanding	selecting and applying mathematics to solve problems
B	Investigating patterns	finding, describing and justifying general rules
C	Communicating	notation, representations and clear lines of reasoning
D	Applying mathematics in real-life contexts	modelling real situations and judging whether answers make sense

Command term	What the examiner wants
State / Write down	a brief answer with no working or explanation
Outline	a brief account or summary
Describe	a detailed account without reasons
Explain	a detailed account including causes, reasons or mechanisms
Calculate / Find	a numerical answer with working shown
Show that	obtain the given result, showing every step
Sketch	a graph or diagram showing the general shape and key features
Justify	give valid reasons or evidence for an answer
Evaluate	weigh strengths and limitations and reach a judgement
Suggest / Predict	propose a reasonable answer from the information given

How to use these notes

Each section teaches one topic in full. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Key point	the one idea you must not forget
Common mistake	an error teachers see every year, and how to avoid it
In the real world	where the idea shows up outside the classroom
Try this	something quick you can do yourself to test the idea
Worked example	full solutions with marks (M1 method, A1 answer, R1 reasoning)
Quick check	short questions to test yourself at the end of each section

1 Counting numbers

Where numbers began.

Think about it

Long before people wrote numbers down, they counted sheep and days with pebbles and notches. Which numbers do you think came first?

Mathematicians sort numbers into families, just as scientists sort living things. Knowing which family a number belongs to tells you how it behaves: whether it can be shared equally, whether it can be negative, whether its decimal stops. In this booklet you will meet natural numbers, whole numbers, integers, rational numbers and irrational numbers, and you will learn to classify any number you are given.

1.1 Natural numbers and whole numbers

The natural numbers (also called counting numbers) are 1, 2, 3, 4, 5, ... They go on forever: there is no biggest natural number, because you can always add 1. If we include zero we get the whole numbers: 0, 1, 2, 3, ... The three dots (...) mean 'and so on forever'.

Some textbooks include 0 in the natural numbers. In this booklet we follow the common MYP convention: natural numbers start at 1, whole numbers start at 0.

Figure 1. Counting numbers on a number line.

Figure 2. Different kinds of numbers in everyday life.

Key point

Natural numbers: 1, 2, 3, ... Whole numbers: 0, 1, 2, 3, ...

WORKED EXAMPLE 1 Identify (basic)

From the list 0, 7, 2.5, 15, −3, write down the natural numbers.

Solution

7 and 15A1

(0 is whole but not natural; 2.5 and −3 are not whole)R1

WORKED EXAMPLE 2 Identify (basic)

Write down the smallest whole number and the smallest natural number.

Solution

smallest whole number: 0A1

smallest natural number: 1A1

WORKED EXAMPLE 3 Explain (standard)

Explain why there is no largest natural number.

Solution

whatever number you choose, you can add 1R1

to get a bigger natural numberR1

1.2 Even and odd numbers

A whole number is even if it can be split into pairs with nothing left over; it is odd if there is always one left over. Even numbers end in 0, 2, 4, 6 or 8. Odd numbers end in 1, 3, 5, 7 or 9.

Calculation	Result	Example
even + even	even	4 + 6 = 10
odd + odd	even	3 + 5 = 8
even + odd	odd	4 + 3 = 7
odd × odd	odd	3 × 5 = 15
even × any whole number	even	4 × 3 = 12

Figure 3. Even numbers make complete pairs; odd numbers have one left over.

WORKED EXAMPLE 4 Classify (basic)

Is 1358 even or odd? Explain.

Solution

evenA1

it ends in 8R1

WORKED EXAMPLE 5 Predict (standard)

Without calculating, is 237 + 459 even or odd?

Solution

odd + oddM1

= evenA1

WORKED EXAMPLE 6 Reason (exam level)

Ravi says: 'If I add three odd numbers, the answer is even.' Is he right? Explain.

Solution

odd + odd = even; then even + odd = oddR1

e.g. 1 + 3 + 5 = 9 which is odd; Ravi is wrongA1

2 Integers: numbers below zero

Going negative.

Think about it

The temperature in Moscow is 5 degrees below zero. How could you write that as a number?

Whole numbers cannot describe everything. Temperatures can fall below zero, a lift can go below the ground floor, and a bank account can be overdrawn. For these we need negative numbers. The integers are all the whole numbers and their negatives: ..., −3, −2, −1, 0, 1, 2, 3, ... Zero is neither positive nor negative.

2.1 Negative numbers in real life

A negative sign in front of a number means 'below zero' or 'less than nothing'. Positive numbers can be written with or without a + sign: +5 and 5 mean the same.

Figure 4. A thermometer showing −2 °C.

Figure 5. Floors in a building: negative floors are below ground.

Figure 6. Heights above and below sea level.

Figure 7. A pocket-money balance that goes below zero.

WORKED EXAMPLE 7 Write (basic)

Write each as an integer: (a) 12 m below sea level (b) a gain of 5 kg (c) a debt of ₹300.

Solution

(a) −12A1

(b) +5A1

WORKED EXAMPLE 8 Read (standard)

Use the pocket-money chart. In which month was the balance lowest, and what was it?

Solution

AprilA1

−₹80A1

WORKED EXAMPLE 9 Difference (standard)

The bird is at +40 m and the submarine is at −80 m. How far apart are they vertically?

Solution

40 − (−80) = 40 + 80M1

= 120 mA1

In the real world

Golf scores, goal difference in football and stock market changes all use negative numbers.

2.2 Ordering integers

On a number line, numbers get bigger as you move right and smaller as you move left. So −2 is bigger than −7, even though 7 looks larger, because −2 is closer to zero on the right.

Figure 8. Integers on a number line.

Figure 9. Morning temperatures in six cities.

WORKED EXAMPLE 10 Order (basic)

Write in order, smallest first: 4, −6, 0, −1, 9.

Solution

−6, −1, 0, 4, 9A2

WORKED EXAMPLE 11 Compare (standard)

Insert < or > between −8 ___ −3.

Solution

−8 < −3A1

−8 is further left on the number lineR1

WORKED EXAMPLE 12 Data (standard)

Using the city chart, find the difference between the warmest and coldest cities.

Solution

24 − (−9)M1

= 33 °CA1

Common mistake

Thinking −9 is bigger than −4 because 9 is bigger than 4. With negatives, the bigger the digit, the SMALLER the number.

3 Rational numbers: fractions and decimals

Numbers in between.

Think about it

Three friends share two pizzas equally. Can you write each person's share using whole numbers only?

Many quantities are between whole numbers: half a cake, 1.5 litres, −0.5 °C. A rational number is any number that can be written as a fraction a/b, where a and b are integers and b is not zero. All integers are rational (for example 5 = 5/1 and −3 = −3/1), and so are all fractions and all decimals that stop or repeat.

3.1 Fractions

A fraction shows parts of a whole. The bottom number (denominator) tells you how many equal parts the whole is split into; the top number (numerator) tells you how many parts you have. Different fractions can be equal in size: these are called equivalent fractions.

Figure 10. Fractions of a pizza.

Figure 11. A fraction wall showing equivalent fractions.

WORKED EXAMPLE 13 Write (basic)

Write 6 as a fraction, and −4 as a fraction.

Solution

6 = 6/1A1

−4 = −4/1A1

WORKED EXAMPLE 14 Equivalent (standard)

Use the fraction wall to name two fractions equal to 1/2.

Solution

2/4A1

3/6A1

WORKED EXAMPLE 15 Share (standard)

Two pizzas are shared equally between three friends. What fraction does each get?

Solution

2 ÷ 3M1

= 2/3 of a pizza eachA1

3.2 Decimals

A fraction can be changed into a decimal by dividing the numerator by the denominator. If the division stops, the decimal is terminating (for example 3/8 = 0.375). If a digit or group of digits repeats for ever, it is recurring (for example 1/3 = 0.333... and 1/6 = 0.1666...). Both kinds are rational.

Figure 12. Fractions and decimals between 0 and 1.

Figure 13. Changing 3/8 into a decimal by division.

Figure 14. Three kinds of decimal.

WORKED EXAMPLE 16 Convert (basic)

Write 3/4 as a decimal.

Solution

3 ÷ 4M1

= 0.75A1

WORKED EXAMPLE 17 Convert (standard)

Write 2/3 as a decimal. Is it terminating or recurring?

Solution

2 ÷ 3 = 0.666...A1

recurring (the 6 repeats)A1

WORKED EXAMPLE 18 Convert (standard)

Write 0.35 as a fraction in its simplest form.

Solution

0.35 = 35/100M1

= 7/20A1

WORKED EXAMPLE 19 Order (exam level)

Put in order, smallest first: 0.7, 2/3, 0.67, 3/4.

Solution

2/3 = 0.666..., 3/4 = 0.75M1

2/3, 0.67, 0.7, 3/4A1

Common mistake

Writing 2/3 = 0.67 exactly. 0.67 is a rounded value; 2/3 = 0.666... goes on for ever.

4 Irrational numbers

Numbers that never settle down.

Think about it

Measure the diagonal of a square tile with sides 1 m. Can you ever write its exact length as a fraction?

Some numbers cannot be written as a fraction of two integers. Their decimals go on for ever WITHOUT repeating. These are called irrational numbers. Famous examples are √2 = 1.41421356..., √3, √5 and π = 3.14159265... The square root of a whole number is irrational unless the whole number is a perfect square (1, 4, 9, 16, ...).

4.1 Square roots

√9 = 3 because 3 × 3 = 9, so √9 is rational. But no fraction multiplied by itself gives exactly 2, so √2 is irrational. We can only find approximations: 1.4² = 1.96, 1.41² = 1.9881, 1.414² = 1.999396. Each extra digit gets closer to 2, but never exactly.

Figure 15. √2 is the length of the diagonal of a 1 m square.

Figure 16. Squares of better and better approximations to √2.

Figure 17. A square-root spiral: each hypotenuse is a new square root.

WORKED EXAMPLE 20 Decide (basic)

Which of these are irrational: √16, √7, √25, √10?

Solution

√16 = 4 and √25 = 5 are rationalR1

√7 and √10 are irrationalA1

WORKED EXAMPLE 21 Estimate (standard)

Between which two whole numbers does √30 lie?

Solution

5² = 25 and 6² = 36M1

so between 5 and 6A1

WORKED EXAMPLE 22 Estimate (exam level)

Without a calculator, is √30 closer to 5 or to 6? Explain.

Solution

30 is 5 away from 25 and 6 away from 36R1

so slightly closer to 5 (√30 ≈ 5.48)A1

4.2 The number π

If you measure round any circle and divide by its diameter, you always get the same number, π (pi). Its decimal never ends and never repeats, so π is irrational. In calculations we use approximations such as 3.14 or 22/7. Remember that these are only close, not exact.

Figure 18. A circle of diameter 1 unit rolled out flat.

WORKED EXAMPLE 23 Calculate (standard)

A wheel has diameter 50 cm. Use π ≈ 3.14 to find the distance round it.

Solution

3.14 × 50M1

= 157 cmA1

WORKED EXAMPLE 24 Explain (standard)

Amira says 22/7 is irrational because it gives π. Explain her mistake.

Solution

22/7 is a fraction of integers, so it is rationalR1

it is only an approximation to π (22/7 = 3.142857..., recurring)R1

In the real world

Engineers use π to design wheels, pipes and satellite orbits. Computers have calculated π to trillions of digits, and no pattern has ever repeated.

5 Special whole numbers

Squares, cubes and primes.

Inside the whole numbers there are some special families that you will study in later topics. Here is a first look.

5.1 Square and cube numbers

A square number is a whole number multiplied by itself: 1, 4, 9, 16, 25, ... You can arrange that many dots in a square. A cube number is a whole number used three times in a multiplication: 1, 8, 27, 64, 125, ... You can build that many small cubes into one big cube.

Figure 19. Square numbers as square arrays of dots.

Figure 20. Cube numbers as stacks of small cubes.

WORKED EXAMPLE 25 List (basic)

List the square numbers between 30 and 70.

Solution

36, 49, 64A2

WORKED EXAMPLE 26 Calculate (standard)

Find 4³ and √81.

Solution

4 × 4 × 4 = 64A1

9 because 9 × 9 = 81A1

5.2 Prime and composite numbers

A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13, ...). A composite number has more than two factors, so it can be arranged as a rectangle in more than one way. 1 is neither prime nor composite, because it has only one factor.

Figure 21. 12 makes several rectangles; 7 makes only a single row.

WORKED EXAMPLE 27 Classify (basic)

Is 21 prime or composite? Explain.

Solution

compositeA1

21 = 3 × 7, so it has more than two factorsR1

WORKED EXAMPLE 28 Reason (standard)

Explain why 2 is the only even prime number.

Solution

every other even number can be divided by 2R1

so it has at least three factors (1, 2, itself)R1

6 Classifying numbers

Putting it all together.

Think about it

Is −7 a natural number, an integer, a rational number, or all three?

The number families fit inside each other like Russian dolls. Every natural number is also a whole number; every whole number is an integer; every integer is rational. Rational and irrational numbers together make up the real numbers, which fill the whole number line. A number can belong to several sets at once.

6.1 The number family diagram

The diagram shows how the sets fit together. Irrational numbers sit outside the rational oval because they can never be written as fractions.

Number	Natural	Whole	Integer	Rational	Irrational
5	✓	✓	✓	✓
0		✓	✓	✓
−7			✓	✓
3/4				✓
0.333...				✓
√3					✓

Figure 22. How the number families fit inside each other.

Figure 23. A flow chart for classifying any number.

WORKED EXAMPLE 29 Classify (basic)

List every set that −7 belongs to.

Solution

integer and rationalA1

(also real; not natural or whole)R1

WORKED EXAMPLE 30 Classify (standard)

Classify √49 and √50.

Solution

√49 = 7: natural, whole, integer, rationalA1

√50 is irrational (50 is not a square number)A1

WORKED EXAMPLE 31 True or false (standard)

True or false: 'Every integer is a whole number.' Give an example.

Solution

falseA1

−2 is an integer but not a whole numberR1

WORKED EXAMPLE 32 Explain (exam level)

Explain why 0.121212... is rational but 0.121121112... is irrational.

Solution

0.121212... repeats the block 12 for ever, so it can be written as a fraction (12/99)R1

0.121121112... never repeats the same block, so it cannotR1

7 Data-based questions and further practice

Applying the ideas.

7.1 Data: sorting a list of numbers

A teacher wrote 21 numbers on the board: 1, 2, 3, 5, 8, 13, 0, −1, −4, −10, −15, −20, ½, 0.2, ¾, 0.333..., 1.5, −2.5, √2, π, √5. The chart shows how many belong to each set.

Figure 24. Counts of numbers in each set (a number can be in more than one).

WORKED EXAMPLE 33 Check (standard)

Check the chart: how many of the 21 numbers are integers?

Solution

natural: 1, 2, 3, 5, 8, 13 (6); plus 0 (whole: 7)M1

plus −1, −4, −10, −15, −20 gives 12 integersA1

WORKED EXAMPLE 34 Explain (standard)

Why do the bars get longer from 'natural' to 'rational'?

Solution

each set contains the one before itR1

so it has the same numbers plus some new onesR1

WORKED EXAMPLE 35 Calculate (exam level)

What fraction of the 21 numbers are irrational? Give it in simplest form.

Solution

3 out of 21M1

= 1/7A1

7.2 Graded practice

WORKED EXAMPLE 36 Recall (basic)

Give an example of an integer that is not a whole number.

Solution

e.g. −5A1

WORKED EXAMPLE 37 Convert (basic)

Write 1/5 as a decimal.

Solution

0.2A1

WORKED EXAMPLE 38 Order (standard)

Order from smallest: −0.5, −2, 1/4, 0, −1/3.

Solution

−1/3 = −0.333...M1

−2, −0.5, −1/3, 0, 1/4A1

WORKED EXAMPLE 39 Estimate (standard)

Between which two integers does √60 lie?

Solution

7² = 49, 8² = 64M1

7 and 8A1

WORKED EXAMPLE 40 Reason (exam level)

Give a number that is rational but not an integer, and lies between −1 and 0.

Solution

e.g. −0.5 or −1/2A1

it is a fraction of integers, but not a whole number or its negativeR1

7.3 Graded practice 2

WORKED EXAMPLE 41 Identify (basic)

Which of these are prime: 9, 11, 15, 17?

Solution

11 and 17A1

WORKED EXAMPLE 42 Calculate (standard)

The temperature was −4 °C at 6 am and rose by 9 degrees by noon. What was it at noon?

Solution

−4 + 9M1

= 5 °CA1

WORKED EXAMPLE 43 Explain (standard)

Is √2 × √2 rational or irrational?

Solution

√2 × √2 = 2M1

2 is rational (two irrationals can give a rational)A1

WORKED EXAMPLE 44 Analyse (exam level)

Leo says: 'All decimals are rational.' Analyse his statement.

Solution

terminating and recurring decimals are rationalR1

but decimals like 1.41421356... that never end or repeat are irrational, so Leo is wrongR1

7.4 Multi-step problem: a trip to the mountains

WORKED EXAMPLE 45 Mountain trip (exam level)

A class starts at a lake 15 m below sea level, climbs to a hut at 1240 m, then to a peak 385 m higher. The temperature at the lake is 18 °C and at the peak is −3 °C. (a) Find the height of the peak. (b) Find the total height climbed. (c) Find the temperature drop. (d) State which type of number each answer is.

Solution

(a) 1240 + 385 = 1625 mA1

(b) 1625 − (−15) = 1640 mM1 A1

(d) all three are natural numbers (so also integers and rational)R1

7.5 Exam technique

•Learn the order of the sets: natural ⊂ whole ⊂ integers ⊂ rational; irrational is separate.
•When asked to classify, list ALL the sets a number belongs to.
•Simplify first: √49 = 7 is rational, even though it is written with a root.
•Show the division when you convert a fraction to a decimal.
•Use the number line to order negative numbers.
•Write recurring decimals with dots: 0.333..., never as 0.33.

7.6 Key terms

Term	Meaning
natural number	counting number: 1, 2, 3, ...
whole number	0, 1, 2, 3, ...
integer	whole numbers and their negatives
rational number	can be written as a/b with integers a, b and b ≠ 0
irrational number	cannot be written as a fraction; decimal never ends or repeats
terminating decimal	decimal that stops
recurring decimal	decimal that repeats for ever
prime number	exactly two factors: 1 and itself
square number	a whole number times itself

7.7 Self-test

Try these without notes, then check the answers.

Quick check

1. Smallest natural number?

2. Is −3 a whole number?

3. 1/4 as a decimal?

4. Is 0.666... rational?

5. Is √36 rational?

6. First four prime numbers?

Answers: 1. 1 2. no (it is an integer) 3. 0.25 4. yes (= 2/3) 5. yes (= 6) 6. 2, 3, 5, 7

7.8 Self-test 2

Students often think…	Actually…
0 is a natural number	0 is a whole number but not a natural number (MYP convention)
−8 is bigger than −3	−8 is further left, so smaller
all roots are irrational	√(perfect square) is rational
3.14 is π	3.14 is an approximation
0.67 = 2/3	2/3 = 0.666... exactly

Topic	Connection
1.4 Prime, composite and square numbers	special numbers
1.6 Integers	operations with negatives
1.7 Rational and irrational numbers	deeper study
Sciences: temperature	negative values

Idea	Remember
Natural / whole	1, 2, 3 ... / 0, 1, 2 ...
Integers	include negatives
Rational	a/b; terminating or recurring
Irrational	√2, π: never end or repeat
Special	square, cube, prime
Classify	list every set

Part	Answer	Marks
(a)	4	A1
	√9 (= 3)	A1
(b)	−5, 0, 4	A1
	√9	A1
(c)	0.666...	A1
	recurring, does not terminate	A1
(d)	√11 and π	A1
	decimals never end or repeat / cannot be written as a fraction	R1
(e)	−5, 0, 0.25, 2/3, √9, π, √11, 4	A2