MYP 2 Mathematics 1.1 Place Value and Number Representation

MYP 2 Mathematics 1.1 Place Value and Number Representation - Full notes

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MYP 2 Mathematics | 1.1 Place Value and Number Representation

MYP 2 MATHEMATICS · CHAPTER 1: NUMBER CONCEPTS & SYSTEMS

1.1

Place Value and Number Representation

Full notes for MYP Year 2 · IB-style

Contents

1	Episode 1: The digit detectives
2	Big numbers: thousands, lakhs and crores
3	Tiny numbers: decimals
4	One number, many costumes
5	Detective challenges
6	Check your skills
7	Common misconceptions and links
8	Exam-style question with markscheme
9	Check your understanding
10	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
find the value of any digit in a number
read and write big numbers in words
use the Indian (lakh, crore) and international systems
understand tenths, hundredths and thousandths
multiply and divide by 10, 100, 1000
show numbers in different forms

Guiding questions

Why does the place of a digit change its value?

How this subtopic is assessed

Criterion A questions ask you to find place values and write numbers. Criterion B asks you to spot patterns, such as multiplying by 10. Criterion C rewards clear number writing. Criterion D uses money, populations and measurements.

Always check the place of every digit!

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 Episode 1: The digit detectives

Where a digit sits decides what it is worth.

Think about it

Would you rather have the 9 in 19 rupees or the 9 in 900 rupees? Why?

Mia and Leo are counting money after the school fair. Leo writes the total as 1968. Mia writes 1698. 'Same digits!' says Leo. 'But not the same money,' laughs Mia. The position of each digit changes its value. This idea is called PLACE VALUE.

We use only ten digits (0 to 9), but by putting them in different places we can write any number in the world: your phone number, India's population, or the tiny width of a hair.

1.1 Every place has a value

Figure 1. Leo and Mia discover place value.

Figure 2. Your mission for this topic.

Figure 3. 1234 shown with blocks.

Figure 4. Five digit cards.

Key point

Each place is worth 10 times the place on its right. The value of a digit = digit × its place.

WORKED EXAMPLE 1 Value (basic)

What is the value of the 9 in 1968 and in 1698?

Solution

1968: 900 (hundreds)A1

1698: 90 (tens)A1

WORKED EXAMPLE 2 Cards (standard)

Use the cards 4, 0, 7, 1, 9 to make the largest and smallest five-digit numbers.

Solution

largest 97 410A1

smallest 10 479 (0 cannot go first)A1

WORKED EXAMPLE 3 Explain (standard)

Why can't 0 go first in the smallest number?

Solution

a number does not start with 0R1

04 179 would really be a four-digit numberR1

2 Big numbers: thousands, lakhs and crores

Reading huge numbers.

A PLACE VALUE CHART helps you read big numbers. In the international system we group digits in threes (thousands, millions). In the Indian system we use thousands, then lakhs and crores: 1 lakh = 1,00,000 and 1 crore = 1,00,00,000. Commas (or small gaps) make big numbers easier to read.

2.1 Place value charts

Figure 5. 4 307 256 in a place value chart.

Figure 6. An abacus showing 3052.

WORKED EXAMPLE 4 Read (basic)

Write 4 307 256 in words.

Solution

four million, three hundred and seven thousand,A1

two hundred and fifty-sixA1

WORKED EXAMPLE 5 Digit (standard)

What is the value of the 3 and of the 0 in 4 307 256?

Solution

3: 300 000A1

0: zero, it holds the ten-thousands placeA1

WORKED EXAMPLE 6 Abacus (standard)

Why is the hundreds rod on the abacus empty?

Solution

3052 has 0 hundredsR1

the empty rod keeps the place, like a zeroR1

Common mistake

Leaving out zeros. Without the 0, 3052 would become 352!

2.2 Indian and international systems

Figure 7. Lakhs and crores.

Figure 8. One number, two ways of grouping.

Figure 9. Countries with the biggest populations.

WORKED EXAMPLE 7 Convert (basic)

How many thousands are in 1 lakh?

Solution

1 lakh = 1,00,000M1

= 100 thousandA1

WORKED EXAMPLE 8 Population (standard)

India's population is about 1.43 billion. About how many crores is that?

Solution

1 billion = 100 croreM1

about 143 croreA1

WORKED EXAMPLE 9 Write (standard)

Write 2,50,00,000 in words using the Indian system.

Solution

two croreA1

fifty lakhA1

Did you know?

A crore is ten million. If you counted one number every second without stopping, counting to one crore would take you almost 4 months!

3 Tiny numbers: decimals

Places after the point.

Think about it

A cricket ball's speed is shown as 142.5 km/h. What does the .5 mean?

Place value also works to the right of the decimal point. The first place is TENTHS (one-tenth = 0.1), then HUNDREDTHS (0.01), then THOUSANDTHS (0.001). Each place is ten times smaller than the place on its left. Multiplying by 10 moves every digit one place LEFT; dividing by 10 moves every digit one place RIGHT.

3.1 Tenths and hundredths

Figure 10. Tenths and hundredths on a 100-square.

Figure 11. Zooming in on a number line.

Figure 12. Decimals on a ruler.

WORKED EXAMPLE 10 Grid (basic)

How many hundredths are in 0.37?

Solution

37 hundredthsA1

3 tenths + 7 hundredthsA1

WORKED EXAMPLE 11 Ruler (standard)

The arrow points at 3.7 cm. Write this in millimetres.

Solution

1 cm = 10 mmM1

37 mmA1

WORKED EXAMPLE 12 Value (standard)

What is the value of the 5 in 142.5?

Solution

5 tenthsA1

= 0.5A1

3.2 Multiplying and dividing by 10

Figure 13. Multiplying by 10 moves digits one place left.

WORKED EXAMPLE 13 ×10 (basic)

Work out 3.45 × 10 and 3.45 × 100.

Solution

34.5A1

345A1

WORKED EXAMPLE 14 ÷10 (standard)

Work out 260 ÷ 10 and 260 ÷ 1000.

Solution

26A1

0.26A1

Common mistake

Saying 'just add a zero' when multiplying decimals by 10. 3.45 × 10 is 34.5, not 3.450!

Brain teaser

I am a decimal between 0 and 1. My tenths digit is double my hundredths digit, and my digits add up to 9. What am I? (Answer: 0.63)

4 One number, many costumes

Different ways to show a number.

The same number can wear many 'costumes': digits (25), words (twenty-five), expanded form (20 + 5), money, tally marks, Roman numerals (XXV) or even binary (11001), the code computers use.

4.1 Expanded form and money

Figure 14. Twenty-five in six different ways.

Figure 15. Writing a number in expanded form.

Figure 16. Place value with Indian currency notes and coins.

WORKED EXAMPLE 15 Expanded (basic)

Write 6207.35 in expanded form.

Solution

6000 + 200 + 7M1

+ 0.3 + 0.05A1

WORKED EXAMPLE 16 Money (standard)

How many ₹100 notes make ₹5000? How many ₹10 coins?

Solution

50 notesA1

500 coinsA1

WORKED EXAMPLE 17 Build (standard)

Write the number: 3 thousands, 0 hundreds, 8 tens and 4 ones.

Solution

3000 + 80 + 4M1

= 3084A1

4.2 Roman numerals and binary

Figure 17. Roman numerals.

Figure 18. Roman numerals on a clock face.

Figure 19. A number shown in binary with light bulbs.

WORKED EXAMPLE 18 Roman (basic)

Write 2026 in Roman numerals.

Solution

MM + XX + VIM1

MMXXVIA1

WORKED EXAMPLE 19 Binary (standard)

What number is binary 1010?

Solution

8 + 0 + 2 + 0M1

= 10A1

WORKED EXAMPLE 20 Zero (exam level)

Why is it hard to do big calculations with Roman numerals?

Solution

Roman numerals have no zero and no place valueR1

so you cannot line digits up in columnsR1

Did you know?

Our digits 0 to 9 came from India! Indian mathematicians, like Brahmagupta around 628 CE, used zero as a real number. The idea travelled to the Arab world and then to Europe.

5 Detective challenges

Puzzles and codes.

Real detectives look for clues. Number detectives look at every digit and its place. Try these challenges and level up!

5.1 Codes and levels

Figure 20. Decode the two secret words.

Figure 21. Level up as you learn.

WORKED EXAMPLE 21 Code (basic)

Crack the code: 1 2 3 4 5 6 3 2 7 5.

Solution

1 = P, 2 = L, 3 = A, 4 = C, 5 = EM1

PLACE VALUEA1

WORKED EXAMPLE 22 Mystery (standard)

I am a 4-digit number. My thousands digit is 5, my hundreds digit is 0, my tens digit is 3 more than my ones digit, and my ones digit is 4. What am I?

Solution

5, 0, 7, 4M1

5074A1

Try this

Ask an adult for an old receipt or electricity bill. Find the biggest number on it, read it aloud, and say the value of each digit!

6 Check your skills

Become a Number master.

6.1 Spot the mistake

Figure 22. Leo's claim.

WORKED EXAMPLE 23 Analyse (standard)

Is Leo right? Explain.

Solution

no: the 7 is in the tenths placeR1

it is worth 0.7 (seven tenths)A1

WORKED EXAMPLE 24 Basic

Write 60 512 in words.

Solution

sixty thousand, five hundred and twelveA2

WORKED EXAMPLE 25 Standard

What is the value of the 4 in 2.045?

Solution

thousandths placeM1

0.004A1

WORKED EXAMPLE 26 Exam level

Write 12,34,56,789 in words (Indian system) and with international commas.

Solution

twelve crore, thirty-four lakh, fifty-six thousand, seven hundred and eighty-nineA1

123,456,789A1

6.2 Mission challenge: the school fair

WORKED EXAMPLE 27 Fair (exam level)

The school fair raised ₹1,25,640. (a) Write this in words. (b) What is the value of the 2? (c) How many ₹100 notes would that be (ignoring the rest)? (d) The fair next year aims to raise 10 times as much. Write the target.

Solution

(a) one lakh, twenty-five thousand, six hundred and forty rupeesA1

(b) ₹20 000A1

(d) ₹12,56,400A1

6.3 Top tips

•Each place is 10 times the place on its right.
•Value of a digit = digit × place.
•Zeros hold places: never leave them out.
•Indian system: thousand, lakh, crore.
•After the point: tenths, hundredths, thousandths.
•×10 moves digits one place left; ÷10 moves them right.

6.4 Key words

Word	Meaning
digit	one of the symbols 0 to 9
place value	the value of a digit's position
expanded form	a number written as a sum of place values
lakh	1,00,000 (one hundred thousand)
crore	1,00,00,000 (ten million)
tenth	one part out of ten: 0.1

6.5 Quick quiz

Try these, then check the answers.

Quick check

1. Value of 6 in 4625?

2. 1 lakh in digits?

3. 0.4 as tenths?

4. 2.3 × 10?

5. XIV in digits?

6. Binary 101?

Answers: 1. 600 2. 1,00,000 3. 4 tenths 4. 23 5. 14 6. 5

6.6 Quick quiz 2

Students often think…	Actually…
7 in 3.75 is worth 7	worth 0.7
leave out zeros	zeros hold places
×10 adds a zero	digits move left
1,00,000 = one thousand	one lakh
tens = tenths	different places

Topic	Connection
1.2 Comparing and ordering	next
1.3 Rounding	using place value
2.3 Decimals	operations
Sciences	measurements

Part	Answer	Marks
(a)	700	A1
(b)	2 hundredths (0.02)	A1
(c)	30 000 + 700 + 40 + 5	M1
	+ 0.6 + 0.02	A1
(d)	307 456.2	A1
(e)	it holds the thousands place (no thousands)	R1

Idea	Remember
Place value	position = value
Zeros	hold places
Indian system	lakh, crore
Decimals	tenths, hundredths
×10	digits move left
Forms	words, expanded, Roman
Binary	computers