NCERT Mathematics Class 6 Chapter 1: Numbers & Comparison-Section 1.1 & 1.2 YouTube Lecture Handouts

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What Will We Learn?

Image of a Numbers and Comparison

Image of a Numbers and Comparison

Image of a Numbers and Comparison

  1. Numbers and Types

  2. Place & Face Value

  3. Positional Number System

  4. Roman Numerals

  5. Shifting Digits, Comparison and Ordering

  6. Indian and International System

  7. SI System

  8. Large numbers

  9. Estimation

  10. Operations and BODMAS

Questions?

  1. What is the place value of 0 in 209?

  2. What is the largest roman numeral?

  3. A notebook has 20 pages. 1 sheet of paper can make 10 pages. How many notebooks can be made from 40 sheets?

  4. Find difference between greatest and smallest number formed using digits 6,2,4,3,1

  5. Out of 40 m cloth, how many shirts can be made if each shirt requires 2 m 15 cm of cloth?

History of Numbers

Image of a History of Numbers

Image of a History of Numbers

Image of a History of Numbers

Image result for chimp counting gif

Image Result for Chimp Counting Gif

Image result for chimp counting gif

Image of a History of Numbers 2

Image of a History of Numbers 2

Image of a History of Numbers 2

Image of a History of Numbers 3

Image of a History of Numbers 3

Image of a History of Numbers 3

  1. Numbers developed as soon as man started counting

    Used for trade, seed and food distribution, taxation etc.

  2. Even mammals can count (upto 3)

  3. Samarians, Egyptians, Greeks, and Indians contributed significantly to mathematics.

  4. Romans developed roman numeral system which was widely used as they conquered the world.

  5. Romans did not contribute to mathematics.

Nominal, Ordinal, and Cardinal

  1. Use of numbers as counting is obvious- cardinal numbers.

  2. Numbers also denote order- for example, 1st, 2nd, 3rd etc. in a race.

    Cant add, subtract ordinal numbers

  3. Numbers are also used as just names. For example 18 for Kholi and for Dhoni is 7.

  4. नाममात्र, क्रमवाचक, गणनसंख्या

Understanding Place Value and Face Value

2211

  • Zero is used as a placeholder to change the position of one.

  • In a decimal system each change in position makes the value of same digit 10 times bigger.

  • Thus the value of the first one is just one while the second 1 although it looks the same is “10”, that is it is 10 times greater.

  • Thus face value of both the ones is 1 while their place value is 1 and 10. Each zero placed to the right of the digit makes its value 10 times.

  • This is same as when you move from one class to next. Your face value remains the same while your value in school by virtue of your position (class) becomes larger.

  • अंकित मूल्य और स्थानीय मान

Non Positional Number System

Image of a Non Positional Number System

Image of a Non Positional Number System

Image of a Non Positional Number System

Image of a Non Positional Number System 2

Image of a Non Positional Number System 2

Image of a Non Positional Number System 2

image of a Non Positional Number System 3

Image of a Non Positional Number System 3

image of a Non Positional Number System 3

4622

  • Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional.

  • Instead of using different positions for 1, 10, 100.

  • गैर स्थितीय संख्या प्रणाली

Positional Number System

image of a Positional Number System 2

Image of a Positional Number System 2

image of a Positional Number System 2

Image of a Positional Number System

Image of a Positional Number System

Image of a Positional Number System

  • As we said position of a number and also its symbol have significance.

  • After a certain count the same digit moves the next position- you move to next class after 1 year.

  • As the number changes position its value changes.

  • In the decimal system the change in position happens after we count to 10. That is each position is 10 times the digit on right.

  • In other systems the change might happen after different counts. For example,

  • Sexagesimal: The Babylonian numeral system, base 60, was the first positional system developed, and its influence is present today in the way time and angles are counted in tallies related to 60, like 60 minutes in an hour, 360 degrees in a circle.

  • In binary system each digit is 2 times the previous one and so on.

  • The problem with positional systems is that they are susceptible to fraud.

  • स्थितीय संख्या प्रणाली

Two Meanings of Zero

  • In positional systems zero is used as a placeholder to change the position of the digits. We could have used –, space etc.

  • Zero also means count of zero.

  • Place value and face value of zero is always 0.

  • Zero was invented by Indians- it is called śūnya in Sanskrit.

  • Aryabhata in 520 A.D., devised a positional decimal number system contained a word, “kha,” for the idea of a placeholder.

  • Brahmagupta, developed zero as an actual independent number, not just a placeholder, and wrote rules for adding and subtracting zero.

  • The Indian writings passed on to al-Khwarizmi and then to Leonardo Fibonacci and others who continued to develop the concept and the number.

  • This Indian numberal system travelled all over the world and hence the current decimal numeral system is also called Hindu-Arabic system

Rules for Roman Numerals

I V X L C D M

1 5 10 50 100 500 1000

  1. Repetition: Repeated symbol => add value,

    1. Max repeat = 3

    2. No repeat for V, L, and D.

  2. Addition: Smaller value to the right of greater value gets added.

  3. Subtraction: Smaller value left of greater value is subtracted Symbols V, L and D are never subtracted

    1. I => V & X only.

    2. X => L, C, and M only.

Romans did not contribute to mathematicians but only to conquests.

For example

  • I=1, II = 2, XX=20 and XXX=30.

  • VI = 5 + 1 = 6, XII = 10 + 2 = 12, and LXV = 50 + 10 + 5 = 65.

  • IV = 5 – 1 = 4, IX = 10 – 1 = 9, XL= 50 – 10 = 40, XC = 100 – 10 = 90

रोमन संख्या

Roman Numerals

V(5) X(10) L(50) C(100) D(500) M(1000)

  1. Repeated symbol (max 3) => add value. Except V, L, and D.

  2. Smaller value to the right of greater value gets added.

  3. Smaller value left of greater value is subtracted. Except V, L and D, I => V & X only, and X => L, M & C only.

  • The current year (2019) is MMXIX

  • 1776 (M+DCC+LXX+VI) = MDCCLXXVI (the date written on the book held by the Statue of Liberty)

  • 1910 is MCMX and not MDCCCCX

  • MMMCMXCIX which in our notation is 3,999

  • V (bar) = 5000 and X (bar) = 10,000

Convert from Roman Numerals

I (1) V(5) X(10) L(50) C(100) D(500) M(1000)

  1. Repeated symbol (max 3) => add value. Except V, L, and D.

  2. Smaller value to the right of greater value gets added.

  3. Smaller value left of greater value is subtracted. Except V, L and D, I => V & X only, and X => L, M & C only.

  • MMXIV is 2014

  • MLXVI is 1066

Writing Large Numbers: Indian and International Numbering Systems

  • 1,000; 10,000; 1,00,000 & Expansion of Numbers

  • 999 + 1 = 1000

  • 9999 + 1 = 10000

  • 99999 + 1 = 1,00,000

  • 45278 = 4 × 10000 + 5 × 1000 + 2 × 100 + 7 × 10 + 8

    • 653275829: Sixty-five crore, thirty-two lakh, seventy-five thousand, eight hundred and twenty nine

    • Short Scale

    Image of a Million first used in 14th Century

    Image of a Million First Used in 14th Century

    Image of a Million first used in 14th Century

    • Million first used in 14th Century

    • 10 Lakhs make a million

    • 10 millions make a crore

    Image of a Million first used in 14th Century 2

    Image of a Million First Used in 14th Century 2

    Image of a Million first used in 14th Century 2

    In international system after quintillion we have sextillion and then septillion. Thus, an n-illion equals 103n + 3

Indian and International Numbering System, Expansion, Number Name

7 2 7 0 5 0 6 2

7 2 7 0 5 0 6 2

  • 7,27,05,062

    Seven crore, twenty-seven lakh, five-thousand and sixty-two

  • 72,705,062

    Seventy-two million, seven hundred and five thousand, sixty-two

Indian and International Numbering System, Expansion, Number Name

6 5 3 2 7 5 8 2 9

6 5 3 2 7 5 8 2 9

  • 65,32,75,829

    Sixty-five crore, thirty-two lakh, seventy-five thousand, eight hundred and twenty-nine

  • 653,275,829

    Six hundred and fifty-three million, two hundred and seventy-five thousand, eight hundred and twenty-nine

Write Numeral

Two crore ninety-lakh fifty-five thousand eight hundred.

Comparing & Ordering

  1. 1473, 89423, 100, 5000, 310

  2. 15623, 15073, 15189, 15800

  3. Find the largest possible 3 digit number using 9, 3, 4

  4. Find the smallest possible 3 digit number using 0, 3, 9

  5. Make the greatest and the smallest 4-digit numbers using any four different digits with digit 9 in once place

  • Heat one end of rod. Particles get energy start vibrating- similar to students sitting at their place- they cant just move- are stuck at one place.

  • Vibration travels to nearby particles and so on ultimately other end becomes hot.

  • Thus some material are good conductors and others are bad- more on that later.

  • Students passing book

Shifting Digits

1473, 89423, 310

Sift the digits to make these numbers smallest and largest

SI Units: M, G, and L

  • How many centimeters make a kilometer?

    image of a SI Units

    Image of a SI Units

    image of a SI Units

    • Different prefixes are used depending on scale. For example, m is ok for measuring distances on track meet. But for measuring larger distances between cities need other unit (km). Similarly for smaller distances need another unit.

    • Same for grams (wheat/rice in kg vs. vegetables in g vs. medicine)

    • Same for liters (medicine bottles vs. bucker vs. overhead tank)

  • A box contains 2,00,000 medicine tablets each weighing 20 mg. What is the total weight of all the tablets in the box in grams and in kilograms?

    image of a SI Units: m, g, and l

    Image of a SI Units: M, G, and L

    image of a SI Units: m, g, and l

    • 4 kg

    • A bus has a speed of 60 km/hour. Find time taken by the bus to reach C from D (2,160,000 m)

    SI Prefixes

    SI Prefixes

    SI Prefixes

    36 hours

BODMAS (Using and Expanding Brackets)

Please Excuse My Dear Aunt Sally

  • (100 + 2) × 2 vs 100 + 2 × 2

  • 48 * 2 + 2 / 12

  • 48÷2∗(9+3) = 288

  • Please Excuse My Dear Aunt Sally

  • 24 ÷ 48 + 54 ÷ 18 × 7 ÷ 14 + 2 × 3 ÷ 18 == (24 / 48) + (((54 / 18) * 7) / 14) + ((2 * 3) / 18)

  • Forty five divided by three times the sum of three and two== 45/(3 * (3 + 2))

  • Write five situations for the following where brackets would be necessary: 5(9 – 4): 5 times difference of 9 and 4. I have 5 crates of 9 apples each. From each crate 4 apples went bad. How many good apples are left.

  • Write five situations for the following where brackets would be necessary.(9 + 2) (6 – 3). Product of sum of 9 and 2 with difference of 6 and 3. I had 9 crates 6 of apples each. My friend gave me 2 more. From each of the crates 3 apples went bad. How many good apples are left?

Practicing Large Numbers: +, -, *, /

A merchant had Rs 78,592 with her. She placed an order for purchasing 40 radio sets at 1200 each. How much money will remain with her after the purchase?

78592-48000=30592

Practicing Large Numbers: +, -, *, /

Instead of multiplying 7656 by 56 student multiplied it by 65? What is the difference between correct and incorrect answers?

  • 7656 * 65 - 7656 * 56 = 497640-428736=68904

  • 7656 * (65-56) = 7656 * (9) = 68904

Practicing Large Numbers: +, -, *, /

A notebook has 20 pages. 1 sheet of paper can make 10 pages. How many notebooks can be made from 40 sheets?

40 * 10 = 400/20 = 20

Practicing Large Numbers: +, -, *, /

Out of 40 m cloth, how many shirts can be made if each shirt requires 2 m 15 cm of cloth?

  • 40 * 100 cm cloth = 4000 cm

  • 215 cm cloth for each shirt

  • 215 * 8 = 1720 and 215 * 9 = 1935

  • Thus number of shirts = 18 and130 cm cloth left

Practicing Large Numbers: +, -, *, /

Find difference between greatest and smallest number formed using digits 6,2,4,3,1 (once and only once)

  • Largest number: 64321

  • Smallest number: 12346

  • 64321-12346 = 51975

Rounding

65437

  • What does rounding to nearest 10’s, 100’s, 1000’s etc. mean?

  • When we use rounding we are estimating.

  • 7805, 65437

Estimating Sums

5,673 – 436

  • There is a compromise- speed vs accuracy

  • Consider trader who receives 13,569, 26,785 but has to pay 37,000. Will he be able to pay?

  • Estimate: 5,673 – 436

  • Round to the greatest place for the smallest number

Estimating Products

    • Try multiplying 216 * 78 = 16848

    • 220 * 80 = 17600

    • 200 * 100 = 20,000

    • 200 * 80 = 16,000

    • Round off each factor to its greatest place, then multiply the rounded off factors.

The Problem of 5’S: Breaking the Tie

Image of a The Problem of 5’s: Breaking the Tie

Image of a the Problem of 5’S: Breaking the Tie

Image of a The Problem of 5’s: Breaking the Tie

  • Tie breaking!

  • Round half up

  • Convergent rounding, statistician’s rounding, or bankers’ rounding

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