CBSE Class 6 Maths Knowing Our Numbers Hots

HOTS

1. The Greek mathematician, Archimedes, lived in the third century BC. He found the existing system of numbers very cumbersome to calculate with and wrote a book called ‘The Sand Reckoner’, in which he devised a method for forming large numbers.
The Greek word for 10,000 was myriad. Archimedes started thinking of a myriad of myriads.
Archimedes called this number an octade.
a. What would we call a myriad of myriads according to the International and the Indian system?
b. Archimedes called a number that had 1 followed by 16 zeros a second octade. How many zeros will follow 1 in myraid of second octade in Indian system of numeration?
Answer: a. One hundred million or ten crore b. 1 followed by 20 zeros

2. Write each of the following quantities in numerals and in words (Indian system). Do not forget the leap years.
a. The number of seconds in a leap year.
b. The number of hours in the year 1900.
c. The number of minutes in the year 2001.
Answer: a. 31622400 seconds ; Three crore sixteen lakh twenty two thousand four hundred
b. 8784 hrs. ; Eight thousand seven hundred eighty four
c. 525600 min. ; Five lakh twenty five thousand six hundred

3. A certain 3 digit number, when rounded off the nearest hundreds gives 600, to the nearest tens gives 570.
The sum of its digits is 18 and it is an odd number. What is thisa number?
Answer: 567

4. Four cities A, B, C and D lie in a straight line on the East-West Highway. You have to discover the positions of these cities on the highway and answer the questions.
City A is 20 km to the West of city D.
City D is 50 km to the East of city C.
City A is 70 km to the West of city B.
a. What is the distance between city A and city C?
b. Which city is at the eastern-most point?
c. Which city is at the western-most point?
d. What is the distance you would travel to go from the western-most point to the eastern-most point?
Answer: a. 30 km b. City B c. City C d. 100 km

5. You are going from City W to City X . At City W, there is a milestone. After travelling for a while, you come across another milestone. How far are City X, City Y and City Z from City W, if they are all on the same line? 

Answer: X is 15 km, Y is 20 km, Z is 30 km

CHALLANGES

1. Form the largest number using the digits 9, 7, 5, 3, 1 each exactly twice. What is the smallest number you can write?

2. Form the largest number using the digits 1,2,3,4,5,6,7,8, each once, such that 5 and 2 are adjacent digits.

3. What is the largest number you can write such that the sum of its digits is 2? What is the smallest number you can write with the sum of the digits 2? (The sum of the digits of a number is called its digital sum.)

4. Prove that the largest 6-digit number with digital sum 50 is divisible by 5.

5. A student was asked to round off 12345 to its nearest hundred. He decided to round off first to the nearest 10 and wrote it as 12350. Since 50 is to be rounded off to 100, he next rounded off 12345 to the nearest 100 as 12400. But 12345 is nearer to 12300 than 12400. Why did he get the wrong answer?

6. What is the largest 8-digit number having distinct digits?

7. What is the largest number having 3 and 5 as its only digits, 3 appearing 3 times and 5 appearing 5 times, yet divisible by 5?

8. Is there the largest number having distinct digits? What is the smallest number having distinct digits?

9. Arrange the digits of 80096589432 to make the resulting number as large as possible. Arrange its digits such that the resulting number is as small as possible.

10. Find the value of 15!. How many digits are there in it? How many zeros are there at its end? (n! means you have to multiply all numbers from 1 to n.
For example, 6! = 1 × 2 × 3 × 4 × 5 × 6 = 720.)

11. Find an 8-digit number formed using 2, 3, 4, 5, 6, 7, 8, 9, each once, such that the resulting number is divisible by 11. Can you find more such numbers?

12. What is the smallest 6-digit number which has 8 in its hundreds place and 6 in its lakhs place (digits may repeat)?

13. What is the smallest 8-digit number which has 7 in its lakh’s place, 4 in its thousand’s place, 3 in its ten’s place and having distinct digits? What is your answer if digits are allowed to repeat?

14. Suppose you take the digital sum of a number. You get another number. Take its digital sum, you get another number. Repeat this process till you get a single digit number. This single digit number is called the digital root of the given number. For example, the digital root of 9986 is 5. (9+9+8+6= 32 and 3+2=5.) Find the digital root of the number 1234567891011121314151617181920, the number obtained by writing 1 to 20 in a row.

15. What is the smallest number having at least 3 digits and having digital root 8?

16. Exploration : List all numbers from 1 to 200 whose digital root is 9. Are these divisible by 9? Check this property for numbers from 201 to 400. (The result is that a number is divisible by 9 if and only if its digital root is 9.)

SUMMARY

1. Counting numbers 1, 2, 3.... are called natural numbers.

2. 1 is the smallest natural number.

3. There is no largest natural number.

4. Any number (however large) can be written by using ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits or figures.

5. The place value of a (non-zero) digit depends upon the place it occupies in the number; the place value of the digit 0 is always 0 regardless of the place it occupies in the number.

6. The face value of a digit in a number is the digit itself, regardless of the place it occupies in the number.

7. If a place is vacant in a number, put 0 at that place. Never put 0 at the extreme left of a number.

8. A single digit or a group of digits representing a number is called numeral.

9. Writing a number in words is called numeration.

10. To read and write numbers, the two systems of numeration in common use are :
a. Indian system
b. International system.

11. In the Indian system, the various periods from the right are:
ones, thousands, lakhs, crores and so on.
While in the International system, periods from the right are:
ones, thousands, millions, billions and so on.

12. In the Indian system, the first comma comes after 3-digits, from the right and next comma comes after every 2-digits; while in the International system, the commas come after every 3-digits from the right.

13. 1 million = 10 lakh, 10 million = 1 crore, 100 million = 10 crore, 1 billion = 100 crore

14. Comparison of numbers. Given two numbers, the number having more digits is greater. If the number of digits is equal, then start comparing digits from the extreme left i.e. the highest place till we get a pair of unequal digits, the number having greater digit is greater.

15. 1 km = 1000 m, lm = 100 cm. 1 cm = 10 mm
1 kg = 1000 g, lg = 1000 mg
1 kL = 1000 L, 1L= 1000 mL

16. Estimation. To estimate (or round off) a number to the nearest
a. Tens:
i. If the digit at ones place is less than 5, then replace ones digit by 0 and keep all other digits as they are.
ii. If the digit at ones place is 5 or greater than 5, then increase the tens digit by 1 and replace the ones digit by 0.

b. Hundreds:
i. If the digit at tens place is less than 5, then replace each of the digits at tens place and ones place by 0. Keep all other digits as they are.
ii. If the digit at tens place is 5 or greater than 5, then increase the digit at hundreds place by 1 and replace each of the digits at tens place and ones place by 0.

c. Thousands :
i. If the digit at hundreds place is less than 5, then replace each of the digits at hundreds place, tens place and ones place by 0. Keep all other digits as they are.
ii. If the digit at hundreds place is 5 or greater than 5, then increase the digit at thousands place by 1 and replace each of the digits at hundreds place, tens place and ones place by 0.

17. Roman Numerals. Seven Roman symbols with their corresponding values in Indian system are: 

18. Certain basic rules to write any number in Roman numerals are:
Rule 1. If a symbol is repeated, its value is added as many times as it occurs.
a. Only symbols I, X, C and M can be repeated.
b. A symbol can be repeated atmost three times.
c. Symbols V, L and D are never repeated.

Rule 2. If a symbol of a smaller value is written to the right of a symbol of a greater value, then its value gets added to the value of the greater symbol.

Rule 3. If a symbol of a smaller value is written to the left of a symbol of a greater value, then its value gets subtracted from the value of the greater symbol.
a. Symbols V, L and D are never written to the left of a symbol of a greater value.
b. I can be subtracted from V and X only.
c. X can be subtracted from L and C only.
d. C can be subtracted from D and M only.

Rule 4. If a symbol of smaller value is written between two symbols of greater values, then its value is always subtracted from a symbol of the greater value which comes immediately after the symbol of the smaller value.

Rule 5. If a bar is placed over a symbol, then its value gets multiplied by 1000.

ERRORANALYSIS

1. Students write the place instead of the place value or period and vice versa.

2. While comparing, some times, instead of counting number of digits first, students start comparing the digit at the highest place.

3. While writing the number, students put commas at the wrong place.

4. While forming numbers, students put commas after every digit.

ACTIVITY I

Using Flash Cards

Place, Place value, face value can be explained using flash cards (0 to 9)
• Take 10 number cards from 0 to 9.
• Pick any 6 cards to form 6 different numbers of 6-digits. Write all these numbers in ascending order. Find out the greatest and the smallest number and write it in the table provided.
• Shuffle the cards and repeat the activity for 7-digit numbers using any 7 cards, 9-digit numbers using any 9 cards and 10-digit numbers using all the 10 cards. 

ACTIVITY II

Riddle

You have to guess the number of mints a large jar contains given the following facts: 

a. The number of mints is an odd three-digit number.
b. The sum of the ones digit and the tens digit is one less than the hundreds digit.
c. When distributed equally among ten children, 3 mints are left.
d. The hundreds digit is 1 less than twice the ones digit.
Hint : When any number is divided by 10, the remainder obtained is the ones digit.
For example, 234 ÷ 10 leaves a remainder of 4.
ANSWER
Riddle
513