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Class 6

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Number Play

Number Play

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Study Action Plan (Sorted by Difficulty)
Collatz Conjecture
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Digit Sums and Patterns
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Estimation Techniques
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Games and Winning Strategies
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Kaprekar Constant
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Number Line and Placement
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Number Usage in Daily Life
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Supercells and Number Arrangement
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Understanding Number Patterns
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Learning Objectives

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Learning Objectives

  • Identify various situations where numbers are used in daily life.
  • Explain the significance of numbers in conveying information and solving problems.
  • Analyze the Collatz conjecture and its implications in number theory.
  • Develop estimation skills for practical applications.
  • Formulate strategies for games involving numbers.
  • Explore patterns in numbers and their properties, including palindromes and digit sums.

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Revision Notes & Summary

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Chapter 3 - Solutions

Number Play

Situations Where Numbers Are Used

  • Time
  • Calendar
  • Counting objects/Marks
  • Measurement of height & weight
  • Money

Collatz Conjecture

  • Definition: Start with any whole number. If the number is even, take half of it; if odd, multiply by 3 and add 1. Repeat until reaching 1.
  • Example Sequences:
    • Starting with 28: 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
    • Starting with 19: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Estimation Examples

  • Estimate the number of students in your school: About 150? 400? A thousand?
  • Estimate the distance between Gandhinagar and Kohima: Approximately 2500 kilometers.

Playing with Number Patterns

  • Always, Sometimes, Never?
    • a. 5-digit number + 5-digit number gives a 5-digit number: Sometimes
    • b. 4-digit number + 2-digit number gives a 4-digit number: Sometimes
    • c. 4-digit number + 2-digit number gives a 6-digit number: Sometimes
    • d. 5-digit number - 5-digit number gives a 5-digit number: Never
    • e. 5-digit number - 2-digit number gives a 3-digit number: Never

Supercells in a Grid

  • Grid Example:
    16,20039,34429,765
    23,60962,87145,306
    19,38150,31938,408
  • Task: Identify the supercell and determine which digits to swap to create more supercells.

Kaprekar Constant

  • Example: For the year 1980, it takes 6 rounds to reach the Kaprekar constant.

Estimation Questions

  • Challenge your classmates: How many hours does a person sleep in their lifetime on average? How many students travel to school by bus?
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

The sequence will eventually reach 1.

B.

The sequence will always reach a cycle other than 1.

C.

The sequence will grow indefinitely without repeating.

D.

The sequence will always reach a number greater than 100.
Correct Answer: A

Solution:

According to the Collatz Conjecture, it is believed that any starting positive integer will eventually reach 1, although this remains unproven.

Chapter Concept:

Collatz Conjecture

A.

27

B.

18

C.

36

D.

54
Correct Answer: C

Solution:

The sequence starts with 123. The sum of the digits of 123 is 1+2+3=61 + 2 + 3 = 6. Multiply by 3 to get 6×3=186 \times 3 = 18. For the next number, the sum of digits of 18 is 1+8=91 + 8 = 9. Multiply by 3 to get 9×3=279 \times 3 = 27. The third number is 27.

Chapter Concept:

Understanding Number Patterns

A.

Multiply by 2 if the number is even; subtract 1 if odd.

B.

Divide by 2 if the number is even; multiply by 3 and add 1 if odd.

C.

Add 1 if the number is even; subtract 2 if odd.

D.

Multiply by 3 if the number is even; divide by 2 if odd.
Correct Answer: B

Solution:

In the Collatz Conjecture, if the number is even, you divide it by 2; if it is odd, you multiply it by 3 and add 1.

Chapter Concept:

Collatz Conjecture

A.

3 iterations

B.

5 iterations

C.

7 iterations

D.

8 iterations
Correct Answer: D

Solution:

Starting with 321, the iterations are: 321 \rightarrow 198 \rightarrow 792 \rightarrow 693 \rightarrow 594 \rightarrow 495. It takes 8 iterations to reach the Kaprekar constant 495.

Chapter Concept:

Kaprekar Constant

A.

0

B.

1

C.

2

D.

3
Correct Answer: C

Solution:

A child says '2' if both the children standing next to them are taller.

Chapter Concept:

Understanding Number Patterns

A.

About 500

B.

About 600

C.

About 700

D.

About 800
Correct Answer: A

Solution:

Paromita estimated about 100 students per class for Class 6. Assuming similar numbers for Classes 7 to 10, the total estimate would be 5 classes x 100 students = 500 students.

Chapter Concept:

Estimation Techniques

A.

81

B.

40

C.

14

D.

1
Correct Answer: B

Solution:

Starting with 27, the sequence will eventually reach 40 as it follows the rule of multiplying by 3 and adding 1 when odd, and dividing by 2 when even.

Chapter Concept:

Collatz Conjecture

A.

59

B.

68

C.

77

D.

95
Correct Answer: A

Solution:

The smallest number whose digits add up to 14 is 59, as 5+9=145 + 9 = 14.

Chapter Concept:

Digit Sums and Patterns

A.

3 iterations

B.

5 iterations

C.

6 iterations

D.

8 iterations
Correct Answer: C

Solution:

Starting with 321, the sequence is: 321 \rightarrow 198 \rightarrow 792 \rightarrow 693 \rightarrow 594 \rightarrow 495. The number 495 repeats after 6 iterations.

Chapter Concept:

Kaprekar Constant

A.

It is always a supercell.

B.

It can never be a supercell.

C.

It is a supercell only if it is at the center of the table.

D.

It depends on the arrangement of other numbers.
Correct Answer: A

Solution:

The largest number in a table is always a supercell because it is greater than all its neighboring cells.

Chapter Concept:

Supercells and Number Arrangement

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Number Play

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True or False

Correct Answer: True

Solution:

The rules of the game state that players take turns adding 1, 2, or 3 to the previous number, and the first player to reach 21 wins.

Chapter Concept :

Games and Winning Strategies

Correct Answer: True

Solution:

By following the steps of rearranging the digits of a 4-digit number to form the largest and smallest numbers and subtracting the smaller from the larger, you will eventually reach the Kaprekar constant 6174.

Chapter Concept :

Kaprekar Constant

Correct Answer: False

Solution:

The largest number in a table will always be a supercell because it is greater than all its neighbouring cells.

Chapter Concept :

Understanding Number Patterns

Correct Answer: True

Solution:

The digit sum of 68 is 6+8=146 + 8 = 14 and the digit sum of 176 is 1+7+6=141 + 7 + 6 = 14. Both sums are equal.

Chapter Concept :

Digit Sums and Patterns

Correct Answer: True

Solution:

The number 2180 is correctly positioned between 2000 and 3000 on a number line, as indicated by the given diagram.

Chapter Concept :

Number Line and Placement

Correct Answer: True

Solution:

The Collatz conjecture suggests that by following the rule of halving even numbers and multiplying odd numbers by 3 and adding 1, the sequence will eventually reach the number 1.

Chapter Concept :

Number Usage in Daily Life

Correct Answer: True

Solution:

The Collatz Conjecture is an unsolved problem in mathematics that suggests any sequence formed by the rule of halving even numbers and applying 3n+13n + 1 to odd numbers will eventually reach 1.

Chapter Concept :

Collatz Conjecture

Correct Answer: True

Solution:

Numbers are indeed used in different contexts to organize our lives, including time, calendar, counting, measurement, and money.

Chapter Concept :

Number Usage in Daily Life

Correct Answer: False

Solution:

A child at the end of a line cannot say '2' because there is no child on one side.

Chapter Concept :

Understanding Number Patterns

Correct Answer: True

Solution:

A supercell is indeed defined as a cell where the number is larger than all its neighboring cells, as described in the text.

Chapter Concept :

Supercells and Number Arrangement

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