Class 6 NCERT bridge course Answers Activity W1.2 The Hailstone Number Game

 Activity W1.2   The Hailstone Number Game

 Step 1: Think of any number 

 Step 2: If the number is odd, triple it and add 1, if the number is even, halve it. 

 Step 3: Continue step 2, based on the resulting number in step 2 and continue the steps

Step 4: Write the pattern generated., 

e.g.,

 Step 1: 5 

 Step 2: 3x5+1= 16 

 Step 3: 16/2 = 8 

 Step 4: 8/2 = 4 

 Step 5: 4/2 =2 

 Step 6: 2/2 =1 

 Step 7: 3x1+1 = 4 

 Step 8: 4/2 = 2 

 Step 9: 2/2 =1 ……………………….

 Resulting pattern is 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, ………………….

 Ask the students to play it in pairs

 Let one child give the number, the other one develops the pattern and vice versa. 

 Discuss all the patterns they developed and ask them to find out the reason for the pattern which is named as hailstone numbers.

Explore a Magical Number Pattern!

Objective:

Students discover a repeating numerical pattern using simple rules and understand the concept of hailstone numbers (also called the Collatz sequence).

How to Play:

1. Step 1: Think of any positive number.

2. Step 2:

   • If the number is odd → multiply it by 3 and add 1.

   • If the number is even → divide it by 2.

3. Step 3: Repeat the process with the new number.

4. Step 4: Observe and record the pattern.

Example: Start with 7

Let's generate the pattern:

1. 7 (odd) → 3×7 + 1 = 22

2. 22 (even) → 22 ÷ 2 = 11

3. 11 (odd) → 3×11 + 1 = 34

4. 34 → 17

5. 17 → 52

6. 52 → 26

7. 26 → 13

8. 13 → 40

9. 40 → 20

10. 20 → 10

11. 10 → 5

12. 5 → 16

13. 16 → 8

14. 8 → 4

15. 4 → 2

16. 2 → 1

17. 1 → 4

18. 4 → 2

19. 2 → 1 → then continues as: 4, 2, 1...

Pattern:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...

What are Hailstone Numbers?

The numbers in this sequence are called hailstone numbers because they rise and fall unpredictably—like hailstones in a storm—before eventually settling into the loop 4 → 2 → 1.

This pattern is part of a famous mathematical problem called the Collatz Conjecture. No matter which number you start with, the pattern always falls to 1!

Classroom Pair Activity:

• Partner A: Chooses a number.

• Partner B: Applies the rules and writes down the sequence.

• Then switch roles!

• Compare patterns: Who reached 1 faster? Which number had more steps?

Questions for Discussion:

• Do all numbers eventually reach 1?

• Which numbers take longer?

• How do odd and even numbers affect the pattern?

Illustrative Image:

A fun and colorful visual representation of the pattern starting from 7:

hailstone number pattern image

(Image shows numbers rising and falling like hailstones, before looping into 4 → 2 → 1.)

Wrap-Up:

This activity builds logical thinking, pattern recognition, and number sense. It's exciting, unpredictable, and a great way to spark mathematical curiosity in your students!

Class 6 NCERT bridge course Answers Activity W1.1 Guess the Secret Number!

  Class 6 NCERT bridge course Answers 

Activity W1.1

Week 1 

Solves puzzles and daily-life problems involving one or more operations on whole numbers (including word puzzles and puzzles from ‘recreational’ areas, such as the construction of magic squares) in finding their own, possibly different, solutions.

 Discovers, recognises, describes, and extends patterns in 2D and 3D shapes. 

 Deduces that shapes having equal areas can have different perimeters and shapes having equal perimeters can have different areas

Week 2 

Deduces that shapes having equal areas can have different perimeters and shapes having equal perimeters can have different areas

Week 3 

Describes location and movement using both common language and mathematical v o c a b u l a r y ;

 understands the notion of map Recognises and creates symmetry (reflection, rotation) in familiar 2D and 3D shapes 

 Understands the definition and formula for the area of a square or rectangle as length times breadth.

Week 4 

Understands the definition and formula for the area of a square or rectangle as length times breadth

 Solves puzzles and daily-life problems involving one or more operations on whole numbers (including word puzzles and puzzles from ‘recreational’ areas, such as the construction of magic squares) in finding their own, possibly different solutions. 

 Selects appropriate methods and tools for computing with whole numbers, such as mental computation, estimation, or paper pencil calculation, in accordance with the context.

Activity W1.1Guess the Secret Number!

Objective:

To help students develop reasoning and questioning skills by guessing a number (between 1 and 30) through a series of logical Yes/No questions.

•  Let the teacher start with the following game:
•  Teacher: I have a number in my mind. 
• It lies between 1 and 30 including both. 
• You are expected to identify the number through a series of questions. 
•  For each of the questions I will reply with either ‘YES’ or ‘NO’ only.
•  Ask the question in such a way that, I can give the answer ‘YES’ or ‘NO’ only. 
• If a child can’t hear the questions they may be written or acted out. 

•  Teacher may ask the students to raise their hands for taking initiative in asking questions and then take up questions one by one with the above-mentioned answers.
• Either one student or a group of 3-4 students may plan their questions and then ask them one by one before reaching a conclusion. 
•  They should tell the class the strategy they tried for getting the number in the teacher’s mind. 
•  If this student or the group of students succeed in locating the number, then the teacher may tell another student or group of students to take over the questioning for some other number. 
• If the earlier student or group of students do not succeed in finding the number, then another student or group can take over the questioning and try to find that number. 
•  The students may be encouraged to ask a varied number of questions. 

Modifications in the game: 

•  The game can be modified by changing the final number 30 to 40, 50, 60, etc.
•  After getting enough exposure to the game, the teacher may ask the students to note the total number of questions asked before identifying the secret number. 
• The teacher may then suggest finding the secret number with fewer and fewer questions. 
• This will motivate the students to improve upon their strategies. 
•  To make the game more interesting, engrossing and competitive, two or three groups of students may be asked to participate. 
• The group which finds the secret number in the least number of steps will be the winner.

Example Game Round:

Teacher's Secret Number (in mind): 18
(Not revealed to students until the end!)

Students begin asking:

1. Is the number greater than 15?
   Teacher: YES

2. Is the number greater than 20?
   Teacher: NO

3. Is the number even?
   Teacher: YES

4. Is the number divisible by 3?
   Teacher: YES

5. Is the number 18?
   Teacher: YES 

Strategy Used:

The group used elimination by:

• First cutting the range in half (greater than 15?).

• Narrowing further by checking if it's greater than 20.

• Then asked about even/odd and divisibility, using math properties to pinpoint the exact number.

Answer:

The number is 18

Teaching Notes & Extension Ideas:

• After each round, ask the students:

  “How many questions did it take to find the number?”
  “Could you have found it in fewer questions?”

• Encourage strategies like:

  • Dividing the range in half each time (binary search method)

  • Asking about divisibility, even/odd, or range (e.g., 10–20)

• You can increase difficulty by changing the range to 1–40, 1–50, etc.

• Make it competitive:
   Groups compete to find the number in the fewest steps

Kanakkadhiharam by (author: korukkaiyoor karinayanar)

 Kanakkadhiharam 
(author: korukkaiyoor kaarinayanar) 

கணக்கதிகாரம் 

எழுதியவர் கொருக்கையூர் காரி நாயனார்

 காரி நாயனார் என்ற புலவரால் கணக்கதிகாரம் என்னும் கணித நூல் 15ஆம் நூற்றாண்டில் எழுதப்பட்டது.

இந்நூலில் ஆறு பிரிவுகளில் 64 வெண்பாக்களும், 45 புதிர் கணக்குகளும் உள்ளன: நிலம் வழி (23 பாக்கள்), பொன் வழி (20 பாக்கள்), நெல் வழி (06 பாக்கள்), அரிசி வழி (02 பாக்கள்), கால் வழி (03 பாக்கள்), கல் வழி (01 பாக்கள்), பொது வழி (05 பாக்கள்) என்ற ஆறுவழிக் கணக்குகளையும் புலவர் அறுபது செய்யுள்களால் உணர்த்தினார் என்பதை:

“ஆதிநிலம் பொன்னெல் லாரிசி யகலிடத்து
நீதிதருங் கால் கல்லே நேரிழையாய் – ஓதி
உறுவதுவாகச் சமைத்தேன் ஒன்றெழியா வண்ணம்
அறுபது காதைக்கே யடைத்து.”

ஆறு வழிக் கணக்கு மட்டுமல்லாது வேறு பல கணக்குகளையும் இந்நூலில் நீங்கள் பார்க்கலாம். இக்கணக்குகள் கற்பவர்க்கு திகைப்பும், வியப்பும், நகைப்பும், நயப்பும் விளைவிக்கும் என்பது திண்ணம்.

தமிழ் எண்கள்: 1. தமிழ் முழு எண்களின் பெயர்கள்; 2. தமிழ் பின்ன எண்களின் பெயர்கள்;

பொழுதுபோக்கு: 3. “மாயசதுர’ கணக்குகள்” – எப்படிக் கூட்டினாலும் ஒரே விடை; 4. வினா-விடைக் கணக்குகள்;

புதிர் கணக்குகள்: 5. பூமியின் அளவு, நிலத்தின் அளவு, நீர் அளவு, சூரியன்-சந்திரன் இடையேயான தொலைவு, மலையின் அளவு;

சூத்திரக் கணக்குகள்: 6. ஒரு படி நெல்லில் எத்தனை நெல் இருக்கும்;  ஒரு பலாப்பழத்தில்  எத்தனை பலாச்சுளை இருக்கும்; ஒரு பரங்கிக்காயில்  எத்தனை விதைகள் இருக்கும்.

கணக்கதிகாரத்தில், வெண்கலம் மற்றும் பித்தளை ஒன்றாகச் சேர்த்து உருக்கி பித்தளை உருவாக்கும் விவரம் பற்றிய ஒரு செய்யுள் காணப்படுகிறது.

கணக்கதிகாரம் செய்யுள் எண்: கக (11)

“எட்டெடை செம்பி லிரெண்டை யீயமிடில்
திட்டமாய் வெண்கலமாஞ் சேர்ந்துருக்கி – லிட்டமுடன்
ஓரேழு செம்பி லொருமூன் றுதுத்தமிடில்
பாரறியப் பித்தளையாம் யார்”

உரை:

எட்டுப்பலஞ் செம்பிலே இரண்டு பலம் ஈயமிட்டுருக்க வெண்கலமாம். ஏழலரைப் பலஞ் செம்பிலே மூன்று பலந் துத்தமிட்டுருக்க பித்தளையாம்.

Two strengths of the eighty -eight copper are the bronze. The seven -year -old is the three -legged brass.

Note: Palam - Antique Tamil Weight (40.8 g).

குறிப்பு: பலம் – பழந்தமிழர் எடை அளவு (40.8 கிராம்).

A garden- lizard climbs up a palm tree of height 32 Cubits. 
It goes up 12 finger Space a day but slips 4 finger space in every move.
How many days will it take to reach the top of the tree?
I cubit = 24 finger Space.

Solution:
32 x 24 finger space = 768
Per day 12 - 4 = 8
Number of days = 768 / 8 = 96 days

A Frog had fallen into lom a well of depth Every time it tried to climb up, it climbed a distance of to cm. and slipped 20 cm. How many times should it try to Come out of that well?

The frog leaped forward to catch the insect away in Em. But It Could cover only 4 m. At second Jump half of the distance am. Third jump half of 2m. If the frog Jumps halb by hally, how many Jumps does the frog require to Catch the insect?

Class 8 NCERT bridge course Answers Activity W6.9 Number Puzzles and Pattern Talk

 Activity W 6.9  Number Puzzles and Pattern Talk

This activity develops number sense, understanding of multiples, special numbers, and pattern recognition.

Teacher may give these situations to students and ask them to discuss about them. 

1. Common Multiples Puzzle

1. In the diagram below, 

"Anant has erased all the numbers except the common multiples."

Find out what those numbers could be and fill in the missing numbers in the empty regions. 

Given common multiples: 72, 48, 24
We need to find two numbers whose common multiples include 72, 48, and 24.

Step 1: Find the LCM of 24, 48, and 72.
Since 72 is a multiple of 24 and 48, we know:

• Possible pair: Multiples of 6 and Multiples of 8 

  
  

Answer:

• Multiple of 6

• Multiple of 8

• Common Multiples: 24, 48, 72

•  Students can verify this:

• LCM(6,8) = 24

• 48 and 72 are higher common multiples

 

2. Special Numbers in Each Box

There are some boxes with four numbers in each box given below. 

"Within each box, say how each number is special compared to the rest."

Let’s look at each set:

Box 1:

5, 7, 12, 35

• 5 and 7 are primes

• 12 is sum of 5 and 7

• 35 is product of 5 and 7

Special: 12 (sum), 35 (product)

Box 2:

3, 8, 11, 24

• 3 + 8 = 11

• 3 × 8 = 24

Special: 11 (sum), 24 (product)

Box 3:

27, 3, 123, 31

• 27 ÷ 3 = 9

• 123 − 31 = 92

• 41 x 3 = 123

Box 4:

17, 27, 44, 65

• 17 + 27 = 44

• 17 × 27 = 459

• 44 + 21 = 65

 3.  The figure on the left shows the puzzle. 

The figure on the right shows the solution of the puzzle. 

Think what the rules can be to solve the puzzle.

Rules

Fill the grid with prime numbers only, so that the product of each row is the number to the right of the row and the product of each column is the number below the column—

(a) Prime Factorization Table

From the solution table (right side), these values are:

We already see the factorization of the numbers in the rightmost column (75, 42, 102):

• The 3 columns before 75/42/102 show the prime factors:

  • 75 → 5 × 5 × 3

  • 42 → 2 × 3 × 7

  • 102 → 17 × 2 × 3

  • 170 → 5 x 2 x 17

  • 30 → 5 x 3 x 2

  • 63 → 3 x 7 x 3

  • 

b)  

Solution

• Row products:

  • 7 x 5 x 3 = 105 

  • 2 x 5 x 2 = 20

  • 2 x 5 x 3 = 30

• Column products:

  • 7 x 2 x 2 = 28

  • 5 x 5 x 5 = 125

  • 3 x 2 x 3 = 18

Solution

Solution

Solution