Class 8 NCERT bridge course Answers Activity W 5.3 Logic Clue Hunt on the Hundred Square

 Activity W 5.3 -  Logic Clue Hunt on the Hundred Square 

For this activity, students work in pairs or small groups. 

The students may draw a hundred square as shown below:

Procedure 

 The following clues may be written on the blackboard: 

 The number is greater than 9. 

 The number is not a multiple of 10. 

 The number is a multiple of 8. ¾ The number is even. 

The number is not a multiple of 11. 

 The number is less than 175. 

 Its ones digit is larger than its tens digit. 

 Its tens digit is odd.

Part A 

 Tell the students 

 How have a number in your mind that is on the hundred squares but you are not going to tell them what it is. 

They have to ask you for any four clues out of the given eight clues. 

With every clue they speak out, you will say just ‘YES’ or ‘NO’. 

 Try to find the set of four clues that help them to find the number in your mind. 

 Give a chance to each group to do this. 

Strategy Tip: Encourage teams to choose clues that narrow the number range quickly.

Part B 

Four of the given clues are true but they do not help in finding the number.

 Find those numbers. 

Reflection 

Consider the questions that led the students being interested and able to progress, and those you needed to clarify. 

Such reflection always helps you engage the students to find mathematics interesting and enjoyable. 

If they do not understand and do something, they are less likely to become involved.

Part A: Which clues help identify the number?

We are given 8 clues. The goal is to identify one specific number using only 4 well-chosen clues. Here’s how you can think through the process:

Let's analyze each clue for usefulness:

Clue Analysis:

1. The number is greater than 9
   ➤ Too broad — eliminates only numbers 1–9.
   Not very useful.

2. The number is not a multiple of 10
   ➤ Removes numbers ending in 0 (e.g., 10, 20, ..., 200).
   Somewhat useful.

3. The number is a multiple of 8
   ➤ Strong clue. Narrows down to numbers like 8, 16, 24, 32, etc.
   Very useful!

4. The number is even
   ➤ All multiples of 8 are even already.
   Redundant if Clue 3 is chosen.

5. The number is not a multiple of 11
   ➤ Excludes numbers like 11, 22, 33, ..., 198.
   Somewhat useful.

6. The number is less than 175
   ➤ Trims the upper end.
   Useful for narrowing down.

7. Its ones digit is larger than its tens digit
   ➤ Powerful filter (e.g., 13, 24, 57, but not 31, 43).
   Very useful!

8. Its tens digit is odd
   ➤ Limits numbers to those with tens digit as 1, 3, 5, 7, or 9.
   Very useful!

 Example: Find the Hidden Number

Let's pick a number that satisfies the following 4 helpful clues:

• Clue 3: Multiple of 8

• Clue 6: Less than 175

• Clue 7: Ones digit > Tens digit

• Clue 8: Tens digit is odd

Let’s test numbers that are:

• Multiples of 8

• Less than 175

• Have ones digit > tens digit

• Tens digit is odd

Example: 136

• Multiple of 8 

• Less than 175 

• Ones digit (6) > Tens digit (3) 

• Tens digit (3) is odd 
  → 136 is a valid hidden number!

Part B: Which clues are always true but not useful?

These clues may be true for many numbers, but don’t help narrow the list:

1. Clue 1: Greater than 9 → Always true for almost all 2- or 3-digit numbers.

2. Clue 2: Not a multiple of 10 → Excludes just a few (10, 20, ..., 200).

3. Clue 4: Even → Already covered by “multiple of 8.”

4. Clue 5: Not a multiple of 11 → Useful only if the number was close to a multiple of 11.

So, the 4 clues that don’t help much, even if true, are:

• Clue 1 (Greater than 9)

• Clue 2 (Not a multiple of 10)

• Clue 4 (Even)

• Clue 5 (Not a multiple of 11)

These clues are logically true for many numbers but don’t help you zero in on the correct number efficiently.

Class 8 NCERT bridge course Answers Activity W 5.2 Number sense

 Activity W 5.2  Number sense

Number sense involves giving meaning to numbers, that is, knowing about how they relate to each other and their relative magnitudes. 

 Having a sense of number is vital for the understanding of numerical aspects of the world. 

Here are some ideas to develop and strengthen students’ sense of numbers. 

Activity 1: Two-Digit Number Trick

Objective: Discover a pattern when reversing and adding two-digit numbers.

Procedure 

  Ask the students to choose a two-digit number. 

 Tell them to reverse the digits to get a new number. 

 Add this new number to the original number. 

 Ask students to check for their divisibility by a number.

 Check if every student gets a number by which the sum obtained in step 3 is divisible. 

 Discuss why this happens! 

Steps:

1. Ask students to choose any two-digit number (e.g., 52).

2. Reverse the digits to form a new number (e.g., 25).

3. Add the original and reversed number (e.g., 52 + 25 = 77).

4. Ask students to check the divisibility of the result.

Challenge Question:

• Is the result always divisible by a specific number?

Answer: yes, The sum is always divisible by 11.

• (Hint: Try with different numbers – 34 + 43 = 77, 61 + 16 = 77...)

✨ Discover: The sum is always divisible by 11. Why does this happen?

Why does this happen?

Answer: 

Let the two-digit number be 10a + b, where a is the tens digit and b is the units digit.

The reverse of the number is 10b + a.

Now add the two:  (10a+b)+(10b+a)=11a+11b=11(a+b)

This sum is clearly divisible by 11, because 11 is a common factor.

Activity 2: Three-Digit Number Difference

Objective: Explore divisibility through subtraction of reversed numbers.

Procedure 

 Ask students to think of a three-digit number. 

 Now, they should make a new number by putting the digits in the reverse order. 

 Subtract the smaller number from the larger one. 

 Ask the students to check by which number the difference so obtained is divisible. Which other multiple of this divisor will divide the difference? 

 Discuss how this happens! 

Steps:

1. Choose any three-digit number (e.g., 741).

2. Reverse the digits (e.g., 147).

3. Subtract the smaller from the larger (e.g., 741 - 147 = 594).

4. Ask: Which number divides this difference?

Question: What is the difference divisible by?

Answer: The difference is always divisible by 99.

Follow-Up:

• Try multiple three-digit numbers. What do you notice?

• Is there a common divisor?

✨ Discover: The difference is always divisible by 99 (or sometimes 9 and 11).

Why does it work?

Let the number be 100a + 10b + c, and the reverse is 100c + 10b + a.

Now subtract the smaller from the larger:

$$(100a+10b+c)-(100c+10b+a)=99a-99c=99(a-c)$$

So the difference is always divisible by 99.

🧠 Bonus: Since 99 = 9 × 11, it's also divisible by 9 and 11.

Activity 3: Rotating Digits of a Three-Digit Number

Objective: Explore rotational patterns in digits and their divisibility.

Procedure 

 Students may be asked to think of any 3-digit number (abc). 

 Now, using this number, students may be asked to form two more 3-digit numbers (cab, bca).

Now, add the three numbers so formed. 

 Students may explore the smallest number by which it will be divisible. 

 Discuss how this happens!

Steps:

1. Think of a three-digit number (e.g., 231).

2. Create two more numbers by rotating the digits:

   • (cab → 312)

   • (bca → 123)

3. Add the three numbers:

   • 231 + 312 + 123 = 666

4. Ask: What is the smallest number that always divides the sum?

Answer: The sum is always divisible by 37 (and also 3 and 9).

✨ Discover: The sum is divisible by 37 (and also 3 and 9)!

Why does it work?

Let the three-digit number be abc:

• Its numerical value is 100a + 10b + c

• Rotation 1 (cab): 100c + 10a + b

• Rotation 2 (bca): 100b + 10c + a

Now add all three:

$$(100a+10b+c)+(100c+10a+b)+(100b+10c+a)=111a+111b+111c=111(a+b+c)$$ $$\text{<br>}\text{<br>}$$

Class 6 NCERT bridge course Answers Activity W3.3 A Treasure Hunt

 Activity W3.3 A Treasure Hunt 

 Provide each student with a copy of the treasure map, which includes coordinates (i.e., pairs of numbers discussed in earlier activity) marking the location of the treasure. 

 Explain the objective of the activity: to use the given coordinates to locate the treasure. 

 Allow students to work individually or in pairs to navigate the map and find the treasure.

Once the treasure is found, celebrate the successful completion of the hunt and discuss the coordinates used to locate the treasure. 

 Encourage students to create their own treasure maps for future activities, incorporating coordinates and landmarks of their choice. 

Creating patterns and designs with rotational and reflection symmetry Symmetry is a property where one shape or arrangement can be transformed into another that looks the same.

Let’s imagine this map uses a simple grid system: the bottom left is (0,0) and the top right is (10,10). Here's how we can place the main points:

• Starting point (where the pirate boy is) — (9,2)

• Pirate Ship — (2,1)

• Skull Rock — (3,5)

• Crocodile Pond — (5,6)

• Lighthouse — (7,9)

• Dragon Cave — (8,6)

• X Marks the Treasure — (6,3)

Path to the Treasure:

1. Start at (9,2) — The boy.

2. Head southwest to the Pirate Ship at (2,1).

3. Move north to the Skull Rock at (3,5).

4. Go northeast to Crocodile Pond at (5,6).

5. Move southeast to reach the Treasure at (6,3).

Named Points:

• A (9,2) — Start

• B (2,1) — Pirate Ship

• C (3,5) — Skull Rock

• D (5,6) — Crocodile Pond

• E (6,3) — Treasure!

So the final treasure is at Point E (6,3).

The path starts at the boy, passes the pirate ship, skull rock, crocodile pond, and finally reaches the treasure at (6,3). 

New Route & Coordinates:

1. Start (Boy) — (8,3)

2. Dragon Cave — (7,7)

3. Snake — (5,7)

4. Water Pond — (5,6)

5. Skull Rock — (3,5)

6. Pirate Ship — (4,2)

7. Lighthouse — (6,8)

8. Crocodile Pond — (2,6)

9. Treasure — (7,4)

New Story Clues:

Clue at Start (8,3)
"Ahoy, young adventurer! Your journey begins where the sun meets the sea. Seek the ship with black sails, it waits for thee!"

"The journey starts at the shore's embrace, where adventure waits at a dragon's place."

"Set sail from the sandy shore, a bony face waits at place four."

🐉 Clue at Dragon Cave (7,7)

"The dragon sleeps but the path's not done, follow the glow of the rising sun!"

"A dragon's roar guides the way, to a slippery friend where secrets lay."

"A fiery friend guards the way, but thirst leads the next display."

Snake (5,🐍 Clue at Snake (5,7)

"Slither and hiss, the serpent's near, climb to the light, the path is clear!"

"Slither past without a sound, where water glimmers, treasure's bound."

"Slither past without delay, crocs await along the bay."

💧 Clue at Water Pond (5,6)

"Cool waters and ripples wide, but snakes nearby know where riches hide!"

"A drink for the weary, cool and clear, look for the skull that whispers fear."

"Cool water to quench your thirst, but to see the light, climb first."

💀 Clue at Skull Rock (3,5)

"A skull of stone watches the shore, to quench your thirst, seek water and more."

"The rock with eyes has seen it all, now sail to the ship that heard the call."

"The silent skull hides no lie, the ship with sails is nearby."

🦜 Clue at Pirate Ship (4,2)
"Ye found the ship, brave and bold, but the treasure's still untold. Beware the crocs — to the swamp you must go, where the water runs slow."

"Yo-ho-ho! But not yet gold — the lighthouse stands, so brave and bold."

"From deck to cave, the dragon's near, brave the beast and show no fear."

💡 Clue at Lighthouse (6,8)

"From this tower the sea is seen, but the dragon guards what's golden and green!"

"High and bright, it lights the way, where crocs are waiting to lead astray."

"From atop the world so high, the slithering snake is nearby."

🐊 Clue at Crocodile Pond (2,6)
"Snap and splash, the crocs do play, the skull-shaped rock points the way!"

"Snap and splash, the final clue! Head to (7,4) for the treasure true."

"Watch your step at the snapping jaws, the treasure waits with pirate laws."

🏴‍☠️ Final Clue at Treasure (7,4)
"X marks the spot, you’ve braved the quest, dig right here for the pirate’s chest!"

"You've solved the riddles, faced each test, dig here now and claim your chest!"

"X marks the spot — dig, and the chest is yours!"

Class 8 NCERT bridge course Answers Activity W 5.1 FRACTALS IN NATURE

 Activities for Week 5 

Activity W 5.1 FRACTALS IN NATURE

 Fractals are all around us in nature. 

They help explain the irregular, repetitive patterns in many things we see every day. 

Students may be made to explore how fractals can represent real-world data using patterns and algorithms. 

Materials Required: 

Graph paper, printouts with different fractal patterns and their corresponding data (such as trees, coastlines, plant structures or any other structure of their choice). 

Procedure 

1. Students may be asked to get a photograph of a tree. 

They may observe the branching. 

They may use this data to create a branching fractal pattern.

 The data shows how each branch divides into smaller branches at a constant angle. 

They may use a repetitive method to create fractal. 

2. This activity may be done for coastline data. The coastline is irregular and jagged, which means it’s a fractal pattern. 

Use this data to draw an irregular coastline that gets more jagged as you zoom in.

3. Another similar activity can be done for the adjoining picture. 

Extension 

Data related to clouds, or mountains, or even river systems can serve as good models for the concept of fractals.