class 8 NCERT bridge course Answers Activity 1W1.2: Dice Math Challenge

 Activity W1.2 Dice Math Challenge 

 Material Required:

• Two large dice (these can be made using cubical cardboard boxes)

Instructions for the Teacher:

• Divide the class into two teams. You can choose creative team names (e.g., “The Math Wizards” and “The Number Ninjas”).
• Draw a line to split the blackboard into two sections and write the team names on each side.
• Each student will take turns throwing the dice for their team.

Steps to Play:

• A student from Team A throws the dice and announces the number that appears on the top.

• The team will multiply the number by itself 3 times (i.e., calculate $number \times number \times number$) and the result will be written under Team A's column on the board.

• A student from Team B takes their turn, following the same steps.

• After each throw, both teams add their new result to their team’s total score on the board.

• Continue until each team has had the set number of turns (e.g., 10 or 12 turns).

• The team whose total score is closest to the target number (500 or 1000, as decided) at the end wins!

                            

SL NO	TEAM A	CUBE	TEAM B	CUBE
1	5	125	2	8
2	1	1	6	216
3	6	216	4	64
4	4	64	3	27
5	2	8	1	1
6	5	125	2	8
7	6	216	5	125
8	3	27	3	27
9	3	27	5	125
10	4	64	5	125
11	1	1	6	216
12	2	8	1	1
13	4	64	2	8
14	3	27	1	1
15	4	64	3	27
	TOTAL	1037	TOTAL	979

TEAM A IS THE WINNER

Based on the above activity, 

Some reflective questions may be discussed, such as: 

1. Number Observation:
   Look at the scores written on the board. Try to find numbers other than seen on the board that are square/cube numbers. 

👉 Question: Can you name a square or cube number that wasn’t written on the board during the game?

ANSWER:
Square numbers (Example: $4, 9, 16, 25, 36, 49, 64, 81, 100, \dots$)

Cube numbers (Example: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \dots$)

2. Cube Building Challenge:

 Students may be given some number of unit cubes say, 8, 10, 25, 27, 30, 50, 64. 

They may be asked to make a bigger cube out of these given number of cubes. 

They may be asked, for which number of cubes they could make a bigger cube and which they could not. Ask them to explain.

You are given unit cubes: 8, 10, 25, 27, 30, 50, 64.

A perfect cube means the total number of cubes can be arranged into a shape where length = width = height (all sides equal). The total must be $n^3$ — a cube number.

Let’s check each one:

Number of Cubes	Can Form a Perfect Cube?	Reason
8	✅ Yes	$2\times 2\times 2=\mathrm{8<span\; class="katex"><br></span> }$$2^3$.
10	❌ No	10 is not a cube number. $2^3 = 8$ and$3^3 = 27$. 10 falls between.
25	❌ No	25 is not a cube number. $2^3 = 8$, $3^3 = 27$. 25 is not in the list.
27	✅ Yes	$3 \times 3 \times 3 = 27$. It is $3^3$.
30	❌ No	30 is not a cube number. $3^3 = 27$ and $4^3=\mathrm{64.\; 30\; falls\; between.}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$
50	❌ No	50 is not a cube number. $3^3 = 27$ and $4^3 = 64$. 50 falls between.
64	✅ Yes	$4\times 4\times 4=\mathrm{64.\; It\; is }$$4^3$.

ANSWER: 

You can make a perfect cube with: 8, 27, and 64. these numbers can form a larger cube.

ANSWER:

 You cannot make a perfect cube with: 10, 25, 30, 50 — these numbers cannot form a larger cube because these numbers are not cube numbers.

 3. Rubik’s Cube Mystery:

Look at a Rubik’s Cube. Guess:
👉 How many small unit cubes make up a complete Rubik’s Cube?

💡 Hint: A Rubik's Cube is usually a cube with equal sides — try imagining how many little cubes fit along each edge and then multiply!

A rubik cube may be shown to students to guess how many small unit cubes have been used to make it.

ANSWER: 

A standard Rubik’s Cube is a 3×3×3 cube.

$$3\times 3\times 3=27$$So, 27 small unit cubes are used to make one complete Rubik's Cube! 

??? Extra Challenge:
If you could design your own cube puzzle, how many small cubes would you choose, and why?

Inclusion of Special Children:

• Pair special children with supportive peer buddies for guidance.

• Encourage them to take active roles: throwing the dice, multiplying the numbers, or writing the scores on the board.

• This teamwork helps build confidence and fosters participation.

class 8 NCERT bridge course Answers Activity 1W1.1: Recreational Puzzle

 Bridge Programme Content: Detailed Activity 

Activities for Week1

Activity W1.1: Recreational Puzzle 

Material Required: 

A calendar of any month, coloured sketch pens and sheet of paper

May 2025

A calendar may be given to students in pairs. 

They have to observe the calendar keenly and write their observations in the notebook.

Let the students do on their own. 

Teacher can suggest question but should not reveal the conclusions before taking responses of students’ observations. 

Teacher may ask students to do the following: 

Draw boxes around square numbers (number obtained when a number is multiplied to itself two times. 9 = 3 × 3 is a square number) and colour them with a single colour. 

Draw circles around cube numbers (number obtained when a number is multiplied to itself three times. 8 = 2 × 2 × 2 is a cube number.) and colour them with a new colour. 

Draw triangle around prime numbers. O Students may be asked to choose any 2 by 2 number square grid from the calendar they have. 

Instructions for Students

1. Observing the Calendar:

Students are given a calendar (in pairs). For example, May 2025 starts on Thursday and has 31 days.

They are asked to observe patterns in the calendar and note down their own observations.
Examples of what students might observe:

• The month starts on a Thursday.

• All weeks have 7 days.

• The 1st, 8th, 15th, 22nd, and 29th are all Thursdays.

• The numbers in the same column increase by 7.

2. Marking Specific Numbers:

Using coloured pens, students mark special types of numbers:

Square Numbers (draw boxes around and colour with one colour):
→ 1, 4, 9, 16, 25

Cube Numbers (draw circles around and colour with a second colour):
→ 1, 8, 27

Prime Numbers (draw triangles around):
→ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

3. Choose any 2x2 number square from the calendar
(For example: Choose the block with 8, 9, 15, 16)

They should try to answer the following questions: 

What is the sum of the numbers in the diagonals of the grid? 

What is the sum of numbers in the diagonals?

• Diagonal 1 (8 + 16) = 24

• Diagonal 2 (9 + 15) = 24

 What magic do you observe? 

 Observation / Magic:

• The sum of both diagonals is the same in a 2x2 grid.

Based on the above activity, some reflective questions may be discussed, such as: 

Reflective Questions & Sample Student Responses

Q1: How do you differentiate between square and cube numbers?

• A square number is a number multiplied by itself once (e.g., 4 = 2 × 2).

• A cube number is a number multiplied by itself twice more (e.g., 8 = 2 × 2 × 2).

Q2: How do you identify prime numbers?

• Prime numbers have exactly two factors: 1 and itself.

• They are not divisible by any other number.

Q3: Why is the sum of diagonals always same in a 2x2 grid?

• Because the opposite corners in a 2x2 square are symmetrical and balance each other.

• The sums (top-left + bottom-right) and (top-right + bottom-left) are always equal due to arithmetic symmetry.

Extension of the Activity 

 Teacher can extend this activity by suggesting students to take a 3 by 3 grid. 

Extension Activity (3x3 Grid)

Take a 3x3 grid (e.g., from 8 to 16):

Why is the sum of diagonals always same for any 2 by 2 grid? 

They may find— 

1. The sum of diagonal numbers. 

Sum of diagonals:

• Diagonal 1 (8 + 16 + 24) = 48

• Diagonal 2 (10 + 16 + 22) = 48

2. The mean of all numbers. 

Mean of all numbers in the grid:
Sum of all numbers: 8+9+10+15+16+17+22+23+24 = 144
Mean = 144 ÷ 9 = 16

Special Observations:

• The center number (16) is the mean.

• Diagonals again have equal sums.

• There is a symmetry in the calendar numbers.

They may be asked to see what special thing they observed. 

Teacher may encourage students to explore more such patterns in the calendar

Encouragement for Exploration

Students can be encouraged to:

• Try different-sized grids (like 4x4).

• Observe symmetry across weeks.

• Explore patterns in odd/even numbers.

• Make their own puzzles from the calendar.

Circumference of the circle

Today we all use the formulas 2πr or πd to calculate the circumference of a circle. Can you believe it if I tell you that this is a Tamil invention??? Look below and you will definitely be surprised.

The formula for calculating the circumference of a circle is set in the Kanakathikaram, Kakkai Padinium etc.

 Crow Patinium

******************************

Song :-

"Vitamore seven, add four to the circle

Sattena Irati Chain

Thikaipana Churtthane"

Setting :-

(Perimeter - P, Diameter - V, Radius - Aa)

= (Vitamore seven) = V/7 

= (Add four to the circle) = V+4V/7

= (Double chain) = 2[V + 4V/7]

(Thikaipana Churtthane)

= 2[V + 4V/7

= 2[11V/7]

= 2x11V/7

= 22/7 x V

According to the present, if 22/7 = π and V=d (diameter)...

Circumference of the circle = 22/7 x V

= πd

(d = 2r) Circumference = 2πr

So today  It is a matter of pride for Tamils ​​that a Tamil has calculated the formulas used to find the circumference of a circle without using π.....

✍ Tamil Forum

Circumference of the circle 

இன்று நாம் அனைவரும் வட்டத்தின் சுற்றளவை கணிப்பதற்கு 2πr அல்லது πd எனும் சூத்திரங்களை பாவிக்கின்றோம். இது தமிழரின் கண்டுபிடிப்பு என்று சொன்னால் உங்களால் நம்ப முடியுமா??? கீழே பாருங்கள் நிச்சயம் ஆச்சரியப்படுவீர்கள்.

வட்டத்தின் சுற்றளவை கணிப்பதற்கு கணக்கதிகாரம், காக்கை பாடினியம் போன்றவற்றில் சூத்திரம் அமைக்கப்பட்டுள்ளது.

காக்கை பாடினியம்

****************************

பாடல் :-

"விட்டமோர் ஏழு செய்து

திகைவர நான்கு சேர்த்து

சட்டென இரட்டி செயின்

திகைப்பன சுற்றுத்தானே"

நிறுவல் :-

(பரிதி - ப, விட்டம் - வி, ஆரை - ஆ என வைத்தால்)

= (விட்டமோர் ஏழு செய்து) = வி/7 

= (திகைவர நான்கு சேர்த்து) = வி+4வி/7

= (சட்டென இரட்டி செயின்) = 2[வி + 4வி/7]

(திகைப்பன சுற்றுத்தானே)

= 2[வி + 4வி/7

= 2[11வி/7]

= 2x11வி/7

= 22/7 x வி

தற்காலத்தின் படி 22/7 = π எனவும் வி=d (விட்டம்) எனவும் கொண்டால்...

வட்டத்தின் சுற்றளவு = 22/7 x வி

= πd

(d = 2r ஆக) சுற்றளவு = 2πr

ஆக இன்று வட்டத்தின் சுற்றளவு காண பயன்படுத்தப்படும் சூத்திரங்களை தமிழன் π பயன்படுத்தாமலே கணித்துள்ளான் என்பது தமிழருக்கு பெருமையே.....

✍ தமிழ் கருத்துக்களம்

 MATHEMATICAL GARDEN IDEAS

An illustration of a mathematical garden that visually represents the spiral root concept using plants and flowers. 

A mathematical garden illustrating the spiral root concept using plants and flowers. 

Creating a spiral root concept in a math garden using plants and flowers sounds fascinating! 

1. Center Point: Start with a central point in your garden.

2. Spiral Layout: Arrange plants and flowers in a spiral pattern, expanding outward from the center.

3. Square Root Spiral: Each segment of the spiral can represent a square root value, such as √2, √3, √4, and so on

Spiral Root Math Garden Layout

1. Central Focal Point (🌼):

   • Start with a vibrant flower or plant at the center of your garden. This represents the origin point (0).

2. Spiral Pathway:

   • Create a path that spirals outward from the center, resembling the shape of a square root spiral.

   • Each turn of the spiral represents an incremental square root value (√1, √2, √3, etc.).

3. Plant Arrangement:

   • First Loop (√1):

     • Plant low-growing flowers like pansies or marigolds close to the center.

   • Second Loop (√2):

     • Use medium-height plants like lavender or sage.

   • Third Loop (√3):

     • Incorporate taller plants such as roses or dahlias.

   • Fourth Loop (√4):

     • Include even taller elements like sunflowers or ornamental grasses.

4. Visualizing Square Roots:

   • The increasing distance from the center symbolizes the increasing value of square roots.

   • The spiral's expansion represents the non-linear growth of square root functions.

5. Pathway Materials:

   • Define the spiral walkway with stones, bricks, or mulch.

   • Consider using different colors or textures for each loop to enhance the visual distinction between square root values.

Metaphorical Elements:

• Growth and Learning:

  • The spiral symbolizes the journey of learning and discovery in mathematics.

  • As one walks the spiral path, they're metaphorically delving deeper into mathematical concepts.

• Connection to Nature:

  • By integrating mathematical concepts with natural elements, you create a space where abstract ideas become tangible.

Additional Enhancements:

• Educational Markers:

  • Place small signs along the path indicating the square root values and perhaps interesting facts or equations.

• Golden Ratio Spiral:

  • For an added mathematical twist, design the spiral based on the Fibonacci sequence or the golden ratio, which occurs frequently in nature.

• Color Gradation:

  • Choose plants whose flower colors transition gradually, creating a gradient effect that enhances the spiral design.

Example Layout Idea:

• Start with white flowers at the center.

• Gradually transition to yellow, then orange, red, purple, and finally blue as you move outward.

• This color shift not only adds visual appeal but can represent the spectrum of light or other mathematical gradients.

Implementing the Spiral:

1. Planning:

   • Use a rope or hose to lay out the spiral shape on the ground before planting.

   • Mark the path with stakes to ensure accuracy.

2. Calculating Distances:

   • Use the spiral equation $r=a\sqrt{\theta }r\; =\; a\; \backslash sqrt\{\backslash theta\}$ where:

     • $rr$ is the radius,

     • $aa$ is a constant that controls the distance between loops,

     • $\theta \backslash theta$ is the angle in radians.

   • This will help you space the loops appropriately according to square root values.

3. Soil Preparation:

   • Ensure the soil is well-prepared with compost and suitable for the plants you've chosen.

   • Consider soil testing to balance pH levels.

Bringing It All Together:

By integrating the spiral root concept into your garden, you create a living embodiment of mathematical principles. This not only makes for a beautiful garden but also a conversation starter and an educational experience.

Let’s Visualize the Concept Further:

Imagine walking through your garden starting at the center:

• Step 1: You see delicate white pansies symbolizing the beginning.

• Step 2: As you follow the spiral, yellow marigolds appear, marking the transition to the next square root value.

• Step 3: Further along, red roses emerge, taller and more prominent.

• Step 4: Nearing the outer loops, majestic sunflowers tower above, representing larger square root values.

Final Thoughts:

Incorporating the spiral root concept blends the beauty of nature with the elegance of mathematics. It's a creative way to explore mathematical ideas while enjoying the pleasures of gardening.

The garden features a spiral pattern with roots intertwining in.

                       🌼

                      /   \

                     🌻     🌸

                    /         \

                   🌷           🌺

                  /               \

                 🌹-----------------🌿

                  \               /

                   🌱           🌵

                    \         /

                     🍄     🌴

                      \     /

                       🌳

The spiral pattern follows a Fibonacci sequence, blending mathematical beauty with nature.