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.

Class 8 NCERT bridge course Answers Activity W 4.6 A fractal

 Activity W 4.6 A fractal

Objective:

 This activity will help students to: 

• Understand the concept of fractals.

• Build a simple fractal (Sierpinski Triangle   or Koch Snowflake) using repeated patterns.

Materials Required: 

Paper, markers, ruler (optional), printouts of basic geometric shapes (triangles, squares, etc.). 

A fractal is a shape or pattern that repeats itself no matter how much you zoom into it. 

Procedure

 Guide students to build fractals step-by-step. 

1. Draw a large equilateral triangle on your paper. 

2. Find the midpoints of each side and join them. Remove the middle triangle so obtained. 

It will look as shown—

 3. Again, join the midpoints of each of the sides of the three shaded triangles. 

Remove the three triangles so obtained. It will look like

4. Ask the students to continue the process for at least two more times. 

Take sufficiently large paper or cardboard for this purpose. 

Reflection

 Ask the students to describe the final structure they obtain. 

ANSWER:

The final structure looks like a big triangle made up of smaller and smaller triangles.

Does the final one looks like a big triangle made up of smaller triangles, and if we zoom in,?

ANSWER:

Yes! If we zoom in, the same pattern keeps repeating — this is the main property of a fractal.

 do we see the same pattern inside? This is a fractal. 

ANSWER:

• A fractal shows self-similarity: no matter how much you zoom in or out, the shape looks the same.

• Pattern Observation:

  • Number of triangles increases at every step:

    • Step 1: 1 large triangle.

    • Step 2: 3 shaded triangles.

    • Step 3: 9 smaller shaded triangles.

    • Step 4: 27 even smaller shaded triangles.

  The number of triangles follows this pattern:
  Number of shaded triangles = 3ⁿ (n = step number)

Extension

 1. Students may try this for different shapes, such as square, rectangle, parallelogram or any other shape of their choice. 

1. Try this fractal pattern with other shapes like:

   • Squares (creates a Sierpinski Carpet),

   • Rectangles,

   • Parallelograms,

   • Hexagons.

Squares – Sierpinski Carpet

• Step 1: Start with one large square.

• Step 2: Divide the square into 9 smaller squares (like a tic-tac-toe grid).

• Remove the middle square.

• Step 3: For each remaining square, repeat the same process: divide into 9 parts, remove the middle one.

• Keep repeating!

👉 Pattern:

• Number of remaining squares grows as 8ⁿ (n = step number).

• Creates a flat, grid-like fractal called the Sierpinski Carpet.

Rectangles

• Step 1: Start with a rectangle.

• Step 2: Divide it into equal smaller rectangles (4, 6, 8 parts or more, depending on shape).

• Remove a central part or selected parts.

• Repeat the same for the remaining rectangles.

Pattern:

• You will get a rectangular fractal where the shape keeps splitting into smaller rectangles, showing self-similarity.

Parallelograms

• Step 1: Start with a parallelogram.

• Step 2: Divide it into smaller, identical parallelograms.

• Remove one or more central parts.

• Repeat the process for each remaining parallelogram.

Pattern:

• The structure will look like a stretched or skewed version of the square fractal, but still self-similar and recursive.

Hexagons

• Step 1: Start with a regular hexagon.

• Step 2: Divide the hexagon into 7 smaller hexagons (1 central + 6 surrounding).

• Remove the central hexagon.

• Repeat the process for each remaining hexagon.

Pattern:

• A repeating honeycomb-like fractal — looks similar to snowflake patterns or beehives!

• The edges form increasingly complex, yet balanced, hexagon-based designs.

No matter the shape — square, rectangle, parallelogram, or hexagon — when you:

1. Divide,

2. Remove parts,

3. Repeat the process...

...you create a fractal: a never-ending, self-similar, and geometric pattern.

2. Students may be asked to associate numbers with each shape. 

Say, number of triangles obtained in each step or the number sides on outer edges in each step (3, 6, ...).

 They need to find a pattern.

Pattern in Outer Edges

• As you remove triangles, the number of sides or edges grows.
• For each step, the outer shape becomes more detailed but still follows a predictable mathematical pattern.

Sierpinski Triangle

• Number of Triangles at Each Step:

  Step	Triangles Formed
  0 (Start)	1
  1	3
  2	9
  3	27
  4	81

  Pattern:
  Number of triangles = 3ⁿ (where n = step number)

   Outer Edges (Visible Triangle Sides):

  Step	Outer Edges
  0	3
  1	6
  2	12
  3	24

  Pattern:
  Outer triangle sides double each step: 3 × 2ⁿ

  Sierpinski Carpet (Square Fractal)

   Number of Squares at Each Step:

  Step	Squares Remaining
  0	1
  1	8
  2	64
  3	512

  Pattern:
  Number of squares = 8ⁿ

   Outer Edges:

  Step	Squares on One Side	Total Outer Sides
  0	1	4
  1	3	12
  2	9	36
  3	27	108

  Pattern:

  • Each side is split into 3ⁿ parts.

  • Total outer sides = 4 × 3ⁿ

  Parallelogram Fractal (Custom logic – similar to square)

  If we divide into 4 parts (2×2 layout) and remove the middle one:

   Number of Parallelograms:

  Step	Shapes
  0	1
  1	3
  2	9
  3	27

  Pattern:
  Just like triangle: 3ⁿ

  Hexagon Fractal

  (Similar to honeycomb — 7 small hexagons, remove the center)

  Number of Hexagons:

  Step	Hexagons Remaining
  0	1
  1	6
  2	36
  3	216

  Pattern:
  Number of hexagons = 6ⁿ

  Summary Table:

  Shape	Shapes per Step	Pattern	Outer Sides Pattern
  Triangle	3ⁿ triangles	Exponential	3 × 2ⁿ
  Square	8ⁿ squares	Exponential	4 × 3ⁿ
  Parallelogram	3ⁿ parallelograms	Exponential	Similar to triangle logic
  Hexagon	6ⁿ hexagons	Exponential	Grows like a honeycomb

• 
  

Conclusion:

A fractal is a fascinating geometric figure where the same pattern repeats itself at every scale. 

The Sierpinski Triangle is a perfect example of this, and by exploring other shapes, we can discover many more fractals in nature and mathematics.

Class 8 NCERT bridge course Answers Activity W 4.5 Building Towers with Blocks

 Activity W 4.5: Building Towers with Blocks 

Material Required 

 Different coloured blocks. 

 Paper and pencil for recording results. 

Explain to the students that each block will represent a number, and stacking blocks will help illustrate exponents.

For example, stacking 2 blocks means 2¹ and stacking 4 blocks means 2² and so on

Procedure & Observation

Ask students to create towers representing different powers of 2: 

 For 2¹  , they stack 2 blocks. 

 For 2² , they stack 4 blocks. 

 For 2³ , they stack 8 blocks. 

Students stack blocks to represent powers of 2:

Exponent	Mathematical Form	Number of Blocks	3-D Shape Formed
2¹	2	2 blocks	Cuboid
2²	4	4 blocks	Cube (if arranged as 2x2x1)
2³	8	8 blocks	Cuboid or Cube (if arranged as 2x2x2)

1. After stacking blocks for different exponents, students may be made to explore what 3-D shape they get. 

For example, 2¹ gives a cuboid, whereas 2² gives a cube and so on. 

Discussion:

What 3-D shape do they get?

• For 2¹ = 2 blocks, students usually get a Cuboid (because two blocks form a rectangular shape).

• For 2² = 4 blocks, if arranged symmetrically (2x2), they can get a Cube.

• For 2³ = 8 blocks, if arranged as 2x2x2, it forms a Perfect Cube.

Conclusion:

• When the blocks can be arranged equally in all three dimensions (length, width, and height), the shape is a cube.

• Otherwise, it remains a cuboid.

They may see for which exponent of 2 they get a cube and for which other a cuboid. 

Can they get any other 3-D shape other than a cube or cuboid?

No, using only simple stacking of square or rectangular blocks, the shapes are usually limited to cuboids or cubes.

Other 3-D shapes like pyramids or spheres cannot be formed unless the block shape or stacking style is changed. 

2. They may observe as to how the number of blocks increases as the exponent increases. 

Is there any pattern?

Is there any pattern in the number of blocks as the exponent increases?

Yes! The number of blocks doubles every time the exponent increases by 1.

Exponent	Number of Blocks
2¹	2
2²	4
2³	8
2⁴	16

Pattern:
As the exponent increases by 1, the number of blocks is multiplied by 2.

Conclusion:
This activity helps visualize how exponents grow and how they relate to 3-D shapes like cubes and cuboids. The number of blocks follows a clear doubling pattern as the exponent increases.

Class 8 NCERT bridge course Answers Activity W 4.4 Exploring Data Through Graphs and Charts

 Activity W 4.4 - Exploring Data Through Graphs and Charts

 Procedure

 A project may be given to students to collect the data from reliable sources. 

Students should be divided into groups of 4–5. 

1. Every group has to collect data on the following topics: 

 Temperature of your city in the month of July for the last 5 years. 

 Literacy rate of any 5 states of India in the last five years. 

How many students of your class like ice-cream among the following: 

vanilla, chocolate cone, butter scotch, strawberry and kesar-pista. 

What is the favourite game among the following: cricket, football, basketball, tennis, badminton and volleyball. 

 Collect data from the students of your class.

2. Each group has to make a table with tally marks. 

3. Each group has to draw a bar graph, line graph and pictograph for the collected data. 

Teacher will provide opportunity to every group to present their work in front of the whole class. 

Here is an example: 

Take population of a country in different decades. 

Represent the data as a pictograph, bar graph and line graph. 

 Pictograph 

😊 = 20 crore people.

Bar Graph

Line Graph

Discuss:

1. What is the difference between these three graphs?

Graph Type	Use	Visual Advantage
Pictograph	Uses icons or symbols to show data.	Makes data fun and easy to understand.
Bar Graph	Uses bars to represent quantities.	Great for comparing groups or categories.
Line Graph	Connects data points to show changes over time.	Best for showing trends and progressions.

• Pictograph:
  A pictograph uses pictures or symbols to represent data. Each symbol stands for a specific number of items. It makes the data easy to read and more visually interesting, especially for younger audiences.

• Bar Graph:
  A bar graph uses rectangular bars (either vertical or horizontal) to show the quantity of different categories. The length of the bar shows how large or small the value is. It is useful for comparing data from different groups.

• Line Graph:
  A line graph uses points connected by lines to show trends over time. It helps to easily spot increases or decreases in the data and is best used for data that changes continuously (like temperature or literacy rates).

2. In which situation could a line graph not be drawn from the data collected by the students and why?

A line graph cannot be drawn for:

• Ice-cream preferences

• Favourite games

Reason:
A line graph is used for continuous data or to show change over time.
Ice-cream flavours and favourite games are examples of categorical data (choices, not numbers that change over time). Since these are simply preferences without any timeline or continuous flow, a line graph would not be appropriate.

• Line Graphs are for time-based or continuous data (like temperature or literacy rate).

• Bar Graphs & Pictographs are great for category-based data (like games or ice-cream).