SAT EXAM PREPARATION 2025-2026

Question:

There are 66 calories in 15 grams of grated Parmesan cheese, and 59% of those calories are from fat.
When measuring Parmesan cheese, 5 grams is equal to 1 tablespoon.

Which of the following is closest to the number of calories from fat per tablespoon of grated Parmesan cheese?

Options:

• A) 3

• B) 8

• C) 9

• D) 13

---

Solution:

Step 1: Calculate total fat calories in 15 grams

$$\text{Fat calories} = 59\% \text{ of } 66 = 0.59 \times 66 = 38.94 \approx 39 \text{ calories}$$

Step 2: Find fat calories per gram

$$\frac{39 \text{ calories}}{15 \text{ grams}} = 2.6 \text{ calories per gram}$$

Step 3: Find fat calories per 1 tablespoon (which is 5 grams)

$$2.6 \times 5 = 13 \text{ calories from fat per tablespoon}$$

---

✅ Correct Answer: D) 13

This is the closest value to the actual fat calories per tablespoon.

Question:

The base of a tree has 10 mushrooms growing from its roots.
The mushroom population doubles every 5 days.

What type of function best models the relationship between the mushroom population and time?

Options:

• A) Decreasing exponential

• B) Decreasing linear

• C) Increasing exponential

• D) Increasing linear

---

Solution:

Let’s understand what’s happening:

• The starting population is 10 mushrooms.

• The population doubles every 5 days, which means it multiplies by 2 repeatedly over time.

This is a classic example of exponential growth, where the population is increasing over time, not decreasing.

Why not linear?

• Linear growth adds a fixed amount each time.

• Exponential growth multiplies (like doubling), so the rate of increase itself increases over time.

---

✅ Correct Answer: C) Increasing exponential

---

Bonus (Equation form):

The function could be modeled as:

$$M(t)=10\cdot {\mathrm{2^}}^{t\mathrm{/}5}$$

Where:

• $M(t)$ is the number of mushrooms after $t$ days,

• 10 is the initial count,

• The exponent $t\mathrm{/}5$ reflects doubling every 5 days.

  Question:
  

  Given the quadratic equation:

  $$x^2 + bx + c = 0$$

  where and are constants.

  If:

  $$\frac{-b+\sqrt{b^2-4c}}{2}=18\text{and}\frac{-b-\sqrt{b^2-4c}}{2}=10$$

  what is one possible value of $\mathrm{x\; ?}$

  ---

  Solution:

  These two expressions are the quadratic formula results for the roots of the equation:

  $$x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$$So, the two solutions are:

  $$x_1 = 18, \quad x_2 = 10$$
  

  Thus, one possible value of x is:

  ✅ Answer: or

  
  

Question:

Solve:

$$3y^2 - 4 = 5y + 1$$

Which of the following is a solution to the equation above?

Options:
A) $\frac{5 - \sqrt{85}}{6}$
B) $\frac{5 - \sqrt{85}}{3}$
C) 5
D) $5 + \sqrt{85}$

---

Solution:

Step 1: Start with the given equation

$$3y^2 - 4 = 5y + 1$$

Step 2: Move all terms to one side to set the equation to 0

$$3y^2 - 5y - 5 = 0$$

Now we solve this quadratic using the quadratic formula:

$$y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-5)}}{2(3)}$$ $$= \frac{5 \pm \sqrt{25 + 60}}{6} = \frac{5 \pm \sqrt{85}}{6}$$

So the two solutions are:

$$y = \frac{5 + \sqrt{85}}{6} \quad \text{or} \quad y = \frac{5 - \sqrt{85}}{6}$$

---

✅ Correct Answer: A) $\frac{5 - \sqrt{85}}{6}$

Here is the question text and a full step-by-step solution based on the image:

---

Question:

A rocket is launched vertically from ground level. The rocket reaches a maximum height of 46.36 meters above the ground after 2.4 seconds, and falls back to the ground after 4.8 seconds.

Which equation best represents the height $s$, in meters, of the rocket $u$ seconds after it is launched?

Options:

• A) $s = -8.05u^2 + 38.64u$

• B) $s = -4.8u^2 + 46.36u$

• C) $s = -u^2 + 46.36$

• D) $s = 8.05u^2 - 38.64u$

---

Solution:

We are told:

• The rocket starts at ground level → initial height = 0

• Maximum height is 46.36 meters at u = 2.4 seconds

• It returns to ground at u = 4.8 seconds

• So the vertex of the parabola is at $u = 2.4$, and the parabola opens downward

General form of a quadratic equation:

$$s = au^2 + bu + c$$

Since it starts at ground level: $c = 0$

Let’s use the vertex form:

$$s = a(u - h)^2 + k$$

Where:

• $h = 2.4$ (time of max height)

• $k = 46.36$ (maximum height)

$$s = a(u - 2.4)^2 + 46.36$$

We also know that at $u = 0$, $s = 0$ (ground level). Plug into the equation:

$$0 = a(0 - 2.4)^2 + 46.36 \Rightarrow 0 = a(5.76) + 46.36 \Rightarrow a = -\frac{46.36}{5.76} \approx -8.05$$

Now plug $a$ into the standard form:

---

✅ Correct Answer: A)

• The rocket is launched from ground level, so the initial height is $0$.

• Using the general form of a quadratic equation for height,

  $$s(t) = -au^2 + bu + c$$

• Since the initial height is 0, $c = 0$

---

At the maximum height:

$$s(2.4) = -a(2.4)^2 + 2.4b = 46.36$$

When it hits the ground:

$$s(4.8) = -a(4.8)^2 + 4.8b = 0$$

---

From these conditions, solve the system of equations:

$$-23.04a + 4.8b = 0 \quad \text{(1)}$$

This gives:

$$b = 4.8a$$

---

Substitute $b = 4.8a$ into:

$$-a(2.4)^2 + 2.4(4.8a) = 46.36\Rightarrow -5.76a + 11.52a = 46.36\Rightarrow 5.76a = 46.36\Rightarrow a = 8.05$$

Then:

$$b = 4.8 \times 8.05 = 38.64$$

---

Thus, the equation is:

$$s(u)=-8.05u^2+38.64u$$

WORKSHEET ch1 class 6

WORKSHEET - Number pattern

1)  1,3,5,7, ________, ___________,  _______

Rule- ________________

2)  2,4,6,8,________, ___________,  _______

Rule- ________________

3) 24,34,44,54, ________, ___________,  _______

Rule- ________________

4) 3, 6,9,12,________, ___________,  _______

Rule- ________________

5) 35,40,45, ________, ___________,  _______

Rule- ________________

6) 11,22, 33, ________, ___________,  _______

Rule- ________________

7) 9,19,29, ________, ___________,  _______

Rule- ________________

WORKSHEET - Number pattern solutions

1)  1,3,5,7, ___9_____, ___11________,  ____13___
Rule- _________odd number_______

2)  2,4,6,8,____10____, ___12________,  _____14__
Rule- _____even number___________

3)  24,34,44,54, 64 ,74, 84
Rule: By adding 10-10 in each number

4) 36,9,12,15 18 21
Rule-By adding 3 in each number.

5) 35,40,45, 50,55,60
Rule: By adding 5-5 in each number.

6) 11,22, 33, 44, 55
Rule-By adding 11-11 in each number.

7)
9,19,29, 39 49,59

Rule: By adding 10-10. in each number.

My Teaching Philosophy in Mathematics

 

My Teaching Philosophy in Mathematics

I believe mathematics is not just a subject of numbers but a language of logic, patterns, and possibilities that helps learners make sense of the world around them. My teaching philosophy is rooted in the idea that every student can develop mathematical thinking when learning is made meaningful, visual, and connected to real life.

In my classroom, I strive to blend conceptual understanding with hands-on experience. I use activities, visual aids, and real-world contexts to build bridges between abstract concepts and tangible situations—like using number lines for rational numbers, cube models for volume, or surveys for data handling. I believe that math should not be memorized but discovered, discussed, and applied.

I design learning experiences that nurture curiosity, collaboration, and confidence. Each concept is introduced through engaging strategies—story-based equations, interactive geometry, peer-led discussions, and application-oriented tasks—so that students not only understand 'how' but also 'why'.

Assessment, for me, is more than evaluation—it's a way to deepen learning. I use open-ended questions, peer activities, and visual tasks to gauge understanding and guide feedback. When gaps appear, I see them as opportunities to personalize support and reframe learning through remedial teaching.

Ultimately, my goal is to empower students with not just mathematical skills, but mathematical thinking—equipping them to reason logically, make decisions confidently, and appreciate the beauty and power of math in everyday life.

a short and brief version of your teaching philosophy in mathematics:

My Teaching Philosophy in Mathematics

I believe mathematics is best learned through real-life connections, hands-on activities, and visual thinking. My goal is to make math meaningful, engaging, and accessible to all learners. I use interactive strategies and practical examples to build strong conceptual understanding and problem-solving skills. Every student can succeed in math when learning is active, collaborative, and rooted in curiosity.

MATH IMAGES

 

Class 6 NCERT Solutions (2024-2025) Ganita Prakash Maths Chapter 1 Patterns in Mathematics

 Class 6 NCERT Solutions (2024-2025) Ganita Prakash Maths Chapter 1 Patterns in Mathematics

1.1 What is Mathematics?

Mathematics is the study of numbers, quantities, shapes, patterns, and logical relationships. It is both a science and a language that helps us describe, analyze, and solve problems in the real world.

Key Aspects of Mathematics:

1. Numbers & Calculations – Arithmetic (addition, subtraction, multiplication, division), algebra, and number theory.

2. Shapes & Spaces – Geometry, trigonometry, and measurements (area, volume, angles).

3. Patterns & Relationships – Algebra, functions, and sequences.

4. Data & Chance – Statistics (averages, graphs) and probability (predicting outcomes).

5. Logic & Reasoning – Problem-solving, proofs, and critical thinking.

Why is Mathematics Important?

• It helps in daily tasks (shopping, cooking, time management).

• It is essential in science, engineering, technology, and finance.

• It improves logical thinking and problem-solving skills.

Mathematics in Nature & Real Life:

• Symmetry in flowers and snowflakes.

• Patterns in weather, music, and art.

• Measurements in construction, medicine, and sports.

In short, mathematics is the foundation of understanding and organizing the world around us!

For example, the understanding of patterns in the motion of stars, planets, and their satellites led humankind to develop the theory of gravitation, allowing us to launch our own satellites and send rockets to the Moon and to Mars; similarly, understanding patterns in genomes has helped in diagnosing and curing diseases—among thousands of other such examples

Figure it Out - Page number 2 

1. Can you think of other examples where mathematics helps us in our everyday lives?

Solution:

Mathematics plays a crucial role in our everyday lives in many ways.

Examples of Mathematics in Daily Life:

1. Budgeting & Shopping:

   • Calculating expenses, discounts, and managing savings.

   • Comparing prices to find the best deals.

2. Time Management:

   • Planning schedules, estimating travel time, and setting alarms.

3. Cooking & Baking:

   • Measuring ingredients, adjusting recipe quantities, and setting cooking times.

4. Travel & Navigation:

   • Calculating distances, fuel consumption, and reading maps/GPS.

5. Home Improvement:

   • Measuring rooms for furniture, paint, or flooring.

   • Calculating areas and volumes for construction.

6. Banking & Finance:

   • Calculating interest on loans/savings, managing investments, and understanding taxes.

7. Sports & Fitness:

   • Keeping score in games, tracking calories, and measuring workout progress.

8. Technology & Gadgets:

   • Using smartphones, computers, and apps that rely on mathematical algorithms.

9. Health & Medicine:

   • Measuring body temperature, blood pressure, and medication dosages.

10. Weather Forecasting:

• Understanding temperature, humidity, and rainfall predictions.

Conclusion:

Mathematics plays a vital role in various professions, such as:

- Helping vegetable sellers calculate change accurately

- Enabling pot makers to craft pots of precise dimensions

- Assisting painters in estimating time and materials needed for a project

- Guiding masons in calculating the required number of bricks

- Aiding doctors in determining the correct dosage of medication

Mathematics is an essential tool in many real-world applications

Mathematics is everywhere—from simple daily tasks to complex decision-making. It helps us solve problems efficiently and make informed choices.

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 2. How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.).

Solution:

Mathematics has been the backbone of human progress, propelling advancements in science, technology, engineering, and society. Here are some key ways it has driven humanity forward:

1. Scientific Discovery & Experimentation

• Physics & Astronomy: Mathematics enabled Newton’s laws of motion, Einstein’s theory of relativity, and space exploration (e.g., calculating rocket trajectories).

• Medicine: Statistical models help in drug trials, epidemiology (e.g., predicting disease spread), and medical imaging (MRI, CT scans).

• Chemistry: Equations predict reactions, helping design new materials, medicines, and energy solutions.

2. Engineering & Infrastructure

• Bridges & Buildings: Calculus and geometry ensure stable structures (e.g., suspension bridges, skyscrapers).

• Transportation: Algorithms optimize traffic flow; equations design safer cars, planes, and high-speed trains.

• Electricity & Electronics: Maxwell’s equations underpin electrical engineering; binary math powers computers.

3. Technology & Communication

• Computers & AI: Binary math, algorithms, and cryptography drive computing and machine learning.

• Internet & Phones: Data compression (e.g., JPEG, MP3) and error-correcting codes enable fast, reliable communication.

• GPS & Navigation: Relies on trigonometry and relativity to pinpoint locations accurately.

4. Economics & Governance

• Finance & Markets: Probability and statistics guide investments, risk assessment, and economic policies.

• Democracy & Voting Systems: Game theory helps design fair voting mechanisms and prevent manipulation.

• Logistics & Supply Chains: Optimization math ensures efficient delivery of goods (e.g., Amazon, FedEx).

5. Everyday Life Innovations

• Clocks & Calendars: Astronomy-based math keeps time accurate (leap years, time zones).

• Consumer Tech: Math designs TVs (pixel algorithms), cameras (image processing), and even bicycles (gear ratios).

• Weather Forecasting: Differential equations and supercomputers predict storms and climate trends.

Conclusion

Mathematics has helped propel humanity forward by Scientific Discovery,  Technology, Economic and growth.

Mathematics is the "language of the universe," allowing us to model reality, solve problems, and innovate. Without it, modern civilization—from smartphones to space travel—would not exist.

 1.2 Patterns in Numbers 

Among the most basic patterns that occur in mathematics are patterns of numbers, particularly patterns of whole numbers: 0, 1, 2, 3, 4, ...
 The branch of Mathematics that studies patterns in whole numbers is called number theory.
Number sequences are the most basic and among the most fascinating types of patterns that mathematicians study. 

Some key number sequences that are studied in Mathematics.

1. All 1’s (1, 1, 1, 1, ...)

Q: What is the next term in the sequence: 1, 1, 1, 1, 1, ...?
A: 1

2. Counting Numbers (1, 2, 3, 4, ...)

Q: What is the 10th term in the sequence of counting numbers?
A: 10

3. Odd Numbers (1, 3, 5, 7, ...)

Q: What is the 6th odd number?
A: 11

4. Even Numbers (2, 4, 6, 8, ...)

Q: What is the sum of the first 3 even numbers?
A: 2 + 4 + 6 = 12

5. Triangular Numbers (1, 3, 6, 10, ...)

Q: How do you generate the next triangular number after 10?
A: Add the next counting number (5) → 10 + 5 = 15

6. Square Numbers (1, 4, 9, 16, ...)

Q: What is the 7th square number?
A: 7² = 49

7. Cube Numbers (1, 8, 27, 64, ...)

Q: What is the cube of 5?
A: 5³ = 125

8. Fibonacci (Virahānka) Numbers (1, 2, 3, 5, 8, 13, ...)

Q: What is the next Fibonacci number after 13?
A: 8 + 13 = 21

9. Powers of 2 (1, 2, 4, 8, 16, ...)

Q: What is 2 raised to the power of 6?
A: 64

10. Powers of 3 (1, 3, 9, 27, ...)

Q: What is 3⁵ (3 to the 5th power)?
A: 243

Bonus Challenge Questions:

1. Q: Which sequence has terms that are sums of the previous two numbers?
   A: Fibonacci (Virahānka) numbers

2. Q: Which sequence represents numbers that can form equilateral triangles?
   A: Triangular numbers

3. Q: What is the difference between consecutive square numbers? (e.g., 4 - 1 = 3, 9 - 4 = 5, etc.)
   A: The differences are consecutive odd numbers.

Figure it Out  Page Number 3

1. Can you recognize the pattern in each of the sequences in Table 1? 

Solution:  Recognizing the Patterns in Each Sequence

Sequence	Pattern (Rule)
1, 1, 1, 1, 1, ...	All 1’s – Every term is 1.
1, 2, 3, 4, 5, ...	Counting numbers – Start at 1, increase by 1 each time.
1, 3, 5, 7, 9, ...	Odd numbers – Start at 1, increase by 2 each time.
2, 4, 6, 8, 10, ...	Even numbers – Start at 2, increase by 2 each time.
1, 3, 6, 10, 15, ...	Triangular numbers – Add the next counting number (e.g., 10 + 5 = 15).
1, 4, 9, 16, 25, ...	Square numbers – Each term is n² (1², 2², 3², ...).
1, 8, 27, 64, 125, ...	Cube numbers – Each term is n³ (1³, 2³, 3³, ...).
1, 2, 3, 5, 8, 13, ...	Fibonacci (Virahānka) numbers – Each term is the sum of the two before it.
1, 2, 4, 8, 16, 32, ...	Powers of 2 – Each term is 2ⁿ⁻¹ (2⁰=1, 2¹=2, 2²=4, ...).
1, 3, 9, 27, 81, 243, ...	Powers of 3 – Each term is 3ⁿ⁻¹ (3⁰=1, 3¹=3, 3²=9, ...).

    

2. Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.

Solution: Extending Each Sequence (Next 3 Numbers) + Rule in My Own Words

(a) All 1’s ( 1, 1, 1, 1, 1, 1, 1 ...)

• Next 3 terms: 1, 1, 1

• Rule: Every number in the sequence is just 1—it never changes!

(b) Counting Numbers ( 1, 2, 3, 4, 5, 6, 7 ...)

• Next 3 terms: 8, 9, 10

• Rule: Start at 1, then keep adding 1 each time.

(c) Odd Numbers (1, 3, 5, 7, 9, 11 , 13 ...)

• Next 3 terms: 15, 17, 19

• Rule: Start at 1, then add 2 each time.

(d) Even Numbers ( 2,4, 6, 8, 10, 12, 14, .......)

• Next 3 terms: 16, 18, 20

• Rule: Start at 2, then add 2 each time.

(e) Triangular Numbers (1, 3, 6, 10, 15, 21, 28 ...)

• Next 3 terms: 36, 45, 55

• Rule: Start at 1, then add 2, then 3, then 4, and so on (each time adding the next counting number).

(f) Square Numbers ( 1, 4, 9, 16, 25,36,49,...)

• Next 3 terms: 64, 81, 100

• Rule: Each number is a perfect square (1×1, 2×2, 3×3, etc.).

(g) Cube Numbers (1, 8, 27, 64, 125, 216, 343, ...)

• Next 3 terms: 512, 729,1000

• Rule: Each number is a perfect cube (1×1×1, 2×2×2, 3×3×3, etc.).

(h) Fibonacci (Virahānka) Numbers (1, 2, 3, 5, 8, 13, 21,...)

• Next 3 terms: 34, 55, 89

• Rule: Start with 1 and 2, then each new number is the sum of the last two.

(i) Powers of 2 (1, 2, 4, 8, 16, 32, 64, ...)

• Next 3 terms: 128, 256, 512

• Rule: Each number is double the previous one (or 2 raised to increasing powers).

(j) Powers of 3 (1, 3, 9, 27, 81, 243, 729, ...)

• Next 3 terms:  2187, 6561, 19683

• Rule: Each number is triple the previous one (or 3 raised to increasing powers).
  
  

  1.3 Visualising Number Sequences

Pictorial representation of some number sequences:

(Pictorial Number Sequences)

1. All 1’s (1, 1, 1, 1, ...) 

Q: If this sequence continues forever, will any term ever be different from 1?

A: No, every term is always 1.

2. Counting Numbers (1, 2, 3, 4, ...)

Q: How would the 5th term in this sequence be represented pictorially?
A: A group of 5 objects (e.g., 5 dots or 5 sticks).

3. Odd Numbers (1, 3, 5, 7, ...)

Q: If the sequence represents "dots in stacked L-shapes," how many dots form the 4th L-shape?
A: 7 dots (since the 4th odd number is 7).

4. Even Numbers (2, 4, 6, 8, ...)

Q: If these numbers represent pairs of shoes, how many shoes are there in the 5th term?
A: 10 shoes (5th even number = 10).

5. Triangular Numbers (1, 3, 6, 10, ...)

Q: If each term forms a triangle with dots, how many dots are added to go from the 3rd to the 4th triangle?
A: 4 dots (3rd term = 6 dots, 4th term = 10 dots → 10 – 6 = 4).

6. Square Numbers (1, 4, 9, 16, ...)

Q: If each square number is a grid of dots (e.g., 1×1, 2×2), how many dots form the 5th square?
A: 25 dots (5 × 5 grid).

7. Cube Numbers (1, 8, 27, 64, ...)

Q: If cubes are represented as 3D stacks of blocks, how many blocks make the 3rd cube?
A: 27 blocks (3 × 3 × 3).

Bonus Challenge Questions

1. Q: Which sequence would form a perfect staircase when drawn as dots?
   A: Triangular numbers (each layer adds one more dot than the last).

2. Q: If "odd numbers" were represented as nested L-shapes, what would the 5th L-shape look like?
   A: A larger L with 9 dots (5th odd number = 9).

3. Q: How are square numbers visually different from triangular numbers?
   A: Squares form equal rows and columns (e.g., 4×4), while triangles form staircase layers (1, 2, 3, ... dots per row).

Figure it Out Page number 5 

1. Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence! 

Solution:

Sequence	Visual Pattern	Next Term Drawing
All 1’s	● (single dot)	●
Counting numbers	● → ●● → ●●● → ●●●● → ●●●●●	●●●●●● (6 dots)
Odd numbers	L-shapes: 1, 3, 5, 7 ,9 dots	●●●●●    ●●●●●● L with 11 dots (6th odd number)
Even numbers	Pairs: ●● → ●●●● → ●●●●●● → ●●●●●●●●	●●●●●●●●●● (10 dots)
Triangular numbers	△ layers: 1, 3, 6, 10 dots	△ with 15 dots (add a row of 5)
Square numbers	1×1, 2×2, 3×3, 4×4 grids	5×5 grid (25 dots)
Cube numbers	1³=1 block, 2³=8 blocks, 3³=27 blocks	4×4×4 cube (64 blocks)

2. Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes? 

Solution:
Triangular numbers: These numbers can be arranged in the shape of a triangle by placing dots in increasing rows. 

Square numbers: These numbers can be arranged in the shape of a square by placing dots in equal rows and columns. 

Cubes: These numbers can be arranged in the shape of a cube by placing dots in three-dimensional layers

Triangular numbers (1, 3, 6, 10...):

Form equilateral triangles with dots:

            ●                  ●               ●                 ●  

                                ● ●            ● ●              ● ●  

                                                  ● ● ●           ● ● ●  

                                                                      ● ● ● ●  

Rule: Add a new row with +1 dot each time.

Square numbers (1, 4, 9, 16...):
Form perfect squares:

            □                □□                □□□            □□□□  

                              □□                □□□            □□□□  

                                                   □□□            □□□□  

                                                                      □□□□  

• Rule: n2 (e.g., 32=9 dots).

• Cube numbers (1, 8, 27, 64...):
  Form 3D cubes with blocks:

• 1 block → 2×2×2 → 3×3×3 → ...  
  

  • Rule: n3 (e.g., 43=64 blocks).

  3. You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!
  3. 36 as Both Triangular and Square

Triangular: 8th triangular number = 36 dots:

●  

● ●  

● ● ●  
● ● ●  ●  

● ● ●  ●  ●  

● ● ●  ●  ●  ●  

● ● ●  ●  ●  ●  ●  

● ● ●  ●  ●  ●  ●  ●

... (8 rows) 

Square: 6×6 grid = 36 dots.
● ● ● ● ● ● 
● ● ● ● ● ● 

● ● ● ● ● ● 
● ● ● ● ● ● 

● ● ● ● ● ● 
● ● ● ● ● ● 

Try with 1 or 1225 (next such number)!

 4. What would you call the following sequence of numbers?

That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence? 

4. Hexagonal Numbers (1, 7, 19, 37...)These are called hexagonal numbers because they can be arranged in a hexagon shape.

Visual: Nested hexagons:

1 dot → 6 dots around it → 12 more → 18 more → ...  

• Next term: 61 (add 24 dots in outer hexagon).

• Rule: 3n2−3n+1.

5. Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
Here is one possible way of thinking about Powers of 2:

5. Visualizing Powers of 2 and 3

Powers of 2: You can visualize the powers of 2 as squares where each subsequent square has twice the number of smaller squares as the previous one. 

Powers of 3: You can visualize the powers of 3 as cubes, where each subsequent cube has three times the number of smaller cubes as the previous one.

• Powers of 2 (1, 2, 4, 8...):
  Double each time:

● → ●● → ●●●● → ●●●●●●●● → ...  

• Powers of 3 (1, 3, 9, 27...):

Triple each time:
● → △ of 3 → 3×3 grid → 3×3×3 cube → ...  

Key Takeaways

1. Same number, different shapes: 36 can be a triangle and a square.

2. Patterns are everywhere: Hexagons, cubes, and grids reveal hidden math.

3. Try it! Draw your own sequences (e.g., pentagonal numbers).

Fun Challenge: Find the next number after 36 that’s both triangular and square! (Hint: It’s 1225.)

1. Square Grid Template

*(For square numbers, powers of 2, and 36-as-square)*

● ● ● ● ● ●  

● ● ● ● ● ●  

● ● ● ● ● ●  

● ● ● ● ● ●  

● ● ● ● ● ●  

● ● ● ● ● ●  

• How to use:

  • Circle a 6×6 grid to show 36 as a square number.

  • Circle 1×1, 2×2, 3×3, etc. for square numbers.

  • For powers of 2: Start with 1 dot, then circle 2, 4, 8, etc.
    

    2. Triangular Grid Template

    *(For triangular numbers and 36-as-triangle)*
    

              ●  

            ●   ●  

          ●   ●   ●  

        ●   ●   ●   ●  

      ●   ●   ●   ●   ●  

    ●   ●   ●   ●   ●   ●  
    

    • How to use:

      • Fill rows to form triangles:

        • 1st row: 1 dot → total = 1

        • 2nd row: 2 dots → total = 3

        • ...

        • 8th row: 8 dots → total = 36 (triangular!)

    3. Hexagonal Grid Template

    (For hexagonal numbers: 1, 7, 19, 37...)
          ●   ●   ●  

        ●   ●   ●   ●  

      ●   ●   ●   ●   ●  

        ●   ●   ●   ●  

          ●   ●   ●  
    

    How to use:

    • 1 dot: Center dot.

    • 7 dots: Center + 6 neighbors (first ring).

    • 19 dots: Add 12 dots in the next ring (total = 7 + 12).

    • Next term (61): Add 24 dots in the outer ring (37 + 24).

4. Cube Template

(For cube numbers: 1, 8, 27...)

Front Layer    Middle Layer   Back Layer  

● ● ●          ● ● ●          ● ● ●  

● ● ●          ● ● ●          ● ● ●  

● ● ●          ● ● ●          ● ● ●  

How to use:

• 1³ = 1: One block in the center.

• 2³ = 8: 2×2×2 blocks (all layers).

• 3³ = 27: 3×3×3 blocks (fill all layers).

• Activity Solutions

  1. Drawing the Next Pictures

  • Triangular numbers: Add a row of 5 dots → 15 dots total.

  • Square numbers: Draw a 5×5 grid → 25 dots.

  • Cubes: Build a 4×4×4 cube → 64 blocks.

  2. Why "Triangular," "Square," "Cubes"?

  • Triangular: Form perfect triangles (e.g., bowling pins).

  • Square: Form perfect squares (like a chessboard).

  • Cubes: Form 3D cubes (like Rubik’s cubes).

  3. 36 as Triangle + Square

  • Triangle: 8 rows (1+2+3+...+8 = 36).

  • Square: 6×6 grid.
    

    Triangular 36       Square 36  

        ●               ● ● ● ● ● ●  

       ● ●             ● ● ● ● ● ●  

      ● ● ●           ● ● ● ● ● ●  

     ● ● ● ●         ● ● ● ● ● ●  

    ● ● ● ● ●       ● ● ● ● ● ●  
    ● ● ● ● ● ● ● ● ● ● ● ●
    ● ● ● ● ● ● ●
    

    
    

    #### **4. Hexagonal Numbers**  

    - **Next number:** 61 (37 + 24).  

    - **Pattern:** Add multiples of 6 (6, 12, 18, 24...).  

    
    

    #### **5. Powers of 2/3 Visualization**  

    - **Powers of 2:** Binary splitting!  
    

    1 → ●
    2 → ● ●
    4 → ● ● ● ●
    8 → ● ● ● ● ● ● ● ●
    - **Powers of 3:** Ternary branching!  
    1 → ●
    3 → ●─●─●
    9 → 3 branches, each splitting into 3
    

    ### **Bonus Challenge**  

    Find the next number after 36 that’s both triangular *and* square!  

    **Hint:** It’s 1225 (triangular = 49th term, square = 35²).  

1.4 Relations among Number Sequences

Example: What happens when we start adding up odd numbers? 
 1 = 1 
 1 + 3 = 4
 1 + 3 + 5 = 9 
 1 + 3 + 5 + 7 = 16 
 1 + 3 + 5 + 7 + 9 = 25
 1 + 3 + 5 + 7 + 9 + 11 = 36 . . . 
This is a really beautiful pattern!

Why does this happen? Do you think it will happen forever?

The answer is that the pattern does happen forever. But why? As mentioned earlier, the reason why the pattern happens is just as important and exciting as the pattern itself

1 + 3 + 5 + 7 + 9 + 11 = 36. Because such a picture can be made for a square of any size, this explains why adding up odd numbers gives square numbers. By drawing a similar picture, can you say

what is the sum of the first 10 odd numbers? 

 Now by imagining a similar picture, or by drawing it partially, as needed, can you say

what is the sum of the first 100 odd numbers?

Another example of such a relation between sequences: Adding up and down Let us look at the following pattern: 

 1 = 1 

 1 + 2 + 1 = 4 

 1 + 2 + 3 + 2 + 1 = 9

 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 

 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 

 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 

This seems to be giving yet another way of getting the square numbers— by adding the counting numbers up and then down

Figure it Out - Page 8

Great! Let's break down the rich mathematical exploration from the image and the questions one by one. The image you uploaded shows a 3D cube with color-coded layers and a circular dot arrangement, helping visualize square numbers and layered additions.

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1. Why does 1 + 2 + 3 + ... + 2 + 1 give square numbers?

The picture shows a cube made from layered L-shapes wrapping around a central yellow cube.
This represents:

• 1 (center)

• Then add 3 sides of 1 cube → total = 1 + 2 = 3

• Add 3 sides of 2 cubes → 1 + 2 + 3 + 2 + 1

• Continue this growing pattern:
  The shape visually builds a square layer-by-layer around a center.

Pattern Rule:

The sequence 1 + 2 + 3 + ... + 3 + 2 + 1 = n²
Because each layer wraps around the previous one symmetrically.

---

2. What's the value of 1 + 2 + ... + 100 + 99 + ... + 2 + 1?

This is a symmetric sum up to 100 and back down:

• Total terms: 199 (1 through 100, then back down through 99 to 1)

• The sum:

$$1 + 2 + \ldots + 99 + 100 + 99 + \ldots + 2 + 1 = 100^2 = 10,000$$

Why?
Using the square pattern: the visual model represents this as a square of side 100.

---

3. Add the All 1’s sequence up and down: What do you get?

• All 1's Up: 1, 1+1 = 2, 1+1+1 = 3, etc. → This gives:
  1, 2, 3, 4, ... = counting numbers.

• All 1's Up and Down:
  1, 2, 3, ..., n, ..., 3, 2, 1 → This gives:
  1 + 2 + 3 + ... + n + ... + 3 + 2 + 1 = n²

So:
Adding the All-1s sequence up gives the counting numbers.
Adding it up and down gives perfect squares.

---

4. Add Counting Numbers Up – What Do You Get?

The sum:

$$1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}$$

These are triangular numbers:

• 1 → 1

• 1 + 2 → 3

• 1 + 2 + 3 → 6

• 1 + 2 + 3 + 4 → 10

• ...

Pictorially: A triangle growing with rows:
Row 1: •
Row 2: ••
Row 3: •••
= triangle!

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5. Add consecutive triangular numbers: 1+3, 3+6, 6+10...

Example:

• 1 + 3 = 4

• 3 + 6 = 9

• 6 + 10 = 16

• 10 + 15 = 25
  → Sequence: 4, 9, 16, 25... = perfect squares

Why?
Each pair of consecutive triangular numbers adds up to a square:

$$T_n + T_{n+1} = (n+1)^2$$

Visual: Two adjoining triangles fill a square.

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6. Powers of 2: 1, 1+2, 1+2+4... then add 1

• Powers of 2 sum:
  1, 3, 7, 15, 31, 63, ...

• Add 1 to each:
  2, 4, 8, 16, 32, 64, ...

So:

1 + 2 + 4 + ... + 2^n = 2^{n+1} - 1
Adding 1 gives 2^{n+1}

Explanation: This is the sum of a geometric series. Visually: binary tree or doubling boxes.

---

7. Multiply triangular numbers by 6 and add 1:

Triangular numbers: 1, 3, 6, 10, 15, 21, ...
Multiply by 6 and add 1:

• 6×1 + 1 = 7

• 6×3 + 1 = 19

• 6×6 + 1 = 37

• 6×10 + 1 = 61

• 6×15 + 1 = 91
  → Sequence: 7, 19, 37, 61, 91...

**This is the sequence of centered hexagonal numbers!

Visual: A hexagon growing around a center point.

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8. Add hexagonal numbers: 1, 1+7, 1+7+19, 1+7+19+37...

Hexagonal numbers:
1, 7, 19, 37, 61, 91, ...
Partial sums:

• 1

• 1 + 7 = 8

• 1 + 7 + 19 = 27

• 1 + 7 + 19 + 37 = 64

• ...

Noticing:
→ 8 = 2² + 2²
→ 27 = 3³
→ 64 = 4³
→ These might approach cube numbers!
Explore further to confirm.

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9. Find Your Own Patterns in Table 1

Here are a few ideas:

• Try differences between terms — are they increasing linearly (arithmetic) or multiplicatively (geometric)?

• Visualize sums in shapes: squares, triangles, pyramids.

• Connect powers (2ⁿ, n², n³) to areas or volumes.

You can find many sequences like:

• Fibonacci numbers

• Centered polygonal numbers

• Figurate numbers

---

Would you like me to turn all these into Google Slides with illustrations or animated explainer videos for your students or YouTube channel? 

1. Can you find a similar pictorial explanation for why adding counting numbers up and down gives square numbers?

 Imagine arranging the numbers in rows, where each row adds one more number and then subtracts one until it reaches back to 1. 

When these numbers are added up, they fill in the squares, forming perfect square grids.

2. By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1? 

This question needs to be answered by students themselves. Here's an explanation for better understanding: When counting up to 100 and then back down, the total sum will form a square number. 

3. What sequence do you get when you start to add the All 1’s sequence up? What sequence do you get when you add the All 1’s sequence up and down?

 Ans:

 All 1’s sequence up: The result is the counting numbers (1, 2, 3, 4, ...). 

All 1’s sequence up and down: The result is triangular numbers (1, 3, 6, 10, ...). 

4. What sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation? 

Ans: When you start adding counting numbers, you get triangular numbers. This question needs to be answered by students themselves. 

5. What happens when you add up pairs of consecutive triangular numbers?

 Ans: Adding pairs of consecutive triangular numbers results in square numbers. 

6. What happens when you start to add up powers of 2 starting with 1, and then add 1 to each of these numbers? 

Ans: The sum of powers of 2 plus 1 results in the next power of 2, forming a new sequence of powers of 2. 

7. What happens when you multiply the triangular numbers by 6 and add 1? Can you explain it with a picture? 

Ans: Multiplying triangular numbers by 6 and adding 1 results in a new sequence. 

8. What happens when you start to add up hexagonal numbers? take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, ... ? Which sequence do you get? Can you explain it using a picture of a cube? 

 Adding up hexagonal numbers forms a new sequence. 

9. Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise? 

Various patterns can be discovered by exploring how sequences like triangular numbers, squares, and cubes relate to each other. For example, triangular numbers added together form square numbers, and visualizing this with diagrams can help explain why this happens. Page 10 

1. Can you recognize the pattern in each of the sequences in Table 3? 

 This question encourages students to observe and identify the patterns on their own. The patterns involve the growth and arrangement of shapes, such as stacking triangles or squares and forming complete graphs.  recognize how each sequence evolves and increases in complexity as more shapes or connections are added. 

2. How many little squares are there in each shape of the sequence of Stacked Squares? 

The number of little squares in each shape follows the pattern of square numbers: 

First shape: 1 square 

Second shape: 4 squares (2x2) 

Third shape: 9 squares (3x3) 

Fourth shape: 16 squares (4x4) 

Explanation: The number of little squares increases as the square of the number of rows or columns in the shape, following the sequence 1, 4, 9, 16, etc. 

3. How many little triangles are there in each shape of the sequence of Stacked Triangles? 

 The number of little triangles in each shape follows the pattern of triangular numbers: 

First shape: 1 triangle 

Second shape: 3 triangles (1+2) 

Third shape: 6 triangles (1+2+3) 

Fourth shape: 10 triangles (1+2+3+4) 

Explanation: The number of little triangles increases according to the triangular number sequence, where each shape adds another row of triangles, following the sequence 1, 3, 6, 10, etc. 

4. How many total line segments are there in each shape of the Koch Snowflake? 

 The number of line segments in each shape of the Koch Snowflake increases as follows: 

First shape: 3 line segments 

Second shape: 12 line segments 

Third shape: 48 line segments 

Fourth shape: 192 line segments 

Explanation: The number of line segments increases by multiplying by 4 each time. 

This results in the sequence 3, 12, 48, 192, which is 3 times the powers of 4 (i.e., 3 × 4^n)