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Questions and Answers
What role does mathematics play in understanding natural patterns?
Which geometric shape is mentioned as part of a simple mandala pattern?
What is the difference between the number sequence 3, 8, 13, 18, 23 and the sequence 1, 4, 9, 16, 25, 36?
What can be considered as a repeating pattern in nature?
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Which of the following best explains a logic pattern as illustrated in the example?
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In the context of identifying relationships in patterns, what does mathematics aid in discovering?
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Which pattern is represented visually as a staggered arrangement of five-pointed stars?
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How does the concept of geometric patterns differ from word patterns?
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What geometric shape do bees use to store honey most efficiently?
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How does the Fibonacci sequence work mathematically?
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What is the approximate value of the golden ratio?
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What role do patterns on a peacock's tail primarily serve?
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Which of the following best describes the symmetry of snowflakes?
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What is the relationship between the Fibonacci sequence and the golden ratio?
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Who is known for discovering the Fibonacci sequence?
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Which property of the golden ratio is often considered aesthetic?
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What describes a situation in which an object is rotated yet appears unchanged?
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How is the angle of rotation calculated for a figure with n-fold rotational symmetry?
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Which of the following figures demonstrates line symmetry?
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In the sequence 3, 6, 12, 24, 48, what is the next number?
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What is the next number in the sequence 1, 4, 10, 22, 46?
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What type of symmetry is displayed by a snowflake with 6-fold symmetry?
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What would be the angle of rotation for a figure with 8-fold symmetry?
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Which series of dots is used to determine a progressive pattern of filling in a grid?
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What is the value of A when P = 505050, r = 5%, and t = 1 year using the formula A = Pert?
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Based on the exponential growth model A = 45e^{0.19t}, what is the population in the year 2000 (t = 5)?
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What is the correct value of Fib(20) in the Fibonacci sequence?
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If Fib(1) + Fib(2) + ... + Fib(10) is calculated, what would be the result?
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When calculating A using P = 240100, r = 11%, and t = 10 years, what is the value of A?
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What is the population of the city in the year 2045 according to the exponential model A = 45e^{0.19t}?
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In the sequence Fib(1) + Fib(2), Fib(1) + Fib(2) + Fib(3), how does the sum pattern behave?
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What is the value of A when P = 247000, t = 17 years, and using unknown r?
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What aspect of the structures designed by the ancient Greeks and Egyptians is influenced by the Golden Ratio?
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Which architectural masterpiece is specifically mentioned as incorporating the Golden Ratio in its design?
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What relationship do the dimensions of ancient structures often reflect?
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How did the ancient Greeks and Egyptians utilize the Fibonacci sequence?
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What visual technique was used in the images of the Parthenon and Roman Colosseum to explain the Golden Ratio?
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Which of the following is NOT a characteristic of structures built using the Golden Ratio?
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In which structure would you find the application of the Golden Ratio primarily in its height-to-length ratio?
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What is the significance of the Golden Ratio in the context of ancient architecture?
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Which element is commonly measured in relation to the Golden Ratio in the architectural masterpieces mentioned?
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What does the harmonious relationship in the dimensions of these structures provide?
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Study Notes
Patterns in Nature and the World
- Patterns are repeated occurrences that exist in nature and the world.
- The cycle of the moon, the changing seasons, and the transmission pattern of a pandemic are examples of patterns.
- Mathematics helps us understand, classify, and exploit patterns.
- Mathematical knowledge assists with generalization, identifying relationships, discovering logical connections, and problem-solving.
Example 1: Logic Pattern
- This example shows a simple grid pattern with three "X" symbols.
- The pattern requires adding a new line in each stage that never touches the last line.
- To continue the pattern, the fourth square in the grid should be filled with an "X."
Example 2: Number Pattern
- This example shows two number sequences.
- The first sequence is 3, 8, 13, 18, 23, ?
- The difference between each term in this sequence is 5; therefore, the next number in the sequence is 28.
- The second sequence is 1, 4, 9, 16, 25, 36, ?
- Each number in this sequence represents the square of consecutive numbers starting with 1 (1^2, 2^2, 3^2...).
- The next number in this sequence is 49 since it represents 7^2.
Geometric and Word Patterns
- Geometric patterns are visual representations of shapes like circles and polygons.
- Word patterns focus on morphological rules, such as pluralizing nouns and conjugating verbs.
- These patterns are found in nature, such as traditional patterns in the Philippines, including Cordillera tattoos, woven mats, and beaded belts.
Self-Assessment Activity 1
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(a) The sequence shows a grid pattern of "X" symbols with the last one rotated.
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The pattern requires adding a row of "X" symbols in each stage to continue.
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The missing square in the grid should be filled with a non-rotated "X."
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(b) The sequence shows a pattern of circles with different symbols.
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The pattern alternates between a full circle with a plus symbol on top and a full circle with a minus symbol on the bottom.
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The next image in the sequence should be a full circle with a minus symbol on the bottom.
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(c) The sequence shows a 3x3 grid of blank squares, with a dot being added in each subsequent image.
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The pattern continues until the final image, creating a grid that is 5 squares wide and 4 squares tall, with all squares filled in.
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2. The next numbers in the sequences are:
- 3, 6, 12, 24, 48, 96 (Each number is multiplied by 2.)
- 1, 4, 10, 22, 46, 94 (Each number is added to the previous number, with an increasing difference.)
- 4, -1, -11, -26, -46, -71 (Each number is decreased by a progressively larger difference.)
Symmetry and Angle of Rotation
- Symmetry occurs when an object can be divided into two identical parts that are mirror images of each other.
- Line Symmetry (also called bilateral symmetry) is when the parts are mirror images about an imaginary line.
- Rotational Symmetry happens when an object can be rotated by a specific angle and still look the same.
- The angle of rotation is the smallest angle an object can be rotated before it matches its original position.
- A figure with n-fold rotational symmetry can be rotated
1/nof a complete turn without changing. - The formula to calculate the angle of rotation is:
Angle of Rotation = 360 / n
Snowflakes and Honeycombs
- Snowflakes exhibit six-fold symmetry due to the effects of atmospheric conditions on ice crystals.
- Honeycombs use a hexagonal pattern for maximum storage space with minimal wax usage.
- Animals use patterns for survival and communication, such as attracting mates and blending in for camouflage.
The Fibonacci Sequence
- The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...) is a pattern where each number is the sum of the two preceding numbers.
- Leonardo Pisano (Fibonacci) discovered the sequence in his 1202 book "Liber Abaci."
- This sequence is often found in nature, such as the arrangement of sunflower seeds.
The Golden Ratio
- The golden ratio (φ) is an irrational number approximately equal to 1.618.
- It's related to the Fibonacci sequence, where dividing consecutive numbers in the sequence approaches the golden ratio.
- The golden ratio is aesthetically pleasing and often found in art, architecture, and nature.
Self-Assessment Activity 2
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1. Find the missing quantity in the formula A = Pert by substituting the given values:
- (a): A = 505,050 * e^(0.05 * 1) ≈ 530,838
- (b): A = 240,100 * e^(0.11 * 10) ≈ 693,555
- (c): P = 786,000 / e^(0.17 * 17) ≈ 178,374
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2. The exponential growth model A = 45e^0.19t describes the population of a city in the Philippines in thousands, t years after 1995.
- 1995: A = 45e^(0.19 * 0) = 45,000
- After 25 years: A = 45e^(0.19 * 25) ≈ 353,792
- 2045: A = 45e^(0.19 * 50) ≈ 2,737,615
Math for Our World
- Mathematics is crucial for understanding and navigating the complexities of the world.
- It plays a significant role in various fields, including science, technology, engineering, finance, and art.
- Mathematical principles help us predict eclipses, develop new drugs, create devices and systems, design structures, manage money, and create art.
Self-Assessment Activity 3
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1. Let Fib(n) be the nth term of the Fibonacci sequence, with Fib(1) = 1, Fib(2) = 1, Fib(3) = 2, and so on.
- (a): Fib(20) = 6,765
- (b): Fib(25) = 75,025
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2. Evaluate the following sums:
- (a): Fib(1) + Fib(2) = 2
- (b): Fib(1) + Fib(2) + Fib(3) = 4
- (c): Fib(1) + Fib(2) + Fib(3) + Fib(4) = 7
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3. The sum of Fib(1) + Fib(2) + ...+ Fib(10) = 143.
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4. The number sequence constructed using the given formula will observe the following pattern: each term in the sequence will be equal to the sum of the terms in the Fibonacci sequence up to the corresponding number.
The Great Works of Man and the Fibonacci Sequence and the Golden Ratio
- Ancient Egyptians and Greeks integrated the Fibonacci sequence and the golden ratio into their architectural masterpieces.
- The Great Pyramid of Giza, the Parthenon, and the Roman Colosseum demonstrate the use of these mathematical principles in their design.
- The proportions of these structures, including height, width, and length, often reflect the golden ratio, creating a harmonious and aesthetically pleasing experience.
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