CASIO Teaching Materials: Real-Life Problems with fx-991CW (Digest Edition)

Introduction

These teaching materials were created with the hope of conveying to many teachers and students the appeal of scientific calculators.

• (1) Change awareness (emphasizing the thinking process) and boost efficiency in learning mathematics: By reducing the time spent on manual calculations, learning can focus on the thinking process, making it more efficient. This reduces aversion to mathematics caused by complicated calculations and allows students to experience the joy of thinking, which is the essence of mathematics.
• (2) Diversification of learning materials and problem-solving methods: Making it possible to do difficult calculations manually allows for diversity in learning materials and problem-solving methods.
• (3) Promoting understanding of mathematical concepts: By using the various functions of the scientific calculator in creative ways, students can deepen their understanding of mathematical concepts through calculations and discussions from different perspectives. This allows for exploratory learning through easy trial and error. Listing and graphing of numerical values by means of tables allows students to discover laws and understand visually.

Features of these teaching materials:

• Makes classes more interesting by using scientific calculators.
• Includes a variety of real-life problems in each unit.
• Allows a deeper understanding of mathematics.
• Enables students to utilize the scientific calculator's functions more skillfully.
• Three degrees of difficulty settings (levels 1 to 3).

Installation Angle of Solar Panels (Level 1)

Diagram Description: A solar panel is depicted with dimensions labeled: Height 'h', Width '2m', Length '5m', Area 'A', and Inclination angle 'θ'. A separate diagram shows a sun ray hitting a panel at an angle of 76°.

• (1) When installing solar panels, power generation efficiency is highest when sunlight is perpendicular to the panel. Given the angle of incidence of the sun is 76°, find the angle 'd' at which the panel is perpendicular to the sunlight.
  Solution: θ = 90° - 76° = 14°
• (2) What should the height [m] of the support column be? (The width of the solar panels is 2m.)
  Solution: h = 2 × sin 14° ≈ 0.48m
• (3) What is the required footprint of the solar panels? (The length of the solar panels is 5m.)
  Solution: A = 5 × 2cos 14° ≈ 9.7m²

Tower and Mountain Height Surveying (Level 2)

Image Description: A cityscape featuring a tall tower and mountains in the background.

• (1) Tower Height: At a point 200 [m] from the tower, the elevation angle θ was 72.5°. Find the height h [m] of the tower.
  Solution: h = 200 × tan 72.5° ≈ 634 [m]
• (2) Mountain Height: Elevation angles were measured at two points A and B. The elevation angle at point A was θ<0xE2><0x82><0x90> = 10.7° and at point B was θ<0xE2><0x82><0x91> = 8.4°. The distance between A and B was x = 5,587m. Find the mountain height h [m].
  Solution: h ≈ 3,776 [m]. (This problem can also be solved using the sine theorem.)

How Many Folds to Reach the Moon? (Level 1)

Image Description: The Moon.

Given that the distance from the Earth to the Moon is 384,400 km, and the thickness of a piece of paper is 0.1 mm, find the number of folds required for the paper's thickness to cover the distance to the Moon.

The thickness after x folds is: 0.1 × 2ˣ mm.

The distance between the Earth and the Moon is 384,400 km, which is equal to 384,400 × 10⁶ mm = 3.844 × 10¹¹ mm.

Find the value of x for which 0.1 × 2ˣ exceeds 3.844 × 10¹¹ mm.

Solution: x ≥ 41.8. The paper will reach the Moon after 42 folds.

The Magnitude of an Earthquake (Level 2)

The magnitude M of an earthquake is defined by the formula: log₁₀ E = 4.8 + 1.5M, where E is the earthquake's energy in joules.

Diagram Description: An Earthquake Magnitude Scale showing different levels of magnitude and associated descriptions (e.g., MICRO, MINOR, LIGHT, MODERATE, STRONG, MAJOR, GREAT).

• (1) Find the energy E [J] of an earthquake with a magnitude of 9.
  Solution: E = 10⁴.⁸⁺¹·⁵×⁹ ≈ 2.0 × 10¹⁸ [J]
• (2) The average energy of a bolt of lightning is 1.5 × 10⁹ [J]. Find the magnitude of an earthquake with the same energy.
  Solution: M ≈ 2.9
• (3) The estimated impact energy of the meteorite that caused the extinction of the dinosaurs is 1.3 × 10²⁴ (J). Find the magnitude of an earthquake with this energy.
  Solution: M ≈ 12.9
• (4) How many times more energy does an earthquake have when the magnitude increases by 1?
  Solution: 32 times

Half-life (Level 3)

The half-life of an element is the time it takes for half of its radioactive nuclei to decay. The relationship is given by: N = N₀ × (1/2)ᵗ/ᵀ, where N is the number of nuclei remaining, N₀ is the initial number, t is time elapsed, and T is the half-life.

Diagram Description: A radioactive symbol.

• (1) Plutonium-239: Half-life = 24,110 years. What percentage of the original material remains after 100 years?
  Solution: ≈ 99.7%
• (2) Carbon-14: Half-life = 5,730 years. After how many years does 1/16 of the original material remain?
  Solution: t = 4 × 5730 = 22920 years
• (3) Ancient Artifact: An artifact contains ¹⁴C. If it has 11/12 of the atmospheric abundance, is it from ancient Egypt?
  Solution: t = 719 years. This artifact appears to be from a more recent time.

Relating Speed and Stopping Distance (Level 1)

The relationship between speed x [km/h] and stopping distance y [m] can be expressed by a quadratic function: y = ax² + bx + c.

Table: Stopping Distances

• (1) Find the equation for the quadratic function.
  Solution: y = (1/100)x² + 5
• (2) Find the stopping distance for speeds of 50 km/h, 60 km/h, and 70 km/h.
  Solutions: 50 km/h: 30 m; 60 km/h: 41 m; 70 km/h: 54 m
• (3) What is the lowest driving speed at which the car will be unable to stop in time if an obstacle is 65 m ahead?
  Solution: ≈ 77.46 km/h

Calculating an Electric Bill (Level 3)

Image Description: An electric meter.

• (1) Express the energy consumption w(x) [kWh] of a 60 [W] lamp used for x hours.
  Solution: w(x) = 0.06x [kWh]
• (2) Express the monthly electric bill f(x) if the base rate is $14.00 and the usage charge is $0.20 per kWh.
  Solution: f(x) = 0.012x + 14 [dollars]
• (3) Find the electric bill (in dollars) if a 60 W lamp is run for 300 hours.
  Solution: $17.6
• (4) What is the minimum number of hours for the monthly electric bill to be at least $20.00?
  Solution: x ≥ 500 [h]
• (5) Express the electric bill g(x) if usage beyond 30 kWh is charged at $0.25 per kWh.
  Solution: g(x) = 0.015x + 12.5
• (6) If the bill was underpaid by $0.60, find the number of hours the lamp was in use.
  Solution: x = 700 [h]

Relating Musical Scales to Geometric Sequences (Level 1)

The range between tones with double frequency is an octave, composed of 12 half steps. Frequencies form a geometric sequence.

Table: Frequencies of Musical Notes

• (1) Find the common ratio for this scale.
  Solution: r = √2
• (2) Find the frequencies of the other tones on the table.
• (3) Find the integer ratios of the frequencies for notes C, E, and G.
  Solution: C:E:G ≈ 4:5:6

The Tower of Hanoi (Level 3)

A puzzle involving moving discs between pegs according to specific rules.

Diagram Description: A visual representation of the Tower of Hanoi puzzle with three pegs and discs of varying sizes.

• (1) Find the number of moves (a<0xE2><0x82><0x99>) for n discs (n=1 to 4).
  Table: Number of Moves

  Number of Discs (n)	1	2	3	4
  Number of Moves (a<0xE2><0x82><0x99>)	1	3	7	15

• (2) Find the recurrence formula relating a<0xE2><0x82><0x99> and a<0xE2><0x82><0x99>₋₁.
  Solution: a<0xE2><0x82><0x99> = 2a<0xE2><0x82><0x99>₋₁ + 1
• (3) Find a<0xE2><0x82><0x99> for n=5 and n=10.
  Solutions: a₅ = 31 moves; a₁₀ = 1023 moves
• (4) Find a<0xE2><0x82><0x99> for n=30.
  Solution: a₃₀ = 1,073,741,823 moves
• (5) Find the formula for the nth term, a<0xE2><0x82><0x99>.
  Solution: a<0xE2><0x82><0x99> = 2ⁿ - 1
• (6) Estimate the time to complete the Tower of Hanoi with 64 discs, assuming one move per second.
  Solution: Approximately 584.5 billion years.

Matching Based on Similarity (Level 2)

To find classmates with similar music preferences, the top 100 songs from playlists of students A, B, C, and D were categorized into genres: rock, pop, and jazz. Vector analysis is used to determine similarity.

Table: Student Music Preferences

Diagram Description: A 3D vector space plotting Rock on the x-axis, Pop on the y-axis, and Jazz on the z-axis.

Analysis: Smaller angles between vectors indicate more similar preferences.

• Most similar pair: BD (Angle ≈ 16.5°)
• Most different pair: AB (Angle ≈ 40.4°)

Finding the Shortest Flight Path for a Drone (Level 2)

A drone delivers to points A (3, 6, 7), B (−8, 5, 4), C (−7, −5, 3), and D (5, 3, −6) in 3D space. The goal is to find the shortest total travel distance for a path starting at A, visiting B, C, and D in any order, and returning to A.

• (1) List all possible flight paths (permutations of B, C, D).
  Paths: ABCDA, ABDCA, ACBDA, ACDBA, ADBCA, ADCBA
• (2) Which flight path has the shortest total travel distance?
  Solution: ACBDA or ADBCA (approx. 54.4 units)
• (3) Find the angles of the turns along the shortest path (ACBDA).
  Angles: ∠ACB ≈ 67.6°, ∠CBD ≈ 83.3°, ∠BDA ≈ 71.1°
• (4) Find the volume of the air space (tetrahedron ABCD).
  Solution: Volume = 312.5

Total Score of University Entrance Exams (Level 1)

Students X, Y, and Z took university entrance exams in Mathematics, English, and Science. Their raw scores and university weightings are used to calculate final scores.

Table 1: Raw Score Table

Total raw scores: X=240, Y=240, Z=240.

Table 2: Subject Weighting

Table 3: Final Total Score (Calculated via matrix multiplication)

Highest Scores: University P: Z, University Q: Y, University R: X.

Cryptography Using Matrices (Level 2)

Information can be transmitted by encrypting numbers assigned to letters using matrices.

Table: Letter to Number Mapping (A=0, B=1, ..., Z=25)

Encryption: Key × Number = Code

• (1) Decode: Given Code [236 235] / [138 138] and Key [7 5] / [4 3], find the Number.
  Solution: Number = [18 15] / [22 26] → Letters: SAVE
• (2) Find Key: If Code [328 133] / [209 85] decodes to "SAVE", find the Key.
  Solution: Key = [3 8] / [2 5]
• (3) Decode: Given Code [224 197] / [142 125] and Key [3 8] / [2 5], find the Number.
  Solution: Number = [22 19] / [16 15] → Letters: LIFE
• (4) Decode: Given Code [66 82 64 30] / [14 45 16 17] / [11 21 34 28] / [53 175 96 96] and Key [1 3 2 0] / [0 1 0 1] / [0 0 1 1] / [0 3 1 5], find the Number.
  Solution: CASINO

Comparing Histograms (Level 1)

Two physical education classes, A and B, had their grip strength measured. We compare the data using histograms.

Class A Data (34 Students): 39, 36, 37, 38, 38, 41, 38, 37, 40, 39, 43, 38, 38, 40, 39, 40, 39, 42, 41, 36, 39, 42, 38, 37, 43, 37, 37, 39, 38, 43, 36, 36, 39, 38 (kg)

Class B Data (34 Students): 36, 30, 31, 40, 33, 34, 35, 45, 35, 46, 26, 36, 48, 39, 39, 36, 30, 45, 41, 42, 42, 43, 44, 35, 46, 37, 48, 49, 48, 36, 46, 39, 45, 41 (kg)

• (1) Mean and Standard Deviation:
  Class A: Mean = 38.85 kg, Standard Deviation = 2.03 kg.
  Class B: Mean = 39.59 kg, Standard Deviation = 5.94 kg.
  Characteristics: Means are similar; Class B has higher standard deviation, indicating more variability.
• (2) Histograms: Histograms show Class A data is clustered near the mean, while Class B data is more scattered.
• (3) Data Trends: Class A shows low variability, suggesting consistent grip strength. Class B shows high variability, indicating diverse grip strength among students.

Linear Regression Analysis (Level 1)

Investigating the relationship between caffeine intake and attention span.

Table: Caffeine Intake vs. Attention Span

Diagram Description: A scatter plot showing caffeine intake versus attention span, with a regression line drawn through the points.

• (1) Regression Equation and Correlation:
  Equation: y = 0.25x + 36.67
  Correlation Coefficient (r): 0.99
• (2) Projection: Projected attention span for 160 mg caffeine intake is 77.2 minutes. The difference between actual and projected span for 100 mg is 2 minutes.
• (3) Correlation: Caffeine intake and attention span are strongly correlated (r=0.99). Note: Avoid overconsumption of caffeine; recommended maximum daily intake is 400 mg.

Number of Correct Answers When Randomly Selecting Test Responses (Level 2)

Consider a test with 10 problems, each with 4 answer choices (1 correct). We calculate the probability of getting X correct answers.

Table: Probability Distribution

• (1) Probability Calculation: P(X) = ¹⁰Cₓ × (1/4)ˣ × (3/4)¹⁰⁻ˣ
• (2) Expected Value: The expected value (np) is 2.5.
• (3) Standard Deviation: The standard deviation (√np(1-p)) is approximately 1.37.

Estimating T-Shirt Production Levels Using Height Distribution (Level 2)

A clothing manufacturer plans to produce 1,000 T-shirts, estimating quantities per size based on men's height distribution.

Table: Recommended Height Range for Shirt Sizes

Men's heights follow a normal distribution with mean (μ) = 173 cm and standard deviation (σ) = 6.0 cm.

Probabilities and Estimated Quantities for 1000 Shirts:

• Size S (Up to 165 cm): Probability ≈ 0.09121 → Quantity ≈ 91 shirts
• Size M (165 to 175 cm): Probability ≈ 0.53935 → Quantity ≈ 539 shirts
• Size L (175 to 185 cm): Probability ≈ 0.34669 → Quantity ≈ 347 shirts
• Size XL (185 cm and up): Probability ≈ 0.02275 → Quantity ≈ 23 shirts

Aging and the Apparent Passage of Time (Level 1)

According to Janet's Law, the apparent length of an interval (y) is proportional to the reciprocal of age (x): y = 1 / (x+1).

• (1) Time Perception Comparison: A 50-year-old perceives time as passing approximately 4.6 times faster than a 10-year-old.
• (2) Experienced Lifetime: For a life expectancy of 85 years, a 15-year-old has experienced 62% of their apparent lifetime.

Cutting a Heart-Shaped Pizza in Half (Level 2)

The task is to cut a heart-shaped pizza into two equal pieces horizontally.

Diagram Description: A heart shape with potential horizontal cut lines indicated.

• (1) Total Area: The total area of the heart-shaped pizza is 16π.
• (2) Optimal Cut: Cutting the pizza horizontally at line (d) provides the most even pieces.

Publisher Information

Publisher: CASIO COMPUTER CO., Ltd.

Date of Publication: 2025/7/31 (2nd edition)

Website: https://edu.casio.com

Copyright © 2025 CASIO COMPUTER CO., LTD.

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