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📚 Filipino · Aralin 3 & 4 (Kompleto at Pinalawak)

Mabisang Komunikasyon Reviewer

5 Elemento, 4 Anyo ng Pagpapahayag, Antas ng Komunikasyon, Awdyens, Pormal vs Impormal, Privacy, Public-Facing Texts, 3 Proseso, at 4 na Konteksto.

Elements of Communication and Netiquette Tips

📌 Visual Infographic: 5 Elemento ng Komunikasyon & Netiquette Safety Guide

Aralin 3.1: 5 Elemento at 4 Anyo ng Pagpapahayag

Tagapagpadala (Sender)

Pinagmumulan ng mensahe o impormasyon.

Mensahe (Message)

Ideya, impormasyon, o damdaming nais iparating.

Daluyan (Channel)

Paraan ng pagpapadala (pasalita, pasulat, online/chat).

Tagatanggap (Receiver)

Tumatanggap at nag-iinterpret sa mensahe.

Tugon (Feedback)

Sagot o reaksyon upang matiyak ang pagkakaunawaan.

1. Pagtatanong

Paraan ng pagkuha ng impormasyon, paglilinaw ng ideya, o paghingi ng paliwanag ("Ano po ang takdang-aralin?").

2. Pagkukuwento

Pagsasalaysay ng karanasan, pangyayari, o sitwasyon upang magbahagi ng impormasyon o magbigay-aliw.

3. Paglalahad ng Obserbasyon

Pagbibigay ng impormasyon batay sa nakita o narinig nang hindi agad nagbibigay ng sariling paghuhusga.

4. Pagpapahayag ng Opinyon

Pagbibigay ng sariling pananaw na may paggalang at may batayan o dahilan.

1. Kahulugan at 5 Elemento ng Komunikasyon

Ang komunikasyon ay ang proseso ng pagpapadala, pagtanggap, at pagpapalitan ng impormasyon, ideya, at damdamin sa pagitan ng dalawa o higit pang tao upang magkaroon ng pagkakaunawaan.

Tagapagpadala (Sender): Ang taong may layuning magbahagi ng kaisipan o impormasyon.
Mensahe (Message): Ang mismong nilalaman ng komunikasyon — maaaring pormal, impormal, pasulat, o pasalita.
Daluyan (Channel): Ang daluyan o teknolohiyang ginagamit tulad ng bibig (pasalita), papel (liham), o internet (chat/video).
Tagatanggap (Receiver): Ang nagpoproseso at umiintindi sa mensahe.
Tugon (Feedback): Ang pagtugon na nagpapatunay kung naging matagumpay ang komunikasyon.

2. Apat na Anyo ng Pagpapahayag (Mula sa Handouts)

1. Pagtatanong: Paraan ng pagkuha ng impormasyon, paglilinaw ng ideya, o paghingi ng paliwanag.
Halimbawa: "Ano po ang takdang-aralin?", "Paano po ninyo nakuha ang sagot?"

2. Pagkukuwento: Pagsasalaysay ng isang karanasan, pangyayari, o sitwasyon upang magbahagi ng impormasyon o magbigay-aliw.

3. Paglalahad ng Obserbasyon: Pagbibigay ng impormasyon batay sa mga nakita, narinig, o naranasan nang hindi agad nagbibigay ng sariling paghuhusga.

4. Pagpapahayag ng Opinyon: Pagbibigay ng sariling pananaw o paniniwala tungkol sa isang paksa. Ito ay dapat na may paggalang at, kung maaari, may batayan o dahilan.

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Quiz: 5 Elemento at Anyo ng Pagpapahayag

Multiple Choice · 1 question at a time · Instant feedback

Aralin 3.2: Mga Antas ng Komunikasyon, Awdyens, at Uri ng Wika

Personal na Komunikasyon

Pakikipag-usap sa sarili (Talaarawan/Diary, Dyornal/Journal, Repleksyon, Goal Setting).

Interpersonal na Komunikasyon

Pakikipagpalitan ng mensahe sa isa o higit pang tao (pakikipag-usap sa kaibigan).

Pampublikong Komunikasyon

Pagpapahayag ng mensahe sa malaking pangkat ng tao (iisa ang tagapagsalita, marami ang tagapakinig).

Awdyens

Ang taong o pangkat na tumatanggap ng mensahe. Mahalagang kilalanin upang makapili ng angkop na tono at wika.

Pormal na Wika

Opisyal at seryosong sitwasyon (Paaralan, Opisina, Talumpati, Liham, Pag-uulat).

Impormal na Wika (Di-pormal)

Pang-araw-araw na usapan sa pamilya at kaibigan (Chat, text, kaswal na usapan).

1. Tatlong Antas ng Komunikasyon

Personal na Komunikasyon: Tumutukoy sa pakikipag-usap na may kaugnayan sa sariling karanasan, damdamin, pananaw, at pang-araw-araw na pakikipag-ugnayan (pakikipag-usap sa sarili tulad ng Talaarawan, Dyornal, at Repleksyon).
Interpersonal na Komunikasyon: Ang pakikipagpalitan ng mensahe sa isa o higit pang tao (hal. pakikipag-usap sa kaibigan o kapamilya).
Pampublikong Komunikasyon: Ang pagpapahayag ng mensahe sa mas malaking pangkat ng mga tao kung saan kadalasan ay iisa ang tagapagsalita at marami ang tagapakinig (hal. talumpati, misa, ulat sa madla).

2. Kahalagahan ng Awdyens

Ang awdyens ay ang taong o pangkat ng mga taong tumatanggap ng ating mensahe. Mahalagang kilalanin ang awdyens upang makapili ng angkop na salita, tono, at paraan ng pagpapahayag.

3. Pormal vs. Impormal (Di-pormal) na Wika

Pormal na Wika: Ginagamit sa mga opisyal at seryosong sitwasyon. Gumagamit ng wastong balarila at magagalang na pananalita.
Ginagamit sa: Paaralan, Opisina, Talumpati, Pagsulat ng liham, at Pag-uulat.

Impormal na Wika (di-pormal): Ginagamit sa pang-araw-araw na pakikipag-usap sa pamilya, kaibigan, at mga taong malapit sa atin. Mas malaya at payak ang paggamit ng mga salita.
Ginagamit sa: Pakikipag-usap sa kaibigan, Chat o text message, at Kaswal na usapan.

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Quiz: Antas ng Komunikasyon at Awdyens

Multiple Choice · 1 question at a time · Instant feedback

Aralin 3.3: Digital Identity, Privacy at Proteksyon sa Datos

Active Footprint

Kusang ibinabahagi online (post, photos, comments, status).

Passive Footprint

Awtomatikong nakokolekta (cookies, IP address, browsing history).

Privacy (Pagkapribado)

Karapatan ng isang tao na panatilihing pribado at ligtas ang kaniyang personal na impormasyon.

Personal Data Protection

Pangangalaga sa personal na impormasyon upang hindi magamit nang walang pahintulot.

Netiquette & 2FA

Wastong asal online at Two-Factor Authentication para sa proteksyon ng account.

Digital Footprint, Privacy at Seguridad

Ang bawat kilos natin sa internet ay lumilikha ng bakas na maaaring maging permanente at magamit ng iba.

Active Footprint: Mga bagay na kusang-loob nating inilalathala (posts, pictures, vlogs).
Passive Footprint: Datos na tahimik na naitatala ng servers (searches, device location, cookies).
Privacy (Pagkapribado): Ang karapatan ng isang tao na panatilihing pribado at ligtas ang kaniyang personal na impormasyon.
Personal Data Protection: Ang pangangalaga sa ating personal na impormasyon upang hindi ito magamit nang walang pahintulot.
Netiquette: Tamang pag-uugali sa internet — mag-isip bago mag-post, iwas cyberbullying, at i-on ang Two-Factor Authentication (2FA).

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Quiz: Digital Identity, Privacy at Netiquette

Multiple Choice · 1 question at a time · Instant feedback

Aralin 3.4: Public-Facing na Teksto at Panlipunang Layunin

Public-Facing na Teksto

Sulatin o mensaheng ginawa upang mabasa ng publiko upang magbigay impormasyon, manghikayat, o magmulat ng kamalayan.

1. Blog

Pormal o impormal na sulatin ng mga karanasan, kaalaman, o opinyon ukol sa partikular na paksa.

2. Komentaryo

Sulatin na nagbibigay ng sariling pananaw, pagsusuri, o opinyon sa isyu o balita.

3. Advocacy Post

Mensahe o post na naglalayong hikayatin ang mga tao na suportahan o makiisa sa isang adbokasiya.

4. Maikling Kolumn

Maikling artikulo na nagpapahayag ng opinyon o pananaw ng may-akda sa napapanahong isyu.

Inclusivity (Inklusibo)

Pantay na paggalang, pagkakataon, at pagtanggap sa lahat anuman ang kasarian, edad, kultura, o pinagmulan.

Equality (Pagkakapantay-pantay)

Paniniwalang ang bawat tao ay may pantay na karapatan at pagtrato nang walang diskriminasyon.

1. Kahulugan ng Public-Facing na Teksto

Ang public-facing na teksto ay mga sulatin o mensaheng ginawa upang mabasa o makita ng publiko. Layunin nitong magbigay ng impormasyon, magpahayag ng opinyon, manghikayat, o magmulat ng kamalayan tungkol sa isang isyu o paksa.

2. Mga Halimbawa ng Public-Facing na Teksto

1. Blog: Isang impormal o pormal na sulatin na naglalaman ng mga karanasan, kaalaman, opinyon, o impormasyon tungkol sa isang partikular na paksa.
2. Komentaryo: Isang sulatin o pahayag na nagbibigay ng sariling pananaw, pagsusuri, o opinyon tungkol sa isang isyu, balita, o pangyayari.
3. Advocacy Post: Isang mensahe o post na naglalayong hikayatin ang mga tao na suportahan o makiisa sa isang adbokasiya o layunin.
4. Maikling Kolumn: Isang maikling artikulo na nagpapahayag ng opinyon o pananaw ng may-akda tungkol sa isang napapanahong isyu.

3. Mga Layunin ng Public-Facing na Teksto

1. Inclusivity (Pagiging Inklusibo): Ang pagiging inklusibo ay ang pagbibigay ng pantay na paggalang, pagkakataon, at pagtanggap sa lahat ng tao anuman ang kanilang kasarian, edad, relihiyon, kultura, kakayahan, o pinagmulan.

2. Equality (Pagkakapantay-pantay): Ang pagkakapantay-pantay ay ang paniniwalang ang bawat tao ay may pantay na karapatan, pagkakataon, at pagtrato sa lipunan nang walang diskriminasyon.

3. Social Awareness (Kamalayang Panlipunan): Pagmumulat sa mga mahahalagang isyu at suliranin ng lipunan upang magbunga ng positibong pagkilos.

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Quiz: Public-Facing na Teksto at Layunin

Multiple Choice · 1 question at a time · Instant feedback

Aralin 4.1: Dinamikong Proseso ng Komunikasyon

Dinamikong Kalikasan

Tuloy-tuloy at interaktibo; bumubuo ng relasyon, damdamin, at lumulutas ng suliranin.

1. Pagbibigay-kahulugan

Decoding batay sa sariling karanasan, kultura, at damdamin.

2. Pakikilahok

Aktibong pakikinig, pagtatanong, at pakikiisa sa talakayan.

3. Pagpapahayag

Paggamit ng salita, kilos, at simbolo upang maipahayag ang sarili.

Ang komunikasyon ay hindi lamang simpleng pagpapadala ng impormasyon; ito ay dinamiko dahil nagbabago ito ayon sa sitwasyon, relasyon, at tugon ng bawat isa.

1. Pagbibigay-kahulugan (Decoding): Ang pag-unawa sa kahulugan ng mensahe batay sa karanasan at emosyon. Halimbawa, ang salitang "Okay." ay maaaring magbago ang kahulugan depende sa tono at konteksto.
2. Pakikilahok (Participation): Ang hindi lamang pananahimik kundi aktibong pakikinig at pakikiisa sa talakayan.
3. Pagpapahayag (Expression): Ang paggamit ng salita, kilos, at ekspresyon ng mukha upang maipahayag ang saloobin.

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Quiz: 3 Proseso ng Komunikasyon

Multiple Choice · 1 question at a time · Instant feedback

Aralin 4.2: 4 na Konteksto, Kamalayang Kultural at Sensibilidad

Personal

Pakikipag-usap sa sarili ("Natatakot akong magsalita sa harap").

Interpersonal

Pakikipag-usap sa ibang tao (hal. pakikipag-usap sa kaibigan).

Sosyal

Komunikasyon sa pangkat o komunidad (hal. pagbibigay ng opinyon sa klase).

Kultural

Komunikasyong isinasaalang-alang ang kultura at paniniwala (hal. paggalang sa paniniwala ng iba).

Kamalayang Kultural

Pag-unawa sa kaugalian ng iba (hal. paggamit ng "po" at "opo").

Sensibilidad

Pag-iisip sa mararamdaman ng kausap ("Mayroon akong ibang pananaw ukol diyan").

Apat na Konteksto ng Komunikasyon (Mula sa Handouts)

Personal: Pakikipag-usap sa sarili. Halimbawa: "Natatakot akong magsalita sa harap".
Interpersonal: Pakikipag-usap sa ibang tao. Halimbawa: Pakikipag-usap sa kaibigan.
Sosyal: Komunikasyon sa pangkat o komunidad. Halimbawa: Pagbibigay ng opinyon sa klase.
Kultural: Komunikasyong isinasaalang-alang ang kultura at paniniwala. Halimbawa: Paggalang sa paniniwala ng ibang tao.

Kamalayang Kultural, Sensibilidad at Hadlang

Kamalayang Kultural: Pag-unawa at paggalang na magkakaiba ang kultura, wika, at tradisyon ng bawat tao.
Sensibilidad: Pagpili ng tamang pananalita upang hindi makasakit (hal. sa halip na "Mali ka!", sabihing "Mayroon akong ibang pananaw ukol diyan").
Hadlang (Noise): Lahat ng balakid sa komunikasyon tulad ng ingay sa paligid, mahinang signal, at matinding galit o emosyon.

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Quiz: 4 na Konteksto at Kamalayang Kultural

Multiple Choice · 1 question at a time · Instant feedback

📖 Official Terms & Definitions Reviewer

General Mathematics: Core Definitions & Concepts

Authoritative definitions, formal terms, classifications, and governing formulas derived directly from Lesson 1 (Payroll, Compensation, Overtime, Taxes & Deductions) and Weeks 6, 7 & 8 (Percentages, Nature Patterns & Sequences).

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Lesson 1.1: Wages, Salaries, and Compensation Schemes – Definitions

Compensation

Definition: The money paid to people for their work. It comes in different forms including wages, salaries, and commissions.

Wages

Definition: Payments based on the number of hours worked or the amount of work completed (common in hourly or factory jobs).

Salaries

Definition: Fixed regular payments, usually given to professionals or office workers who are paid to complete tasks rather than track time.

Commissions

Definition: Earnings based on how much a person sells, commonly seen in real estate, insurance, and similar sales fields.

Payroll

Definition: The business process of tracking hours, salary, deductions, and pay periods to ensure fair and accurate employee payments.

GRESMA Method

Definition: A 5-step problem solving framework: Given, Required, Equation, Solution, and Answer.

Formal Definitions of Common Pay Periods

Weekly Salary

Definition: Paid once a week, resulting in exactly 52 paychecks per year. Computation: Annual Salary ÷ 52.

Biweekly Salary

Definition: Paid once every two weeks, resulting in exactly 26 paychecks per year. Computation: Annual Salary ÷ 26.

Semi-Monthly Salary

Definition: Paid twice a month (e.g., the 15th and 30th), resulting in 24 paychecks per year. Computation: Annual Salary ÷ 24.

Monthly Salary

Definition: Paid once a month, resulting in exactly 12 paychecks per year. Computation: Annual Salary ÷ 12.

GRESMA Problem-Solving Structure (Defined)

1. Given (G): Identify all known quantities and facts stated in the problem.
2. Required (R): State explicitly what unknown value or quantity is being asked for.
3. Equation (E): Write down the exact mathematical formula or equation that relates the given to the required.
4. Solution (S): Perform the step-by-step arithmetic and algebraic computations.
5. Answer (A): Express the final result with appropriate units of measurement and currency (e.g., ₱).

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Quiz: Payroll & Compensation Definitions

Multiple Choice · Test Terms, Pay Periods & GRESMA Steps

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Lesson 1.2: Overtime (OT) Pay – Article 87 Definitions & Rates

Overtime Pay

Definition: Additional compensation mandated by Article 87 of the Philippine Labor Code for work performed beyond 8 regular hours per day.

Regular Day OT

Definition: Work rendered beyond 8 hours on an ordinary working day (+25% premium). Rate: 1.25 × Hourly Rate.

Rest / Special Day OT

Definition: Work performed on a scheduled rest day or special non-working holiday (+30% on top of 130% base). Rate: 1.69 × Hourly Rate.

Regular Holiday OT

Definition: Overtime work rendered during an official regular holiday (+30% on top of 200% base). Rate: 2.60 × Hourly Rate.

Holiday on Rest Day OT

Definition: Overtime rendered on a regular holiday that coincides with the employee's rest day (+30% on 260% base). Rate: 3.38 × Hourly Rate.

Article 87 Philippine Labor Code Definitions Table

1. Regular Day Overtime: Work rendered beyond 8 hours on a regular working day. Formula: Hourly Rate × 1.25 × OT Hours.
2. Rest Day or Special Day Overtime: Work rendered on a designated rest day or special non-working day. Formula: Hourly Rate × 1.69 × OT Hours.
3. Regular Holiday Overtime: Work rendered on an official national regular holiday. Formula: Hourly Rate × 2.60 × OT Hours.
4. Holiday Falling on Rest Day Overtime: Work rendered on a regular holiday that simultaneously falls on a scheduled rest day. Formula: Hourly Rate × 3.38 × OT Hours.

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Quiz: Article 87 Overtime Definitions

Multiple Choice · Rates, Legal Bases & Multipliers

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Lesson 1.3: Piecework & Commission Schemes – Definitions

Piecework

Definition: Compensation paid strictly according to the number of completed items produced rather than time spent. Formula: Gross = Rate Per Item × Number of Items.

Differential Piecework

Definition: A piecework system where workers earn one base rate per unit up to a quota, and a higher incentive rate for all units produced beyond that quota.

Commission

Definition: Payment based directly on a percentage of sales generated. Formula: Commission = Commission Rate × Amount of Sales.

Straight Commission

Definition: A commission structure where the percentage rate remains constant for all sales regardless of total volume.

Variable Commission

Definition: A commission structure where the percentage rate increases as sales volume grows past predetermined milestone tiers.

Comparative Definitions of Incentive Systems

• Piecework vs Differential Piecework: Standard piecework maintains a flat fee per item regardless of quantity (e.g. ₱50/sleeve). Differential piecework rewards high production by paying a premium rate per item once the target quota is exceeded (e.g. ₱10 for first 100 boxes, ₱15 for excess boxes).

• Straight vs Variable Commission: Straight commission applies a single rate (e.g., 5% on all real estate sales). Variable commission escalates (e.g., 3% on first ₱1M, 5% on next ₱2M, 8% thereafter).

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Quiz: Piecework & Commission Definitions

Multiple Choice · Quotas, Straight & Variable Rates

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Lesson 1.4: Philippine Taxes & 12% VAT – Definitions of 11 Tax Types

Taxes

Definition: Amounts of money collected by the government from individuals and businesses to generate revenue and support public resources (education, healthcare, infrastructure).

Excise Tax

Definition: Tax imposed on specific goods such as alcohol, fuel, and tobacco.

Capital Gains Tax (CGT)

Definition: Tax imposed on profit earned from selling assets like land, buildings, or stocks.

Documentary Stamp Tax (DST)

Definition: Tax levied on legal documents, promissory notes, and formal loan transactions.

Percentage Tax

Definition: Business tax for small enterprises that are not registered for Value-Added Tax (VAT).

Donor's Tax

Definition: Tax imposed on gratuitous transfers of property or money given as a gift.

Complete Definitions of All 11 Philippine Tax Types

Real Property Tax (RPT)

Definition: Annual local ad valorem tax imposed on the ownership of land, buildings, and real estate improvements.

Business Tax

Definition: Local tax imposed by cities or municipalities on the privilege of operating a commercial business.

Transfer Tax

Definition: Tax imposed upon the legal transfer of real property ownership from one party to another.

Income Tax

Definition: Tax imposed on the earnings, wages, and profits of individuals and corporate entities.

Value-Added Tax (VAT)

Definition: Indirect consumption tax (12% in the Philippines) added to the price of goods and services at each stage of distribution.

Withholding Tax

Definition: Tax deducted in advance by employers or payors from compensation before releasing payment to the payee.

Value-Added Tax (VAT) Classification Definitions

• VAT-Exclusive Price: The initial base price of a good or service before the 12% tax is added. VAT = 12% × Base Price.
• VAT-Inclusive Price: The total retail price paid by consumers containing both the base price and the 12% VAT. Price Before VAT = Total ÷ 1.12.

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Quiz: 11 Philippine Tax Types & 12% VAT

Multiple Choice · Definitions, Classifications & VAT Formulas

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Lesson 1.5: 2025 Mandatory Deductions & Net Income – Definitions

Gross Income

Definition: The total unadjusted earnings of an employee prior to any deductions (Basic Pay + Overtime Pay).

Taxable Income

Definition: Gross income minus statutory mandatory deductions (SSS, PhilHealth, Pag-IBIG) that is subject to the tax brackets.

Net Income (Take-Home Pay)

Definition: The actual cash received by an employee after subtracting all statutory deductions and withholding tax.

SSS Contribution (2025)

Definition: Social security contribution mandated at 15% under RA 11199 (10% employer share, 5% employee share).

Pag-IBIG Contribution

Definition: Mandatory national housing fund contribution capped at ₱200.00 employee share (matched by ₱200.00 employer share).

PhilHealth Contribution

Definition: Mandatory health insurance premium set at 5% of basic monthly salary (divided equally between employer and employee).

2025 Income Tax Brackets (Formal Definitions)

• Not over ₱250,000 Annual Taxable Income: Defined as 0% Tax (Completely Tax-Exempt).
• ₱250,001 to ₱400,000: Defined as 15% of the excess over ₱250,000.
• ₱400,001 to ₱800,000: Defined as ₱22,500 + 20% of the excess over ₱400,000.
• ₱800,001 to ₱2,000,000: Defined as ₱102,500 + 25% of the excess over ₱800,000.
• ₱2,000,001 to ₱8,000,000: Defined as ₱402,500 + 30% of the excess over ₱2,000,000.
• Over ₱8,000,000: Defined as ₱2,202,500 + 35% of the excess over ₱8,000,000.

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Quiz: Deductions & Net Pay Definitions

Multiple Choice · SSS RA 11199, Pag-IBIG, PhilHealth & 2025 Brackets

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Week 6: Percentages in Business & Finance – Definitions

Inflation

Definition: The general increase in prices of goods and services over time, which causes the purchasing power of money to decline.

Mark-Up

Definition: The amount added to the cost price of an item to make a profit: Mark-Up = Selling Price - Cost Price.

Discount

Definition: The amount reduced from the original regular price of an item during a sale: Discount = Original Price - Net Price.

Cost Price (CP)

Definition: The original purchase price or production expense paid by the business to acquire an item.

Selling Price (SP)

Definition: The final price charged to the consumer after mark-up is added: SP = Cost Price + Mark-Up.

Net Price

Definition: The actual final amount paid by the customer after discount deductions: Net Price = Original - Discount.

Profit and Loss Terminology (Defined)

• Profit (Gain): Defined as the positive financial gain achieved when total revenue/selling price exceeds total cost (SP > CP). Formula: Profit = Selling Price − Cost Price.
• Loss: Defined as the financial deficit incurred when an item is sold for less than its acquisition cost (SP < CP). Formula: Loss = Cost Price − Selling Price.
• Mark-Up Percentage: The mark-up expressed as a fraction of cost: (Mark-Up / CP) × 100.
• Discount Percentage: The discount expressed as a fraction of original price: (Discount / Original) × 100.
• Profit Percentage: (Profit / Cost Price) × 100 | Loss Percentage: (Loss / Cost Price) × 100.

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Quiz: Business Percentages Definitions

Multiple Choice · Inflation, Mark-Up, Discount, Profit & Loss

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Week 7: Patterns in Life and Nature – Definitions

Pattern

Definition: A repeated or regular arrangement of shapes, numbers, designs, or events that follow a recognizable rule.

Sequence

Definition: An ordered list of numbers (terms) that follow a specific governing mathematical rule explaining how each term is obtained.

Fibonacci Sequence

Definition: A sequence where each term is the sum of the two preceding terms (Fn = Fn-1 + Fn-2), starting with 0 and 1.

Fractals

Definition: Infinitely complex, self-similar repeating geometric patterns found in nature (trees, ferns, lightning, rivers).

Tessellations

Definition: Repeating arrangements of geometric shapes fitted closely together without any gaps or overlapping edges (honeycombs).

Symmetry

Definition: Balanced proportion across an axis: Bilateral (2 mirror halves) vs. Radial (radiates from center).

Special Sequence Classifications (Defined)

• Arithmetic Sequence (Progression): Defined as a sequence in which the difference between consecutive terms is constant (the common difference \(d\)).
• Geometric Sequence (Progression): Defined as a sequence formed by multiplying each term by a constant multiplier (the common ratio \(r\)).
• Harmonic Sequence: Defined as a progression formed by taking the reciprocals of an arithmetic progression (\(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\dots\)).
• Sequence of Perfect Squares: Defined as \(a_n = n^2\) (\(1, 4, 9, 16, 25, 36\dots\)).
• Constant Sequence: Defined as a sequence where all terms are identical (\(c, c, c\dots\)), possessing \(d=0\) and \(r=1\).

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Quiz: Patterns & Fibonacci Definitions

Multiple Choice · Fractals, Spirals, Symmetry & Special Sequences

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Week 8: Arithmetic & Geometric Sequences in Real Life – Definitions

Arithmetic Sequence

Definition: A mathematical model for linear steady growth with fixed constant increments added over equal intervals: an = a1 + (n - 1)d.

Common Difference (d)

Definition: The constant value added in each step of an arithmetic sequence: d = an - an-1.

Geometric Sequence

Definition: A mathematical model for exponential growth or decay through constant multiplication over equal intervals: an = a1 · r^(n - 1).

Common Ratio (r)

Definition: The constant multiplier in a geometric sequence: r = an ÷ an-1.

Growth Multiplier

Definition: The common ratio for exponential growth where r = 1 + Growth Rate (e.g. 5% growth → r=1.05).

Depreciation Multiplier

Definition: The common ratio for asset depreciation where r = 1 - Depreciation Rate (e.g. 15% depreciation → r=0.85).

Real-World Sequence Applications (Defined)

• Installment Plan: Defined as a structured financial arrangement where a buyer pays fixed periodic increments over time, modeled arithmetically.
• Asset Depreciation: Defined as the gradual reduction in economic value of tangible property (e.g. phones, laptops, cars) over time due to age and wear, modeled geometrically (\(0 < r < 1\)).
• Exponential Growth: Defined as rapid growth where the quantity increases by a fixed proportion per unit of time (e.g. bacteria doubling, population growth, compound interest where \(r > 1\)).

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