Class 8 Maths · Chapter 12
Short answer: Class 8 Maths Chapter 12 'Tales by Dots and Lines' (Ganita Prakash Part 2, naye rationalised syllabus 2026-27) me bar graphs, line graphs aur Cartesian plane ke coordinates cover hote hain — ye chapter data ko visually 'padhna' sikhata hai aur aage coordinate geometry ki neev banta hai.
Chapter 12 'Tales by Dots and Lines' Ganita Prakash Class 8 Part 2 ka ek visually interesting chapter hai jo data ko graphs ke through 'kahani' ki tarah padhna sikhata hai. Is chapter me bar graphs, line graphs aur Cartesian plane (coordinates) ke basics cover hote hain — yaani ki numbers ko dots aur lines me convert karke unse insights nikalna. Ye chapter aage chalkar coordinate geometry aur statistics ki neev banta hai, isliye concepts (quadrants, plotting points, slope ki intuition) ko sirf ratna nahi, samajhna zaroori hai. Neeche har exercise question ka step-by-step CBSE marking-scheme style solution diya gaya hai, saath hi common mistakes aur exam-oriented tips bhi.
Chapter 12 Summary — 5 Minute Revision
'Tales by Dots and Lines' Class 8 Ganita Prakash Part 2 ka Chapter 12 hai jo graphs aur data visualization ke fundamentals sikhata hai. Chapter me teen core skills develop hoti hain: (1) bar graphs se categorical data compare karna, (2) line graphs se continuous data (jaise time ke saath change) ka trend dikhana, aur (3) Cartesian plane par points ko coordinates (x, y) ke through plot karna aur quadrants pehchaanna.
Ye chapter conceptually important hai kyunki ye students ko numbers se seedha 'insight' nikalna sikhata hai — jaise kaun sa interval me sabse tezi growth hui, ya kaunsa object sabse pehle rukta hai. Ye skills aage Class 9-10 ke coordinate geometry aur statistics chapters ki direct neev banate hain.
Practice ke liye har question ko pehle table banakar, phir sahi scale choose karke, aur axes ko properly label karke solve karna chahiye — yehi CBSE marking scheme me full marks dilata hai.
In-Text Questions — Solutions
Page ke shuru me diya gaya ek activity — do dosto ki cycling speed ka comparison graph se — is activity ka purpose kya hai?
Ye activity students ko real-life scenario (do cyclists) ke through introduce karti hai ki kaise ek hi graph par do alag data sets ko plot karke unki relative speed compare ki ja sakti hai. Steeper line = faster speed — ye visual intuition build karta hai jo baad me formal distance-time graphs padhne me kaam aata hai.Chapter me diya gaya table-to-graph conversion example (jaise temperature vs time data) se kya seekhne ko milta hai?
Ye example dikhata hai ki raw tabular data ko visual graph me convert karne se trend (badhna, ghatna, ya constant rehna) turant pakad me aata hai — jo sirf numbers dekhkar itni jaldi samajh nahi aata. Yeh 'data se insight' nikalne ka core skill hai.Cartesian plane introduce karte waqt diya gaya 'treasure hunt / grid location' type example kis concept ko simplify karta hai?
Ye example x aur y coordinates ki abstract idea ko ek relatable real-world grid (jaise map par location dhundna) se jodta hai, taaki students easily samajh sakein ki har point ko do numbers (x, y) se uniquely describe kiya ja sakta hai.In-text me diya explore box jisme students se poocha jaata hai 'kya har graph ek straight line hoga?' — iska answer kya hai?
Nahi, har graph straight line nahi hota. Jab rate of change constant ho (jaise fixed speed) tab hi straight line milti hai; agar speed/rate change ho raha ho (jaise rukna, phir tez chalna) toh graph curved ya multiple straight segments wala (piecewise) hoga — jaise Q5 exercise ke Rider B ka case.Exercise Questions — Solutions (Q1–Q6)
Q1. Ek car 3 ghante me 180 km chalti hai constant speed se. (a) Ek line graph banao jisme x-axis par time (hours) aur y-axis par distance (km) ho. (b) Graph se 2.5 ghante me total distance nikaalo.
Step 1: Speed = 180/3 = 60 km/h (constant), isliye graph ek straight line hoga jo origin (0,0) se guzregi.
Table: (0,0), (1,60), (2,120), (3,180)
Step 2: In points ko plot karke origin se judne wali straight line kheecho — x-axis par 'Time (hours)' aur y-axis par 'Distance (km)' label karo, suitable scale lo (jaise 1 unit = 30 km).
Step 3: Graph se x = 2.5 par vertical line kheecho, jahan line ko cross kare wahan se horizontal line kheecho y-axis tak.
2.5 ghante me distance = 60 × 2.5 = 150 km
Graph reading is calculation se match karegi kyunki relationship direct proportion hai.
Q2. Neeche diye gaye data se bar graph banao — ek school ke 5 sections (A, B, C, D, E) me students ki sankhya: 40, 35, 50, 30, 45.
Step 1: X-axis par 'Sections' (A, B, C, D, E) aur Y-axis par 'Number of students' lo.
Step 2: Scale choose karo — 1 unit = 5 students (kyunki numbers 30 se 50 ke beech hain).
Step 3: Har section ke liye equal-width bar kheecho, uniform gap rakhte hue:
Section A → 40, B → 35, C → 50, D → 30, E → 45
Step 4: Graph ko title do jaise 'Number of Students in Each Section' aur bars ke upar values likho clarity ke liye.
Comparison se clear hota hai ki Section C sabse zyada (50) aur Section D sabse kam (30) students rakhta hai.
Q3. Ek point P ki coordinates (4, −3) hai. Ye Cartesian plane ke kis quadrant me hoga? Reason ke saath batao.
Step 1: Quadrant rule yaad karo:
Quadrant I: (+, +) | Quadrant II: (−, +) | Quadrant III: (−, −) | Quadrant IV: (+, −)
Step 2: Point P(4, −3) me x-coordinate = +4 (positive) aur y-coordinate = −3 (negative).
Step 3: (+, −) combination sirf Quadrant IV me hota hai.
Isliye point P Cartesian plane ke Quadrant IV me sthit hai.
Q4. Ek pot me plant ki height har hafte record ki gayi: Week 1 = 2 cm, Week 2 = 5 cm, Week 3 = 9 cm, Week 4 = 12 cm, Week 5 = 14 cm. (a) Line graph banao. (b) Kaunse interval me growth sabse tezi se hui?
Step 1: X-axis par 'Week' (1 to 5) aur Y-axis par 'Height (cm)' lo, scale 1 unit = 2 cm.
Step 2: Points plot karo: (1,2), (2,5), (3,9), (4,12), (5,14) aur unhe straight lines se jodo.
Step 3: Har interval ki growth nikaalo (slope compare karo):
Week1→2: 5−2=3 cm | Week2→3: 9−5=4 cm | Week3→4: 12−9=3 cm | Week4→5: 14−12=2 cm
Answer: Sabse tezi growth Week 2 se Week 3 ke beech hui (4 cm), kyunki us segment ka slope sabse steep (upar ki taraf sabse tez) hai graph par.
Q5. Do bike riders A aur B ki journey ka travel graph diya hai — A: 0 se 4 ghante me 0 se 80 km (straight line), B: 0 se 2 ghante ruka rehta hai fir 2 se 4 ghante me 0 se 80 km jaata hai. Dono graphs ek hi axes par kaise dikhenge, aur kaun pehle pahuchega?
Step 1: Rider A ka graph origin (0,0) se seedha (4,80) tak jaayega — ek straight sloping line, uniform speed dikhata hai.
Step 2: Rider B ka graph pehle (0,0) se (2,0) tak horizontal line hoga (kyunki wo ruka hua hai, distance nahi badh raha), fir (2,0) se (4,80) tak steep sloping line hoga (usko kam time me zyada distance cover karni hai).
Step 3: Dono lines same graph par plot karne se comparison saaf dikhta hai — B ki dusri segment ka slope A ki poori line se zyada steep hoga (B ki speed = 80/2 = 40 km/h vs A ki speed = 80/4 = 20 km/h in that phase).
Answer: Dono 4 ghante me hi 80 km pahunchte hain (graphs ek hi point (4,80) par milte hain), isliye A aur B saath (same time) pahunchte hain, halaanki B ki actual riding speed zyada thi kyunki wo bich me ruka tha.
Q6. Ek dukaan me pichle 4 saal ka annual profit (₹ lakh me): 2022→10, 2023→15, 2024→12, 2025→20. Is data ko double bar graph ki jagah simple bar graph se represent karo aur batao trend kya keh raha hai.
Step 1: X-axis par 'Year' (2022–2025) aur Y-axis par 'Profit (₹ lakh)' lo, scale 1 unit = 2 lakh.
Step 2: Bars kheecho: 2022→10, 2023→15, 2024→12, 2025→20, equal width aur equal spacing ke saath.
Step 3: Trend analysis: 2022 se 2023 tak profit badha (+5), 2023 se 2024 tak profit gira (−3), 2024 se 2025 tak dobara badha (+8).
Conclusion: Overall trend upward hai (10 se 20 lakh, yani double) lekin beech me 2024 me ek dip aayi thi — isse pata chalta hai ki growth steady nahi balki fluctuating rahi, aur business ko 2024 ke dip ka reason analyse karna chahiye.
Important Equations — Ek Nazar Me
| Concept | Rule / Formula | Note |
|---|---|---|
| Cartesian plane point | P(x, y) | x = horizontal distance (abscissa), y = vertical distance (ordinate) from origin |
| Quadrant signs | I(+,+) II(−,+) III(−,−) IV(+,−) | Anticlockwise order starting top-right |
| Origin | O(0, 0) | Dono axes ka intersection point |
| Slope / steepness (informal) | Steepness ∝ (change in y) ÷ (change in x) | Jitna zyada steep line, utni fast rate of change (jaise speed) |
| Direct proportion graph | y = kx (origin se guzarti straight line) | Constant speed/rate wale scenarios (Q1) |
| Bar graph scale | 1 unit (grid) = fixed value (jaise 5, 10, 50 units) | Data range ke hisaab se convenient scale chuno |
| Horizontal segment on distance-time graph | Δdistance = 0 while Δtime > 0 | Object stationary/at rest (Q5) |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Coordinates likhte waqt x aur y ka order ulta kar dena — hamesha yaad rakho (x, y) format hai, pehle x-coordinate phir y-coordinate.
- Quadrant ka sign-rule galat yaad rakhna — Quadrant II aur IV ko confuse kar dena. Practice karo: I(+,+), II(−,+), III(−,−), IV(+,−).
- Bar graph banate waqt bars ke beech unequal gap chhod dena ya alag-alag width ki bars kheech dena — hamesha uniform width aur equal spacing rakho.
- Line graph me axes ko label karna bhool jaana (units ke saath) — bina label ke graph incomplete mana jaata hai aur marks katte hain.
- Scale galat choose karna — agar data 10 se 100 tak hai aur scale 1 unit = 1 rakh diya, toh graph bahut bada ho jaayega. Data range dekhkar convenient scale chuno.
- Horizontal (flat) segment ko distance-time graph me 'time nahi bit raha' samajh lena — jabki iska sahi matlab hai object stationary hai lekin time continue ho raha hai.
Board-Style Important Questions
- Note: Ganita Prakash (naya rationalised 2-part format) 2026-27 session se laagu hua hai, isliye is specific chapter number/naam ke exact board-exam-pattern previous year questions abhi publicly verify nahi ho paye is research session me — official rationalisation ke baad ka pehla cycle hone ki wajah se historical PYQ data seedhe is chapter naam se available nahi hai.
- Purane NCERT syllabus me graphs/coordinates se related concepts alag chapter naam/number ke under aate the — agar aapke school ka test purane pattern ka reference le raha hai toh apne teacher/school se latest exam pattern confirm kar lena sahi rahega.
- Chapter-end exercises se practice questions (jaise upar diye 6 questions) hi abhi ke liye sabse reliable practice source hain jab tak official board sample papers is naye syllabus ke hisaab se release nahi hote.
- Graphs, Cartesian plane aur coordinate-based questions historically CBSE Class 8 assessments me regularly aate rahe hain (chapter number/naam alag hone ke bawajood), isliye concept practice priority par rakhni chahiye — exact weightage claim nahi ki ja sakti.
Aksar Poochhe Jaane Wale Sawaal
Ganita Prakash Class 8 me 'Tales by Dots and Lines' kaunsa chapter number hai?
Naye rationalised Ganita Prakash (2026-27 session) syllabus me ye Chapter 12 hai, jo Part 2 (chapters 8–14) ka hissa hai. Ye chapter graphs — bar graphs, line graphs aur Cartesian plane par coordinates ke basics cover karta hai.
Class 8 Maths ka naya NCERT book ka naam kya hai (2026-27 session ke liye)?
Naye NCERT ne purani single-book format hata kar Class 8 Maths ko do parts me diya hai — 'Ganita Prakash Part 1' (Chapters 1–7) aur 'Ganita Prakash Part 2' (Chapters 8–14), total 14 chapters. Ye NCF 2023 ke aligned naya curriculum hai.
Line graph aur bar graph me kya difference hai?
Line graph continuous data (jaise time ke saath temperature ya distance) ko dikhane ke liye use hota hai — points ko lines se jodkar trend dikhaya jaata hai. Bar graph discrete/categorical data (jaise alag-alag sections ke students) compare karne ke liye use hota hai, jahan har category ka apna separate bar hota hai. Dono hi is chapter ka core concept hain.
Cartesian plane ke 4 quadrants ke sign rules kya hain?
Quadrant I me dono coordinates positive (+, +) hote hain, Quadrant II me x negative aur y positive (−, +), Quadrant III me dono negative (−, −), aur Quadrant IV me x positive aur y negative (+, −) hota hai. Origin (0,0) par dono axes intersect karte hain.
Kya ye chapter board exam ke liye important hai?
Graphs aur data representation ka concept Class 8 se lekar higher classes (coordinate geometry, statistics) tak carry hota hai, isliye conceptual clarity zaroori hai. Exact marks-weightage school-to-school aur exam-pattern ke hisaab se alag ho sakti hai, isliye apne school ke latest exam pattern/circular se confirm kar lena best rahega.
Travel graph (distance-time graph) padhte waqt sabse common confusion kya hoti hai?
Students aksar horizontal line (flat segment) ko 'no movement in time' ki jagah 'no time passing' samajh lete hain — jabki horizontal segment ka matlab hai time toh badh raha hai lekin distance same reh raha hai (yaani object ruka hua hai, jaise Q5 solution me Rider B ka pehla segment). Slope ki steepness ko speed se directly correlate karna seekhna is chapter ka key skill hai.
Class 8 Maths — Saare Chapters
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