Tales by Dots and Lines — Class 8 Mathematics (Ganita Prakash)
"A picture is worth a thousand words. A well-drawn graph is worth a thousand calculations."
1. About the Chapter
'Tales by Dots and Lines' is about visualising mathematics — turning abstract numbers into pictures we can SEE. The chapter covers:
- Coordinate plane (Cartesian coordinates)
- Plotting points and lines
- Graphs of relationships (proportional, linear)
- Bar graphs, histograms, pie charts, line graphs
- Reading and interpreting graphs
Key Idea
Mathematics is not just numbers — it is patterns visualised. Graphs make complex relationships clear at a glance.
2. The Coordinate Plane (Cartesian System)
Introduction
Named after René Descartes (1596-1650, French mathematician/philosopher) — though the idea has roots in Indian and Islamic mathematical traditions.
Components
- x-axis: horizontal line
- y-axis: vertical line
- Origin (O): where they meet (0, 0)
- Four quadrants:
- Q1 (top-right): +x, +y
- Q2 (top-left): −x, +y
- Q3 (bottom-left): −x, −y
- Q4 (bottom-right): +x, −y
Plotting a Point (x, y)
- x = horizontal distance from origin (right = +, left = −)
- y = vertical distance from origin (up = +, down = −)
- (3, 5) = 3 right, 5 up
- (−2, 4) = 2 left, 4 up
- (−1, −3) = 1 left, 3 down
- (5, −2) = 5 right, 2 down
Special Points
- Origin: (0, 0)
- On x-axis: y = 0, e.g., (5, 0), (−3, 0)
- On y-axis: x = 0, e.g., (0, 4), (0, −2)
3. Plotting Linear Relationships
Direct Proportion: y = kx
- Goes through ORIGIN (0, 0)
- Constant slope = k
- All points lie on a straight LINE
Example
y = 2x:
- x=0 → y=0
- x=1 → y=2
- x=2 → y=4
- x=−1 → y=−2
- All points (0,0), (1,2), (2,4), (−1,−2) lie on a straight line.
General Linear: y = mx + c
- m = slope (how steep)
- c = y-intercept (where it crosses y-axis)
- Goes through (0, c)
- If c = 0: it's a direct proportion
Constant Function: y = k
- Horizontal line at height k
Vertical Line: x = k
- Vertical line at distance k from y-axis
4. Bar Graphs
Definition
A bar graph uses rectangular bars of various heights to represent data.
When to Use
- Categorical data (e.g., subjects, months)
- Comparing distinct categories
Example
Heights of students in a class (categorical: name; numerical: height):
- Rohan: 150 cm
- Priya: 145 cm
- Aman: 152 cm
- Sneha: 148 cm
Draw bars for each name with height proportional to their value.
Types
- Vertical bar graph (most common)
- Horizontal bar graph
- Double bar graph (compare two sets)
- Stacked bar graph (parts of whole)
5. Histograms
Definition
A histogram is like a bar graph BUT for continuous data grouped into intervals (classes).
Difference from Bar Graph
- Bars in histogram TOUCH each other (no gaps) — because data is continuous
- Bars in bar graph have gaps — discrete data
Example
Marks of 50 students:
- 0-10: 5 students
- 10-20: 12 students
- 20-30: 20 students
- 30-40: 10 students
- 40-50: 3 students
Histogram bars cover each interval with height = frequency.
6. Pie Charts (Circle Graphs)
Definition
A circle divided into sectors, each representing a fraction of the whole.
Key Formula
Angle for category = (Frequency / Total) × 360°
Example
Time spent in 24 hours:
- Sleep: 8 hours → (8/24) × 360° = 120°
- Study: 6 hours → (6/24) × 360° = 90°
- Eat: 3 hours → (3/24) × 360° = 45°
- Other: 7 hours → (7/24) × 360° = 105°
- Total: 360° ✓
When to Use
- Showing parts of a whole
- Comparing relative proportions
- NOT good for exact comparisons (use bar graph)
7. Line Graphs
Definition
A line graph connects data points with line segments to show trends over time.
When to Use
- Continuous data over time
- Showing trends (increasing, decreasing, constant)
- Multiple datasets on same graph
Example
Temperature recorded at 6 AM each day:
- Day 1: 18°C
- Day 2: 20°C
- Day 3: 22°C
- Day 4: 19°C
- Day 5: 23°C
Plot (1, 18), (2, 20), (3, 22), (4, 19), (5, 23) and connect with lines.
8. Scatter Plots
Definition
A scatter plot shows individual data points on a coordinate plane to reveal relationships between two variables.
When to Use
- Investigating correlation
- Identifying patterns or outliers
Example
Height vs Weight of 20 students:
- Plot (height, weight) for each
- If points trend UP-AND-RIGHT, positive correlation
- If points trend DOWN-RIGHT, negative correlation
- If points scattered randomly, no correlation
9. Reading Graphs — Key Skills
From a Graph, Find
- Maximum/minimum value
- Mean/Median (approximation)
- Trends (rising, falling, constant)
- Specific values at specific points
- Comparisons between categories or time periods
Practice Questions
Given a bar graph showing monthly sales:
- Which month had highest sales?
- What is the total for the year?
- By what percentage did sales increase from March to April?
10. Worked Examples
Example 1: Plot Points
Plot (3, 4), (−2, 5), (−3, −2), (4, −1).
- All four quadrants — practice student should plot carefully.
Example 2: Direct Proportion Graph
Graph y = 3x for x from −3 to 3.
- Points: (−3, −9), (−2, −6), (−1, −3), (0, 0), (1, 3), (2, 6), (3, 9)
- All on a straight line through origin, slope = 3.
Example 3: Bar Graph Calculation
A bar graph shows: Apples 25, Bananas 40, Oranges 30, Mangoes 35. Total fruits? 25 + 40 + 30 + 35 = 130. Percentage of mangoes? (35/130) × 100 ≈ 26.9%.
Example 4: Pie Chart
Plot a pie chart showing the budget of a family:
- Food: ₹4000 → (4000/12000)×360 = 120°
- Rent: ₹3000 → 90°
- Transport: ₹1000 → 30°
- Education: ₹2000 → 60°
- Savings: ₹2000 → 60°
- Total: 360° ✓
Example 5: Histogram
Plot a histogram for ages 15-19: 5 students, 20-24: 12, 25-29: 18, 30-34: 8, 35-39: 4.
- Continuous data, bars touch each other.
- Tallest bar at 25-29 (mode group).
11. Common Mistakes
-
Mixing up coordinate order
- (3, 5) is NOT same as (5, 3)
- Always: (x, y) — first horizontal, then vertical
-
Wrong quadrant
- Negative x → left
- Negative y → down
- (−3, −2) is in Q3 (bottom-left)
-
Bar graph vs histogram
- Bar graph: discrete; gaps between bars
- Histogram: continuous; no gaps
-
Pie chart angles
- Must add to 360°
- Use ratio formula
-
Reading the wrong axis
- x-axis is horizontal (independent variable)
- y-axis is vertical (dependent variable)
12. Real-World Applications
Business
- Sales reports use bar graphs
- Market share via pie charts
- Stock prices via line graphs
Science
- Experimental data via scatter plots
- Temperature over time via line graphs
- Distribution of measurements via histograms
Sports
- Cricket: bar graphs of runs per over
- Football: scatter plots of shots vs goals
- Marathon: line graphs of time vs distance
News and Media
- Election results via pie charts
- Population growth via line graphs
- Weather forecasts via bar graphs
13. Conclusion
'Tales by Dots and Lines' teaches you that mathematics is visual. Behind every formula is a picture; behind every dataset is a story to tell.
Master the basic graphs — bar, histogram, pie, line, scatter — and you'll be able to:
- Read newspapers, reports, and research papers fluently
- Create clear visualisations of your own data
- Understand complex relationships at a glance
- Spot patterns and outliers immediately
This is also a foundation for Class 9 Coordinate Geometry, Class 10 Statistics, and all data science.
