TEACHGYAN • CLASS 8 MATHEMATICS • GANITA PRAKASH PART II
Chapter 13: Algebra Play
NCERT-aligned detailed notes • Concepts • Worked examples • MCQs • Competency • HOTS
📚 NCERT Topic-by-Topic Explanation
1. Think-of-a-number tricks
A think-of-a-number trick can be understood by replacing the unknown number with a variable. For example, choose x, double it to 2x, add 6 to get 2x+6, and then reverse the operations. Algebra reveals why the trick works for every allowed starting number.
🧮 Worked Example
Start with the simplest case, apply the rule once, record the result, and repeat the same rule to identify the pattern.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
2. Number pyramids
Number pyramids can be analysed by assigning variables to entries and expressing each next entry through addition or another stated rule. The puzzle becomes an algebraic structure rather than guesswork.
🧮 Worked Example
Example: For data 5, 7, 8, 10, the mean is (5+7+8+10)÷4=30÷4=7.5. The total deviation from 7.5 is zero, showing the balancing interpretation.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
3. Algebraic patterns
Grid problems invite students to look for relationships between positions and values. A good strategy is to record the pattern systematically before forming a general rule.
🧮 Worked Example
Example: Let a number be x. If a trick says multiply by 3 and add 6, the result is 3x+6. Reversing the steps gives x=(result−6)÷3.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
4. Grid puzzles
Largest-product problems develop reasoning about how a fixed total can be split to maximise a product. Testing organised possibilities can lead to a general observation and justification.
🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
5. Largest product problems
Divisibility tricks should be explained by place value and algebra, not memorised as magic rules. For example, the digit-sum test for divisibility by 3 follows from writing a number in place-value form.
🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
6. Divisibility tricks and algebraic reasoning
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🧮 Worked Example
Example: A parallelogram has base 12 cm and perpendicular height 7 cm. Its area is 12×7=84 cm². The perpendicular height, not the sloping side, is used.
💡 Common Mistake
Do not apply a remembered rule without checking the conditions, units, diagram, scale or stated pattern. Explain why the chosen method matches the problem.
🧠 Concept-Based MCQs
- If x is increased by 5, the new expression is: A) x−5 B) 5x C) x+5 D) x/5
- Algebra helps explain number tricks because: A) variables represent unknown/general numbers B) variables remove reasoning C) every answer is guessed D) symbols have no meaning
- A divisibility rule should ideally be supported by: A) place-value reasoning B) guessing C) drawing only D) memorisation alone
Answers: 1-B, 2-B, 3-A.
📌 Competency-Based Case Study
A learner is given an unfamiliar diagram, data display or puzzle and remembers formulas but is unsure which idea to use. The first step is to identify the structure and explain the relationship before calculating.
- What information is given?
- Which NCERT concept applies?
- Show the reasoning step by step.
- How can the answer be checked?
🔥 HOTS & Competition Corner
- Find a pattern and state a rule that predicts the next stage.
- Create a counterexample to a tempting but false claim.
- Solve the problem in two different ways and compare the methods.
- Design your own NCERT-style challenge question.
✍️ Exam-Ready Practice
- Explain the central concept in your own words.
- Solve a direct application problem.
- Solve an unfamiliar competency-based problem.
- Explain one common mistake and correct it.
- Write a complete 5-mark solution with working and conclusion.
🏆 Five-Level Mastery
| Level | Skill |
|---|---|
| 1 Recall | Definitions, terms and formulas |
| 2 Understand | Explain concepts and interpret examples |
| 3 Apply | Solve routine and real-life problems |
| 4 Analyse | Detect errors, compare methods and justify |
| 5 Create | Build a pattern, puzzle, model or problem |
🔁 One-Minute Revision
Concept → Condition → Representation → Method → Example → Check → Application.
✅ Mastery Checklist
- ☐ I can explain every topic.
- ☐ I can solve standard problems.
- ☐ I can apply the concept in unfamiliar situations.
- ☐ I can justify my method.
- ☐ I can find and correct mistakes.