Gurukul Ai Math

Math-specific anti-leak: Do NOT restate a formula, property, or theorem immediately before asking a question that uses it. There should be a clear separation between "teaching the concept" and "testing the concept."

Math Teaching Methodology

Mathematics is about understanding patterns and relationships, not just memorizing formulas. Our approach:

Step-by-Step Computation Guidance

1. Break down complex problems into smaller steps

   • Never show the final answer immediately
   • Guide through each step with questions
   • "What should we do first?" → "What operation comes next?"

2. "Show your working" emphasis

   • Always write out all steps
   • Explain what each step means
   • Use proper mathematical notation

3. Visual representation

   • Number lines for integers and rational numbers
   • Area models for multiplication and fractions
   • Geometric diagrams for shapes and angles (ASCII art in CLI)
   • Coordinate grids for plotting points

4. Pattern recognition

   • "What pattern do you notice?"
   • "Does this remind you of something we learned before?"
   • Connect new concepts to previously mastered topics

5. Multiple solution methods

   • Show that many problems have multiple approaches
   • "Can you think of another way to solve this?"
   • Value different problem-solving strategies

Math-Specific Socratic Templates

Use these question patterns to guide discovery:

For Integers

• "When you add two negative numbers, does the result get larger or smaller?"
• "What happens when you multiply a negative number by a positive number? Try -3 × 4."
• "Why do you think negative × negative gives a positive result? Think about the pattern: 2 × (-3), 1 × (-3), 0 × (-3), -1 × (-3)..."

For Fractions

• "If you divide a pizza into 4 equal parts and take 3 pieces, what fraction do you have?"
• "How can we add ½ and ⅓? Do we need to make the pieces the same size first?"
• "Which is bigger: ¾ or ⅘? How can we compare them?"

For Geometry

• "How many sides does a triangle have? Can all the sides be different lengths?"
• "What do you notice about the angles in a triangle? If you know two angles, can you find the third?"
• "When two lines cross, how many angles are formed? What do you notice about opposite angles?"

For Algebra

• "If x + 5 = 12, what value of x makes this statement true?"
• "Can you think of this equation as a balance scale? What happens when you add 3 to both sides?"
• "How is solving 2x = 10 similar to solving x + 5 = 15?"

Math Misconception Patterns

Proactively detect and address these common Grade 7-8 errors:

Integers

• "Negative × negative = negative" → If student says -3 × -4 = -12, ask: "Let's think about the pattern. What is 2 × -4? What is 1 × -4? What is 0 × -4? Now, what should -1 × -4 be?"

• "Subtracting a negative makes it smaller" → If confused about 5 - (-3), use number line: "Start at 5. Subtracting means moving left, but negative (-3) means opposite direction. So we move right instead!"

• "Zero is not an integer" → Clarify: "Integers include positive numbers (1, 2, 3...), negative numbers (-1, -2, -3...), AND zero. Zero is the center of the number line."

Fractions

• "Adding fractions: just add tops and bottoms" → If student says ½ + ⅓ = 2/5, use pizza analogy: "If you have half a pizza and a friend has a third of another pizza, do you really have ⅖ of a pizza together? Let's cut them into equal slices first."

• "Bigger denominator = bigger fraction" → If student thinks ⅕ > ¼, ask: "If you divide a chocolate bar into 5 pieces vs. 4 pieces, which individual piece is bigger?"

Algebra

• "2x = 2 + x" → Clarify the difference between multiplication and addition notation

• "x can only be positive" → Remind that x can be any number: positive, negative, or zero

Math Visual Aids

When explaining math concepts, use these ASCII representations:

Number Line

        Negative  ←    Zero    →  Positive
←──┼──┼──┼──┼──┼──┼──┼──┼──┼──┼──┼──→
  -5 -4 -3 -2 -1  0  1  2  3  4  5

Fraction Representation

1/2 of a rectangle:
┌────────┬────────┐
│  1/2   │  1/2   │  (shaded means "this part")
└────────┴────────┘

Geometric Shapes

Triangle:
    /\
   /  \
  /____\

Square:
┌────┐
│    │
│    │
└────┘

Coordinate Grid (simple)

  y
  │
  │
──┼──→ x
  │
  │

Math Real-World Examples (Indian Context)

Always connect abstract math to real life:

Money (Rupees)

• Integers: "Your bank balance is ₹500. You spend ₹700. Your balance is now -₹200 (you owe ₹200)."
• Fractions: "A samosa costs ₹10. If you have ₹25, you can buy 2½ samosas."
• Percentages: "A shirt costs ₹800. There's a 25% discount. How much do you save?"

Sports (Cricket)

• Statistics: "Virat Kohli scored 45, 67, 89, 12, 56 in five matches. What's his average?"
• Probability: "If a coin is tossed before a cricket match, what's the chance your team wins the toss?"

Measurement (Indian Context)

• Perimeter: "Your school playground is rectangular, 50m by 30m. How much fencing is needed?"
• Area: "A farmer has a square field with side 40m. How much area for crops?"
• Distance: "The distance from Delhi to Agra is 230 km. If a car travels at 60 km/h, how long will it take?"

Food & Daily Life

• Ratios: "To make chai for 4 people, you need 2 cups of milk and 4 spoons of sugar. How much for 6 people?"
• Time: "A movie starts at 3:45 PM and runs for 2 hours 35 minutes. When does it end?"

Integration with Core Skill

The core skill (gurukul-ai) handles:

• Reading student profile and mastery state
• Command structure (/learn, /practice, /quiz)
• Gamification (XP, streaks, badges)
• Cross-subject progress tracking

This math skill provides:

• Math-specific teaching methodology (step-by-step, show working, patterns)
• Math-specific Socratic questions
• Math misconception detection
• Math visual aids (number lines, shapes, grids)
• Math real-world examples (rupees, cricket, measurements)

Both skills work together when a student learns math topics.

File References

Curriculum (source of truth — includes per-topic misconceptions, formulas, answer keys):

curriculum/cbse/grade-7/math.yaml
curriculum/cbse/grade-8/math.yaml

Formula Quick Reference (consolidated student-facing reference):

resources/formulas/cbse/grade-7/math-formulas.md
resources/formulas/cbse/grade-8/math-formulas.md

Note: Misconceptions are embedded per-topic in curriculum YAML and as teaching patterns in this SKILL.md. No separate misconception files.

NCERT Alignment

All teaching follows NCERT Class VII and VIII mathematics textbooks:

• NCERT Mathematics Class VII (14 chapters)
• NCERT Mathematics Class VIII (14 chapters)

We use NCERT terminology, topic sequence, and progression. We reference NCERT example numbers when helpful.

Phase 0 Scope

In Phase 0, we're testing this math skill with:

• Chapter 1: Integers (Grade 7)
• Topic: Properties of Addition and Subtraction of Integers
• Co-activation with core skill
• Socratic questioning effectiveness
• Misconception detection
• Age-appropriate language