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Complete Example: 2(3x−5)−4(2x−3)2(3x-5)-4(2x-3)

See how our three-stage approach transforms complex algebraic problems into manageable learning.

Stage 1: Definition Mastery

Students first learn what each multiplication means in plain language.

2(3x−5)2(3x-5)
= "Add (3x-5) itself twice"
4(2x−3)4(2x-3)
= "Add (2x-3) four times"

Stage 2: Logical Development

Using their definition knowledge, students work through the logical steps systematically.

2(3x−5)=(3x−5)+(3x−5)=3x+3x−5−5=6x−102(3x-5) = (3x-5)+(3x-5) = 3x+3x-5-5 = 6x-10
4(2x−3)=(2x−3)+(2x−3)+(2x−3)+(2x−3)=8x−124(2x-3) = (2x-3)+(2x-3)+(2x-3)+(2x-3) = 8x-12
2(3x−5)−4(2x−3)=(6x−10)−(8x−12)2(3x-5)-4(2x-3) = (6x-10)-(8x-12)
=6x−10−8x+12= 6x-10-8x+12
=6x−8x−10+12= 6x-8x-10+12
=−2x+2= -2x+2

Stage 3: Fluency Training

After mastering stages 1 and 2, students practice until they can solve complex problems mentally in one step.

2(3x−5)−4(2x−3)2(3x-5)-4(2x-3)
↓
2×3−4×2=−22\times3-4\times2=-2
2×(−5)−4×(−3)=22\times(-5)-4\times(-3)=2
↓
−2x+2-2x+2
Goal: One-step mental calculation of complex expressions

This fluency training is achieved through our interactive Trainer app, which provides personalized practice sessions.

Learn about Trainer App →

Real Problems, Real Solutions

See how SeedTree identifies and fixes common student errors

a+b≠a+b\sqrt{a} + \sqrt{b} \neq \sqrt{a + b} Proof

By squaring both numbers, students should be able to prove why a+b\sqrt{a} + \sqrt{b} is not equal to a+b\sqrt{a + b}

📝Example Question

Prove that 2+3\sqrt{2} + \sqrt{3} is not equal to 5\sqrt{5}
❌
Current Approach

Most students get confused with this after memorizing:

a×b=ab\sqrt{a} \times \sqrt{b} = \sqrt{ab}

They incorrectly apply multiplication rules to addition

✅
Improved Approach

SeedTree guides students to:

The Coefficient of x: 1. Square (2+3)2(\sqrt{2} + \sqrt{3})^2
The Constant: 2. Expand to 2+26+32 + 2\sqrt{6} + 3
3. Compare with (5)2=5(\sqrt{5})^2 = 5
∴ They are not equal

12x3=2x−3\frac{1}{2x^3} = 2x^{-3}

Only the index form moves to the numerator, not the entire denominator including multiplier

❌
Common Mistake
12x3=2x−3\frac{1}{2x^3} = 2x^{-3}

Incorrectly moving entire denominator

✅
Correct Method
12x3=12×1x3=12x−3\frac{1}{2x^3} = \frac{1}{2}\times\frac{1}{x^3}= \frac{1}{2}x^{-3}

Only the index form moves up

Why this matters: Understanding the rules of exponents is crucial for algebra and calculus. SeedTree helps students see why rules work, not just memorize them.

Build deep mathematical understanding step-by-step.

Our three-stage approach: master definitions, develop logical reasoning, then build computational fluency.

Trainer

Interactive platform where students practice fluency training through personalized sessions to achieve one-step mental calculation of complex expressions.

Worknote

Structured practice platform for building computational fluency after conceptual mastery is achieved.