Is Mathematics Beautiful?

Titin Suryati Sukmadewi

SMA NEGERI 1 SUMEDANG, WEST JAVA, INDONESIA

The Beauty of Math How to show it to our students?

I don’t know exactly

1. Different solution with different perspective

 From this quadratic sequence:  5, 8, 13, 20, 29, …  Find the n-th term

 5, 8, 13, 20, 29, …   3 5 7 9 2 2 2 𝑓 𝑛 = 𝑎𝑛 2 + 𝑏𝑛 + 𝑐 𝑓 1 = 𝑎 + 𝑏 + 𝑐 = 5 𝑓 2 = 4𝑎 + 2𝑏 + 𝑐 = 8 𝑓 3 = 9𝑎 + 3𝑏 + 𝑐 = 13 𝑎 = 1, 𝑏 = 0, 𝑐 = 4 3𝑎 + 𝑏 = 3 5𝑎 + 𝑏 = 5 𝑓 𝑛 = 𝑛 2 + 4

  

Different Perspective

5, 8, 13, 20, 29, …; Find the n-th term Find the quadratic function equation which through points (1, 5), (2, 8), (3, 13) Solution: 𝑓 𝑥 = 𝛼 𝑥 − 𝑥 1 𝑥 − 𝑥 2 + 𝑚𝑥 + 𝑐 𝑓 𝑥 = 𝛼 𝑥 − 1 𝑥 − 2 + (3𝑥 + 2) 13 = 𝛼 3 − 1 3 − 2 + (3.3 + 2) 2𝛼 = 2 𝛼 = 1 𝑓 𝑥 = 𝑥 − 1 𝑥 − 2 + (3𝑥 + 2) 𝑓 𝑥 = 𝑥 2 + 4

Is procedural skill important?

  Conceptual understanding in mathematics, along with procedural skill, is much more powerful than procedural skill alone.

Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

http://www.corestandards.org/Math

2. SMART SOLUTION

  Procedural Task 1 : Find the value of:  Procedural Task 2: lim 𝑥→2 5𝑥 − 1 − 3𝑥 + 3 4 − 𝑥 2 cos 8𝑥 − 1 lim 𝑥→0 sin 2𝑥 tan 3𝑥

 Solution task 1: lim 𝑥→2 5𝑥 − 1 − 3𝑥 + 3 4 − 𝑥 2 5𝑥 − 1 + 3𝑥 + 3 5𝑥 − 1 + 3𝑥 + 3 5𝑥 − 1 − (3𝑥 + 3) = lim 𝑥→2 (2 − 𝑥)(2 + 𝑥) 5𝑥 − 1 + 3𝑥 + 3 5𝑥 − 1 − (3𝑥 + 3) = lim 𝑥→2 (2 − 𝑥)(2 + 𝑥) 5𝑥 − 1 + 3𝑥 + 3 = lim 𝑥→2 2(𝑥 − 2) −(𝑥 − 2)(2 + 𝑥) 5𝑥 − 1 + 3𝑥 + 3 2 = lim 𝑥→2 −(2 + 𝑥) 2 = − 2 + 2 1 = − 12 5𝑥 − 1 + 3𝑥 + 3 9 + 9 𝒂 − 𝒃 = 𝒂 − 𝒃 𝒂 + 𝒃

 Solution task 2: cos 8𝑥 − 1 lim 𝑥→0 sin 2𝑥 tan 3𝑥 = lim 𝑥→0 − 1 − cos 8𝑥 sin 2𝑥 tan 3𝑥 = lim 𝑥→0 − 2 𝑠𝑖𝑛 2 4𝑥 sin 2𝑥 tan 3𝑥 = lim 𝑥→0 (−2) sin 4𝑥 sin 2𝑥 sin 4𝑥 tan 3𝑥 = (−2) 4 2 4 3 = − 16 3

 Fast solution task 1: lim 𝑥→2 5𝑥 − 1 − 3𝑥 + 3 4 − 𝑥 2 (5 − 3) = lim 𝑥→2 (−2𝑥)(2)(3) 2 = (−4)(2)(3) 1 = − 12  Fast solution task 2: cos 8𝑥 − 1 lim 𝑥→0 sin 2𝑥 tan 3𝑥 8 2 /2 = − (2)(3) = − 32 6 = − 16 3 𝒂 − 𝒃 = 𝒂 − 𝒃 𝒂 + 𝒃 𝒂 − 𝒃 = 𝟎 → 𝒂 + 𝒃 = 𝟐 𝒂

  Provide me to modify the previous task: Task 3: lim 𝑥→2 3 − 𝑥 2 + 5 3𝑥 − 2 − 𝑥 + 2  Fast solution task 3: lim 𝑥→2 3 − 𝑥 2 + 5 3𝑥 − 2 − 𝑥 + 2 = lim 𝑥→2 0 − 2𝑥 2 (2) (3 − 1)(2)(3) = − 4 3

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