摘要 現行高中課本的公式與定理中,有許多的證明過程十份繁複,雖然嚴謹但有些缺乏直觀。本文主要的內容在於如何將課本中的公式與定理,以較淺顯易懂的方式來呈現,並與各版本課本上所使用的證明方式做比較。高中課程中,比較適合以圖形來呈現的單元集中在數列級數與三角函數。 級數求和的公式,大部份都能利用方塊的重新排列,組合成一個新的圖形,來證明等式的成立。 三角函數原本就是以直角三角形的邊角關係出發,因此以圖形來呈現最為貼切。這個部份的證明,主要是利用三角函數來表示線段長或面積,再利用線段長或面積的相等來證明等式的成立。 最後收納了兩個高中很重要的不等式,算幾不等式及柯西不等式,尤其是找到一個比傳統更直觀的算幾不等式證明法。
Abstract There are many formulas and theorems in high school textbooks. The main content of this paper is how to present these formulas and theorems in an obvious way. It contents three parts, summation of series, trigonometric functions, and inequality. Most of the series summation can be shown by rearrangement of squares or cubes. The trigonometric function originally defined by a right-angled triangle, so it is appropriate to present in a graphical form. The last part contains two important inequalities in high school, including Arithmetic-Geometric Mean Inequality and Cauchy Inequalities.