📚 Review Set 3A - Non-Calculator: Essential IB Maths Strategies | 复习题集 3A - 无计算器：IB 数学核心策略
The IB Mathematics non-calculator paper demands fluency, precision and a strong conceptual foundation. Review Set 3A – Non-Calculator collects exercises that target the most crucial techniques you must perform manually, from simplifying algebraic fractions to integrating polynomial functions. Mastering this set not only builds speed and accuracy but also deepens your understanding of why mathematical procedures work. Below, we explore the main topic areas and share effective strategies to tackle each question type.
IB 数学无计算器试卷对运算流畅度、精确性和扎实的概念基础都有很高要求。复习题集 3A – 无计算器汇总了必须手动完成的最关键技巧的练习题，从化简代数分式到对多项式函数积分。掌握这套题集不仅能提高速度和准确率，还能加深你对数学运算过程背后原因的理解。下面我们将逐一探讨主要专题领域，并分享应对各类题型的有效策略。
1. Simplifying Algebraic Expressions | 化简代数表达式
Start by collecting like terms carefully. For example, 3x + 2y – fies to 2x + 6y. When expanding brackets, apply the distributive law methodically: (x+3)(x–2) = x² – 2x + 3x – 6 = x² + x – 6. Watch for sign errors – p to avoid losing negative signs, especially when subtracting a bracketed expression: -(2x–5) becomes -2x + 5.
首先仔细合并同类项。例如，3x + 2y – x + 4y 化简为 2x + 6y。展开括号时，有条理地应用分配律：(x+3)(x–2) = x² – 2x + 3x – 6 = x² + x – 6。注意符号错误——这是无计算器运算中最常见的错误。务必写出每个中间步骤，以避免丢失负号，尤其是在减去一个括号表达式时：-(2x–5) 变为 -2x + 5。
For rational expressions, factorise both numerator and denominator first. E.g., (x²–4)/(x–2) = ((x–2)(x+2))/(x– = x+2, provided x ≠ 2. Cancelling common factors is only valid when the factor is not zero – ≠ 0. Remember, missing the domain condition can lose marks even if the algebra is correct.
对于有理式，先对分子分母进行因式分解。例如，(x²–4)/(x–2) = ((x–2)(x+2))/(x–2) = x+2，前提是 x ≠ 2。约去公因式只有在该因式不为零时才有效——务必标明定义域限制。另一个典型化简是 (x²+3x)/(x) = x+3，x ≠ 0。记住，即使代数运算正确，缺少定义域条件也会失分。
2. Solving Linear and Quadratic Equations by Hand | 手算求解一次与二次方程
For linear equations, systematically isolate the variable. With 2(3x–1) = 4x+6, first expand: 6x– hen 2x = 8 → x = 4. Check your solution by substituting it back into the original equation – le non-calculator habit. Avoid skipping steps; write the balancing operation on each line.
求解一次方程时，要系统地将变量分离。对于 2(3x–1) = 4x+6，先展开：6x–2 = 4x+6，然后 2x = 8 → x = 4。将解代回原方程检验——这是一个可靠的无计算器习惯。不要跳步，在每一行写出等式两侧的平衡运算。
Quadratic equations should first be approached by factorisation. Example: x²–5x+6=0 → (x–2)(x–3)=0 → x=2 or x=3. If factors are not obvious, complete the square: x²–4x–5=0 → (x–2)² – 9 = 0 → (x–2)² = 9 → x–2 = ±3 → x = 5 or –1. The quadratic formula x = (–b ± √b²–4ac) / (2a) must be memorised and used when factorisation is impractical. Practice handling the discriminant without a calculator, simplifying square roots where possible, e.g., √49 = 7, √48 = 4√3. For simultaneous equations, use elimination: 2x+y=7 and x–y=2 → add to get 3x=9 → x=3, y=1. Always verify both original equations.
二次方程应首先尝试因式分解。例如：x²–5x+6=0 → (x–2)(x–3)=0 → x=2 或 x=3。如果因式不明显，可用配方法：x²–4x–5=0 → (x–2)² – 9 = 0 → (x–2)² = 9 → x–2 = ±3 → x = 5 或 –1。二次公式 x = (–b ± √b²–4ac) / (2a) 必须熟记，并在无法因式分解时使用。练习在无计算器条件下处理判别式，尽可能化简平方根，例如 √49 = 7，√48 = 4√3。对于联立方程，用消元法：2x+y=7 与 x–y=2 → 相加得 3x=9 → x=3，y=1。务必用两个原方程进行验证。
3. Manipulating Surds and Rationalizing Denominators | 根
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