📚 IB Maths: Casio fx-CG20 Common Mistakes Summary | IB 数学：casio fx-cg20 易错点总结
The Casio fx-CG20 is a powerful graphing calculator widely used in the IB Mathematics exams, especially in Papers 2 and 3. While it can solve complex problems rapidly, small operational oversights often lead to severe mark loss. This article highlights the most common mistakes students make with this calculator and provides practical advice to avoid them.
Casio fx-CG20 是 IB 数学考试（特别是试卷二和试卷三）中广泛使用的图形计算器。虽然它能快速解决复杂问题，但操作上小小的疏忽常常导致严重失分。本文总结了学生使用这台计算器时最常见的错误，并给出实用建议来规避这些陷阱。
1. Angle Mode Errors | 角度模式错误
A classic pitfall: the calculator is in Degree mode when an exam question requires Radians, or vice versa. Trigonometric functions, calculus with trig limits, and complex number polar forms all rely on the correct angle unit. For instance, evaluating sin(π/6) on a Radian-mode machine correctly yields 0.5, but if the CG20 is unwittingly set to Degrees, you will obtain sin(0.5235…) which is completely wrong.
一个经典陷阱：考试要求弧度，计算器却处于角度模式，或反之。三角函数、涉及三角函数的微积分以及复数极坐标形式都依赖于正确的角度单位。例如，在弧度模式下计算 sin(π/6) 会正确给出 0.5，但如果 CG20 不知何时被设成了角度模式，你会得到 sin(0.5235…) 这样一个完全错误的值。
Always check the Status Bar or press SHIFT → SET UP to verify the Angle setting before starting any trigonometric computation. When working with IB Paper 2 questions on trigonometric equations or calculus, Radian mode should be the default.
开始任何三角运算之前，务必查看状态栏或按 SHIFT → SET UP 检查角度设置。做 IB 试卷二的三角方程或微积分题目时，弧度模式应是默认状态。
2. Graph Window Settings | 图形窗口设置
Many students sketch a function only to see a blank screen or a tiny, unrecognisable segment. This happens when the View Window (V-Window) settings such as Xmin, Xmax, Ymin, and Ymax are inappropriate for the function’s range. For example, y = tan x with default window of −3.6 ≤ x ≤ 3.6 and −2.5 ≤ y ≤ 2.5 will display sudden vertical jumps at the asymptotes, which may be misinterpreted as part of the graph.
许多学生绘制函数图像时只看到一个空白屏幕或极小、无法辨认的片段。这是因为视窗（V-Window）设置（如 Xmin、Xmax、Ymin、Ymax）与函数的值域不匹配。例如，y = tan x 在默认窗 −3.6 ≤ x ≤ 3.6、−2.5 ≤ y ≤ 2.5 下会于渐近线处出现突兀的垂直线段，容易误以为是图形的一部分。
Use SHIFT → F3 (V-Window) → F1 (INIT) to reset, then adjust based on the function’s period and asymptotes. For trigonometric graphs, set Xmin and Xmax to cover at least two full periods, and use the F6 (DRAW) menu to auto-scale when needed. Always verify that key points (roots, turning points) are visible before copying the sketch onto your answer sheet.
使用 SHIFT → F3 (V-Window) → F1 (INIT) 进行重置，然后根据函数的周期和渐近线进行调整。绘制三角函数图像时，将 Xmin 和 Xmax 设得至少覆盖两个完整周期，必要时使用 F6 (DRAW) 菜单自动缩放。在将草图搬到答题纸上之前，务必确认关键点（零点、极值点）可见。
3. Equation Solver Pitfalls | 方程求解常见陷阱
The CG20 offers several solving tools, but misusing them can lead to missed solutions. When you press MENU → A: EQUA and choose F3: Solver, the calculator requires an initial guess. If the guess is far from a particular root, the solver may converge to an unintended root or fail entirely. For trigonometric equations with infinite solutions, using the Solver without specifying a domain often returns just one principal value.
CG20 提供多种求解工具，但误用会导致漏解。按 MENU → A: EQUA 并选择 F3: Solver 时，计算器需要一个初始猜测值。若该猜测值离某个根太远，可能会收敛到另一个根甚至求解失败。对于有无限多解的三角方程，未指定定义域就直接使用 Solver 常常只返回一个主值。
For polynomial equations, always use the dedicated Polynomial mode (F1 or F2) instead of the generic Solver; it returns all real and complex roots. For trig equations, first restrict the domain by entering inequalities as conditions in the Solver, or use the GRAPH module to visualise all intersections before solving numerically.
对多项式方程，务必使用专用的 Polynomial 模式（F1 或 F2），而非通用 Solver；它能给出所有实根和复根。对于三角方程，可先在 Solver 中输入不等式条件来限定定义域，或利用 GRAPH 模块可视化所有交点再进行数值求解。
4. Area Calculation vs. Definite Integral | 面积计算与定积分
A very common IB exam mistake is to use the definite integral ∫ f(x)dx directly as the area under a curve, without checking whether f(x) is negative on part of the interval. The integral yields a signed net area, so if a curve crosses the x‑axis, negative and positive contributions cancel out. For example, evaluating ∫₋₁² x³ dx gives a net value of 3.75, but the true enclosed area is ∫₋₁⁰ |x³| dx + ∫₀² x³ dx = 1/4 + 4 = 4.25.
IB 考试中一个极常见的错误是直接将定积分 ∫ f(x)dx 当作曲线下方面积，而没有检查 f(x) 在区间部分是否取负值。积分给出的是带符号的面积净值，因此若曲线穿越 x 轴，正负面积会相互抵消。例如计算 ∫₋₁² x³ dx 得净值为 3.75，但实际包围的面积是 ∫₋₁⁰ |x³| dx + ∫₀² x³ dx = 1/4 + 4 = 4.25。
Always sketch the function using the graph module or analyse its sign manually. Use the integral tool with the absolute value function, or break the integral at x‑intercepts. On the CG20 you can find roots with G-SOLVE → ROOT and then integrate piecewise; the calculator can handle ∫(|f(x)|, a, b) by entering Abs f(x) as the integrand.
务必先用图形模块绘制函数或手动分析其符号。使用绝对值函数的积分工具，或在 x 轴交点处分割积分。在 CG20 上可通过 G-SOLVE → ROOT 找到零点，然后分段积分；也可以直接在积分式中输入 Abs f(x) 来求 ∫(|f(x)|, a, b)。
5. Statistics Data Entry Mistakes | 统计数据输入错误
When inputting data into the Statistics module, students frequently forget to specify a frequency list. By default, the CG20 assumes a frequency of 1 for each data point. If you have a frequency column (e.g. List 2) but do not set it in SET (F2), the calculator will treat all entries in List 1 as individual raw data, leading to wildly incorrect means, medians, and standard deviations.
在统计模块中输入数据时，学生常忘记指定频数列表。CG20 默认每个数据点频数为 1。如果你有频数列（如 List 2）却未在 SET (F2) 中设置，计算器会把 List 1 中的所有条目都当作单个原始数据，导致均值、中位数、标准差等结果完全错误。
Also, beware of unintentional duplicates when entering data manually. After performing a 1‑Variable calculation, always check the sample size n displayed on the screen to verify it matches your expectation. In 2‑Variable statistics, ensure both XList and YList are correctly assigned and that their lengths are equal; otherwise a Dimension Error will appear.
另外，手动输入数据时要警惕无意中造成的重复录入。完成单变量计算后，始终检查屏幕上显示的样本量 n 是否与预期一致。在双变量统计中，要确保 XList 和 YList 分配正确且长度相等，否则会出现维数错误。
6. Complex Number Mode | 复数模式
The CG20 often defaults to ‘Real’ mode. If a student tries to compute √(−1) or solve an equation yielding complex roots while in Real mode, the calculator will return a ‘Non-Real’ error. Even in ‘a+bi’ mode, polar angle display depends on the current angle unit: in Radian mode, the argument is given in radians; in Degree mode, in degrees.
CG20 常默认为 ‘Real’ 模式。若学生处于 Real 模式却试图计算 √(−1) 或求解会产生复根的方程，计算器会报 ‘Non-Real’ 错误。即使在 ‘a+bi’ 模式下，极坐标辐角的显示也取决于当前角度单位：弧度模式下辐角以弧度给出，角度模式下则以度给出。
Always check the Complex Mode setting by pressing SHIFT → SET UP and scrolling down to ‘Complex Mode’. Select ‘a+bi’ for Cartesian form or ‘r∠θ’ for polar form as needed. When working with IB questions on complex numbers (especially roots of unity), set Angle to Radians and Complex Mode to ‘a+bi’ to obtain consistent algebraic expressions.
务必按 SHIFT → SET UP 并下滑至 ‘Complex Mode’ 检查复数模式设置。根据需要选择 ‘a+bi’（笛卡尔形式）或 ‘r∠θ’（极坐标形式）。处理 IB 复数题目（特别是单位根）时，应将角度设为弧度，复数模式设为 ‘a+bi’，以便得到一致的代数表达式。
7. Matrix Dimension Mismatch | 矩阵维数错误
Matrix operations are common in IB Mathematics HL (especially the topic of linear transformations). A ‘Dimension Error’ occurs almost exclusively because the student has defined matrices of incompatible sizes, or has tried to multiply Mat A (m×n) by Mat B (p×q) without ensuring n = p. Another subtle mistake is forgetting that matrices are stored as global variables; an old Mat A from a previous calculation may still reside in memory and silently corrupt a new operation.
矩阵运算在 IB 数学 HL（特别是线性变换专题）中很常见。’Dimension Error’ 几乎总是因为学生定义了尺寸不匹配的矩阵，或在未确保 n = p 的情况下尝试将 Mat A (m×n) 与 Mat B (p×q) 相乘。另一个不易察觉的错误是忘记矩阵已作为全局变量存储；此前计算中留下的旧 Mat A 可能依然在内存中，悄悄破坏新的运算结果。
Before any matrix task, enter the RUN·MAT menu, press OPTN → F2 (MAT) and delete old matrices. Always confirm the dimensions using F6 (DIM). When solving systems of linear equations, use the simultaneous equation solver in the EQUA menu as a safer alternative to manual matrix inversion.
进行任何矩阵操作前，进入 RUN·MAT 菜单，按 OPTN → F2 (MAT) 并删除旧矩阵。务必使用 F6 (DIM) 确认尺寸。求解线性方程组时，使用 EQUA 菜单中的联立方程求解器比手工矩阵求逆更安全。
8. List Operations Gone Wrong | 列表操作失误
In the Statistics, Run-Matrix, and Spreadsheet modules, the CG20’s six built-in list memories (List 1 – List 26) are globally shared. If you do not clear a list before a new task, remnants of previous work can bias regression, generate incorrect summary statistics, or cause ‘List Empty’ errors when you try to access a misremembered name. A common mistake is to use List 1 as data but then reference List 2 without realising it still holds old frequency values from an earlier question.
在统计、运行-矩阵和电子表格模块中，CG20 的六个内置列表存储区（List 1 – List 26）是全局共享的。如果在开始新任务前不清空列表，残留的旧数据会使回归结果产生偏差，生成错误的汇总统计量，或在你试图访问记错的列表名时引发 ‘List Empty’ 错误。一个常见错误是将 List 1 用作数据，然后引用了 List 2，却没意识到它仍然存放着之前题目的旧频数值。
Develop the habit of pressing OPTN → F1 (LIST) → F6 (more) → F4 (ClrList) and specifying the lists you intend to use. Alternatively, reset the entire list memory from the MEMORY menu. When handling large data sets, double‑check references in the SET menu to ensure XList and Freq are pointing to fresh, empty lists.
养成按 OPTN → F1 (LIST) → F6（翻页）→ F4 (ClrList) 并指定将使用的列表的习惯。也可以从 MEMORY 菜单重置全部列表存储。处理大型数据集时，在 SET 菜单中仔细核对引用，确保 XList 和 Freq 指向全新的空列表。
9. Numerical Derivative Misuse | 数值求导误用
The d/dx function on the CG20 (in the RUN·MAT menu) computes a numerical derivative using a small step h. The default tolerance can occasionally produce small rounding errors, especially near points of discontinuity, sharp turns, or when the function oscillates rapidly. Students sometimes blindly trust the output even when the derivative does not exist, or forget that the calculator gives an instantaneous rate of change at a single point, not an algebraic formula for the derivative function.
CG20 的 d/dx 功能（在 RUN·MAT 菜单中）使用一个小步长 h 来计算数值导数。默认容差偶尔会产生小的舍入误差，特别是在间断点、尖点附近或函数快速振荡时。学生有时盲目相信输出值，即使导数根本不存在，或者忘记计算器给出的是单点处的瞬时变化率，而非导函数的代数公式。
For IB questions asking for f'(a), the CG20 is suitable, but always confirm that f(x) is differentiable at x = a by analysing the original function first. If the answer seems unstable, reduce the step increment using OPTN → F2 (CALC) → F4 (d/dx) and then specify a smaller tolerance. Do not use d/dx to wander across an interval; for that, build a table of values or use the graph module’s tangent line feature.
对于要求 f'(a) 的 IB 题目，CG20 是合适的，但务必先分析原函数确认 f(x) 在 x = a 处可导。若结果似乎不稳定，可通过 OPTN → F2 (CALC) → F4 (d/dx) 并指定一个更小的容差来减小步长。不要用 d/dx 漫游整个区间；要观察整体导数行为，应建立数值表或使用图形模块的切线功能。
10. TVM Finance Errors | TVM 金融计算错误
In IB Mathematics: Applications and Interpretation (AI) HL, the Time Value of Money (TVM) solver is essential. The three most persistent errors are: confusing ‘Begin’ vs ‘End’ payment modes (affect annuities due vs ordinary annuities), entering interest rates as percentages instead of decimals, and disregarding the sign convention (cash inflows positive, outflows negative). For example, using PMT as a positive value for both a loan and the subsequent repayment will yield an incorrect N or FV.
在 IB 数学：应用与解释 (AI) HL 中，资金时间价值 (TVM) 求解器不可或缺。三类最常见的错误是：混淆 ‘Begin’ 与 ‘End’ 付款模式（影响期初年金与普通年金）、以百分数而非小数输入利率，以及忽视符号约定（现金流入为正，现金流出为负）。例如，将贷款额和随后的还款额都设为正值 PMT，将得出错误的 N 或 FV。
Open the TVM solver via MENU → F (Financial) → F2: Compound Interest. Always set Payment: End for ordinary annuities unless the question explicitly states payments occur at the beginning of each period. Convert an annual interest rate to the per-period rate before entering I%: if payments are monthly, divide the annual rate by 12. Finally, consistently assign signs: money moving away from you (deposits, loan payments) should be negative; money you receive (loan principal, investment withdrawals) positive.
通过 MENU → F (Financial) → F2: Compound Interest 打开 TVM 求解器。除非题目明确说明付款发生在每期期初，否则普通年金一律设为 Payment: End。在输入 I% 之前将年利率转换为每期利率：若按月付款，则将年利率除以 12。最后，始终如一地分配符号：离开你的钱（存款、偿还贷款）为负；你收到的钱（贷款本金、投资提取）为正。
| Parameter / 参数 | Sign Convention / 符号约定 |
|---|---|
| Loan principal (PV) / 贷款本金 | Positive (you receive money) / 正 |
| Regular payments (PMT) / 定期还款 | Negative (money leaving you) / 负 |
| Future value (FV) of an investment / 投资终值 | Typically positive (you withdraw) / 通常正 |
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