Angles, lengths and area · Foundation · 角度、长度与面积 · 基础
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| scale factor/skeɪl ˈfæktə/ | 相似比 | xiāng sì bǐ |
Does volume scale like length?
- A model has lengths one third of the real object. How much smaller are its area and volume?
- This lesson studies scale factor 相似比: The multiplier that relates corresponding lengths in similar shapes.
体积是否随长度同比例缩放？
- 某模型的尺寸是实物的三分之一。其表面积和体积分别缩小了多少倍？
- 本课学习相似比（scale factor）：关联相似图形对应边长的倍数。
Choose the mathematical structure
- Use angle facts with a stated reason. Similar shapes have equal corresponding angles and proportional corresponding lengths. Areas of rectangles and triangles come from their dimensions; compound shapes can be split into simpler parts.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 利用角度性质并结合理由进行推导。相似图形的对应角相等，对应边成比例。矩形和三角形的面积由其维度决定；组合图形可分割为简单部分求解。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines scale factor? · 以下哪项描述正确定义了比例因子？
The multiplier that relates corresponding lengths in similar shapes. · 关联相似图形对应长度的乘数。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A rectangle of length 8 cm and width 5 cm has area A=LW=40 cm². A triangle on the same base and height has area A=bh/2=20 cm². For a pentagon, the interior-angle sum is (5-2)×180=540°.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
长为8 cm、宽为5 cm的矩形面积为 A=LW=40 cm²。同底等高的三角形面积为 A=bh/2=20 cm²。五边形内角和为 (5-2)×180=540°。
Angles, lengths and area · 角度、长度与面积
Use angle facts with a stated reason · 结合陈述理由使用角度定理
Explain why the triangle has half the rectangle area when base and height match. · 解释为何当底和高相等时，三角形的面积是矩形的一半。
Find the area of a rectangle of sides 8 cm and 5 cm. · 求边长分别为 8 cm 和 5 cm 的矩形的面积。
Rectangle area=8×5=40 cm². · 矩形面积 = 8 × 5 = 40 cm²。
Test a tempting shortcut
- Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A triangle and rectangle with the same base and height have the same area. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 仅凭对应角相等只能判定相似，不能判定全等。请使用顺序对应的边长。在计算面积或体积前，先将长度单位换算一致，或将换算因子的平方/立方正确应用于面积/体积计算。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
同底等高的三角形和矩形面积相等。该说法是错误的。请解释其违反了哪个定义或假设。
Find the area of a triangle of base 8 cm and height 5 cm. · 求底为 8 cm、高为 5 cm 的三角形的面积。
Triangle area=8×5/2=20 cm². · 三角形面积 = 8 × 5 ÷ 2 = 20 cm²。
A triangle and rectangle with the same base and height have the same area. · 同底等高的三角形与矩形具有相同的面积。
Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly. · 仅相等的角只能确定相似性，不能确定大小。需按相同顺序使用对应边长。在计算面积或体积前应先转换线性单位，或正确对换算因子进行平方/立方运算。
Interpret a new situation
- Use a labelled sketch and appropriate units. This Foundation/Core lesson does not test area/volume scale factors or advanced circle-theorem proofs.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 使用带标注的草图及适当的单位。本基础/核心课程不考查面积/体积缩放因子或复杂的圆定理证明。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Find the sum of interior angles of a pentagon in degrees. · 求五边形内角和（单位：度）。
Interior-angle sum=(5-2)×180=540°. · 内角和 = (5 - 2) × 180 = 540°。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 8300 · Foundation · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- 8300 · 基础 · 3.4。在布置拓展任务前，请先匹配目标层级和教学大纲要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
关联相似图形对应边长的倍数。请选择关系式，展示计算步骤，检查其适用假设并解读结果。