📚 Mathematics G4 Teacher’s Guide | 数学 G4 教师用书
This guide is designed to support Grade 4 mathematics teachers in planning, teaching, and assessing key concepts with confidence. It brings together curriculum expectations, practical strategies, and ready-to-use examples for a full academic year.
本指南旨在支持四年级数学教师自信地规划、教学和评估核心概念。它整合了课程要求、实用策略和现成示例，覆盖整个学年。
1. Course Overview and Learning Objectives | 课程概览与学习目标
Grade 4 mathematics builds on earlier number work and introduces more abstract reasoning with larger numbers, fractions, and geometric properties.
四年级数学建立在早期数感基础上，引入更大数字、分数和几何性质的更抽象推理。
Teachers should map the year into units that balance procedural fluency with conceptual understanding and real-world applications.
教师应将全年划分为若干单元，在程序熟练度、概念理解和实际应用之间取得平衡。
- Core topics: place value to 1,000,000, multi-digit operations, fractions and decimals, geometry, measurement, and data.
- 核心主题：百万以内位值、多位数运算、分数和小数、几何、测量和数据。
Each unit should begin with a diagnostic check to identify prior knowledge and end with a summative task linked to the learning objectives.
每个单元应以诊断性检查开始，找出已有知识，并以与学习目标相关的总结性任务结束。
2. Number Sense and Place Value | 数感与位值
Grade 4 students should read, write, compare, and round whole numbers up to 1,000,000.
四年级学生应能读、写、比较和取整到 1,000,000 以内的整数。
Use place-value charts and base-ten blocks to show that each digit is ten times the value of the digit to its right.
使用位值表和十进制积木展示每个数位是其右边数位价值的十倍。
Key notation: 245,306 = 200,000 + 40,000 + 5,000 + 300 + 6.
关键记法：245,306 = 200,000 + 40,000 + 5,000 + 300 + 6。
Rounding to the nearest 10, 100, or 1,000 should be practised on number lines before moving to abstract rules.
取整到最近的 10、100 或 1,000 应先在数轴上练习，再过渡到抽象规则。
3. Addition and Subtraction Strategies | 加法与减法策略
Students should fluently add and subtract multi-digit numbers using the standard algorithm and explain regrouping.
学生应使用标准竖式熟练进行多位数加减法，并能解释进位和退位。
Encourage estimation first: for 6,482 + 3,905, a reasonable estimate is 6,500 + 3,900 = 10,400.
鼓励先估算：对于 6,482 + 3,905，合理估算为 6,500 + 3,900 = 10,400。
Word problems should include compare, part-whole, and change structures so pupils learn to choose the correct operation.
应用题应包含比较、部分-整体和变化结构，让学生学会选择正确运算。
4. Multiplication and Division Foundations | 乘法与除法基础
Grade 4 focuses on multiplying up to four-digit numbers by one-digit numbers and two-digit by two-digit numbers.
四年级重点包括四位数乘一位数以及两位数乘两位数。
Use area models and partial products before the standard algorithm. For 27 × 34, show 27 × 30 = 810 and 27 × 4 = 108, so 810 + 108 = 918.
在标准算法前使用面积模型和部分积。例如 27 × 34，展示 27 × 30 = 810 和 27 × 4 = 108，因此 810 + 108 = 918。
Division work includes finding whole-number quotients and remainders, with models such as sharing and grouping.
除法包括求整数商和余数，并用平均分和分组模型展示。
5. Fractions and Decimals | 分数与小数
Students compare fractions with different numerators and denominators, generate equivalent fractions, and add/subtract fractions with like denominators.
学生比较不同分子和分母的分数，生成等值分数，并进行同分母分数的加减。
Use fraction strips and number lines to show that 1/2 = 2/4 = 3/6 and that 3/8 + 2/8 = 5/8.
使用分数条和数轴展示 1/2 = 2/4 = 3/6，以及 3/8 + 2/8 = 5/8。
Decimal work introduces tenths and hundredths: 0.6 = 6/10 and 0.25 = 25/100.
小数学习引入十分位和百分位：0.6 = 6/10，0.25 = 25/100。
6. Geometry: Shapes, Angles, and Symmetry | 几何：图形、角与对称
Students classify shapes by their properties, including parallel and perpendicular lines, and identify right, acute, and obtuse angles.
学生根据性质对图形分类，包括平行线和垂直线，并识别直角、锐角和钝角。
Symmetry work involves drawing lines of symmetry and recognising that a square has four lines of symmetry while a rectangle has two.
对称学习包括画对称轴，并认识正方形有四条对称轴，而长方形有两条。
Angle notation such as ∠ABC and the degree symbol ° should be introduced informally through turns and protractors.
角记法如 ∠ABC 和度数符号 ° 应通过旋转和量角器以非正式方式引入。
7. Measurement and Data Handling | 测量与数据处理
Grade 4 covers length, mass, volume, time, and money within the metric system, including conversions such as 1 km = 1,000 m.
四年级覆盖公制中的长度、质量、体积、时间和货币，包括换算如 1 千米 = 1,000 米。
Area and perimeter are distinguished: perimeter is the distance around a shape; area is the number of square units inside it.
区分周长和面积：周长是图形外圈的长度；面积是图形内部的平方单位数。
Data work includes creating and interpreting line plots, bar graphs, and picture graphs, with attention to scale and labels.
数据学习包括制作和解读折线图、条形图和象形图，注意刻度和标签。
8. Problem Solving and Mathematical Reasoning | 问题解决与数学推理
Teachers should model a four-step approach: understand the problem, devise a plan, carry out the plan, and check the answer.
教师应示范四步法：理解问题、制定计划、执行计划和检查答案。
Encourage multiple strategies, such as drawing a diagram, making a table, working backwards, or solving a simpler problem.
鼓励多种策略，如画图、列表、倒推或先解决更简单的问题。
Reasoning tasks should ask pupils to justify why a statement is always, sometimes, or never true, using mathematical language.
推理任务应要求学生使用数学语言判断一个陈述是总是、有时还是从不成立，并说明理由。
9. Differentiated Instruction for Mixed Abilities | 差异化教学
Plan three tiers of tasks: support with concrete manipulatives, core with visual representations, and extension with abstract challenges.
规划三层任务：支持层使用具体教具，核心层使用视觉表征，拓展层进行抽象挑战。
For struggling learners, reduce the number of steps and provide worked examples; for advanced learners, pose open-ended investigations.
对学习困难的学生，减少步骤并提供范例；对学有余力的学生，提出开放式探究任务。
Use flexible grouping so students can move between groups based on daily formative assessment rather than fixed labels.
采用灵活分组，让学生根据日常形成性评估在各组之间流动，而不是固定标签。
10. Assessment and Feedback | 评估与反馈
Formative assessment should be ongoing through observation, questioning, exit tickets, and mini whiteboards.
形成性评估应持续进行，通过观察、提问、出门票和迷你白板实现。
Summative assessments should align with learning objectives and include both fluency items and multi-step problems.
总结性评估应与学习目标一致，包括熟练度题目和多步问题。
Feedback should be specific and actionable: instead of ‘good work’, write ‘you correctly renamed 3/4 as 6/8; next try ordering mixed fractions on a number line.’
反馈应具体且可操作：不要写’做得好’，而应写’你正确地将 3/4 转化为 6/8；接下来尝试在数轴上排列带分数’。
11. Using Manipulatives and Visual Models | 使用教具与视觉模型
Concrete tools such as base-ten blocks, fraction circles, geoboards, and number lines bridge hands-on experience to abstract symbols.
十进制积木、分数圆、几何板和数轴等具体工具在动手经验和抽象符号之间架起桥梁。
Visual models include bar models for word problems, area models for multiplication, and tape diagrams for fractions.
视觉模型包括应用题的条形模型、乘法的面积模型和分数的带条图。
For example, a bar model for ‘Sam has 45 stickers; he gives 1/3 to a friend’ shows the whole divided into three equal parts of 15.
例如，’山姆有 45 张贴纸；他给了朋友 1/3’的条形模型将整体分为三个相等的部分，每部分 15。
12. Home-School Connection and Digital Tools | 家校连接与数字工具
Share unit overviews and maths vocabulary with families so they can support practice at home without introducing conflicting methods.
与家庭分享单元概览和数学词汇，使他们能在家支持练习，而不会引入冲突的方法。
Digital tools such as virtual manipulatives, interactive number lines, and quiz apps can supplement but not replace hands-on learning.
虚拟教具、交互式数轴和测验应用等数字工具可以补充但不能取代动手学习。
Set short, focused home tasks that reinforce fluency, such as 10 minutes of multiplication facts or a real-life measurement challenge.
布置短小集中的家庭任务以巩固熟练度，例如 10 分钟乘法口诀练习或一个实际测量挑战。
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