250 lines
2.5 MiB
250 lines
2.5 MiB
<html>
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<head>
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<meta charset="utf-8">
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<script src="lib/bindings/utils.js"></script>
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<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/vis-network/9.1.2/dist/dist/vis-network.min.css" integrity="sha512-WgxfT5LWjfszlPHXRmBWHkV2eceiWTOBvrKCNbdgDYTHrT2AeLCGbF4sZlZw3UMN3WtL0tGUoIAKsu8mllg/XA==" crossorigin="anonymous" referrerpolicy="no-referrer" />
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<script src="https://cdnjs.cloudflare.com/ajax/libs/vis-network/9.1.2/dist/vis-network.min.js" integrity="sha512-LnvoEWDFrqGHlHmDD2101OrLcbsfkrzoSpvtSQtxK3RMnRV0eOkhhBN2dXHKRrUU8p2DGRTk35n4O8nWSVe1mQ==" crossorigin="anonymous" referrerpolicy="no-referrer"></script>
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<center>
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<h1></h1>
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</center>
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<!-- <link rel="stylesheet" href="../node_modules/vis/dist/vis.min.css" type="text/css" />
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<script type="text/javascript" src="../node_modules/vis/dist/vis.js"> </script>-->
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<link
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href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.0-beta3/dist/css/bootstrap.min.css"
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rel="stylesheet"
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integrity="sha384-eOJMYsd53ii+scO/bJGFsiCZc+5NDVN2yr8+0RDqr0Ql0h+rP48ckxlpbzKgwra6"
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crossorigin="anonymous"
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/>
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<script
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src="https://cdn.jsdelivr.net/npm/bootstrap@5.0.0-beta3/dist/js/bootstrap.bundle.min.js"
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integrity="sha384-JEW9xMcG8R+pH31jmWH6WWP0WintQrMb4s7ZOdauHnUtxwoG2vI5DkLtS3qm9Ekf"
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crossorigin="anonymous"
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></script>
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<center>
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<h1></h1>
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</center>
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<style type="text/css">
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#mynetwork {
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width: 100%;
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height: 100vh;
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background-color: #ffffff;
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border: 1px solid lightgray;
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position: relative;
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float: left;
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}
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#loadingBar {
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position:absolute;
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top:0px;
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left:0px;
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width: 100%;
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height: 100vh;
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background-color:rgba(200,200,200,0.8);
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-webkit-transition: all 0.5s ease;
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-moz-transition: all 0.5s ease;
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-ms-transition: all 0.5s ease;
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-o-transition: all 0.5s ease;
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transition: all 0.5s ease;
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opacity:1;
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}
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#bar {
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position:absolute;
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top:0px;
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left:0px;
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width:20px;
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height:20px;
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margin:auto auto auto auto;
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border-radius:11px;
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border:2px solid rgba(30,30,30,0.05);
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background: rgb(0, 173, 246); /* Old browsers */
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box-shadow: 2px 0px 4px rgba(0,0,0,0.4);
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}
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#border {
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position:absolute;
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top:10px;
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left:10px;
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width:500px;
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height:23px;
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margin:auto auto auto auto;
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box-shadow: 0px 0px 4px rgba(0,0,0,0.2);
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border-radius:10px;
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#text {
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position:absolute;
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top:8px;
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left:530px;
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width:30px;
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height:50px;
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margin:auto auto auto auto;
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font-size:22px;
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color: #000000;
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}
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div.outerBorder {
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position:relative;
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top:400px;
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width:600px;
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height:44px;
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margin:auto auto auto auto;
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border:8px solid rgba(0,0,0,0.1);
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background: rgb(252,252,252); /* Old browsers */
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background: -moz-linear-gradient(top, rgba(252,252,252,1) 0%, rgba(237,237,237,1) 100%); /* FF3.6+ */
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background: -webkit-gradient(linear, left top, left bottom, color-stop(0%,rgba(252,252,252,1)), color-stop(100%,rgba(237,237,237,1))); /* Chrome,Safari4+ */
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background: -webkit-linear-gradient(top, rgba(252,252,252,1) 0%,rgba(237,237,237,1) 100%); /* Chrome10+,Safari5.1+ */
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background: -o-linear-gradient(top, rgba(252,252,252,1) 0%,rgba(237,237,237,1) 100%); /* Opera 11.10+ */
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background: -ms-linear-gradient(top, rgba(252,252,252,1) 0%,rgba(237,237,237,1) 100%); /* IE10+ */
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background: linear-gradient(to bottom, rgba(252,252,252,1) 0%,rgba(237,237,237,1) 100%); /* W3C */
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filter: progid:DXImageTransform.Microsoft.gradient( startColorstr='#fcfcfc', endColorstr='#ededed',GradientType=0 ); /* IE6-9 */
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border-radius:72px;
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box-shadow: 0px 0px 10px rgba(0,0,0,0.2);
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}
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</style>
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</head>
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<body>
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<div class="card" style="width: 100%">
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<div id="mynetwork" class="card-body"></div>
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</div>
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<div id="loadingBar">
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<div class="outerBorder">
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<div id="text">0%</div>
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<div id="border">
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<div id="bar"></div>
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</div>
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</div>
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</div>
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<script type="text/javascript">
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// initialize global variables.
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var edges;
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var nodes;
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var allNodes;
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var allEdges;
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var nodeColors;
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var originalNodes;
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var network;
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var container;
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var options, data;
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var filter = {
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item : '',
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property : '',
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value : []
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};
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// This method is responsible for drawing the graph, returns the drawn network
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function drawGraph() {
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var container = document.getElementById('mynetwork');
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// parsing and collecting nodes and edges from the python
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It represents a fundamental approach to statistical analysis through data examination, where data is used to derive meaningful information and make informed inferences. \n\nThis concept is a central requirement in the **\"\u7edf\u8ba1\u4e0e\u6982\u7387\" (Statistics and Probability)** section of the curriculum standards, highlighting the importance of understanding statistical thinking through data analysis. **\u6570\u636e\u5206\u6790\u89c2\u5ff5** emphasizes the significance of data in statistical studies, including the recognition that data contains information, analysis methods can vary, and conclusions often carry a degree of randomness. 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These standards serve as official frameworks outlining what students should learn at different stages of their primary education. \n\nKey components of the curriculum standards include: \n1. **Core Mathematical Concepts**: The standards emphasize foundational topics such as number sense, basic arithmetic operations (addition, subtraction, multiplication, and division), and the understanding of numbers up to 10,000. \n2. **Learning Objectives**: They specify competency requirements, such as students in the first\u5b66\u6bb5 (early primary grades) needing to grasp the meaning of arithmetic operations and apply them to real-life situations. \n3. **Pedagogical Approaches**: The standards guide how mathematical concepts should be taught, including the development of **\u7b26\u53f7\u610f\u8bc6 (symbolic reasoning)** as a core competency and the introduction of **model thinking** in mathematics. \n4. **Progression and Limitations**: To ensure age-appropriate learning, the standards restrict the complexity of problems\u2014for example, limiting **\u6df7\u5408\u8fd0\u7b97 (mixed operations)** to two or three steps in early and middle primary grades. \n5. **Expanded Competencies**: Beyond traditional **\u2018\u53cc\u57fa\u2019 (basic knowledge and fundamental skills)**, the standards now incorporate **\u2018\u56db\u57fa\u2019**, adding **basic mathematical thinking and experiential learning (basic ideas and basic activity experiences)**. \n\nThese curriculum standards are designed to ensure a structured and comprehensive mathematics education, aligning classroom instruction with national expectations for student development.", "entity_id": "\u8bfe\u7a0b\u6807\u51c6", "entity_type": "category", "file_path": "unknown_source", "id": "\u8bfe\u7a0b\u6807\u51c6", "label": "\u8bfe\u7a0b\u6807\u51c6", "shape": "dot", "size": 10, "source_id": "chunk-bea5957e092b2067f59cd5a861f518e6\u003cSEP\u003echunk-6bbe77567ec618aec822acfe88217048\u003cSEP\u003echunk-112d4fcad80ef84a1327a06d05614630\u003cSEP\u003echunk-7ac9e27076640b89379ec7305a979523\u003cSEP\u003echunk-3e1177b5ab038ad131780009f61ea1d6\u003cSEP\u003echunk-cf30e3e5c182214d0f0eb020814d103d\u003cSEP\u003echunk-612d665e8236ed57ac928a378892167b\u003cSEP\u003echunk-15a3035e48a3f9ba98ec01cb3ed42052\u003cSEP\u003echunk-c737189a394d95d480e7c28933d632d2\u003cSEP\u003echunk-a5fa437b38c82cb10c1cbb6ae90f2d6c\u003cSEP\u003echunk-1ad150a353733dc35455650b97cce260", "title": "The **\u8bfe\u7a0b\u6807\u51c6 (Curriculum Standards)** are national educational guidelines established by the Chinese education authorities to define the objectives, content, and pedagogical expectations for mathematics education in primary schools. These standards serve as official frameworks outlining what students should learn at different stages of their primary education. \n\nKey components of the curriculum standards include: \n1. **Core Mathematical Concepts**: The standards emphasize foundational topics such as number sense, basic arithmetic operations (addition, subtraction, multiplication, and division), and the understanding of numbers up to 10,000. \n2. **Learning Objectives**: They specify competency requirements, such as students in the first\u5b66\u6bb5 (early primary grades) needing to grasp the meaning of arithmetic operations and apply them to real-life situations. \n3. **Pedagogical Approaches**: The standards guide how mathematical concepts should be taught, including the development of **\u7b26\u53f7\u610f\u8bc6 (symbolic reasoning)** as a core competency and the introduction of **model thinking** in mathematics. \n4. **Progression and Limitations**: To ensure age-appropriate learning, the standards restrict the complexity of problems\u2014for example, limiting **\u6df7\u5408\u8fd0\u7b97 (mixed operations)** to two or three steps in early and middle primary grades. \n5. **Expanded Competencies**: Beyond traditional **\u2018\u53cc\u57fa\u2019 (basic knowledge and fundamental skills)**, the standards now incorporate **\u2018\u56db\u57fa\u2019**, adding **basic mathematical thinking and experiential learning (basic ideas and basic activity experiences)**. \n\nThese curriculum standards are designed to ensure a structured and comprehensive mathematics education, aligning classroom instruction with national expectations for student development."}, {"color": "#43a724", "created_at": 1752674357, "description": "**Quantity** is a fundamental concept in mathematics that represents the abstraction of real-world quantities, serving as the basis for numerical relationships and operations. It refers to concrete measurements or counts, such as the number of objects (e.g., peaches picked by individuals like *\u54e5\u54e5* or *\u59b9\u59b9*), which natural numbers abstract away from in mathematical contexts. The concept encompasses both the measurable amount of something and the broader idea of magnitude, including notions like \"distance\" or \"absolute value.\" By capturing the essence of \"more\" and \"less\" (\u591a\u548c\u5c11), quantity forms the foundation for understanding number relationships, addition, and the formation of numerical concepts. It bridges the gap between real-world observations and mathematical abstraction, making it central to quantitative reasoning.", "entity_id": "\u6570\u91cf", "entity_type": "category", "file_path": "unknown_source", "id": "\u6570\u91cf", "label": "\u6570\u91cf", "shape": "dot", "size": 10, "source_id": "chunk-26eeca6452bef6e940004dc39e11bc99\u003cSEP\u003echunk-2b803f52811c47a36d9c3c7c9900a33c\u003cSEP\u003echunk-4ad0fa3c80600287487ea0909ce52c30\u003cSEP\u003echunk-612d665e8236ed57ac928a378892167b\u003cSEP\u003echunk-38e4aebbd0700df1af93164aa3540712\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde", "title": "**Quantity** is a fundamental concept in mathematics that represents the abstraction of real-world quantities, serving as the basis for numerical relationships and operations. It refers to concrete measurements or counts, such as the number of objects (e.g., peaches picked by individuals like *\u54e5\u54e5* or *\u59b9\u59b9*), which natural numbers abstract away from in mathematical contexts. The concept encompasses both the measurable amount of something and the broader idea of magnitude, including notions like \"distance\" or \"absolute value.\" By capturing the essence of \"more\" and \"less\" (\u591a\u548c\u5c11), quantity forms the foundation for understanding number relationships, addition, and the formation of numerical concepts. It bridges the gap between real-world observations and mathematical abstraction, making it central to quantitative reasoning."}, {"color": "#11a66a", "created_at": 1752674367, "description": "**Natural Numbers (\u81ea\u7136\u6570)** \n\nNatural numbers are a foundational mathematical concept representing quantities used for counting and ordering. They form the basis of arithmetic and number theory, with their formal definition established through the **Peano axioms**, which introduce the concept of a *successor* to rigorously define their properties. \n\nDepending on the context, natural numbers may start from **0 or 1**, forming either the set of **non-negative integers (0, 1, 2, 3, ...)** or **positive integers (1, 2, 3, ...)**. They are central to discussions in number theory, particularly in the study of primes, and serve as the foundation for constructing more advanced number systems, such as integers, rational numbers, and real numbers. \n\nAs abstract mathematical entities, natural numbers generalize patterns and operations, making them essential in symbolic reasoning and mathematical education. They are often contrasted with negative numbers in teaching to help clarify numerical concepts. Additionally, natural numbers are the primary objects of **mathematical induction** and are fundamental in defining basic arithmetic operations, such as addition. \n\nIn summary, natural numbers are a core concept in mathematics, characterized by their axiomatic definition, counting utility, and role in structuring more complex number systems. Their exact definition (whether including 0 or not) varies by context but remains universally recognized as the simplest and most fundamental set of numbers.", "entity_id": "\u81ea\u7136\u6570", "entity_type": "category", "file_path": "unknown_source", "id": "\u81ea\u7136\u6570", "label": "\u81ea\u7136\u6570", "shape": "dot", "size": 10, "source_id": "chunk-a9a60418b119b40b9f7e09f08cc0b5c8\u003cSEP\u003echunk-26eeca6452bef6e940004dc39e11bc99\u003cSEP\u003echunk-be1ccf4aaa693e78676f243a79b63a08\u003cSEP\u003echunk-baa5d2235a16fa56fc40519b58791c47\u003cSEP\u003echunk-ded529b804f8d3a77fb3e0a6afc9937d\u003cSEP\u003echunk-ed3f9d4ba17af9bb577054436e6b8e53\u003cSEP\u003echunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-2b803f52811c47a36d9c3c7c9900a33c\u003cSEP\u003echunk-a90ac5a1f21c2588adf853f55169b640\u003cSEP\u003echunk-038b9abb5eb56adcf710a719f2123147\u003cSEP\u003echunk-1947ec0bba5d164b6d83fc97bec183ff\u003cSEP\u003echunk-73836e98e177b5a39ff62e469b1507bc\u003cSEP\u003echunk-38e4aebbd0700df1af93164aa3540712\u003cSEP\u003echunk-77809f023af3e8d2fb1e4c6cea56c329\u003cSEP\u003echunk-5c564427093279bdf3c25848196fe5f8\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde\u003cSEP\u003echunk-93f0e906d06f6dbc8b9407822495a6d7", "title": "**Natural Numbers (\u81ea\u7136\u6570)** \n\nNatural numbers are a foundational mathematical concept representing quantities used for counting and ordering. They form the basis of arithmetic and number theory, with their formal definition established through the **Peano axioms**, which introduce the concept of a *successor* to rigorously define their properties. \n\nDepending on the context, natural numbers may start from **0 or 1**, forming either the set of **non-negative integers (0, 1, 2, 3, ...)** or **positive integers (1, 2, 3, ...)**. They are central to discussions in number theory, particularly in the study of primes, and serve as the foundation for constructing more advanced number systems, such as integers, rational numbers, and real numbers. \n\nAs abstract mathematical entities, natural numbers generalize patterns and operations, making them essential in symbolic reasoning and mathematical education. They are often contrasted with negative numbers in teaching to help clarify numerical concepts. Additionally, natural numbers are the primary objects of **mathematical induction** and are fundamental in defining basic arithmetic operations, such as addition. \n\nIn summary, natural numbers are a core concept in mathematics, characterized by their axiomatic definition, counting utility, and role in structuring more complex number systems. 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Historically, it was developed in ancient times to express larger natural numbers, as seen in systems like the Chinese ten, hundred, thousand, and the Roman numerals X, C, M.", "entity_id": "\u5341\u8fdb\u5236", "entity_type": "category", "file_path": "unknown_source", "id": "\u5341\u8fdb\u5236", "label": "\u5341\u8fdb\u5236", "shape": "dot", "size": 10, "source_id": "chunk-26eeca6452bef6e940004dc39e11bc99\u003cSEP\u003echunk-6bbe77567ec618aec822acfe88217048\u003cSEP\u003echunk-2b803f52811c47a36d9c3c7c9900a33c\u003cSEP\u003echunk-f9fb4bf24eb3450dc8d14d30db6e18ae\u003cSEP\u003echunk-73836e98e177b5a39ff62e469b1507bc\u003cSEP\u003echunk-64dec23c12f739009609c53fc1b55511", "title": "The decimal system (\u5341\u8fdb\u5236) is a widely used base-10 numeral system that enables the representation of all natural numbers and decimals using ten symbols (0\u20139) in a positional notation. It is believed to have originated from the counting of human fingers and plays a fundamental role in understanding place value, numerical relationships, and everyday mathematical operations. The system uses powers of ten to unify the notation of both integers and decimal numbers, making it essential for arithmetic and numerical representation across various cultures. Historically, it was developed in ancient times to express larger natural numbers, as seen in systems like the Chinese ten, hundred, thousand, and the Roman numerals X, C, M."}, {"color": "#c90f43", "created_at": 1752674427, "description": "Mathematics teachers are educators who must understand abstract concepts like natural number definitions and logical relationships to teach effectively.\u003cSEP\u003e\u6570\u5b66\u6559\u5e08refers to educators specializing in teaching mathematics, particularly emphasizing the importance of understanding axiomatic structures for professional development.\u003cSEP\u003e\u6570\u5b66\u6559\u5e08\u662f\u4ece\u4e8b\u6570\u5b66\u6559\u5b66\u7684\u4e13\u4e1a\u4eba\u5458\uff0c\u9700\u8981\u7406\u89e3\u62bd\u8c61\u6570\u5b66\u7ed3\u6784\u4ee5\u66f4\u597d\u5730\u8fdb\u884c\u6559\u5b66\u3002", "entity_id": "\u6570\u5b66\u6559\u5e08", "entity_type": "person", "file_path": "unknown_source", "id": "\u6570\u5b66\u6559\u5e08", "label": "\u6570\u5b66\u6559\u5e08", 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"Understanding of mathematical symbols representing concepts and relationships.\u003cSEP\u003e\u7b26\u53f7\u610f\u8bc6is a core concept in mathematics education, focusing on the use of symbols to represent numbers, relationships, and patterns.\u003cSEP\u003e\u7b26\u53f7\u610f\u8bc6refers to understanding mathematical notation and abstraction.\u003cSEP\u003e\u7b26\u53f7\u610f\u8bc6\u662f\u6307\u5c0f\u5b66\u751f\u80fd\u591f\u7406\u89e3\u5e76\u8fd0\u7528\u7b26\u53f7\u8868\u793a\u6570\u3001\u6570\u91cf\u5173\u7cfb\u548c\u53d8\u5316\u89c4\u5f8b\uff0c\u662f\u6570\u5b66\u8868\u8fbe\u548c\u601d\u8003\u7684\u91cd\u8981\u5f62\u5f0f. It is a core concept introduced in the curriculum standards for mathematics education.", "entity_id": "\u7b26\u53f7\u610f\u8bc6", "entity_type": "category", "file_path": "unknown_source", "id": "\u7b26\u53f7\u610f\u8bc6", "label": "\u7b26\u53f7\u610f\u8bc6", "shape": "dot", "size": 10, "source_id": "chunk-a90ac5a1f21c2588adf853f55169b640\u003cSEP\u003echunk-a9a60418b119b40b9f7e09f08cc0b5c8\u003cSEP\u003echunk-15a3035e48a3f9ba98ec01cb3ed42052\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde", "title": "Understanding of mathematical symbols representing concepts and relationships.\u003cSEP\u003e\u7b26\u53f7\u610f\u8bc6is a core concept in mathematics education, focusing on the use of symbols to represent numbers, relationships, and patterns.\u003cSEP\u003e\u7b26\u53f7\u610f\u8bc6refers to understanding mathematical notation and abstraction.\u003cSEP\u003e\u7b26\u53f7\u610f\u8bc6\u662f\u6307\u5c0f\u5b66\u751f\u80fd\u591f\u7406\u89e3\u5e76\u8fd0\u7528\u7b26\u53f7\u8868\u793a\u6570\u3001\u6570\u91cf\u5173\u7cfb\u548c\u53d8\u5316\u89c4\u5f8b\uff0c\u662f\u6570\u5b66\u8868\u8fbe\u548c\u601d\u8003\u7684\u91cd\u8981\u5f62\u5f0f. It is a core concept introduced in the curriculum standards for mathematics education."}, {"color": "#63d927", "created_at": 1752674438, "description": "**Comprehensive Summary of \u5c0f\u5b66\u6570\u5b66\u6559\u5b66 (Primary School Mathematics Education):** \n\n\u5c0f\u5b66\u6570\u5b66\u6559\u5b66 refers to the foundational stage of mathematics education aimed at primary school students, focusing on the development of basic mathematical knowledge, conceptual understanding, and problem-solving skills. It emphasizes intuitive learning, real-world applications, and the gradual abstraction of mathematical ideas. \n\nKey aspects of \u5c0f\u5b66\u6570\u5b66\u6559\u5b66 include: \n- **Conceptual Foundations:** The curriculum helps students grasp fundamental concepts such as numbers (e.g., natural numbers through correspondence methods), basic operations (e.g., fraction division), and Euclidean geometry (e.g., points, lines, angles, and shapes), using descriptive and intuitive definitions rather than abstract formalisms like the Peano axioms. \n- **Contextual Learning:** Teachers employ real-world scenarios and models (e.g., geometric or physical models) to illustrate mathematical principles, ensuring students connect theory to practical applications. \n- **Staged Progression:** \n - **First Stage (Lower Grades):** Focuses on selecting appropriate units for estimation problems and establishing number-quantity relationships. \n - **Second Stage (Upper Grades):** Shifts toward method selection for problem-solving and deeper conceptual analysis, including direct judgment of mathematical propositions. \n- **Skill Development:** The pedagogy cultivates logical thinking, abstraction (e.g., via situational activities), and model-based reasoning, while avoiding overly complex theoretical content unsuitable for the age group. \n\nIn summary, \u5c0f\u5b66\u6570\u5b66\u6559\u5b66 serves as a bridge between concrete experiences and formal mathematics, prioritizing accessibility, relevance, and foundational mastery for young learners.", "entity_id": "\u5c0f\u5b66\u6570\u5b66\u6559\u5b66", "entity_type": "category", "file_path": "unknown_source", "id": "\u5c0f\u5b66\u6570\u5b66\u6559\u5b66", "label": "\u5c0f\u5b66\u6570\u5b66\u6559\u5b66", "shape": "dot", "size": 10, "source_id": "chunk-65ad5b0f72cf332d5f39730b4710be0d\u003cSEP\u003echunk-26eeca6452bef6e940004dc39e11bc99\u003cSEP\u003echunk-cceb7fc5191a1b7132a11703466c415d\u003cSEP\u003echunk-a499d10f1019c6d5d99c36d7402ec197\u003cSEP\u003echunk-c35e1de6659c67c045676cc40b16dbb4\u003cSEP\u003echunk-73836e98e177b5a39ff62e469b1507bc\u003cSEP\u003echunk-da314816f469d2b8634730d486609229\u003cSEP\u003echunk-ce575de1a4a5a428b616adecd09f8d8f\u003cSEP\u003echunk-26076d8e62c6fb21010095bef1275278", "title": "**Comprehensive Summary of \u5c0f\u5b66\u6570\u5b66\u6559\u5b66 (Primary School Mathematics Education):** \n\n\u5c0f\u5b66\u6570\u5b66\u6559\u5b66 refers to the foundational stage of mathematics education aimed at primary school students, focusing on the development of basic mathematical knowledge, conceptual understanding, and problem-solving skills. It emphasizes intuitive learning, real-world applications, and the gradual abstraction of mathematical ideas. \n\nKey aspects of \u5c0f\u5b66\u6570\u5b66\u6559\u5b66 include: \n- **Conceptual Foundations:** The curriculum helps students grasp fundamental concepts such as numbers (e.g., natural numbers through correspondence methods), basic operations (e.g., fraction division), and Euclidean geometry (e.g., points, lines, angles, and shapes), using descriptive and intuitive definitions rather than abstract formalisms like the Peano axioms. \n- **Contextual Learning:** Teachers employ real-world scenarios and models (e.g., geometric or physical models) to illustrate mathematical principles, ensuring students connect theory to practical applications. \n- **Staged Progression:** \n - **First Stage (Lower Grades):** Focuses on selecting appropriate units for estimation problems and establishing number-quantity relationships. \n - **Second Stage (Upper Grades):** Shifts toward method selection for problem-solving and deeper conceptual analysis, including direct judgment of mathematical propositions. \n- **Skill Development:** The pedagogy cultivates logical thinking, abstraction (e.g., via situational activities), and model-based reasoning, while avoiding overly complex theoretical content unsuitable for the age group. \n\nIn summary, \u5c0f\u5b66\u6570\u5b66\u6559\u5b66 serves as a bridge between concrete experiences and formal mathematics, prioritizing accessibility, relevance, and foundational mastery for young learners."}, {"color": "#e5b0a9", "created_at": 1752674410, "description": "Mathematics Education is a vital branch of educational theory and practice focused on the teaching and learning of mathematics. It emphasizes the development of logical reasoning, including both inductive and deductive methods, while fostering a deep understanding of mathematical concepts and the nature of mathematics itself. A key aspect of mathematics education is guiding students to think independently from the ground up, cultivating good habits of thought, and highlighting the importance of axioms and postulates as the foundation of logical reasoning.\n\nThis field involves the transmission of mathematical knowledge and thinking methods, with a strong emphasis on helping students understand mathematical ideas through real-world contexts and accumulating valuable thinking experiences. Mathematics education is sensitive to policy and research developments and plays a crucial role in nurturing students\u0027 mathematical literacy and creativity, ultimately contributing to the development of innovative talent.\n\nIt also acknowledges the importance of age-appropriate methods at different learning stages to help students grasp the relationships between numbers and quantities. Furthermore, it recognizes the significance of respecting the original intentions of classical mathematicians, such as Euclid, by ensuring that geometric concepts are not misapplied beyond experiential space. 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It emphasizes the development of logical reasoning, including both inductive and deductive methods, while fostering a deep understanding of mathematical concepts and the nature of mathematics itself. A key aspect of mathematics education is guiding students to think independently from the ground up, cultivating good habits of thought, and highlighting the importance of axioms and postulates as the foundation of logical reasoning.\n\nThis field involves the transmission of mathematical knowledge and thinking methods, with a strong emphasis on helping students understand mathematical ideas through real-world contexts and accumulating valuable thinking experiences. Mathematics education is sensitive to policy and research developments and plays a crucial role in nurturing students\u0027 mathematical literacy and creativity, ultimately contributing to the development of innovative talent.\n\nIt also acknowledges the importance of age-appropriate methods at different learning stages to help students grasp the relationships between numbers and quantities. Furthermore, it recognizes the significance of respecting the original intentions of classical mathematicians, such as Euclid, by ensuring that geometric concepts are not misapplied beyond experiential space. 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It is a critical phase in the curriculum where students build upon foundational knowledge and progress toward more complex mathematical concepts. \n\nKey components of the curriculum at this stage include: \n- **Advanced Geometric Operations:** Students learn complex geometric transformations, including concepts related to area and volume, extending their prior understanding of length. \n- **Statistical Understanding:** They are introduced to averages (means), learning how to calculate them and interpret their real-world significance. \n- **Decimal and Fraction Mastery:** The curriculum emphasizes a deeper comprehension of decimals, fractions, and percentages, including their interrelationships and arithmetic operations (e.g., addition, subtraction, multiplication, and division). \n- **Mathematical Application:** Skills developed in this stage are designed to enhance problem-solving abilities and practical application in everyday contexts. \n\nThis phase serves as a bridge between basic arithmetic and more advanced mathematical reasoning, ensuring students acquire both computational proficiency and conceptual clarity. \n\n*(Note: The term \"\u7b2c\u4e8c\u5b66\u6bb5\" is consistently translated here as \"the second academic stage\" to align with the educational context, though it specifically denotes the later primary years in the Chinese curriculum.)*"}, {"color": "#6718cd", "created_at": 1752674544, "description": "**\u7b2c\u4e00\u5b66\u6bb5 (First Stage of Primary Education in China)** \n\n\u7b2c\u4e00\u5b66\u6bb5 refers to the initial stage of primary education in China, typically covering grades 1-2 or 1-3, as part of the foundational phase of the country\u0027s basic education system. 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During this stage, students are introduced to essential mathematical concepts, including: \n- **Numbers and Arithmetic:** Learning numbers up to 10,000, counting, sequencing, and basic arithmetic operations. \n- **Geometric Transformations:** Basic recognition of shapes and spatial transformations. \n- **Measurement:** Understanding fundamental concepts like length and its units. \n- **Fractions and Decimals:** Preliminary exposure to fractions and decimals, including comparisons in contextual settings. \n\nThis stage serves as a critical foundation for future learning, emphasizing numeracy, problem-solving, and conceptual understanding. While some sources specify grades 1-2, others include grade 3, reflecting slight variations in curriculum interpretations. 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understanding fractions, including recognizing the whole (1), equal partitioning, and identifying the unit fraction.", "entity_id": "\u6559\u5b66\u8981\u70b9", "entity_type": "category", "file_path": "unknown_source", "id": "\u6559\u5b66\u8981\u70b9", "label": "\u6559\u5b66\u8981\u70b9", "shape": "dot", "size": 10, "source_id": "chunk-447d851ef33dcb6cb0712f2ef985e50e", "title": "\u6559\u5b66\u8981\u70b9refers to the essential teaching points for understanding fractions, including recognizing the whole (1), equal partitioning, and identifying the unit fraction."}, {"color": "#7eaf4a", "created_at": 1752674560, "description": "\u5c0f\u6570\u7684\u8ba4\u8bc6refers to the educational topic of introducing and understanding decimal numbers in mathematics, particularly in the context of primary school curriculum.", "entity_id": "\u5c0f\u6570\u7684\u8ba4\u8bc6", "entity_type": "category", "file_path": "unknown_source", "id": "\u5c0f\u6570\u7684\u8ba4\u8bc6", "label": 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"\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570distinguishes between numbers expressible as fractions and those that aren\u0027t.", "entity_id": "\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570", "entity_type": "category", "file_path": "unknown_source", "id": "\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570", "label": "\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570", "shape": "dot", "size": 10, "source_id": "chunk-a9a60418b119b40b9f7e09f08cc0b5c8", "title": "\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570distinguishes between numbers expressible as fractions and those that aren\u0027t."}, {"color": "#736d32", "created_at": 1752674550, "description": "\u674e\u5b81\u5b81is a teacher at Changchun Shuxun Primary School and the author of a teaching design on understanding 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magnitude of the number 1000 in different contexts.", "entity_id": "\u591a", "entity_type": "category", "file_path": "unknown_source", "id": "\u591a", "label": "\u591a", "shape": "dot", "size": 10, "source_id": "chunk-612d665e8236ed57ac928a378892167b", "title": "\u591arefers to a large quantity used in the\u6570\u611fdevelopment activity, helping students grasp the magnitude of the number 1000 in different contexts."}, {"color": "#f61110", "created_at": 1752674566, "description": "Liu Yanping is the educator from Northeast Normal University Affiliated Primary School who designed the teaching material about averages.\u003cSEP\u003e\u5218\u8273\u5e73is a teacher from Northeast Normal University Affiliated Elementary School who designed a lesson on understanding the meaning of averages.\u003cSEP\u003e\u5218\u8273\u5e73is an educator from Northeast Normal University Affiliated Elementary School, involved in designing a mathematics lesson focused on number sense and addition.", "entity_id": "\u5218\u8273\u5e73", "entity_type": "person", "file_path": "unknown_source", "id": "\u5218\u8273\u5e73", "label": "\u5218\u8273\u5e73", "shape": "dot", "size": 10, "source_id": "chunk-2a4876e1595a657157d330fa9e38f2f6\u003cSEP\u003echunk-612d665e8236ed57ac928a378892167b\u003cSEP\u003echunk-1d698eb4c5d9af871fb0b911bee4b4cc", "title": "Liu Yanping is the educator from Northeast Normal University Affiliated Primary School who designed the teaching material about averages.\u003cSEP\u003e\u5218\u8273\u5e73is a teacher from Northeast Normal University Affiliated Elementary School who designed a lesson on understanding the meaning of averages.\u003cSEP\u003e\u5218\u8273\u5e73is an educator from Northeast Normal University Affiliated Elementary School, involved in designing a mathematics lesson focused on number sense and addition."}, {"color": "#4a28aa", "created_at": 1752674566, "description": "**Northeast Normal University Affiliated Elementary School (\u4e1c\u5317\u5e08\u5927\u9644\u5c0f)** is an elementary school affiliated with Northeast Normal University in China. It serves as an educational institution where innovative teaching methods are developed and implemented by teachers such as **Liu Yanping**, **\u9ad8\u4fca\u751f (Gao Junsheng)**, **\u6731\u9896 (Zhu Ying)**, and **\u8d75\u8273\u8f89 (Zhao Yanhui)**. The school is known for its involvement in educational content design, including topics such as statistical graphs and measurement concepts (e.g., length, area, and volume). Additionally, it is the institution where the discussed math lesson and teaching designs were created and implemented, with some of these designs being authored by its faculty. Overall, Northeast Normal University Affiliated Elementary School is a hub for pedagogical innovation and serves as the professional base for its accomplished teaching staff.", "entity_id": "\u4e1c\u5317\u5e08\u5927\u9644\u5c0f", "entity_type": "organization", "file_path": "unknown_source", "id": "\u4e1c\u5317\u5e08\u5927\u9644\u5c0f", "label": "\u4e1c\u5317\u5e08\u5927\u9644\u5c0f", "shape": "dot", "size": 10, "source_id": "chunk-6bbe77567ec618aec822acfe88217048\u003cSEP\u003echunk-76d274903e04367b66aca88ea0b789b4\u003cSEP\u003echunk-7ac9e27076640b89379ec7305a979523\u003cSEP\u003echunk-612d665e8236ed57ac928a378892167b\u003cSEP\u003echunk-d49b7902492c618044ad70c5d4b3ba7f\u003cSEP\u003echunk-2a4876e1595a657157d330fa9e38f2f6\u003cSEP\u003echunk-9010a16ee414b4840a9bed251664d62b\u003cSEP\u003echunk-a5fa437b38c82cb10c1cbb6ae90f2d6c\u003cSEP\u003echunk-1ad150a353733dc35455650b97cce260\u003cSEP\u003echunk-1d698eb4c5d9af871fb0b911bee4b4cc", "title": "**Northeast Normal University Affiliated Elementary School (\u4e1c\u5317\u5e08\u5927\u9644\u5c0f)** is an elementary school affiliated with Northeast Normal University in China. It serves as an educational institution where innovative teaching methods are developed and implemented by teachers such as **Liu Yanping**, **\u9ad8\u4fca\u751f (Gao Junsheng)**, **\u6731\u9896 (Zhu Ying)**, and **\u8d75\u8273\u8f89 (Zhao Yanhui)**. The school is known for its involvement in educational content design, including topics such as statistical graphs and measurement concepts (e.g., length, area, and volume). Additionally, it is the institution where the discussed math lesson and teaching designs were created and implemented, with some of these designs being authored by its faculty. Overall, Northeast Normal University Affiliated Elementary School is a hub for pedagogical innovation and serves as the professional base for its accomplished teaching staff."}, {"color": "#5a8bb4", "created_at": 1752674536, "description": "The **\u6559\u5b66\u8bbe\u8ba1 (Instructional Design)** refers to the systematic planning and structuring of educational activities and lessons aimed at achieving specific learning objectives in mathematics. It encompasses a variety of conceptual and practical approaches tailored to different topics, including measurement, geometric transformations, decimals, statistical concepts, and mathematical operations like addition. \n\nThe design integrates both theoretical and hands-on learning to cultivate students\u0027 **\u6570\u611f (number sense)** and **\u7b26\u53f7\u610f\u8bc6 (symbolic awareness)**, ensuring a deeper comprehension of mathematical principles. It includes detailed lesson plans with activities, examples, and contextual scenarios to enhance student understanding. Additionally, the instructional strategies outlined in the design are applicable across primary and secondary education, providing structured guidance for teachers to facilitate effective learning. \n\nKey focuses of the **\u6559\u5b66\u8bbe\u8ba1** include: \n- Developing foundational skills in measurement and geometric transformations. \n- Strengthening conceptual and procedural knowledge of decimals and addition. \n- Fostering statistical literacy through structured activities and real-world applications. \n- Implementing teaching strategies that align with curriculum goals for\u4e2d\u5c0f\u5b66 (primary and secondary) education. \n\nThis comprehensive approach ensures that lessons are engaging, methodical, and tailored to students\u0027 cognitive development in mathematics.", "entity_id": "\u6559\u5b66\u8bbe\u8ba1", "entity_type": "event", "file_path": "unknown_source", "id": "\u6559\u5b66\u8bbe\u8ba1", "label": "\u6559\u5b66\u8bbe\u8ba1", "shape": "dot", "size": 10, "source_id": "chunk-112d4fcad80ef84a1327a06d05614630\u003cSEP\u003echunk-f9fb4bf24eb3450dc8d14d30db6e18ae\u003cSEP\u003echunk-abfbf7c5f139cdcc4cd4ac64b19841c7\u003cSEP\u003echunk-612d665e8236ed57ac928a378892167b\u003cSEP\u003echunk-15a3035e48a3f9ba98ec01cb3ed42052\u003cSEP\u003echunk-2abc65aa9bf3cc848d8609779c8072da", "title": "The **\u6559\u5b66\u8bbe\u8ba1 (Instructional Design)** refers to the systematic planning and structuring of educational activities and lessons aimed at achieving specific learning objectives in mathematics. It encompasses a variety of conceptual and practical approaches tailored to different topics, including measurement, geometric transformations, decimals, statistical concepts, and mathematical operations like addition. \n\nThe design integrates both theoretical and hands-on learning to cultivate students\u0027 **\u6570\u611f (number sense)** and **\u7b26\u53f7\u610f\u8bc6 (symbolic awareness)**, ensuring a deeper comprehension of mathematical principles. It includes detailed lesson plans with activities, examples, and contextual scenarios to enhance student understanding. Additionally, the instructional strategies outlined in the design are applicable across primary and secondary education, providing structured guidance for teachers to facilitate effective learning. \n\nKey focuses of the **\u6559\u5b66\u8bbe\u8ba1** include: \n- Developing foundational skills in measurement and geometric transformations. \n- Strengthening conceptual and procedural knowledge of decimals and addition. \n- Fostering statistical literacy through structured activities and real-world applications. \n- Implementing teaching strategies that align with curriculum goals for\u4e2d\u5c0f\u5b66 (primary and secondary) education. \n\nThis comprehensive approach ensures that lessons are engaging, methodical, and tailored to students\u0027 cognitive development in mathematics."}, {"color": "#bb3000", "created_at": 1752674378, "description": "**\u52a0\u6cd5 (Addition)** is a fundamental arithmetic operation widely used in mathematics and early education. It involves combining two or more quantities to produce a sum, forming the basis for calculating averages, among other applications. Addition is rigorously defined using mathematical axioms and serves as the foundation for more complex operations, such as multiplication, while its inverse operation is subtraction. The operation has multiple interpretation methods in mathematics, reflecting its versatility and importance. Introduced early in math education, addition is a core concept that underpins many other mathematical principles and calculations. \n\n*(Note: The original descriptions included a mix of English and Chinese, but the summary has been consolidated into a coherent English overview while preserving all key details.)*", "entity_id": "\u52a0\u6cd5", "entity_type": "category", "file_path": "unknown_source", "id": "\u52a0\u6cd5", "label": "\u52a0\u6cd5", "shape": "dot", "size": 10, "source_id": "chunk-a5fa437b38c82cb10c1cbb6ae90f2d6c\u003cSEP\u003echunk-baa5d2235a16fa56fc40519b58791c47\u003cSEP\u003echunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892\u003cSEP\u003echunk-2dcd5d60b98cfa8a05fa69795c6edd6e\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde", "title": "**\u52a0\u6cd5 (Addition)** is a fundamental arithmetic operation widely used in mathematics and early education. It involves combining two or more quantities to produce a sum, forming the basis for calculating averages, among other applications. Addition is rigorously defined using mathematical axioms and serves as the foundation for more complex operations, such as multiplication, while its inverse operation is subtraction. The operation has multiple interpretation methods in mathematics, reflecting its versatility and importance. Introduced early in math education, addition is a core concept that underpins many other mathematical principles and calculations. \n\n*(Note: The original descriptions included a mix of English and Chinese, but the summary has been consolidated into a coherent English overview while preserving all key details.)*"}, {"color": "#a7373c", "created_at": 1752674378, "description": "Arithmetic operation understood as the inverse of addition in number systems.\u003cSEP\u003eSubtraction is an arithmetic operation, but in the context of equation solving, it is contrasted with systematic methods like transposition.\u003cSEP\u003eSubtraction is the inverse operation of addition, representing the removal of one quantity from another.\u003cSEP\u003e\u51cf\u6cd5\u662f\u52a0\u6cd5\u7684\u9006\u8fd0\u7b97\uff0c\u901a\u8fc7\u52a0\u6cd5\u5b9a\u4e49\uff0c\u7528\u4e8e\u89e3\u51b3\u6570\u7684\u5dee\u503c\u95ee\u9898\u3002", "entity_id": "\u51cf\u6cd5", "entity_type": "category", "file_path": "unknown_source", "id": "\u51cf\u6cd5", "label": "\u51cf\u6cd5", "shape": "dot", "size": 10, "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde\u003cSEP\u003echunk-5b8e148c0868493680a6eec71a90d808\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892", "title": "Arithmetic operation understood as the inverse of addition in number systems.\u003cSEP\u003eSubtraction is an arithmetic operation, but in the context of equation solving, it is contrasted with systematic methods like transposition.\u003cSEP\u003eSubtraction is the inverse operation of addition, representing the removal of one quantity from another.\u003cSEP\u003e\u51cf\u6cd5\u662f\u52a0\u6cd5\u7684\u9006\u8fd0\u7b97\uff0c\u901a\u8fc7\u52a0\u6cd5\u5b9a\u4e49\uff0c\u7528\u4e8e\u89e3\u51b3\u6570\u7684\u5dee\u503c\u95ee\u9898\u3002"}, {"color": "#ea2607", "created_at": 1752674387, "description": "Multiplication (\u4e58\u6cd5) is a fundamental arithmetic operation initially defined on natural numbers as a simplified form of repeated addition. It preserves key properties such as commutativity and distributivity, which remain consistent when the operation is extended to integers and other mathematical domains. While multiplication in natural numbers can be intuitively understood as repeated addition, its definition becomes more complex in the context of integers, where it incorporates broader rules to maintain these properties. Additionally, multiplication can be transformed into division in certain problem-solving contexts, demonstrating its versatility in mathematical operations. Overall, multiplication serves as a foundational concept in arithmetic, with applications extending beyond natural numbers and integers into more advanced mathematical fields. \n\n(Note: Contradictions were resolved by acknowledging the dual perspectives\u2014multiplication as repeated addition in natural numbers versus its generalized definition in integers\u2014while emphasizing its consistent properties across domains.)", "entity_id": "\u4e58\u6cd5", "entity_type": "category", "file_path": "unknown_source", "id": "\u4e58\u6cd5", "label": "\u4e58\u6cd5", "shape": "dot", "size": 10, "source_id": "chunk-0756c8764e46b050990c725e6fb7d82d\u003cSEP\u003echunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892\u003cSEP\u003echunk-f0d039ecac6d9749d604feadbbc70472\u003cSEP\u003echunk-a5fa437b38c82cb10c1cbb6ae90f2d6c\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde", "title": "Multiplication (\u4e58\u6cd5) is a fundamental arithmetic operation initially defined on natural numbers as a simplified form of repeated addition. It preserves key properties such as commutativity and distributivity, which remain consistent when the operation is extended to integers and other mathematical domains. While multiplication in natural numbers can be intuitively understood as repeated addition, its definition becomes more complex in the context of integers, where it incorporates broader rules to maintain these properties. Additionally, multiplication can be transformed into division in certain problem-solving contexts, demonstrating its versatility in mathematical operations. Overall, multiplication serves as a foundational concept in arithmetic, with applications extending beyond natural numbers and integers into more advanced mathematical fields. \n\n(Note: Contradictions were resolved by acknowledging the dual perspectives\u2014multiplication as repeated addition in natural numbers versus its generalized definition in integers\u2014while emphasizing its consistent properties across domains.)"}, {"color": "#54cae5", "created_at": 1752674438, "description": "0\u662f\u52a0\u6cd5\u8fd0\u7b97\u7684\u5355\u4f4d\u5143\uff0c\u6ee1\u8db3a + 0 = a\uff0c\u5728\u4e58\u6cd5\u4e2d0\u4e58\u4ee5\u4efb\u4f55\u6570\u5747\u4e3a0\u3002\u003cSEP\u003eZero is the additive identity element crucial for defining negative numbers and integer operations.", "entity_id": "0", "entity_type": "category", "file_path": "unknown_source", "id": "0", "label": "0", "shape": "dot", "size": 10, "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892", "title": "0\u662f\u52a0\u6cd5\u8fd0\u7b97\u7684\u5355\u4f4d\u5143\uff0c\u6ee1\u8db3a + 0 = a\uff0c\u5728\u4e58\u6cd5\u4e2d0\u4e58\u4ee5\u4efb\u4f55\u6570\u5747\u4e3a0\u3002\u003cSEP\u003eZero is the additive identity element crucial for defining negative numbers and integer operations."}, {"color": "#961047", "created_at": 1752674438, "description": "The total quantity model describes the relationship between a total quantity and its parts, using addition to represent the sum of parts. It is also referred to as the addition model.\u003cSEP\u003eThe total quantity model is a mathematical model that abstracts scenarios involving total quantities and their distributions, often combined with other models like the distance model.", "entity_id": "\u603b\u91cf\u6a21\u578b", "entity_type": "category", "file_path": "unknown_source", "id": "\u603b\u91cf\u6a21\u578b", "label": "\u603b\u91cf\u6a21\u578b", "shape": "dot", "size": 10, "source_id": "chunk-5b8e148c0868493680a6eec71a90d808\u003cSEP\u003echunk-0756c8764e46b050990c725e6fb7d82d", "title": "The total quantity model describes the relationship between a total quantity and its parts, using addition to represent the sum of parts. It is also referred to as the addition model.\u003cSEP\u003eThe total quantity model is a mathematical model that abstracts scenarios involving total quantities and their distributions, often combined with other models like the distance model."}, {"color": "#684036", "created_at": 1752674485, "description": "\u516c\u74069 is the mathematical induction axiom that ensures the validity of recursive definitions and proofs for all natural numbers.", "entity_id": "\u516c\u74069", "entity_type": "category", "file_path": "unknown_source", "id": "\u516c\u74069", "label": "\u516c\u74069", "shape": "dot", "size": 10, "source_id": "chunk-baa5d2235a16fa56fc40519b58791c47", "title": "\u516c\u74069 is the mathematical induction axiom that ensures the validity of recursive definitions and proofs for all natural numbers."}, {"color": "#2f23b9", "created_at": 1752674485, "description": "\u5927\u5c0f\u5173\u7cfbrefers to the formal definition of greater than or less than relations on natural numbers using addition.", "entity_id": "\u5927\u5c0f\u5173\u7cfb", "entity_type": "category", "file_path": "unknown_source", "id": "\u5927\u5c0f\u5173\u7cfb", "label": "\u5927\u5c0f\u5173\u7cfb", "shape": "dot", "size": 10, "source_id": "chunk-baa5d2235a16fa56fc40519b58791c47", "title": "\u5927\u5c0f\u5173\u7cfbrefers to the formal definition of greater than or less than relations on natural numbers using addition."}, {"color": "#197930", "created_at": 1752674387, "description": "A mathematical operation used in computing the average by dividing the sum of values by the number of observations.\u003cSEP\u003eArithmetic operation understood as the inverse of multiplication.\u003cSEP\u003e\u9664\u6cd5(division) is a mathematical operation used in reformulating speed-distance-time 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"title": "A mathematical operation used in computing the average by dividing the sum of values by the number of observations.\u003cSEP\u003eArithmetic operation understood as the inverse of multiplication.\u003cSEP\u003e\u9664\u6cd5(division) is a mathematical operation used in reformulating speed-distance-time problems.\u003cSEP\u003e\u9664\u6cd5\u662f\u4e58\u6cd5\u7684\u9006\u8fd0\u7b97\uff0c\u901a\u8fc7\u4e58\u6cd5\u5b9a\u4e49\uff0c\u7528\u4e8e\u89e3\u51b3\u6570\u7684\u5206\u914d\u95ee\u9898\uff0c\u5e76\u53ef\u5c06\u6574\u6570\u96c6\u5408\u6269\u5145\u5230\u6709\u7406\u6570\u96c6\u5408\u3002\u003cSEP\u003e\u9664\u6cd5\u662f\u6570\u5b66\u8fd0\u7b97\u7684\u57fa\u672c\u5f62\u5f0f\u4e4b\u4e00\uff0c\u5728\u6559\u5b66\u4e2d\u5f3a\u8c03\u5176\u4e0e\u4e58\u6cd5\u7684\u9006\u5173\u7cfb\uff0c\u7279\u522b\u662f\u5728\u5206\u6570\u9664\u6cd5\u4e2d\u7684\u5e94\u7528\u3002"}, {"color": "#335223", "created_at": 1752674438, "description": "A fundamental property of multiplication stating that the order of operands does not affect the result.\u003cSEP\u003eCommutative property (a\u00d7b=b\u00d7a) is essential for extending multiplication to integers.\u003cSEP\u003e\u4ea4\u6362\u5f8b\u662f\u4e58\u6cd5\u8fd0\u7b97\u7684\u57fa\u672c\u6027\u8d28\u4e4b\u4e00\uff0c\u8868\u793aa \u00d7 b = b \u00d7 a\uff0c\u662f\u4e58\u6cd5\u63a8\u5e7f\u5230\u6574\u6570\u96c6\u5408\u7684\u5173\u952e\u5de5\u5177\u3002\u003cSEP\u003e\u4ea4\u6362\u5f8b\u662f\u56db\u5219\u8fd0\u7b97\u7684\u57fa\u672c\u6cd5\u5219\u4e4b\u4e00\uff0c\u89c4\u5b9aa + b = b + a\u548ca \u00d7 b = b \u00d7 a\uff0c\u4f46\u5728\u51cf\u6cd5\u548c\u9664\u6cd5\u4e2d\u4e0d\u6210\u7acb\u3002", "entity_id": "\u4ea4\u6362\u5f8b", "entity_type": "category", "file_path": "unknown_source", "id": "\u4ea4\u6362\u5f8b", "label": "\u4ea4\u6362\u5f8b", "shape": "dot", "size": 10, "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892\u003cSEP\u003echunk-f0d039ecac6d9749d604feadbbc70472\u003cSEP\u003echunk-5bbd74eb38646ddcfc35611725fd4482", "title": "A fundamental property of multiplication stating that the order of operands does not affect the result.\u003cSEP\u003eCommutative property (a\u00d7b=b\u00d7a) is essential for extending multiplication to integers.\u003cSEP\u003e\u4ea4\u6362\u5f8b\u662f\u4e58\u6cd5\u8fd0\u7b97\u7684\u57fa\u672c\u6027\u8d28\u4e4b\u4e00\uff0c\u8868\u793aa \u00d7 b = b \u00d7 a\uff0c\u662f\u4e58\u6cd5\u63a8\u5e7f\u5230\u6574\u6570\u96c6\u5408\u7684\u5173\u952e\u5de5\u5177\u3002\u003cSEP\u003e\u4ea4\u6362\u5f8b\u662f\u56db\u5219\u8fd0\u7b97\u7684\u57fa\u672c\u6cd5\u5219\u4e4b\u4e00\uff0c\u89c4\u5b9aa + b = b + a\u548ca \u00d7 b = b \u00d7 a\uff0c\u4f46\u5728\u51cf\u6cd5\u548c\u9664\u6cd5\u4e2d\u4e0d\u6210\u7acb\u3002"}, {"color": "#eece2f", "created_at": 1752674438, "description": "Distributive property links addition and multiplication during integer operations extension.\u003cSEP\u003e\u5206\u914d\u7387\u662f\u4e58\u6cd5\u8fd0\u7b97\u7684\u53e6\u4e00\u57fa\u672c\u6027\u8d28\uff0c\u8868\u793a(a + b) \u00d7 c = (a \u00d7 c) + (b \u00d7 c)\uff0c\u4e0e\u4ea4\u6362\u5f8b\u5171\u540c\u6784\u6210\u4e58\u6cd5\u7684\u7b97\u7406\u6838\u5fc3\u3002", "entity_id": "\u5206\u914d\u7387", "entity_type": "category", "file_path": "unknown_source", "id": "\u5206\u914d\u7387", "label": "\u5206\u914d\u7387", "shape": "dot", "size": 10, "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892", "title": "Distributive property links addition and multiplication during integer operations extension.\u003cSEP\u003e\u5206\u914d\u7387\u662f\u4e58\u6cd5\u8fd0\u7b97\u7684\u53e6\u4e00\u57fa\u672c\u6027\u8d28\uff0c\u8868\u793a(a + b) \u00d7 c = (a \u00d7 c) + (b \u00d7 c)\uff0c\u4e0e\u4ea4\u6362\u5f8b\u5171\u540c\u6784\u6210\u4e58\u6cd5\u7684\u7b97\u7406\u6838\u5fc3\u3002"}, {"color": "#f6b53f", "created_at": 1752674438, "description": "1\u662f\u4e58\u6cd5\u8fd0\u7b97\u7684\u5355\u4f4d\u5143\uff0c\u6ee1\u8db3a \u00d7 1 = a\uff0c\u662f\u4e58\u6cd5\u9006\u8fd0\u7b97(\u9664\u6cd5)\u7684\u57fa\u7840\u3002", "entity_id": "1", "entity_type": "category", "file_path": "unknown_source", "id": "1", "label": "1", "shape": "dot", "size": 10, "source_id": "chunk-4be9177c327d58831959006a2ea25892", "title": "1\u662f\u4e58\u6cd5\u8fd0\u7b97\u7684\u5355\u4f4d\u5143\uff0c\u6ee1\u8db3a \u00d7 1 = a\uff0c\u662f\u4e58\u6cd5\u9006\u8fd0\u7b97(\u9664\u6cd5)\u7684\u57fa\u7840\u3002"}, {"color": "#48f6cc", "created_at": 1752674438, "description": "The distance model describes the relationship between distance, speed, and time, using multiplication. It can also be applied to other real-world problems like pricing or array counting.\u003cSEP\u003eThe distance model is a mathematical model that abstracts real-world scenarios involving speed, distance, and time, often used in primary education.\u003cSEP\u003e\u8def\u7a0b\u6a21\u578brefers to the mathematical model of distance = speed \u00d7 time being taught to students.\u003cSEP\u003e\u8def\u7a0b\u6a21\u578b\u662f\u63cf\u8ff0\u7269\u4f53\u8fd0\u52a8\u8def\u5f84\u7684\u6570\u5b66\u6a21\u578b\uff0c\u901a\u5e38\u4e0d\u8003\u8651\u89c2\u5bdf\u8005\u7684\u4f4d\u7f6e\u548c\u8fd0\u52a8\u72b6\u6001\u3002\u003cSEP\u003e\u8def\u7a0b\u6a21\u578b\u662f\u63cf\u8ff0\u8ddd\u79bb\u3001\u901f\u5ea6\u548c\u65f6\u95f4\u4e09\u8005\u4e4b\u95f4\u5173\u7cfb\u7684\u6570\u5b66\u6a21\u578b\uff0c\u901a\u5e38\u8868\u793a\u4e3ax = vt\uff0c\u7528\u4e8e\u89e3\u91ca\u73b0\u5b9e\u4e16\u754c\u4e2d\u7684\u8fd0\u52a8\u73b0\u8c61\u3002", "entity_id": "\u8def\u7a0b\u6a21\u578b", "entity_type": "category", "file_path": "unknown_source", "id": "\u8def\u7a0b\u6a21\u578b", "label": "\u8def\u7a0b\u6a21\u578b", "shape": "dot", "size": 10, "source_id": "chunk-0756c8764e46b050990c725e6fb7d82d\u003cSEP\u003echunk-5b8e148c0868493680a6eec71a90d808\u003cSEP\u003echunk-77809f023af3e8d2fb1e4c6cea56c329\u003cSEP\u003echunk-d67f6fd57e334174ea64d85cdb089d77\u003cSEP\u003echunk-9a3d6b7b471c153a635a854a31395ed5", "title": "The distance model describes the relationship between distance, speed, and time, using multiplication. It can also be applied to other real-world problems like pricing or array counting.\u003cSEP\u003eThe distance model is a mathematical model that abstracts real-world scenarios involving speed, distance, and time, often used in primary education.\u003cSEP\u003e\u8def\u7a0b\u6a21\u578brefers to the mathematical model of distance = speed \u00d7 time being taught to students.\u003cSEP\u003e\u8def\u7a0b\u6a21\u578b\u662f\u63cf\u8ff0\u7269\u4f53\u8fd0\u52a8\u8def\u5f84\u7684\u6570\u5b66\u6a21\u578b\uff0c\u901a\u5e38\u4e0d\u8003\u8651\u89c2\u5bdf\u8005\u7684\u4f4d\u7f6e\u548c\u8fd0\u52a8\u72b6\u6001\u3002\u003cSEP\u003e\u8def\u7a0b\u6a21\u578b\u662f\u63cf\u8ff0\u8ddd\u79bb\u3001\u901f\u5ea6\u548c\u65f6\u95f4\u4e09\u8005\u4e4b\u95f4\u5173\u7cfb\u7684\u6570\u5b66\u6a21\u578b\uff0c\u901a\u5e38\u8868\u793a\u4e3ax = vt\uff0c\u7528\u4e8e\u89e3\u91ca\u73b0\u5b9e\u4e16\u754c\u4e2d\u7684\u8fd0\u52a8\u73b0\u8c61\u3002"}, {"color": "#a858a8", "created_at": 1752674526, "description": "The set of positive integers used as the foundational domain for defining arithmetic operations like multiplication.", "entity_id": "\u81ea\u7136\u6570\u96c6\u5408N", "entity_type": "category", "file_path": "unknown_source", "id": "\u81ea\u7136\u6570\u96c6\u5408N", "label": "\u81ea\u7136\u6570\u96c6\u5408N", "shape": "dot", "size": 10, "source_id": "chunk-f0d039ecac6d9749d604feadbbc70472", "title": "The set of positive integers used as the foundational domain for defining arithmetic operations like multiplication."}, {"color": "#5ead0b", "created_at": 1752674526, "description": "The extended set of numbers including positive, negative, and zero, to which multiplication is expanded from natural numbers.", "entity_id": "\u6574\u6570\u96c6\u5408Z", "entity_type": "category", "file_path": "unknown_source", "id": "\u6574\u6570\u96c6\u5408Z", "label": "\u6574\u6570\u96c6\u5408Z", "shape": "dot", "size": 10, "source_id": "chunk-f0d039ecac6d9749d604feadbbc70472", "title": "The extended set of numbers including positive, negative, and zero, to which multiplication is expanded from natural numbers."}, {"color": "#ac4dca", "created_at": 1752674438, "description": "A fundamental property relating multiplication and addition, stating that multiplication distributes over addition.\u003cSEP\u003e\u5206\u914d\u5f8b\u662f\u56db\u5219\u8fd0\u7b97\u7684\u57fa\u672c\u6cd5\u5219\u4e4b\u4e00\uff0c\u89c4\u5b9aa \u00d7 (b + c) = a \u00d7 b + a \u00d7 c\uff0c\u5728\u6709\u7406\u6570\u96c6\u5408\u4e2d\u6210\u7acb\u3002", "entity_id": "\u5206\u914d\u5f8b", "entity_type": "category", "file_path": "unknown_source", "id": "\u5206\u914d\u5f8b", "label": "\u5206\u914d\u5f8b", "shape": "dot", "size": 10, "source_id": "chunk-f0d039ecac6d9749d604feadbbc70472\u003cSEP\u003echunk-5bbd74eb38646ddcfc35611725fd4482", "title": "A fundamental property relating multiplication and addition, stating that multiplication distributes over addition.\u003cSEP\u003e\u5206\u914d\u5f8b\u662f\u56db\u5219\u8fd0\u7b97\u7684\u57fa\u672c\u6cd5\u5219\u4e4b\u4e00\uff0c\u89c4\u5b9aa \u00d7 (b + c) = a \u00d7 b + a \u00d7 c\uff0c\u5728\u6709\u7406\u6570\u96c6\u5408\u4e2d\u6210\u7acb\u3002"}, {"color": "#9f98b1", "created_at": 1752674526, "description": "A mathematical process such as addition or multiplication that combines numbers according to defined rules.", "entity_id": "\u8fd0\u7b97", "entity_type": "category", "file_path": "unknown_source", "id": "\u8fd0\u7b97", "label": "\u8fd0\u7b97", "shape": "dot", "size": 10, "source_id": "chunk-f0d039ecac6d9749d604feadbbc70472", "title": "A mathematical process such as addition or multiplication that combines numbers according to defined rules."}, {"color": "#4b6486", "created_at": 1752674566, "description": "A choir group used as an example to demonstrate multiplication through organized rows and columns.", "entity_id": "\u5408\u5531\u5c0f\u7ec4", "entity_type": "category", "file_path": "unknown_source", "id": "\u5408\u5531\u5c0f\u7ec4", "label": "\u5408\u5531\u5c0f\u7ec4", "shape": "dot", "size": 10, "source_id": "chunk-a5fa437b38c82cb10c1cbb6ae90f2d6c", "title": "A choir group used as an example to demonstrate multiplication through organized rows and columns."}, {"color": "#4493f9", "created_at": 1752674438, "description": "\u9644\u5f55\u7684\u8bdd\u989821\u901a\u8fc7\u5177\u4f53\u4f8b\u5b50\u8bf4\u660e\u5982\u4f55\u501f\u52a9\u4e58\u6cd5\u7406\u89e3\u9664\u6cd5\uff0c\u9002\u7528\u4e8e\u5b9e\u9645\u6559\u5b66\u573a\u666f\u3002", "entity_id": "\u9644\u5f55\u7684\u8bdd\u989821", "entity_type": "category", "file_path": "unknown_source", "id": "\u9644\u5f55\u7684\u8bdd\u989821", "label": "\u9644\u5f55\u7684\u8bdd\u989821", "shape": "dot", "size": 10, "source_id": "chunk-4be9177c327d58831959006a2ea25892", "title": "\u9644\u5f55\u7684\u8bdd\u989821\u901a\u8fc7\u5177\u4f53\u4f8b\u5b50\u8bf4\u660e\u5982\u4f55\u501f\u52a9\u4e58\u6cd5\u7406\u89e3\u9664\u6cd5\uff0c\u9002\u7528\u4e8e\u5b9e\u9645\u6559\u5b66\u573a\u666f\u3002"}, {"color": "#6fee45", "created_at": 1752674438, "description": "\u7cbe\u7b97refers to exact calculation in mathematics, emphasizing its role in developing abstract thinking and precision.\u003cSEP\u003e\u7cbe\u7b97\u662f\u57fa\u4e8e\u6570\u7684\u7cbe\u786e\u8ba1\u7b97\uff0c\u672c\u8d28\u4e0a\u662f\u5bf9\u4e8e\u6570\u7684\u8fd0\u7b97\u3002", "entity_id": "\u7cbe\u7b97", "entity_type": "category", "file_path": "unknown_source", "id": "\u7cbe\u7b97", "label": "\u7cbe\u7b97", "shape": "dot", "size": 10, "source_id": "chunk-5358c62a1cf279262024e04d2bacfab8\u003cSEP\u003echunk-5c564427093279bdf3c25848196fe5f8", "title": "\u7cbe\u7b97refers to exact calculation in mathematics, emphasizing its role in developing abstract thinking and precision.\u003cSEP\u003e\u7cbe\u7b97\u662f\u57fa\u4e8e\u6570\u7684\u7cbe\u786e\u8ba1\u7b97\uff0c\u672c\u8d28\u4e0a\u662f\u5bf9\u4e8e\u6570\u7684\u8fd0\u7b97\u3002"}, {"color": "#6e835c", "created_at": 1752674438, "description": "\u6cd5\u56fd\u8111\u79d1\u5b66\u5bb6refers to French neuroscientists who studied brain activity during exact calculations and estimations, linking them to different cognitive functions.", "entity_id": "\u6cd5\u56fd\u8111\u79d1\u5b66\u5bb6", "entity_type": "organization", "file_path": "unknown_source", "id": "\u6cd5\u56fd\u8111\u79d1\u5b66\u5bb6", "label": "\u6cd5\u56fd\u8111\u79d1\u5b66\u5bb6", "shape": "dot", "size": 10, "source_id": "chunk-5358c62a1cf279262024e04d2bacfab8", "title": "\u6cd5\u56fd\u8111\u79d1\u5b66\u5bb6refers to French neuroscientists who studied brain activity during exact calculations and estimations, linking them to different cognitive functions."}, {"color": "#45945a", "created_at": 1752674427, "description": "**\u6570\u5b66\u7d20\u517b (Mathematical Literacy)** is a comprehensive competency developed through mathematics education, encompassing abstract and intuitive abilities, reasoning skills (including deductive and inductive reasoning), and the accumulation of knowledge, skills, ideas, and practical experience. 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By engaging with these diverse mathematical domains, students develop a well-rounded mathematical proficiency essential for academic and real-world applications. \n\n*(Note: Contradictions, if any, were resolved by synthesizing descriptions into a unified definition, emphasizing breadth and practical application.)*", "entity_id": "\u6570\u5b66\u7d20\u517b", "entity_type": "category", "file_path": "unknown_source", "id": "\u6570\u5b66\u7d20\u517b", "label": "\u6570\u5b66\u7d20\u517b", "shape": "dot", "size": 10, "source_id": "chunk-55fb2bb0a567e6a6d49fb80e347a185e\u003cSEP\u003echunk-b8cb0b3bb1cff9a2ccf4623209e1bea6\u003cSEP\u003echunk-cf30e3e5c182214d0f0eb020814d103d\u003cSEP\u003echunk-27faa903a9798367feb7512a5e9c6a66\u003cSEP\u003echunk-5358c62a1cf279262024e04d2bacfab8\u003cSEP\u003echunk-d49b7902492c618044ad70c5d4b3ba7f", "title": "**\u6570\u5b66\u7d20\u517b (Mathematical Literacy)** is a comprehensive competency developed through mathematics education, encompassing abstract and intuitive abilities, reasoning skills (including deductive and inductive reasoning), and the accumulation of knowledge, skills, ideas, and practical experience. It is a fundamental educational objective in basic education, fostering students\u0027 logical thinking and application abilities. \n\nThis literacy is cultivated through various approaches, including geometric classification, standardization exercises, statistical and probability learning, and the study of graphical motion and recording. It serves as one of the core goals of mathematical instruction, integrating both theoretical understanding and hands-on problem-solving skills. By engaging with these diverse mathematical domains, students develop a well-rounded mathematical proficiency essential for academic and real-world applications. \n\n*(Note: Contradictions, if any, were resolved by synthesizing descriptions into a unified definition, emphasizing breadth and practical application.)*"}, {"color": "#5943ae", "created_at": 1752674438, "description": "\u53cc\u4fa7\u9876\u53f6\u4e0b\u90e8is the brain region activated during estimation, associated with spatial perception.", "entity_id": "\u53cc\u4fa7\u9876\u53f6\u4e0b\u90e8", "entity_type": "category", "file_path": "unknown_source", "id": "\u53cc\u4fa7\u9876\u53f6\u4e0b\u90e8", "label": "\u53cc\u4fa7\u9876\u53f6\u4e0b\u90e8", "shape": "dot", "size": 10, "source_id": "chunk-5358c62a1cf279262024e04d2bacfab8", "title": "\u53cc\u4fa7\u9876\u53f6\u4e0b\u90e8is the brain region activated during estimation, associated with spatial perception."}, {"color": "#ccd62b", "created_at": 1752674566, "description": 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"\u51d1\u6574\u8ba1\u7b97\u662f\u4f30\u7b97\u7684\u5173\u952e\u65b9\u6cd5\u4e4b\u4e00\uff0c\u7528\u4e8e\u7b80\u5316\u8ba1\u7b97\u5e76\u5224\u65ad\u4e0a\u754c\u548c\u4e0b\u754c\u3002"}, {"color": "#7d8130", "created_at": 1752674560, "description": "\u5c0f\u5b66\u9636\u6bb5refers to the elementary or primary school level in education, where foundational mathematical concepts like length, area, and volume are introduced to students.\u003cSEP\u003e\u5c0f\u5b66\u9636\u6bb5refers to the elementary school level, where students are introduced to foundational mathematical laws such as the commutative property of multiplication.\u003cSEP\u003e\u5c0f\u5b66\u9636\u6bb5refers to the primary education level in China, where the curriculum includes the introduction and development of decimal understanding as outlined in national standards.\u003cSEP\u003e\u5c0f\u5b66\u9636\u6bb5\u662f\u4f30\u7b97\u6559\u5b66\u5b9e\u65bd\u7684\u4e3b\u8981\u6559\u80b2\u9636\u6bb5\uff0c\u5f3a\u8c03\u5b9e\u9645\u80cc\u666f\u4e0e\u601d\u7ef4\u8bad\u7ec3\u3002", "entity_id": "\u5c0f\u5b66\u9636\u6bb5", "entity_type": "category", "file_path": "unknown_source", "id": "\u5c0f\u5b66\u9636\u6bb5", "label": "\u5c0f\u5b66\u9636\u6bb5", "shape": "dot", "size": 10, "source_id": "chunk-5d6da01e75262470fd3f87fabfbe1110\u003cSEP\u003echunk-a499d10f1019c6d5d99c36d7402ec197\u003cSEP\u003echunk-9010a16ee414b4840a9bed251664d62b\u003cSEP\u003echunk-f9fb4bf24eb3450dc8d14d30db6e18ae", "title": "\u5c0f\u5b66\u9636\u6bb5refers to the elementary or primary school level in education, where foundational mathematical concepts like length, area, and volume are introduced to students.\u003cSEP\u003e\u5c0f\u5b66\u9636\u6bb5refers to the elementary school level, where students are introduced to foundational mathematical laws such as the commutative property of multiplication.\u003cSEP\u003e\u5c0f\u5b66\u9636\u6bb5refers to the primary education level in China, where the curriculum includes the introduction and development of decimal understanding as outlined in national standards.\u003cSEP\u003e\u5c0f\u5b66\u9636\u6bb5\u662f\u4f30\u7b97\u6559\u5b66\u5b9e\u65bd\u7684\u4e3b\u8981\u6559\u80b2\u9636\u6bb5\uff0c\u5f3a\u8c03\u5b9e\u9645\u80cc\u666f\u4e0e\u601d\u7ef4\u8bad\u7ec3\u3002"}, {"color": "#61ff63", "created_at": 1752674393, "description": "**Summary of \u65b9\u7a0b (Equations):** \n\n\u65b9\u7a0b (equations) are mathematical statements that express equality between two expressions, typically involving unknown quantities (variables). They serve as fundamental tools in algebra and are widely used to model and solve problems by describing quantitative relationships. Equations can represent various mathematical models, enabling the combination and resolution of different systems through techniques such as transposition and variable isolation. \n\nAs symbolic equalities containing unknowns, equations are essential for explaining mathematical concepts, including the logic behind operations like fraction division, where unknown variables can be introduced to derive solutions. Their versatility and power lie in their ability to formalize relationships between quantities, making them a cornerstone of mathematical problem-solving and theoretical exploration. \n\nIn summary, \u65b9\u7a0b (equations) are mathematical expressions of equality with variables, functioning as key analytical tools in both practical applications and abstract algebraic studies.", "entity_id": "\u65b9\u7a0b", "entity_type": "category", "file_path": "unknown_source", "id": "\u65b9\u7a0b", "label": "\u65b9\u7a0b", "shape": "dot", "size": 10, "source_id": "chunk-a9a60418b119b40b9f7e09f08cc0b5c8\u003cSEP\u003echunk-0756c8764e46b050990c725e6fb7d82d\u003cSEP\u003echunk-5b8e148c0868493680a6eec71a90d808\u003cSEP\u003echunk-ce575de1a4a5a428b616adecd09f8d8f\u003cSEP\u003echunk-5c564427093279bdf3c25848196fe5f8\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde\u003cSEP\u003echunk-68021b83de51425f98b10395ef797752", "title": "**Summary of \u65b9\u7a0b (Equations):** \n\n\u65b9\u7a0b (equations) are mathematical statements that express equality between two expressions, typically involving unknown quantities (variables). They serve as fundamental tools in algebra and are widely used to model and solve problems by describing quantitative relationships. Equations can represent various mathematical models, enabling the combination and resolution of different systems through techniques such as transposition and variable isolation. \n\nAs symbolic equalities containing unknowns, equations are essential for explaining mathematical concepts, including the logic behind operations like fraction division, where unknown variables can be introduced to derive solutions. Their versatility and power lie in their ability to formalize relationships between quantities, making them a cornerstone of mathematical problem-solving and theoretical exploration. \n\nIn summary, \u65b9\u7a0b (equations) are mathematical expressions of equality with variables, functioning as key analytical tools in both practical applications and abstract algebraic studies."}, {"color": "#13bfe8", "created_at": 1752674438, "description": "\u5b57\u6bcd\u8868\u793a\u6570refers to the concept of using letters to represent numbers or variables in mathematics, which is a key component of developing symbol awareness in students.\u003cSEP\u003e\u5b57\u6bcd\u8868\u793a\u6570refers to using letters to represent numbers, a key aspect of symbolic awareness.", "entity_id": "\u5b57\u6bcd\u8868\u793a\u6570", "entity_type": "category", "file_path": "unknown_source", "id": "\u5b57\u6bcd\u8868\u793a\u6570", "label": "\u5b57\u6bcd\u8868\u793a\u6570", "shape": "dot", "size": 10, "source_id": "chunk-a90ac5a1f21c2588adf853f55169b640\u003cSEP\u003echunk-15a3035e48a3f9ba98ec01cb3ed42052", "title": "\u5b57\u6bcd\u8868\u793a\u6570refers to the concept of using letters to represent numbers or variables in mathematics, which is a key component of developing symbol awareness in students.\u003cSEP\u003e\u5b57\u6bcd\u8868\u793a\u6570refers to using letters to represent numbers, a key aspect of symbolic awareness."}, {"color": "#a60068", "created_at": 1752674438, "description": "\u52a0\u6cd5\u4ea4\u6362\u5f8bis a mathematical rule explained using symbolic representation.\u003cSEP\u003e\u52a0\u6cd5\u4ea4\u6362\u5f8b\u662f\u6570\u5b66\u57fa\u672c\u5b9a\u5f8b\u4e4b\u4e00\uff0c\u6307\u51faa + b = b + a\u5bf9\u6240\u6709\u81ea\u7136\u6570\u6210\u7acb\u3002\u003cSEP\u003e\u52a0\u6cd5\u4ea4\u6362\u5f8b\u662f\u6570\u5b66\u57fa\u672c\u5b9a\u5f8b\uff0c\u8868\u660e\u52a0\u6cd5\u8fd0\u7b97\u4e2d\u6570\u7684\u987a\u5e8f\u4e0d\u5f71\u54cd\u7ed3\u679c\uff0c\u8868\u793a\u4e3aa + b = b + a\u3002", 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His work laid the foundation for the axiomatization of modern mathematics, emphasizing the importance of logical relationships between axioms over their concrete meanings. \n\nA key aspect of his approach was the introduction of the **\"Side-Angle-Side\" (SAS) theorem as an axiom**, allowing geometry to be freed from reliance on physical interpretations. His efforts significantly advanced the development of the **axiomatic method**, influencing subsequent mathematical and logical research. \n\nWidely recognized for his influence on mathematical philosophy, Hilbert regarded mathematical objects as abstract entities, prioritizing their structural and relational properties. 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It is characterized by its reliance on strict logical deduction, systematic progression from assumptions to conclusions, and its role in upholding the precision and coherence of mathematical knowledge. \n\nKey aspects include: \n- **Logical Rigor**: Proofs employ deductive reasoning to ensure conclusions follow necessarily from premises. \n- **Foundation in Axioms**: They build upon accepted axioms and previously proven theorems. \n- **Methods**: Techniques like contradiction, direct/indirect reasoning, and induction are widely used. \n- **Universality**: The process transcends cultural or linguistic boundaries (e.g., termed \"mathematical proof\" in both Chinese and English contexts). \n\nIn summary, a mathematical proof is a definitive, structured demonstration of a proposition\u0027s truth, central to the integrity and advancement of mathematical theory."}, {"color": "#420e6c", "created_at": 1752674526, "description": 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{"color": "#d17193", "created_at": 1752674438, "description": "\u221a2 is an irrational number, commonly represented as 1.4142135... in decimal form, and cannot be expressed as a fraction, distinguishing it from rational numbers. It is a mathematical constant that represents the square root of 2 and is famously known as the first discovered irrational number. \u221a2 plays a significant role in mathematics, particularly in demonstrating the existence of irrational numbers and serving as a classic example of proof by contradiction. It also has a geometric interpretation as the length of the diagonal of a square with side length 1. Historically, \u221a2 played a pivotal role in ancient Greek mathematics, challenging the prevailing belief that all quantities could be expressed as ratios of integers. Its irrationality can be rigorously proven using properties of odd and even numbers, highlighting the importance of mathematical rigor and logical reasoning.", "entity_id": "\u221a2", "entity_type": "category", "file_path": "unknown_source", "id": "\u221a2", "label": "\u221a2", "shape": "dot", "size": 10, "source_id": "chunk-65ad5b0f72cf332d5f39730b4710be0d\u003cSEP\u003echunk-dc9988d8dd524d81a661c65645bca716\u003cSEP\u003echunk-1947ec0bba5d164b6d83fc97bec183ff\u003cSEP\u003echunk-5dace47f09692ec14aeb85d6e02520fc\u003cSEP\u003echunk-7f6551045ead5f4c49f6ae5d346c0299\u003cSEP\u003echunk-64dec23c12f739009609c53fc1b55511", "title": "\u221a2 is an irrational number, commonly represented as 1.4142135... in decimal form, and cannot be expressed as a fraction, distinguishing it from rational numbers. It is a mathematical constant that represents the square root of 2 and is famously known as the first discovered irrational number. \u221a2 plays a significant role in mathematics, particularly in demonstrating the existence of irrational numbers and serving as a classic example of proof by contradiction. It also has a geometric interpretation as the length of the diagonal of a square with side length 1. Historically, \u221a2 played a pivotal role in ancient Greek mathematics, challenging the prevailing belief that all quantities could be expressed as ratios of integers. Its irrationality can be rigorously proven using properties of odd and even numbers, highlighting the importance of mathematical rigor and logical reasoning."}, {"color": "#2da7eb", "created_at": 1752674438, "description": "The Fundamental Theorem of Algebra states that every non-zero polynomial equation has as many roots as its degree in the complex number system.\u003cSEP\u003e\u9ad8\u65af\u63d0\u51fa\u7684\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\uff0c\u7528\u4ee3\u6570\u56e0\u5b50\u4e58\u79ef\u65b9\u6cd5\u6784\u5efa\u9ad8\u6b21\u65b9\u7a0b\u89e3\u7684\u7406\u8bba\u57fa\u7840\u3002", "entity_id": "\u4ee3\u6570\u57fa\u672c\u5b9a\u7406", "entity_type": "category", "file_path": "unknown_source", "id": "\u4ee3\u6570\u57fa\u672c\u5b9a\u7406", "label": "\u4ee3\u6570\u57fa\u672c\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-65ad5b0f72cf332d5f39730b4710be0d\u003cSEP\u003echunk-fc1e60e87ee195f9f93e8e10e72c275a", "title": "The Fundamental Theorem of Algebra states that every non-zero polynomial equation has as many roots as its degree in the complex number system.\u003cSEP\u003e\u9ad8\u65af\u63d0\u51fa\u7684\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\uff0c\u7528\u4ee3\u6570\u56e0\u5b50\u4e58\u79ef\u65b9\u6cd5\u6784\u5efa\u9ad8\u6b21\u65b9\u7a0b\u89e3\u7684\u7406\u8bba\u57fa\u7840\u3002"}, {"color": "#f24ed2", "created_at": 1752674526, "description": "Pafnuty Chebyshev was a Russian mathematician who proved Bertrand\u0027s conjecture in 1850. The result is now known as the Bertrand-Chebyshev theorem.", "entity_id": "\u5207\u6bd4\u96ea\u592b", "entity_type": "person", "file_path": "unknown_source", "id": "\u5207\u6bd4\u96ea\u592b", "label": "\u5207\u6bd4\u96ea\u592b", "shape": "dot", "size": 10, "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "Pafnuty Chebyshev was a Russian mathematician who proved Bertrand\u0027s conjecture in 1850. The result is now known as the Bertrand-Chebyshev theorem."}, {"color": "#3c6b02", "created_at": 1752674526, "description": "Goldbach\u0027s conjecture is one of the oldest and best-known unsolved problems in number theory. 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It was first conjectured by Bertrand and proved by Chebyshev.", "entity_id": "\u4f2f\u7279\u5170-\u5207\u6bd4\u96ea\u592b\u5b9a\u7406", "entity_type": "category", "file_path": "unknown_source", "id": "\u4f2f\u7279\u5170-\u5207\u6bd4\u96ea\u592b\u5b9a\u7406", "label": "\u4f2f\u7279\u5170-\u5207\u6bd4\u96ea\u592b\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "The Bertrand-Chebyshev theorem states that for any natural number n \u003e 1, there exists at least one prime p such that n \u003c p \u003c 2n. It was first conjectured by Bertrand and proved by Chebyshev."}, {"color": "#36ccbc", "created_at": 1752674526, "description": "This is a Mersenne prime, a very large prime number with over 9 million digits. It represents the current largest known prime number mentioned in the text.", "entity_id": "2^30402457^-1", "entity_type": "category", "file_path": "unknown_source", "id": "2^30402457^-1", "label": "2^30402457^-1", "shape": "dot", "size": 10, "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "This is a Mersenne prime, a very large prime number with over 9 million digits. It represents the current largest known prime number mentioned in the text."}, {"color": "#9b45ff", "created_at": 1752674526, "description": "\u6570\u8bba\u662f\u6570\u5b66\u7684\u4e00\u4e2a\u5206\u652f\uff0c\u4e3b\u8981\u7814\u7a76\u6574\u6570\u7684\u6027\u8d28\uff0c\u5c24\u5176\u662f\u7d20\u6570\u548c\u5b8c\u6ee1\u6570\u7b49\uff0c\u4e5f\u4e0e\u73b0\u4ee3\u5bc6\u7801\u5b66\u5bc6\u5207\u76f8\u5173\u3002", "entity_id": "\u6570\u8bba", "entity_type": "category", "file_path": "unknown_source", "id": "\u6570\u8bba", "label": "\u6570\u8bba", "shape": "dot", "size": 10, "source_id": "chunk-5dace47f09692ec14aeb85d6e02520fc", "title": "\u6570\u8bba\u662f\u6570\u5b66\u7684\u4e00\u4e2a\u5206\u652f\uff0c\u4e3b\u8981\u7814\u7a76\u6574\u6570\u7684\u6027\u8d28\uff0c\u5c24\u5176\u662f\u7d20\u6570\u548c\u5b8c\u6ee1\u6570\u7b49\uff0c\u4e5f\u4e0e\u73b0\u4ee3\u5bc6\u7801\u5b66\u5bc6\u5207\u76f8\u5173\u3002"}, {"color": "#dd5f5c", "created_at": 1752674526, "description": 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systems.\u003cSEP\u003e\u4e8c\u8fdb\u5236\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\u5e38\u7528\u7684\u6570\u5236\u7cfb\u7edf\uff0c\u7528\u4e8e\u8868\u793a\u81ea\u7136\u6570\u548c\u8fdb\u884c\u8ba1\u7b97\uff0c\u4e0e\u5341\u8fdb\u5236\u7b49\u4ef7\u3002"}, {"color": "#45c3b5", "created_at": 1752674427, "description": "The duodecimal system is a base-12 numeral system, historically used in timekeeping and ancient calendars.\u003cSEP\u003e\u5341\u4e8c\u8fdb\u5236\u662f\u4e00\u79cd\u4ee512\u4e3a\u57fa\u6570\u7684\u8bb0\u6570\u7cfb\u7edf,\u53ef\u80fd\u4e0e\u6708\u5386\u7684\u4f7f\u7528\u6709\u5173\u3002", "entity_id": "\u5341\u4e8c\u8fdb\u5236", "entity_type": "category", "file_path": "unknown_source", "id": "\u5341\u4e8c\u8fdb\u5236", "label": "\u5341\u4e8c\u8fdb\u5236", "shape": "dot", "size": 10, "source_id": "chunk-2b803f52811c47a36d9c3c7c9900a33c\u003cSEP\u003echunk-1e23a044ee65399fb94239aba274f94c", "title": "The duodecimal system is a base-12 numeral system, historically used in timekeeping and ancient calendars.\u003cSEP\u003e\u5341\u4e8c\u8fdb\u5236\u662f\u4e00\u79cd\u4ee512\u4e3a\u57fa\u6570\u7684\u8bb0\u6570\u7cfb\u7edf,\u53ef\u80fd\u4e0e\u6708\u5386\u7684\u4f7f\u7528\u6709\u5173\u3002"}, {"color": "#d4d165", "created_at": 1752674427, "description": "Concept of place value in numbers, fundamental to understanding large numbers.\u003cSEP\u003ePlace value positions that determine a digit\u0027s numerical value in the decimal system.\u003cSEP\u003ePositional notation is the system where digit placement determines value magnitude in numeral systems.", "entity_id": "\u6570\u4f4d", "entity_type": "category", "file_path": "unknown_source", "id": "\u6570\u4f4d", "label": "\u6570\u4f4d", "shape": "dot", "size": 10, "source_id": "chunk-c737189a394d95d480e7c28933d632d2\u003cSEP\u003echunk-2b803f52811c47a36d9c3c7c9900a33c\u003cSEP\u003echunk-5a9059a6de66861a29258b33a73bfdc4", "title": "Concept of place value in numbers, fundamental to understanding large numbers.\u003cSEP\u003ePlace value positions that determine a digit\u0027s numerical value in the decimal system.\u003cSEP\u003ePositional notation is the system where digit placement determines value magnitude in numeral systems."}, {"color": "#79ad26", "created_at": 1752674438, "description": "\u7ebf\u6027\u7ec4\u5408(linear combination) in mathematics uses bases like 10^n to represent numbers, connecting algebra and geometry.", "entity_id": "\u7ebf\u6027\u7ec4\u5408", "entity_type": "category", "file_path": "unknown_source", "id": "\u7ebf\u6027\u7ec4\u5408", "label": "\u7ebf\u6027\u7ec4\u5408", "shape": "dot", "size": 10, "source_id": "chunk-64dec23c12f739009609c53fc1b55511", "title": "\u7ebf\u6027\u7ec4\u5408(linear combination) in mathematics uses bases like 10^n to represent numbers, connecting algebra and geometry."}, {"color": "#3ffc14", "created_at": 1752674438, "description": "\u5206\u6570\u5355\u4f4d(fractional units) are subdivided parts (e.g., 0.1, 0.01) critical for understanding decimal multiplication.\u003cSEP\u003e\u5206\u6570\u5355\u4f4drefers to the unit fraction (e.g., 1/2, 1/3) that serves as the building block for understanding more complex fractions like 2/3 or 3/4.", "entity_id": "\u5206\u6570\u5355\u4f4d", "entity_type": "category", "file_path": "unknown_source", "id": "\u5206\u6570\u5355\u4f4d", "label": "\u5206\u6570\u5355\u4f4d", "shape": "dot", "size": 10, "source_id": "chunk-447d851ef33dcb6cb0712f2ef985e50e\u003cSEP\u003echunk-64dec23c12f739009609c53fc1b55511", "title": "\u5206\u6570\u5355\u4f4d(fractional units) are subdivided parts (e.g., 0.1, 0.01) critical for understanding decimal multiplication.\u003cSEP\u003e\u5206\u6570\u5355\u4f4drefers to the unit fraction (e.g., 1/2, 1/3) that serves as the building block for understanding more complex fractions like 2/3 or 3/4."}, {"color": "#438211", "created_at": 1752674438, "description": "6.75\u5143is a decimal representation of currency (6 yuan, 7 jiao, 5 fen), illustrating practical use of decimals in daily life.", "entity_id": "6.75\u5143", "entity_type": "category", "file_path": "unknown_source", "id": "6.75\u5143", "label": "6.75\u5143", "shape": "dot", "size": 10, "source_id": "chunk-64dec23c12f739009609c53fc1b55511", "title": "6.75\u5143is a decimal representation of currency (6 yuan, 7 jiao, 5 fen), illustrating practical use of decimals in daily life."}, {"color": "#27941f", "created_at": 1752674485, "description": "China (\u4e2d\u56fd) is a country with a profound historical and cultural legacy, particularly in the fields of astronomy, mathematics, and scientific innovation. In ancient times, China developed sophisticated astronomical and mathematical systems, including the solar year calculation and the 60-year cyclical calendar. The civilization is also known for its use of the 24 solar terms (\u4e8c\u5341\u56db\u8282\u6c14) and the lunisolar calendar (\u9634\u9633\u5408\u5386), which incorporates intercalary months to align the lunar cycle with natural seasonal changes. \n\nChina is recognized as one of the key origins of the decimal numeral system, pioneering the use of digits like ten (\u5341), hundred (\u767e), and thousand (\u5343) to denote place value. It was among the earliest civilizations to conceptualize decimal notation, significantly contributing to the development of positional numeral systems. \n\nThe country has been home to many influential ancient thinkers and mathematicians, including Han Fei (\u97e9\u975e), Zhao Junqing (\u8d75\u541b\u537f), and Zhao Shuang (\u8d75\u723d), as well as later scholars such as Wang Chong (\u738b\u5145), Wu Wenjun (\u5434\u6587\u4fca), and Xu Guangqi (\u5f90\u5149\u542f), whose works have left lasting impacts on mathematical and philosophical traditions. \n\nIn modern times, China continues to be a prominent player in global mathematics research and academic achievement. Additionally, the nation has demonstrated excellence in international sports, as evidenced by its impressive medal tally in the 30th London Olympics\u201438 gold, 27 silver, and 23 bronze medals (88 total). \n\nThis summary integrates all provided descriptions into a cohesive overview, highlighting China\u0027s historical contributions to science and mathematics, its cultural heritage, and its contemporary achievements.", "entity_id": "\u4e2d\u56fd", "entity_type": "geo", "file_path": "unknown_source", "id": "\u4e2d\u56fd", "label": "\u4e2d\u56fd", "shape": "dot", "size": 10, "source_id": "chunk-26eeca6452bef6e940004dc39e11bc99\u003cSEP\u003echunk-be1ccf4aaa693e78676f243a79b63a08\u003cSEP\u003echunk-1e23a044ee65399fb94239aba274f94c\u003cSEP\u003echunk-71b68912b88bbd9937923ad7dc8e4bff\u003cSEP\u003echunk-bdbdfd36281dc6f99de8edf24b12b07e\u003cSEP\u003echunk-2a4876e1595a657157d330fa9e38f2f6\u003cSEP\u003echunk-4de08126c7eb1529ef6198b623197116", "title": "China (\u4e2d\u56fd) is a country with a profound historical and cultural legacy, particularly in the fields of astronomy, mathematics, and scientific innovation. In ancient times, China developed sophisticated astronomical and mathematical systems, including the solar year calculation and the 60-year cyclical calendar. The civilization is also known for its use of the 24 solar terms (\u4e8c\u5341\u56db\u8282\u6c14) and the lunisolar calendar (\u9634\u9633\u5408\u5386), which incorporates intercalary months to align the lunar cycle with natural seasonal changes. \n\nChina is recognized as one of the key origins of the decimal numeral system, pioneering the use of digits like ten (\u5341), hundred (\u767e), and thousand (\u5343) to denote place value. It was among the earliest civilizations to conceptualize decimal notation, significantly contributing to the development of positional numeral systems. \n\nThe country has been home to many influential ancient thinkers and mathematicians, including Han Fei (\u97e9\u975e), Zhao Junqing (\u8d75\u541b\u537f), and Zhao Shuang (\u8d75\u723d), as well as later scholars such as Wang Chong (\u738b\u5145), Wu Wenjun (\u5434\u6587\u4fca), and Xu Guangqi (\u5f90\u5149\u542f), whose works have left lasting impacts on mathematical and philosophical traditions. \n\nIn modern times, China continues to be a prominent player in global mathematics research and academic achievement. Additionally, the nation has demonstrated excellence in international sports, as evidenced by its impressive medal tally in the 30th London Olympics\u201438 gold, 27 silver, and 23 bronze medals (88 total). \n\nThis summary integrates all provided descriptions into a cohesive overview, highlighting China\u0027s historical contributions to science and mathematics, its cultural heritage, and its contemporary achievements."}, {"color": "#a8576d", "created_at": 1752674427, "description": "The sexagesimal system is a base-60 numeral system, traditionally used in astronomy and time 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c\u00b2\u3002\u8d39\u9a6c\u5927\u5b9a\u7406\u662f\u5bf9\u52fe\u80a1\u5b9a\u7406\u5728n\u22653\u65f6\u7684\u63a8\u5e7f\u731c\u60f3\u3002", "entity_id": "\u52fe\u80a1\u5b9a\u7406", "entity_type": "category", "file_path": "unknown_source", "id": "\u52fe\u80a1\u5b9a\u7406", "label": "\u52fe\u80a1\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-f97ce783119eb62752133fb048ac4a2f\u003cSEP\u003echunk-cceb7fc5191a1b7132a11703466c415d\u003cSEP\u003echunk-f2e7a029222fb754d59f5338d783171d\u003cSEP\u003echunk-c1f0f38352bf492f96567cff86102e58", "title": 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"title": "\u5e73\u79fbis a type of rigid motion where a figure is moved in a specific direction without rotation or reflection, preserving all distances and shape.\u003cSEP\u003e\u5e73\u79fb\u662f\u4e00\u79cd\u521a\u4f53\u53d8\u6362\uff0c\u4fdd\u6301\u56fe\u5f62\u4e2d\u4efb\u610f\u4e24\u70b9\u95f4\u7684\u8ddd\u79bb\u4e0d\u53d8\u3002\u003cSEP\u003e\u5e73\u79fb\u662f\u4e00\u79cd\u521a\u4f53\u8fd0\u52a8\uff0c\u4fdd\u6301\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\uff0c\u662f\u51e0\u4f55\u5b66\u4e2d\u7684\u57fa\u672c\u8fd0\u52a8\u4e4b\u4e00\u3002\u003cSEP\u003e\u5e73\u79fb\u662f\u4e00\u79cd\u56fe\u5f62\u8fd0\u52a8\u65b9\u5f0f\uff0c\u6307\u56fe\u5f62\u6cbf\u67d0\u4e00\u65b9\u5411\u79fb\u52a8\u4e00\u5b9a\u8ddd\u79bb\u800c\u4e0d\u6539\u53d8\u65b9\u5411\u6216\u5927\u5c0f\u3002"}, {"color": "#dfa609", "created_at": 1752674446, "description": 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"chunk-c1f0f38352bf492f96567cff86102e58", "title": "\u6b63\u591a\u8fb9\u5f62\u662f\u6240\u6709\u8fb9\u957f\u76f8\u7b49\u4e14\u6240\u6709\u5185\u89d2\u76f8\u7b49\u7684\u591a\u8fb9\u5f62\uff0c\u5177\u6709\u9ad8\u5ea6\u7684\u5bf9\u79f0\u6027\u3002"}, {"color": "#7ad06e", "created_at": 1752674452, "description": "In Euclidean geometry, the **\u70b9** (point) is the most fundamental abstract concept, serving as the basic building block for constructing geometric figures. Defined as having no size, a point represents a location in space without length, area, or volume. It is an idealized abstraction derived from real-world objects and is characterized solely by its position. Euclid, in his *Elements*, described the point as \"that which has no part,\" emphasizing its indivisible nature. Later, mathematician David Hilbert further formalized the concept, arguing that points should be defined symbolically rather than through physical attributes. Collectively, these descriptions establish the point as a primitive, dimensionless entity foundational to geometric theory. \n\n(Note: The output is in English, as requested, and consolidates all provided descriptions into a coherent summary, resolving any minor repetitions by synthesizing shared ideas.)", "entity_id": "\u70b9", "entity_type": "category", "file_path": "unknown_source", "id": "\u70b9", "label": "\u70b9", "shape": "dot", "size": 10, "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d\u003cSEP\u003echunk-d11cf0f89eb1830b6b1b56a3ccb1fa8d\u003cSEP\u003echunk-ff90db28e0dfaf333678ed9b64d39c95\u003cSEP\u003echunk-cea5b1b910550ead490e3df2cb11b540\u003cSEP\u003echunk-77809f023af3e8d2fb1e4c6cea56c329\u003cSEP\u003echunk-c1f0f38352bf492f96567cff86102e58", "title": "In Euclidean geometry, the **\u70b9** (point) is the most fundamental abstract concept, serving as the basic building block for constructing geometric figures. Defined as having no size, a point represents a location in space without length, area, or volume. It is an idealized abstraction derived from real-world objects and is characterized solely by its position. Euclid, in his *Elements*, described the point as \"that which has no part,\" emphasizing its indivisible nature. Later, mathematician David Hilbert further formalized the concept, arguing that points should be defined symbolically rather than through physical attributes. Collectively, these descriptions establish the point as a primitive, dimensionless entity foundational to geometric theory. \n\n(Note: The output is in English, as requested, and consolidates all provided descriptions into a coherent summary, resolving any minor repetitions by synthesizing shared ideas.)"}, {"color": "#55cf48", "created_at": 1752674459, "description": "In Euclidean geometry, a **\u7ebf (line)** is a fundamental geometric object composed of an infinite series of points, extending indefinitely without width. It is one of the basic constructs of geometric systems, alongside points and planes. A line possesses length but lacks width, as defined by Euclid\u2019s description of it as \"having length without breadth.\" This concept encompasses straight lines, rays, and line segments, abstracted from the edges of real-world objects. Mathematically, a line represents a continuous arrangement of points in space and serves as a core element in the study of geometry. It can manifest as either straight or curved, though its defining characteristic remains its one-dimensional nature. Collectively, lines, points, and planes form the foundational elements of spatial geometry. \n\nThe provided descriptions consistently emphasize the line\u2019s role as a basic geometric entity, devoid of width and composed of infinite points, with no contradictions among the statements.", "entity_id": "\u7ebf", "entity_type": "category", "file_path": "unknown_source", "id": "\u7ebf", "label": "\u7ebf", "shape": "dot", "size": 10, "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d\u003cSEP\u003echunk-d11cf0f89eb1830b6b1b56a3ccb1fa8d\u003cSEP\u003echunk-ff90db28e0dfaf333678ed9b64d39c95\u003cSEP\u003echunk-cea5b1b910550ead490e3df2cb11b540\u003cSEP\u003echunk-77809f023af3e8d2fb1e4c6cea56c329\u003cSEP\u003echunk-c1f0f38352bf492f96567cff86102e58", "title": "In Euclidean geometry, a **\u7ebf (line)** is a fundamental geometric object composed of an infinite series of points, extending indefinitely without width. It is one of the basic constructs of geometric systems, alongside points and planes. A line possesses length but lacks width, as defined by Euclid\u2019s description of it as \"having length without breadth.\" This concept encompasses straight lines, rays, and line segments, abstracted from the edges of real-world objects. Mathematically, a line represents a continuous arrangement of points in space and serves as a core element in the study of geometry. It can manifest as either straight or curved, though its defining characteristic remains its one-dimensional nature. Collectively, lines, points, and planes form the foundational elements of spatial geometry. \n\nThe provided descriptions consistently emphasize the line\u2019s role as a basic geometric entity, devoid of width and composed of infinite points, with no contradictions among the statements."}, {"color": "#f97d8d", "created_at": 1752674466, "description": "In Euclidean geometry, **\u9762** (surface) is a two-dimensional geometric object formed by the movement of a line, serving as a fundamental structure in plane geometry. It is a geometric concept without thickness, abstracted from the surfaces of real-world objects, and is regarded as a two-dimensional geometric element. As a basic unit in geometry, **\u9762** was formalized by Hilbert as part of a symbolic system. Defined by Euclid, it possesses length and width but no depth. Structurally, **\u9762** consists of an infinite number of lines, forming a two-dimensional graphical framework essential for describing shapes and spatial relationships. In summary, **\u9762** is a length-and-width-defined, infinitely line-composed geometric entity central to Euclidean and modern geometric theories.", "entity_id": "\u9762", "entity_type": "category", "file_path": "unknown_source", "id": "\u9762", "label": "\u9762", "shape": "dot", "size": 10, "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d\u003cSEP\u003echunk-d11cf0f89eb1830b6b1b56a3ccb1fa8d\u003cSEP\u003echunk-ff90db28e0dfaf333678ed9b64d39c95\u003cSEP\u003echunk-cea5b1b910550ead490e3df2cb11b540\u003cSEP\u003echunk-77809f023af3e8d2fb1e4c6cea56c329\u003cSEP\u003echunk-c1f0f38352bf492f96567cff86102e58", "title": "In Euclidean geometry, **\u9762** (surface) is a two-dimensional geometric object formed by the movement of a line, serving as a fundamental structure in plane geometry. It is a geometric concept without thickness, abstracted from the surfaces of real-world objects, and is regarded as a two-dimensional geometric element. As a basic unit in geometry, **\u9762** was formalized by Hilbert as part of a symbolic system. Defined by Euclid, it possesses length and width but no depth. Structurally, **\u9762** consists of an infinite number of lines, forming a two-dimensional graphical framework essential for describing shapes and spatial relationships. In summary, **\u9762** is a length-and-width-defined, infinitely line-composed geometric entity central to Euclidean and modern geometric theories."}, {"color": "#dbebb0", "created_at": 1752674466, "description": "\u4f53\u662f\u4e09\u7ef4\u7684\u51e0\u4f55\u6982\u5ff5\uff0c\u62bd\u8c61\u81ea\u73b0\u5b9e\u7269\u4f53\u672c\u8eab\uff0c\u5177\u6709\u957f\u5ea6\u3001\u5bbd\u5ea6\u548c\u9ad8\u5ea6\u3002\u003cSEP\u003e\u4f53\u662f\u7531\u9762\u7ec4\u6210\u7684\u4e09\u7ef4\u51e0\u4f55\u56fe\u5f62\uff0c\u6709\u957f\u5ea6\u3001\u5bbd\u5ea6\u548c\u539a\u5ea6\u3002", "entity_id": "\u4f53", "entity_type": "category", "file_path": "unknown_source", "id": "\u4f53", "label": "\u4f53", "shape": "dot", "size": 10, "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d\u003cSEP\u003echunk-c1f0f38352bf492f96567cff86102e58", "title": "\u4f53\u662f\u4e09\u7ef4\u7684\u51e0\u4f55\u6982\u5ff5\uff0c\u62bd\u8c61\u81ea\u73b0\u5b9e\u7269\u4f53\u672c\u8eab\uff0c\u5177\u6709\u957f\u5ea6\u3001\u5bbd\u5ea6\u548c\u9ad8\u5ea6\u3002\u003cSEP\u003e\u4f53\u662f\u7531\u9762\u7ec4\u6210\u7684\u4e09\u7ef4\u51e0\u4f55\u56fe\u5f62\uff0c\u6709\u957f\u5ea6\u3001\u5bbd\u5ea6\u548c\u539a\u5ea6\u3002"}, {"color": "#f3fd8e", "created_at": 1752674473, "description": "**\u89d2** is a fundamental geometric concept originally defined by Euclid in *Elements* as the inclination between two intersecting lines. It is formed by two rays sharing a common endpoint and is used to describe the degree of inclination between the lines or the change in direction. In geometry, angles serve as a basic element for analyzing the relationships between shapes and measuring directional changes. \n\nWhile angles are commonly introduced in elementary mathematics, their precise definition can be challenging at that level. Importantly, the size of an angle is not determined by the lengths of its sides but rather by the chord or arc length of the corresponding unit circle. Thus, angles are essential in describing geometric configurations independently of scale. \n\nIn summary, **\u89d2** is a core geometric figure consisting of two rays with a shared origin, widely applied in mathematics to quantify inclination, directional variation, and spatial relationships between lines and shapes.", "entity_id": "\u89d2", "entity_type": "category", "file_path": "unknown_source", "id": "\u89d2", "label": "\u89d2", "shape": "dot", "size": 10, "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d\u003cSEP\u003echunk-b8cb0b3bb1cff9a2ccf4623209e1bea6\u003cSEP\u003echunk-4ad0fa3c80600287487ea0909ce52c30\u003cSEP\u003echunk-cea5b1b910550ead490e3df2cb11b540\u003cSEP\u003echunk-77809f023af3e8d2fb1e4c6cea56c329\u003cSEP\u003echunk-c1f0f38352bf492f96567cff86102e58", "title": "**\u89d2** is a fundamental geometric concept originally defined by Euclid in *Elements* as the inclination between two intersecting lines. It is formed by two rays sharing a common endpoint and is used to describe the degree of inclination between the lines or the change in direction. In geometry, angles serve as a basic element for analyzing the relationships between shapes and measuring directional changes. \n\nWhile angles are commonly introduced in elementary mathematics, their precise definition can be challenging at that level. Importantly, the size of an angle is not determined by the lengths of its sides but rather by the chord or arc length of the corresponding unit circle. Thus, angles are essential in describing geometric configurations independently of scale. \n\nIn summary, **\u89d2** is a core geometric figure consisting of two rays with a shared origin, widely applied in mathematics to quantify inclination, directional variation, and spatial relationships between lines and shapes."}, {"color": "#0601c3", "created_at": 1752674473, "description": "\u5355\u4f4d\u5706\u662f\u534a\u5f84\u4e3a1\u7684\u5706\uff0c\u7528\u4e8e\u7edf\u4e00\u5ea6\u91cf\u89d2\u7684\u5927\u5c0f\uff0c\u901a\u8fc7\u5f26\u957f\u6216\u5f27\u957f\u6bd4\u8f83\u89d2\u7684\u5927\u5c0f\u3002", "entity_id": "\u5355\u4f4d\u5706", "entity_type": "category", "file_path": "unknown_source", "id": "\u5355\u4f4d\u5706", "label": "\u5355\u4f4d\u5706", "shape": "dot", "size": 10, "source_id": "chunk-b8cb0b3bb1cff9a2ccf4623209e1bea6", "title": 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"\u8d39\u9a6c\u662f\u6cd5\u56fd\u6570\u5b66\u5bb6\uff0c\u4ed6\u5728\u9605\u8bfb\u300a\u7b97\u672f\u300b\u65f6\u63d0\u51fa\u4e86\u8457\u540d\u7684\u8d39\u9a6c\u5927\u5b9a\u7406\uff0c\u5373\u5173\u4e8ea^n + b^n = c^n\u5728n\u22653\u65f6\u65e0\u6574\u6570\u89e3\u7684\u731c\u60f3\u3002"}, {"color": "#090e4c", "created_at": 1752674526, "description": "\u65b9\u7a0b\u6574\u6570\u89e3\u95ee\u9898\u662f\u7814\u7a76\u65b9\u7a0b\u7684\u6b63\u6574\u6570\u89e3\uff0c\u4e22\u756a\u56fe\u5bf9\u6b64\u8fdb\u884c\u4e86\u7cfb\u7edf\u7814\u7a76\uff0c\u540e\u4eba\u5c06\u6b64\u7c7b\u95ee\u9898\u79f0\u4e3a\u4e22\u756a\u56fe\u95ee\u9898\u3002", "entity_id": "\u65b9\u7a0b\u6574\u6570\u89e3\u95ee\u9898", "entity_type": "category", "file_path": "unknown_source", "id": "\u65b9\u7a0b\u6574\u6570\u89e3\u95ee\u9898", "label": "\u65b9\u7a0b\u6574\u6570\u89e3\u95ee\u9898", "shape": "dot", "size": 10, "source_id": "chunk-f2e7a029222fb754d59f5338d783171d", "title": "\u65b9\u7a0b\u6574\u6570\u89e3\u95ee\u9898\u662f\u7814\u7a76\u65b9\u7a0b\u7684\u6b63\u6574\u6570\u89e3\uff0c\u4e22\u756a\u56fe\u5bf9\u6b64\u8fdb\u884c\u4e86\u7cfb\u7edf\u7814\u7a76\uff0c\u540e\u4eba\u5c06\u6b64\u7c7b\u95ee\u9898\u79f0\u4e3a\u4e22\u756a\u56fe\u95ee\u9898\u3002"}, {"color": "#3e9d5b", "created_at": 1752674526, "description": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u662f\u5f62\u5982ax\u00b2 + bx + c = 0\u7684\u4ee3\u6570\u65b9\u7a0b\uff0c\u97e6\u8fbe\u901a\u8fc7\u5f15\u5165\u5b57\u6bcd\u8868\u793a\u6cd5\uff0c\u63a8\u5bfc\u51fa\u5176\u6839\u4e0e\u7cfb\u6570\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5373\u97e6\u8fbe\u5b9a\u7406\u3002", "entity_id": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b", "entity_type": "category", "file_path": "unknown_source", "id": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b", "label": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b", "shape": "dot", "size": 10, "source_id": "chunk-f2e7a029222fb754d59f5338d783171d", "title": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u662f\u5f62\u5982ax\u00b2 + bx + c = 0\u7684\u4ee3\u6570\u65b9\u7a0b\uff0c\u97e6\u8fbe\u901a\u8fc7\u5f15\u5165\u5b57\u6bcd\u8868\u793a\u6cd5\uff0c\u63a8\u5bfc\u51fa\u5176\u6839\u4e0e\u7cfb\u6570\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5373\u97e6\u8fbe\u5b9a\u7406\u3002"}, {"color": "#e8037a", "created_at": 1752674526, "description": "\u4ee3\u6570\u5b66\u53d1\u5c55\u662f\u6570\u5b66\u53f2\u7684\u91cd\u8981\u5206\u652f\uff0c\u8d77\u6e90\u4e8e\u53e4\u5e0c\u814a\u6570\u5b66\u5bb6\u4e22\u756a\u56fe\u7684\u5de5\u4f5c\uff0c\u5e76\u9010\u6b65\u6f14\u53d8\u4e3a\u73b0\u4ee3\u4ee3\u6570\u4f53\u7cfb\u3002", "entity_id": "\u4ee3\u6570\u5b66\u53d1\u5c55", "entity_type": "category", "file_path": "unknown_source", "id": "\u4ee3\u6570\u5b66\u53d1\u5c55", "label": "\u4ee3\u6570\u5b66\u53d1\u5c55", "shape": "dot", "size": 10, "source_id": "chunk-ce575de1a4a5a428b616adecd09f8d8f", "title": "\u4ee3\u6570\u5b66\u53d1\u5c55\u662f\u6570\u5b66\u53f2\u7684\u91cd\u8981\u5206\u652f\uff0c\u8d77\u6e90\u4e8e\u53e4\u5e0c\u814a\u6570\u5b66\u5bb6\u4e22\u756a\u56fe\u7684\u5de5\u4f5c\uff0c\u5e76\u9010\u6b65\u6f14\u53d8\u4e3a\u73b0\u4ee3\u4ee3\u6570\u4f53\u7cfb\u3002"}, {"color": "#314be6", "created_at": 1752674526, "description": "\u6000\u5c14\u65af\u662f\u82f1\u56fd\u6570\u5b66\u5bb6\uff0c\u4ed6\u57281993\u5e74\u6210\u529f\u8bc1\u660e\u4e86\u8d39\u9a6c\u5927\u5b9a\u7406\uff0c\u5e76\u4e8e1995\u5e74\u53d1\u8868\u957f\u8fbe130\u9875\u7684\u8bba\u6587\u3002", "entity_id": "\u6000\u5c14\u65af", "entity_type": "person", "file_path": "unknown_source", "id": "\u6000\u5c14\u65af", "label": "\u6000\u5c14\u65af", "shape": "dot", "size": 10, "source_id": "chunk-f2e7a029222fb754d59f5338d783171d", "title": "\u6000\u5c14\u65af\u662f\u82f1\u56fd\u6570\u5b66\u5bb6\uff0c\u4ed6\u57281993\u5e74\u6210\u529f\u8bc1\u660e\u4e86\u8d39\u9a6c\u5927\u5b9a\u7406\uff0c\u5e76\u4e8e1995\u5e74\u53d1\u8868\u957f\u8fbe130\u9875\u7684\u8bba\u6587\u3002"}, {"color": "#0658d5", "created_at": 1752674566, "description": "\u897f\u8499\u00b7\u8f9b\u683c\u662f\u300a\u8d39\u9a6c\u5927\u5b9a\u7406\u300b\u7684\u4f5c\u8005\uff0c\u79d1\u666e\u4f5c\u5bb6\u3002", "entity_id": "\u8f9b\u683c", "entity_type": "person", "file_path": "unknown_source", "id": "\u8f9b\u683c", "label": "\u8f9b\u683c", "shape": "dot", "size": 10, "source_id": "chunk-bbe8d346dcd3b2ff7c7f361b5b5adb4a", "title": "\u897f\u8499\u00b7\u8f9b\u683c\u662f\u300a\u8d39\u9a6c\u5927\u5b9a\u7406\u300b\u7684\u4f5c\u8005\uff0c\u79d1\u666e\u4f5c\u5bb6\u3002"}, {"color": "#0b9039", "created_at": 1752674526, "description": 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c/a\u3002\u003cSEP\u003e\u97e6\u8fbe\u5b9a\u7406\u662f\u63cf\u8ff0\u591a\u9879\u5f0f\u65b9\u7a0b\u6839\u4e0e\u7cfb\u6570\u4e4b\u95f4\u5173\u7cfb\u7684\u6570\u5b66\u5b9a\u7406\uff0c\u662f\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\u7684\u91cd\u8981\u5e94\u7528\uff0c\u7531\u6cd5\u56fd\u6570\u5b66\u5bb6\u97e6\u8fbe\u63d0\u51fa\u3002", "entity_id": "\u97e6\u8fbe\u5b9a\u7406", "entity_type": "event", "file_path": "unknown_source", "id": "\u97e6\u8fbe\u5b9a\u7406", "label": "\u97e6\u8fbe\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-f97ce783119eb62752133fb048ac4a2f\u003cSEP\u003echunk-f2e7a029222fb754d59f5338d783171d", "title": "\u97e6\u8fbe\u5b9a\u7406\u63cf\u8ff0\u4e86\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u7684\u6839\u4e0e\u7cfb\u6570\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5373x\u2081 + x\u2082 = -b/a\u548cx\u2081\u00b7x\u2082 = c/a\u3002\u003cSEP\u003e\u97e6\u8fbe\u5b9a\u7406\u662f\u63cf\u8ff0\u591a\u9879\u5f0f\u65b9\u7a0b\u6839\u4e0e\u7cfb\u6570\u4e4b\u95f4\u5173\u7cfb\u7684\u6570\u5b66\u5b9a\u7406\uff0c\u662f\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\u7684\u91cd\u8981\u5e94\u7528\uff0c\u7531\u6cd5\u56fd\u6570\u5b66\u5bb6\u97e6\u8fbe\u63d0\u51fa\u3002"}, {"color": "#00ca82", "created_at": 1752674526, "description": "\u7b1b\u5361\u5c14\u662f17\u4e16\u7eaa\u6cd5\u56fd\u54f2\u5b66\u5bb6\u548c\u6570\u5b66\u5bb6\uff0c\u89e3\u6790\u51e0\u4f55\u7684\u521b\u59cb\u4eba\uff0c\u5bf9\u4ee3\u6570\u7b26\u53f7\u7cfb\u7edf\u8fdb\u884c\u4e86\u8fdb\u4e00\u6b65\u6539\u8fdb\uff0c\u4f7f\u7528a\u3001b\u3001c\u8868\u793a\u5df2\u77e5\u91cf\uff0cx\u3001y\u3001z\u8868\u793a\u672a\u77e5\u91cf\u3002\u003cSEP\u003e\u7b1b\u5361\u5c14\u662f\u6cd5\u56fd\u54f2\u5b66\u5bb6\u548c\u6570\u5b66\u5bb6\uff0c\u5c06\u201c\u4ee5\u592a\u201d\u6982\u5ff5\u5f15\u5165\u79d1\u5b66\u9886\u57df\u3002", "entity_id": "\u7b1b\u5361\u5c14", "entity_type": "person", 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numbers", "source_id": "chunk-112d4fcad80ef84a1327a06d05614630", "title": "\u6570\u8f74\u662f\u89e3\u91ca\u8d1f\u6570\u548c\u76f8\u53cd\u6570\u7684\u5e38\u7528\u5de5\u5177\uff0c\u4f46\u4e0d\u80fd\u7528\u4e8e\u5b9a\u4e49\u8d1f\u6570\u672c\u8eab\u3002", "to": "\u6570\u8f74", "width": 8.0}, {"created_at": 1752674566, "description": "Fractions form the basis of rational numbers, which were initially thought to encompass all measurable quantities.\u003cSEP\u003eRational numbers are defined as ratios of integers, hence they can all be expressed as fractions.", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "fractional form,mathematical concepts,number classification,number systems", "source_id": "chunk-fc1e60e87ee195f9f93e8e10e72c275a\u003cSEP\u003echunk-f0d039ecac6d9749d604feadbbc70472", "title": "Fractions form the basis of rational numbers, which were initially thought to encompass all measurable quantities.\u003cSEP\u003eRational numbers are defined as ratios of integers, hence they can all be expressed as fractions.", "to": "\u6709\u7406\u6570", "width": 18.0}, {"created_at": 1752674566, "description": "Finite decimals are shown to be equivalent to fractions with denominators that are powers of ten.", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "decimal representation,fractional equivalence", "source_id": "chunk-f0d039ecac6d9749d604feadbbc70472", "title": "Finite decimals are shown to be equivalent to fractions with denominators that are powers of ten.", "to": "\u6709\u9650\u5c0f\u6570", "width": 10.0}, {"created_at": 1752674566, "description": "Infinite repeating decimals are shown to be expressible as fractions through the use of geometric series.", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "decimal expansion,fractional equivalence", "source_id": "chunk-f0d039ecac6d9749d604feadbbc70472", "title": "Infinite repeating decimals are shown to be expressible as fractions through the use of geometric series.", "to": "\u65e0\u9650\u5faa\u73af\u5c0f\u6570", "width": 10.0}, {"created_at": 1752674566, "description": "\u6bd4\u4f8b\u5173\u7cfb\u901a\u5e38\u901a\u8fc7\u5206\u6570\u5f62\u5f0f\u8868\u793a\u4e24\u4e2a\u6570\u91cf\u4e4b\u95f4\u7684\u6bd4\u7387\uff0c\u5982\u0027\u9e45\u662f\u9e2d\u5b50\u6570\u91cf\u76841/3\u0027\u3002", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "\u6570\u5b66\u6982\u5ff5,\u6bd4\u7387\u8868\u8fbe", "source_id": "chunk-ce575de1a4a5a428b616adecd09f8d8f", "title": "\u6bd4\u4f8b\u5173\u7cfb\u901a\u5e38\u901a\u8fc7\u5206\u6570\u5f62\u5f0f\u8868\u793a\u4e24\u4e2a\u6570\u91cf\u4e4b\u95f4\u7684\u6bd4\u7387\uff0c\u5982\u0027\u9e45\u662f\u9e2d\u5b50\u6570\u91cf\u76841/3\u0027\u3002", "to": "\u6bd4\u4f8b\u5173\u7cfb", "width": 9.0}, {"created_at": 1752674566, "description": "\u5206\u6570\u548c\u5c0f\u6570\u662f\u8868\u793a\u540c\u4e00\u7c7b\u6570\u91cf\u7684\u4e0d\u540c\u65b9\u5f0f\uff0c\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u59821/2 = 0.5\u3002", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "conversion,numerical representation", "source_id": "chunk-5f9c1f0cf6bfb67dfcbdfbdf95b7c389", "title": "\u5206\u6570\u548c\u5c0f\u6570\u662f\u8868\u793a\u540c\u4e00\u7c7b\u6570\u91cf\u7684\u4e0d\u540c\u65b9\u5f0f\uff0c\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u59821/2 = 0.5\u3002", "to": "\u5c0f\u6570", "width": 9.0}, {"created_at": 1752674566, "description": "\u5206\u6570\u53ef\u4ee5\u901a\u8fc7\u6362\u7b97\u8f6c\u5316\u4e3a\u767e\u5206\u6570\uff0c\u59821/2 = 50%\uff0c\u7528\u4e8e\u5b9e\u9645\u95ee\u9898\u4e2d\u7684\u6bd4\u8f83\u548c\u5e94\u7528\u3002", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "numerical equivalence,practical use", "source_id": "chunk-5f9c1f0cf6bfb67dfcbdfbdf95b7c389", "title": "\u5206\u6570\u53ef\u4ee5\u901a\u8fc7\u6362\u7b97\u8f6c\u5316\u4e3a\u767e\u5206\u6570\uff0c\u59821/2 = 50%\uff0c\u7528\u4e8e\u5b9e\u9645\u95ee\u9898\u4e2d\u7684\u6bd4\u8f83\u548c\u5e94\u7528\u3002", "to": "\u767e\u5206\u6570", "width": 8.0}, {"created_at": 1752674566, "description": "\u7b2c\u4e8c\u5b66\u6bb5\u662f\u5b66\u751f\u6df1\u5165\u7406\u89e3\u548c\u5e94\u7528\u5206\u6570\u7684\u5173\u952e\u5b66\u4e60\u9636\u6bb5\u3002", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "\u5b66\u4e60\u9636\u6bb5,\u6570\u5b66\u5185\u5bb9", "source_id": "chunk-447d851ef33dcb6cb0712f2ef985e50e", "title": "\u7b2c\u4e8c\u5b66\u6bb5\u662f\u5b66\u751f\u6df1\u5165\u7406\u89e3\u548c\u5e94\u7528\u5206\u6570\u7684\u5173\u952e\u5b66\u4e60\u9636\u6bb5\u3002", "to": "\u7b2c\u4e8c\u5b66\u6bb5", "width": 9.0}, {"created_at": 1752674566, "description": "\u7b2c\u4e00\u5b66\u6bb5\u662f\u5b66\u751f\u521d\u6b65\u8ba4\u8bc6\u5206\u6570\u6982\u5ff5\u7684\u5b66\u4e60\u9636\u6bb5\u3002", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": 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"chunk-447d851ef33dcb6cb0712f2ef985e50e", "title": "\u5206\u6570are introduced through the concept of\u5e73\u5747\u5206(equal partitioning), where a whole is divided into equal parts and each part represents a fraction.", "to": "\u5e73\u5747\u5206", "width": 10.0}, {"created_at": 1752674566, "description": "\u6559\u5b66\u8981\u70b9include key ideas that help students understand\u5206\u6570, such as identifying the whole, equal division, and recognizing unit fractions.", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "conceptual clarity,teaching strategy", "source_id": "chunk-447d851ef33dcb6cb0712f2ef985e50e", "title": "\u6559\u5b66\u8981\u70b9include key ideas that help students understand\u5206\u6570, such as identifying the whole, equal division, and recognizing unit fractions.", "to": "\u6559\u5b66\u8981\u70b9", "width": 9.0}, {"created_at": 1752674566, "description": "\u5206\u6570and\u5c0f\u6570\u7684\u8ba4\u8bc6are both used to express part-whole relationships, and understanding one helps in understanding the other.", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "conceptual parallel,mathematical relationship", "source_id": "chunk-f9fb4bf24eb3450dc8d14d30db6e18ae", "title": "\u5206\u6570and\u5c0f\u6570\u7684\u8ba4\u8bc6are both used to express part-whole relationships, and understanding one helps in understanding the other.", "to": "\u5c0f\u6570\u7684\u8ba4\u8bc6", "width": 8.0}, {"created_at": 1752674566, "description": "\u6574\u4f531 is the foundational concept in understanding\u5206\u6570, as it represents the whole that is divided into equal parts.", "file_path": "unknown_source", "from": "\u5206\u6570", "keywords": "mathematical basis,part-whole relationship", "source_id": "chunk-f9fb4bf24eb3450dc8d14d30db6e18ae", "title": "\u6574\u4f531 is the foundational concept in understanding\u5206\u6570, as it represents the whole that is divided into equal parts.", "to": "\u6574\u4f531", "width": 9.0}, {"created_at": 1752674566, "description": "\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570are distinguished through their\u5c0f\u6570representations.", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "decimal properties,number classification", "source_id": "chunk-a9a60418b119b40b9f7e09f08cc0b5c8", "title": "\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570are distinguished through their\u5c0f\u6570representations.", "to": "\u6709\u7406\u6570\u4e0e\u65e0\u7406\u6570", "width": 8.0}, {"created_at": 1752674566, "description": "\u5c0f\u6570can be converted into\u767e\u5206\u6570and vice versa, forming a key part of the curriculum in the second learning stage.\u003cSEP\u003e\u5c0f\u6570\u53ef\u4ee5\u901a\u8fc7\u4e58\u4ee5100\u8f6c\u5316\u4e3a\u767e\u5206\u6570\uff0c\u59820.5 = 50%\uff0c\u7528\u4e8e\u8868\u793a\u6bd4\u4f8b\u5173\u7cfb\u3002", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "conversion,numerical representation,proportional representation", "source_id": "chunk-447d851ef33dcb6cb0712f2ef985e50e\u003cSEP\u003echunk-5f9c1f0cf6bfb67dfcbdfbdf95b7c389", "title": "\u5c0f\u6570can be converted into\u767e\u5206\u6570and vice versa, forming a key part of the curriculum in the second learning stage.\u003cSEP\u003e\u5c0f\u6570\u53ef\u4ee5\u901a\u8fc7\u4e58\u4ee5100\u8f6c\u5316\u4e3a\u767e\u5206\u6570\uff0c\u59820.5 = 50%\uff0c\u7528\u4e8e\u8868\u793a\u6bd4\u4f8b\u5173\u7cfb\u3002", "to": "\u767e\u5206\u6570", "width": 18.0}, {"created_at": 1752674566, "description": "\u674e\u5b81\u5b81\u53c2\u4e0e\u4e86\u6709\u5173\u5c0f\u6570\u6559\u5b66\u8bbe\u8ba1\u7684\u8ba8\u8bba,\u4f53\u73b0\u4e86\u5176\u5728\u6570\u5b66\u6559\u5b66\u4e2d\u7684\u4e13\u4e1a\u8d21\u732e\u3002", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "\u6559\u5b66\u8bbe\u8ba1,\u6570\u5b66\u5185\u5bb9", "source_id": "chunk-447d851ef33dcb6cb0712f2ef985e50e", "title": 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particularly in understanding how numbers relate to real-world measurements and quantities.", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "conceptual development,numerical literacy", "source_id": "chunk-6bbe77567ec618aec822acfe88217048", "title": "Decimals are used as part of developing number sense, particularly in understanding how numbers relate to real-world measurements and quantities.", "to": "\u6570\u611f", "width": 7.0}, {"created_at": 1752674566, "description": "Decimals are an extension of the decimal (base-ten) system, and understanding this system is essential to grasping decimal numbers.", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "numerical structure,place value", "source_id": "chunk-6bbe77567ec618aec822acfe88217048", "title": "Decimals are an extension of the decimal (base-ten) system, and understanding this system is essential to grasping decimal numbers.", "to": "\u5341\u8fdb\u5236", "width": 10.0}, {"created_at": 1752674566, "description": "The monetary units yuan, jiao, and fen provide a concrete example of the decimal system and are used to introduce and reinforce the concept of decimals.", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "numerical representation,real-world application", "source_id": "chunk-6bbe77567ec618aec822acfe88217048", "title": "The monetary units yuan, jiao, and fen provide a concrete example of the decimal system and are used to introduce and reinforce the concept of decimals.", "to": "\u5143\u3001\u89d2\u3001\u5206", "width": 9.0}, {"created_at": 1752674566, "description": "Shopping activities are used as a practical context to help students visualize and understand decimal numbers in everyday life.", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "contextual learning,practical math", "source_id": "chunk-6bbe77567ec618aec822acfe88217048", "title": "Shopping activities are used as a practical context to help students visualize and understand decimal numbers in everyday life.", "to": "\u8d2d\u7269\u6d3b\u52a8", "width": 8.0}, {"created_at": 1752674566, "description": "The real-life value of decimals is emphasized in teaching to help students understand their relevance in daily activities such as shopping and measurement.", "file_path": "unknown_source", "from": "\u5c0f\u6570", "keywords": "contextual learning,practical application", "source_id": "chunk-6bbe77567ec618aec822acfe88217048", "title": "The real-life value of decimals is emphasized in teaching to help students understand their relevance in daily activities such as shopping and measurement.", "to": "\u73b0\u5b9e\u751f\u6d3b\u4e2d\u7684\u4ef7\u503c\u4e0e\u4f5c\u7528", "width": 9.0}, {"created_at": 1752674566, "description": "\u6570\u611fis a core concept in \u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6\u300b, linking abstract numbers to real-world contexts.", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "education policy,mathematical literacy", "source_id": "chunk-64dec23c12f739009609c53fc1b55511", "title": "\u6570\u611fis a core concept in \u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6\u300b, linking abstract numbers to real-world contexts.", "to": "\u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6\u300b", "width": 8.0}, {"created_at": 1752674566, "description": "\u6570\u611f\u5305\u62ec\u5bf9\u8fd0\u7b97\u7ed3\u679c\u7684\u76f4\u89c9\u5224\u65ad\uff0c\u4f30\u7b97\u5219\u662f\u5bf9\u6570\u91cf\u7684\u8fd1\u4f3c\u8ba1\u7b97\uff0c\u4e8c\u8005\u90fd\u5f3a\u8c03\u6570\u4e0e\u73b0\u5b9e\u80cc\u666f\u7684\u8054\u7cfb\u3002", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "\u6570\u5b66\u7d20\u517b,\u73b0\u5b9e\u5e94\u7528", "source_id": "chunk-5c564427093279bdf3c25848196fe5f8", "title": "\u6570\u611f\u5305\u62ec\u5bf9\u8fd0\u7b97\u7ed3\u679c\u7684\u76f4\u89c9\u5224\u65ad\uff0c\u4f30\u7b97\u5219\u662f\u5bf9\u6570\u91cf\u7684\u8fd1\u4f3c\u8ba1\u7b97\uff0c\u4e8c\u8005\u90fd\u5f3a\u8c03\u6570\u4e0e\u73b0\u5b9e\u80cc\u666f\u7684\u8054\u7cfb\u3002", "to": "\u4f30\u7b97", "width": 8.0}, {"created_at": 1752674566, "description": "\"1000 is used as a key number in\u6570\u611fdevelopment to help students understand large quantities and transition from concrete to abstract numerical reasoning.\"\\\u003cSEP\u003eThe number 1000 is used as a vehicle to explore and develop number sense through real-world investigations and discussions.", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "contextual learning,numerical understanding,real-world context", "source_id": "chunk-612d665e8236ed57ac928a378892167b\u003cSEP\u003echunk-6bbe77567ec618aec822acfe88217048", "title": "\"1000 is used as a key number in\u6570\u611fdevelopment to help students understand large quantities and transition from concrete to abstract numerical reasoning.\"\\\u003cSEP\u003eThe number 1000 is used as a vehicle to explore and develop number sense through real-world investigations and discussions.", "to": "1000", "width": 17.0}, {"created_at": 1752674566, "description": "Understanding the decimal system is foundational to developing number sense, particularly in how numbers relate to each other and to quantities.", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "numerical reasoning,system understanding", "source_id": "chunk-6bbe77567ec618aec822acfe88217048", "title": "Understanding the decimal system is foundational to developing number sense, particularly in how numbers relate to each other and to quantities.", "to": "\u5341\u8fdb\u5236", "width": 8.0}, {"created_at": 1752674566, "description": "The teaching design analysis outlines strategies to help students develop number sense by linking abstract numbers to real-life situations.", "file_path": 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natural number addition contributes to the development of\u6570\u611fby helping students connect mathematical operations with real-world contexts and intuitive reasoning.\"\\", "to": "\u81ea\u7136\u6570\u7684\u52a0\u6cd5", "width": 9.0}, {"created_at": 1752674566, "description": "\"The\u7efc\u5408\u4e0e\u5b9e\u8df5\u8bfeis specifically designed to enhance students\u0027 \u6570\u611fthrough hands-on, real-life contextual learning and exploration of numerical concepts.\"\\\u003cSEP\u003e\"\u7efc\u5408\u4e0e\u5b9e\u8df5\u8bfeis designed specifically to build\u6570\u611fthrough contextual and experiential learning activities that help students connect numbers to real-life situations.\"\\", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "cognitive development,conceptual development,pedagogical strategy", "source_id": "chunk-612d665e8236ed57ac928a378892167b", "title": "\"The\u7efc\u5408\u4e0e\u5b9e\u8df5\u8bfeis specifically designed to enhance students\u0027 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are implemented to enhance early mathematics learning.\"\\", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "educational environment,instructional focus", "source_id": "chunk-612d665e8236ed57ac928a378892167b", "title": "\"\u4e1c\u5317\u5e08\u5927\u9644\u5c0fis the educational setting where\u6570\u611f-focused lessons, such as the one designed by\u5218\u8273\u5e73, are implemented to enhance early mathematics learning.\"\\", "to": "\u4e1c\u5317\u5e08\u5927\u9644\u5c0f", "width": 9.0}, {"created_at": 1752674566, "description": "\"According to the\u8bfe\u7a0b\u6807\u51c6, \u6570\u611fis a key learning objective in the\u7b2c\u4e00\u5b66\u6bb5, guiding how foundational math concepts are taught and assessed.\"\\", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "conceptual focus,standards-based learning", "source_id": "chunk-612d665e8236ed57ac928a378892167b", "title": "\"According to the\u8bfe\u7a0b\u6807\u51c6, \u6570\u611fis a key learning objective in the\u7b2c\u4e00\u5b66\u6bb5, guiding how foundational math concepts are taught and assessed.\"\\", "to": "\u8bfe\u7a0b\u6807\u51c6", "width": 10.0}, {"created_at": 1752674566, "description": "\"\u6559\u5b66\u8bbe\u8ba1for\u6570\u611fdevelopment involves creating structured lessons that use contextual and comparative examples to build numerical intuition.\"\\", "file_path": "unknown_source", "from": "\u6570\u611f", "keywords": "cognitive development,lesson planning", "source_id": "chunk-612d665e8236ed57ac928a378892167b", "title": "\"\u6559\u5b66\u8bbe\u8ba1for\u6570\u611fdevelopment involves creating structured lessons that use contextual and comparative examples to build numerical intuition.\"\\", "to": "\u6559\u5b66\u8bbe\u8ba1", "width": 9.0}, {"created_at": 1752674566, "description": "Subtraction is defined as the inverse operation of addition in the set of integers.\u003cSEP\u003eSubtraction is fundamentally understood as the inverse operation of addition.\u003cSEP\u003e\u51cf\u6cd5\u662f\u52a0\u6cd5\u7684\u9006\u8fd0\u7b97\uff0c\u901a\u8fc7\u52a0\u6cd5\u5b9a\u4e49\uff0c\u6ee1\u8db3a - b = c \u2194 a = b + c\u3002", "file_path": "unknown_source", "from": "\u52a0\u6cd5", "keywords": "arithmetic,inverse operations,\u8fd0\u7b97\u5173\u7cfb,\u9006\u8fd0\u7b97", "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892", "title": "Subtraction is defined as the inverse operation of addition in the set of integers.\u003cSEP\u003eSubtraction is fundamentally understood as the inverse operation of addition.\u003cSEP\u003e\u51cf\u6cd5\u662f\u52a0\u6cd5\u7684\u9006\u8fd0\u7b97\uff0c\u901a\u8fc7\u52a0\u6cd5\u5b9a\u4e49\uff0c\u6ee1\u8db3a - b = c \u2194 a = b + c\u3002", "to": "\u51cf\u6cd5", "width": 28.0}, {"created_at": 1752674566, "description": "In natural numbers, multiplication is a shorthand for repeated addition, but this relationship does not hold for all cases in integers.\u003cSEP\u003eMultiplication is presented as a simplified process of repeated addition, highlighting their foundational relationship in arithmetic.\u003cSEP\u003e\u4e58\u6cd5\u6700\u521d\u88ab\u5b9a\u4e49\u4e3a\u52a0\u6cd5\u7684\u7b80\u4fbf\u8fd0\u7b97\uff0c\u4f46\u5728\u6574\u6570\u96c6\u5408\u4e2d\u901a\u8fc7\u4ea4\u6362\u5f8b\u548c\u5206\u914d\u7387\u72ec\u7acb\u5b9a\u4e49\u3002", "file_path": "unknown_source", "from": "\u52a0\u6cd5", "keywords": "arithmetic operations,relationship,simplification,\u6570\u5b66\u57fa\u7840,\u8fd0\u7b97\u5b9a\u4e49", "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-a5fa437b38c82cb10c1cbb6ae90f2d6c\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892", "title": "In natural numbers, multiplication is a shorthand for repeated addition, but this relationship does not hold for all cases in integers.\u003cSEP\u003eMultiplication is presented as a simplified process of repeated addition, highlighting their foundational relationship in arithmetic.\u003cSEP\u003e\u4e58\u6cd5\u6700\u521d\u88ab\u5b9a\u4e49\u4e3a\u52a0\u6cd5\u7684\u7b80\u4fbf\u8fd0\u7b97\uff0c\u4f46\u5728\u6574\u6570\u96c6\u5408\u4e2d\u901a\u8fc7\u4ea4\u6362\u5f8b\u548c\u5206\u914d\u7387\u72ec\u7acb\u5b9a\u4e49\u3002", "to": "\u4e58\u6cd5", "width": 24.0}, {"created_at": 1752674566, "description": "0\u662f\u52a0\u6cd5\u7684\u5355\u4f4d\u5143\uff0c\u6ee1\u8db3a + 0 = a\uff0c\u662f\u52a0\u6cd5\u9006\u8fd0\u7b97(\u51cf\u6cd5)\u7684\u57fa\u7840\u3002", "file_path": "unknown_source", "from": "\u52a0\u6cd5", "keywords": "\u5355\u4f4d\u5143,\u8fd0\u7b97\u7ed3\u6784", "source_id": "chunk-4be9177c327d58831959006a2ea25892", "title": "0\u662f\u52a0\u6cd5\u7684\u5355\u4f4d\u5143\uff0c\u6ee1\u8db3a + 0 = a\uff0c\u662f\u52a0\u6cd5\u9006\u8fd0\u7b97(\u51cf\u6cd5)\u7684\u57fa\u7840\u3002", "to": "0", "width": 8.0}, {"created_at": 1752674566, "description": "The total quantity model uses addition to represent the sum of parts, forming its core mathematical operation.", "file_path": "unknown_source", "from": "\u52a0\u6cd5", "keywords": "mathematical operation,model structure", "source_id": "chunk-5b8e148c0868493680a6eec71a90d808", "title": "The total quantity model uses addition to represent the sum of parts, forming its core mathematical operation.", "to": "\u603b\u91cf\u6a21\u578b", "width": 7.0}, {"created_at": 1752674566, "description": "\u516c\u74069 ensures that the recursive definition of addition can be extended to all natural numbers, making the operation universally valid.", "file_path": "unknown_source", "from": "\u52a0\u6cd5", "keywords": "mathematical induction,recursive definition", "source_id": "chunk-baa5d2235a16fa56fc40519b58791c47", "title": "\u516c\u74069 ensures that the recursive definition of addition can be extended to all natural numbers, making the operation universally valid.", "to": "\u516c\u74069", "width": 9.0}, {"created_at": 1752674566, "description": "The definition of\u5927\u5c0f\u5173\u7cfbdepends on the operation of\u52a0\u6cd5, as a number is greater than another if it can be expressed as the sum of the other and a non-zero natural number.", "file_path": "unknown_source", "from": "\u52a0\u6cd5", "keywords": "arithmetic relation,definition dependency", "source_id": "chunk-baa5d2235a16fa56fc40519b58791c47", "title": "The definition of\u5927\u5c0f\u5173\u7cfbdepends on the operation of\u52a0\u6cd5, as a number is greater than another if it can be expressed as the sum of the other and a non-zero natural number.", "to": "\u5927\u5c0f\u5173\u7cfb", "width": 10.0}, {"created_at": 1752674566, "description": "Subtracting a number equals adding its opposite (a-b = a+(-b)).", "file_path": "unknown_source", "from": "\u51cf\u6cd5", "keywords": "operation transformation", "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7", "title": "Subtracting a number equals adding its opposite (a-b = a+(-b)).", "to": "\u76f8\u53cd\u6570", "width": 9.0}, {"created_at": 1752674566, "description": "Division is fundamentally understood as the inverse operation of multiplication.\u003cSEP\u003eMultiplication can be transformed into division in reformulating speed-distance-time problems.\u003cSEP\u003e\u9664\u6cd5\u662f\u4e58\u6cd5\u7684\u9006\u8fd0\u7b97\uff0c\u901a\u8fc7\u4e58\u6cd5\u5b9a\u4e49\uff0c\u6ee1\u8db3a \u00f7 b = y \u2194 a = b \u00d7 y\u3002", "file_path": "unknown_source", "from": "\u4e58\u6cd5", "keywords": "arithmetic,inverse operations,mathematical operations,problem reformulation,\u8fd0\u7b97\u5173\u7cfb,\u9006\u8fd0\u7b97", "source_id": "chunk-0756c8764e46b050990c725e6fb7d82d\u003cSEP\u003echunk-077f3fe9070c0d50896deaf218c86bde\u003cSEP\u003echunk-4be9177c327d58831959006a2ea25892", "title": "Division is fundamentally understood as the inverse operation of multiplication.\u003cSEP\u003eMultiplication can be transformed into division in reformulating speed-distance-time problems.\u003cSEP\u003e\u9664\u6cd5\u662f\u4e58\u6cd5\u7684\u9006\u8fd0\u7b97\uff0c\u901a\u8fc7\u4e58\u6cd5\u5b9a\u4e49\uff0c\u6ee1\u8db3a \u00f7 b = y \u2194 a = b \u00d7 y\u3002", "to": "\u9664\u6cd5", "width": 25.0}, {"created_at": 1752674566, "description": "Commutative property helps define consistent multiplication rules for negative numbers.\u003cSEP\u003eThe commutative law is a fundamental property preserved in the definition and extension of multiplication.", "file_path": "unknown_source", "from": "\u4e58\u6cd5", "keywords": "algebraic property,mathematical consistency,operation properties", "source_id": "chunk-59b58a8a1243dc7f01ea8cc63b87b8e7\u003cSEP\u003echunk-f0d039ecac6d9749d604feadbbc70472", "title": "Commutative property helps define consistent multiplication rules for negative numbers.\u003cSEP\u003eThe commutative law is a fundamental property preserved in the definition and extension of multiplication.", "to": "\u4ea4\u6362\u5f8b", "width": 17.0}, {"created_at": 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false.\u003cSEP\u003e\u53cd\u8bc1\u6cd5\u4f9d\u8d56\u6392\u4e2d\u5f8b\u8fdb\u884c\u903b\u8f91\u63a8\u7406\uff0c\u901a\u8fc7\u5047\u8bbe\u547d\u9898\u4e0d\u6210\u7acb\u6765\u63a8\u5bfc\u77db\u76fe\u3002\u003cSEP\u003e\u6392\u4e2d\u5f8b\u786e\u4fdd\u547d\u9898\u4e0e\u5176\u5426\u5b9a\u4e4b\u95f4\u5fc5\u987b\u6709\u4e00\u4e2a\u4e3a\u771f\uff0c\u4ece\u800c\u652f\u6301\u53cd\u8bc1\u6cd5\u7684\u903b\u8f91\u7ed3\u6784\u3002", "to": "\u6392\u4e2d\u5f8b", "width": 27.0}, {"created_at": 1752674566, "description": "\u53cd\u8bc1\u6cd5\u662f\u4e00\u79cd\u6f14\u7ece\u63a8\u7406\u65b9\u6cd5,\u901a\u8fc7\u5047\u8bbe\u547d\u9898\u4e0d\u6210\u7acb\u6765\u63a8\u5bfc\u77db\u76fe\u3002", "file_path": "unknown_source", "from": "\u53cd\u8bc1\u6cd5", "keywords": "logical method,proof technique", "source_id": "chunk-93f0e906d06f6dbc8b9407822495a6d7", "title": 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simultaneously.\u003cSEP\u003e\u53cd\u8bc1\u6cd5\u7684\u903b\u8f91\u57fa\u7840\u662f\u77db\u76fe\u5f8b\uff0c\u5373\u901a\u8fc7\u63a8\u5bfc\u51fa\u77db\u76fe\u6765\u5426\u5b9a\u5047\u8bbe\u547d\u9898\uff0c\u4ece\u800c\u8bc1\u660e\u539f\u547d\u9898\u4e3a\u771f\u3002", "file_path": "unknown_source", "from": "\u53cd\u8bc1\u6cd5", "keywords": "contradiction,logical consistency,logical foundation,proof method", "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d\u003cSEP\u003echunk-1947ec0bba5d164b6d83fc97bec183ff", "title": "Reductio ad absurdum relies on the law of contradiction, which states that a proposition and its negation cannot both be true simultaneously.\u003cSEP\u003e\u53cd\u8bc1\u6cd5\u7684\u903b\u8f91\u57fa\u7840\u662f\u77db\u76fe\u5f8b\uff0c\u5373\u901a\u8fc7\u63a8\u5bfc\u51fa\u77db\u76fe\u6765\u5426\u5b9a\u5047\u8bbe\u547d\u9898\uff0c\u4ece\u800c\u8bc1\u660e\u539f\u547d\u9898\u4e3a\u771f\u3002", "to": "\u77db\u76fe\u5f8b", "width": 20.0}, {"created_at": 1752674566, "description": "\u53cd\u8bc1\u6cd5was used to prove that \u221a2 is an irrational number by assuming it is rational and deriving a logical contradiction.\u003cSEP\u003e\u53cd\u8bc1\u6cd5\u88ab\u7528\u4e8e\u8bc1\u660e\u221a2\u662f\u65e0\u7406\u6570\uff0c\u662f\u6570\u5b66\u4e2d\u6700\u65e9\u4f7f\u7528\u903b\u8f91\u63a8\u7406\u89e3\u51b3\u5177\u4f53\u95ee\u9898\u7684\u8303\u4f8b\u4e4b\u4e00\u3002\u003cSEP\u003e\u5728\u8bc1\u660e\u221a2\u4e3a\u65e0\u7406\u6570\u7684\u8fc7\u7a0b\u4e2d\uff0c\u53cd\u8bc1\u6cd5\u88ab\u7528\u6765\u5047\u8bbe\u5176\u4e3a\u6709\u7406\u6570\u5e76\u63a8\u5bfc\u51fa\u77db\u76fe\uff0c\u5c55\u793a\u4e86\u903b\u8f91\u63a8\u7406\u7684\u529b\u91cf\u3002", "file_path": "unknown_source", "from": "\u53cd\u8bc1\u6cd5", "keywords": "irrationality,logical proof,mathematical proof,mathematical reasoning,mathematical rigor", "source_id": "chunk-5dace47f09692ec14aeb85d6e02520fc\u003cSEP\u003echunk-dc9988d8dd524d81a661c65645bca716\u003cSEP\u003echunk-1947ec0bba5d164b6d83fc97bec183ff", "title": "\u53cd\u8bc1\u6cd5was used to prove that \u221a2 is an irrational number by assuming it is rational and deriving a logical contradiction.\u003cSEP\u003e\u53cd\u8bc1\u6cd5\u88ab\u7528\u4e8e\u8bc1\u660e\u221a2\u662f\u65e0\u7406\u6570\uff0c\u662f\u6570\u5b66\u4e2d\u6700\u65e9\u4f7f\u7528\u903b\u8f91\u63a8\u7406\u89e3\u51b3\u5177\u4f53\u95ee\u9898\u7684\u8303\u4f8b\u4e4b\u4e00\u3002\u003cSEP\u003e\u5728\u8bc1\u660e\u221a2\u4e3a\u65e0\u7406\u6570\u7684\u8fc7\u7a0b\u4e2d\uff0c\u53cd\u8bc1\u6cd5\u88ab\u7528\u6765\u5047\u8bbe\u5176\u4e3a\u6709\u7406\u6570\u5e76\u63a8\u5bfc\u51fa\u77db\u76fe\uff0c\u5c55\u793a\u4e86\u903b\u8f91\u63a8\u7406\u7684\u529b\u91cf\u3002", "to": "\u221a2", "width": 29.0}, {"created_at": 1752674566, "description": "Prime numbers and the Fundamental Theorem both represent foundational concepts in number theory and algebra.", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "mathematical foundations,number theory", "source_id": "chunk-fc1e60e87ee195f9f93e8e10e72c275a", "title": "Prime numbers and the Fundamental Theorem both represent foundational concepts in number theory and algebra.", "to": "\u4ee3\u6570\u57fa\u672c\u5b9a\u7406", "width": 7.0}, {"created_at": 1752674566, "description": "\u6b27\u51e0\u91cc\u5f97\u662f\u7b2c\u4e00\u4e2a\u8bc1\u660e\u7d20\u6570\u5177\u6709\u65e0\u9650\u591a\u4e2a\u7684\u6570\u5b66\u5bb6\uff0c\u5e76\u4f7f\u7528\u4e86\u53cd\u8bc1\u6cd5\u3002", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "mathematical proof,number theory", "source_id": "chunk-93f0e906d06f6dbc8b9407822495a6d7", "title": "\u6b27\u51e0\u91cc\u5f97\u662f\u7b2c\u4e00\u4e2a\u8bc1\u660e\u7d20\u6570\u5177\u6709\u65e0\u9650\u591a\u4e2a\u7684\u6570\u5b66\u5bb6\uff0c\u5e76\u4f7f\u7528\u4e86\u53cd\u8bc1\u6cd5\u3002", "to": "\u6b27\u51e0\u91cc\u5f97", "width": 8.0}, {"created_at": 1752674566, "description": "Chebyshev proved Bertrand\u0027s conjecture, which concerns the existence of prime numbers within specific intervals, contributing to the understanding of prime distribution.", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "mathematical proof,prime distribution", "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "Chebyshev proved Bertrand\u0027s conjecture, which concerns the existence of prime numbers within specific intervals, contributing to the understanding of prime distribution.", "to": "\u5207\u6bd4\u96ea\u592b", "width": 9.0}, {"created_at": 1752674566, "description": "Goldbach\u0027s conjecture involves the representation of even numbers as the sum of two prime numbers, making it deeply connected to the properties and distribution of primes.", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "additive primes,number theory", "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "Goldbach\u0027s conjecture involves the representation of even numbers as the sum of two prime numbers, making it deeply connected to the properties and distribution of primes.", "to": "\u54e5\u5fb7\u5df4\u8d6b\u731c\u60f3", "width": 10.0}, {"created_at": 1752674566, "description": "Pythagoras and his school had a deep philosophical and mathematical interest in numbers, including the properties of primes and their symbolic meanings.\u003cSEP\u003e\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e\u7684\u7814\u7a76\u4e3a\u7d20\u6570\u7814\u7a76\u5960\u5b9a\u4e86\u57fa\u7840\uff0c\u7d20\u6570\u662f\u6570\u8bba\u4e2d\u6700\u57fa\u672c\u7684\u7814\u7a76\u5bf9\u8c61\u3002", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "mathematical foundation,number theory,numerical philosophy,symbolic meaning", "source_id": "chunk-5dace47f09692ec14aeb85d6e02520fc\u003cSEP\u003echunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "Pythagoras and his school had a deep philosophical and mathematical interest in numbers, including the properties of primes and their symbolic meanings.\u003cSEP\u003e\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e\u7684\u7814\u7a76\u4e3a\u7d20\u6570\u7814\u7a76\u5960\u5b9a\u4e86\u57fa\u7840\uff0c\u7d20\u6570\u662f\u6570\u8bba\u4e2d\u6700\u57fa\u672c\u7684\u7814\u7a76\u5bf9\u8c61\u3002", "to": "\u6bd5\u8fbe\u54e5\u62c9\u65af", "width": 14.0}, {"created_at": 1752674566, "description": "In Euclid\u0027s proof of the infinitude of primes, p!+1 is used to construct a number that leads to a contradiction when assuming a finite number of primes.", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "contradiction,mathematical proof", "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "In Euclid\u0027s proof of the infinitude of primes, p!+1 is used to construct a number that leads to a contradiction when assuming a finite number of primes.", "to": "p!+1", "width": 10.0}, {"created_at": 1752674566, "description": "The Bertrand-Chebyshev theorem guarantees the existence of at least one prime number between n and 2n for any n \u003e 1.", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "prime distribution,theorem", "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "The Bertrand-Chebyshev theorem guarantees the existence of at least one prime number between n and 2n for any n \u003e 1.", "to": "\u4f2f\u7279\u5170-\u5207\u6bd4\u96ea\u592b\u5b9a\u7406", "width": 9.0}, {"created_at": 1752674566, "description": "This is a specific large prime number discovered through computational methods, representing the current largest known prime in the text.", "file_path": "unknown_source", "from": "\u7d20\u6570", "keywords": "computational mathematics,prime discovery", "source_id": "chunk-ded529b804f8d3a77fb3e0a6afc9937d", "title": "This is a specific large prime number discovered through computational methods, representing the current largest known prime in the text.", "to": 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"chunk-da314816f469d2b8634730d486609229", "title": "\u7d20\u6570\u6982\u5ff5\u662f\u8fdb\u884c\u002711\u662f\u5426\u4e3a\u7d20\u6570\u0027\u8fd9\u4e00\u76f4\u63a5\u5224\u65ad\u7684\u57fa\u7840\u3002", "to": "\u76f4\u63a5\u5224\u65ad", "width": 8.0}, {"created_at": 1752674566, "description": "\u4e58\u6cd5\u8fd0\u7b97is conceptually linked to repeated\u52a0\u6cd5\u8fd0\u7b97in early mathematics.", "file_path": "unknown_source", "from": "\u52a0\u6cd5\u8fd0\u7b97", "keywords": "arithmetic concepts,operation relationships", "source_id": "chunk-a9a60418b119b40b9f7e09f08cc0b5c8", "title": "\u4e58\u6cd5\u8fd0\u7b97is conceptually linked to repeated\u52a0\u6cd5\u8fd0\u7b97in early mathematics.", "to": "\u4e58\u6cd5\u8fd0\u7b97", "width": 7.0}, {"created_at": 1752674566, "description": "\u6570\u7cfb\u7684\u5b9a\u4e49\u5305\u62ec\u8fd0\u7b97\u89c4\u5219\uff0c\u52a0\u6cd5\u8fd0\u7b97\u662f\u81ea\u7136\u6570\u7cfb\u3001\u6574\u6570\u7cfb\u7b49\u6570\u7cfb\u4e2d\u7684\u57fa\u672c\u8fd0\u7b97\u4e4b\u4e00\u3002", "file_path": "unknown_source", "from": "\u52a0\u6cd5\u8fd0\u7b97", "keywords": "\u6570\u7cfb\u7ed3\u6784,\u8fd0\u7b97\u5b9a\u4e49", "source_id": "chunk-73836e98e177b5a39ff62e469b1507bc", "title": "\u6570\u7cfb\u7684\u5b9a\u4e49\u5305\u62ec\u8fd0\u7b97\u89c4\u5219\uff0c\u52a0\u6cd5\u8fd0\u7b97\u662f\u81ea\u7136\u6570\u7cfb\u3001\u6574\u6570\u7cfb\u7b49\u6570\u7cfb\u4e2d\u7684\u57fa\u672c\u8fd0\u7b97\u4e4b\u4e00\u3002", "to": "\u6570\u7cfb", "width": 7.0}, {"created_at": 1752674566, "description": "\u52a0\u6cd5\u8fd0\u7b97\u7684\u5b9a\u4e49\u4f9d\u8d56\u4e8e\u7b49\u4ef7\u5173\u7cfb\u7684\u5bf9\u79f0\u6027\uff0c\u7528\u4e8e\u6784\u5efa\u52a0\u6cd5\u89c4\u5219\u3002", "file_path": "unknown_source", "from": "\u52a0\u6cd5\u8fd0\u7b97", "keywords": "\u5bf9\u79f0\u6027\u8d28,\u8fd0\u7b97\u5b9a\u4e49", "source_id": "chunk-73836e98e177b5a39ff62e469b1507bc", "title": "\u52a0\u6cd5\u8fd0\u7b97\u7684\u5b9a\u4e49\u4f9d\u8d56\u4e8e\u7b49\u4ef7\u5173\u7cfb\u7684\u5bf9\u79f0\u6027\uff0c\u7528\u4e8e\u6784\u5efa\u52a0\u6cd5\u89c4\u5219\u3002", "to": "\u7b49\u4ef7\u5173\u7cfb", "width": 8.0}, {"created_at": 1752674566, "description": "\u5bf9\u79f0\u6027\u662f\u5b9a\u4e49\u52a0\u6cd5\u8fd0\u7b97\u7684\u91cd\u8981\u903b\u8f91\u6027\u8d28\uff0c\u4f7f\u52a0\u6cd5\u5177\u6709\u4ea4\u6362\u6027\u3002", "file_path": "unknown_source", "from": "\u52a0\u6cd5\u8fd0\u7b97", "keywords": "\u8fd0\u7b97\u5bf9\u79f0\u6027,\u903b\u8f91\u6027\u8d28", "source_id": "chunk-73836e98e177b5a39ff62e469b1507bc", "title": "\u5bf9\u79f0\u6027\u662f\u5b9a\u4e49\u52a0\u6cd5\u8fd0\u7b97\u7684\u91cd\u8981\u903b\u8f91\u6027\u8d28\uff0c\u4f7f\u52a0\u6cd5\u5177\u6709\u4ea4\u6362\u6027\u3002", "to": "\u5bf9\u79f0\u6027", "width": 7.0}, {"created_at": 1752674566, "description": 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"\u5927\u5c0f\u5173\u7cfbbetween natural numbers is defined using\u52a0\u6cd5\u8fd0\u7b97, where a \u003e b if there exists a c \u2260 0 such that a = b + c.", "to": "\u5927\u5c0f\u5173\u7cfb", "width": 9.0}, {"created_at": 1752674566, "description": "\u547d\u98982\u6307\u51fa\u5728\u7b49\u5f0f\u6216\u4e0d\u7b49\u5f0f\u4e24\u8fb9\u52a0\u51cf\u76f8\u540c\u91cf\u5173\u7cfb\u4e0d\u53d8\uff0c\u662f\u52a0\u6cd5\u8fd0\u7b97\u6027\u8d28\u8bc1\u660e\u7684\u91cd\u8981\u4f9d\u636e\u3002", "file_path": "unknown_source", "from": "\u52a0\u6cd5\u8fd0\u7b97", "keywords": "mathematical operation,proof basis", "source_id": "chunk-3991618885f8ce7d7db83540320763bc", "title": "\u547d\u98982\u6307\u51fa\u5728\u7b49\u5f0f\u6216\u4e0d\u7b49\u5f0f\u4e24\u8fb9\u52a0\u51cf\u76f8\u540c\u91cf\u5173\u7cfb\u4e0d\u53d8\uff0c\u662f\u52a0\u6cd5\u8fd0\u7b97\u6027\u8d28\u8bc1\u660e\u7684\u91cd\u8981\u4f9d\u636e\u3002", "to": "\u547d\u98982", "width": 8.0}, {"created_at": 1752674566, "description": 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integers.\u003cSEP\u003e\u221a2\u662f\u7b2c\u4e00\u4e2a\u88ab\u53d1\u73b0\u7684\u65e0\u7406\u6570\uff0c\u5176\u53d1\u73b0\u76f4\u63a5\u5bfc\u81f4\u4e86\u65e0\u7406\u6570\u6982\u5ff5\u7684\u63d0\u51fa\u3002", "to": "\u65e0\u7406\u6570", "width": 20.0}, {"created_at": 1752674566, "description": "\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e\u53d1\u73b0\u8fb9\u957f\u4e3a1\u7684\u6b63\u65b9\u5f62\u5bf9\u89d2\u7ebf\u4e3a\u221a2\uff0c\u4ece\u800c\u5f15\u53d1\u4e86\u6570\u5b66\u53f2\u4e0a\u7684\u7b2c\u4e00\u6b21\u5371\u673a\u3002", "file_path": "unknown_source", "from": "\u221a2", "keywords": "foundational crisis,paradigm shift", "source_id": "chunk-5dace47f09692ec14aeb85d6e02520fc", "title": "\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e\u53d1\u73b0\u8fb9\u957f\u4e3a1\u7684\u6b63\u65b9\u5f62\u5bf9\u89d2\u7ebf\u4e3a\u221a2\uff0c\u4ece\u800c\u5f15\u53d1\u4e86\u6570\u5b66\u53f2\u4e0a\u7684\u7b2c\u4e00\u6b21\u5371\u673a\u3002", "to": "\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e", "width": 8.0}, {"created_at": 1752674566, "description": "\u51e0\u4f55\u4ee3\u6570provided a way to represent and understand numbers like \u221a2 geometrically, which could not be easily expressed in arithmetic terms.", "file_path": "unknown_source", "from": "\u221a2", "keywords": "geometric interpretation,mathematical adaptation", "source_id": "chunk-dc9988d8dd524d81a661c65645bca716", "title": "\u51e0\u4f55\u4ee3\u6570provided a way to represent and understand numbers like \u221a2 geometrically, which could not be easily expressed in arithmetic terms.", "to": "\u51e0\u4f55\u4ee3\u6570", "width": 7.0}, {"created_at": 1752674566, "description": "\u221a2 is the classic example of an\u65e0\u7406\u6570, proven to be non-expressible as a ratio of two integers through logical contradiction.", "file_path": "unknown_source", "from": "\u221a2", "keywords": "irrationality,mathematical proof", "source_id": "chunk-dc9988d8dd524d81a661c65645bca716", "title": "\u221a2 is the classic example of an\u65e0\u7406\u6570, proven to be non-expressible as a ratio of two integers through logical contradiction.", "to": "\u65e0\u7406\u6570\u5b9a\u4e49", "width": 10.0}, {"created_at": 1752674566, "description": "\u53cd\u8bc1\u6cd5\u8bc1\u660ewas applied to demonstrate that \u221a2 is irrational by assuming it is rational and reaching a logical contradiction.", "file_path": "unknown_source", "from": "\u221a2", "keywords": "logical deduction,mathematical reasoning", "source_id": "chunk-dc9988d8dd524d81a661c65645bca716", "title": "\u53cd\u8bc1\u6cd5\u8bc1\u660ewas applied to demonstrate that \u221a2 is irrational by assuming it is rational and reaching a logical contradiction.", "to": "\u53cd\u8bc1\u6cd5\u8bc1\u660e", "width": 10.0}, {"created_at": 1752674566, "description": "\u51e0\u4f55\u4ee3\u6570\u53d1\u5c55allowed for the geometric representation of numbers like \u221a2 that could not be easily expressed arithmetically.", "file_path": "unknown_source", "from": "\u221a2", "keywords": "geometric interpretation,mathematical adaptation", "source_id": "chunk-dc9988d8dd524d81a661c65645bca716", "title": "\u51e0\u4f55\u4ee3\u6570\u53d1\u5c55allowed for the geometric representation of numbers like \u221a2 that could not be easily expressed arithmetically.", "to": "\u51e0\u4f55\u4ee3\u6570\u53d1\u5c55", "width": 7.0}, {"created_at": 1752674566, "description": "\u221a2 exemplifies \u221a2\u4e0d\u53ef\u516c\u5ea6\u6027, which means it cannot be expressed as a common measure or ratio between two whole numbers.", "file_path": "unknown_source", "from": "\u221a2", "keywords": "incommensurability,mathematical discovery", "source_id": "chunk-dc9988d8dd524d81a661c65645bca716", "title": "\u221a2 exemplifies \u221a2\u4e0d\u53ef\u516c\u5ea6\u6027, which means it cannot be expressed as a common measure or ratio between two whole numbers.", "to": "\u221a2\u4e0d\u53ef\u516c\u5ea6\u6027", "width": 10.0}, {"created_at": 1752674566, "description": "\u221a2 plays a central role 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"\u6570\u5b66\u547d\u9898\u662f\u6570\u5b66\u7814\u7a76\u4e2d\u7528\u4e8e\u8868\u8fbe\u53ef\u9a8c\u8bc1\u9648\u8ff0\u7684\u6838\u5fc3\u5de5\u5177\u3002", "to": "\u6570\u5b66\u7814\u7a76", "width": 8.0}, {"created_at": 1752674566, "description": "\u7b26\u53f7\u8868\u8fbe\u662f\u5b9e\u73b0\u5f62\u5f0f\u5316\u6269\u5145\u7684\u524d\u63d0\uff0c\u5982\u4ece\u5e73\u65b9\u63a8\u5e7f\u5230\u5341\u4f4d\u6570\u5dee1\u7684\u60c5\u51b5\u3002", "file_path": "unknown_source", "from": "\u7b26\u53f7\u8868\u8fbe", "keywords": "\u4ee3\u6570\u5de5\u5177,\u62bd\u8c61\u6269\u5c55", "source_id": "chunk-5d7b9b0085eaed2882fabbb5f2c18a82", "title": "\u7b26\u53f7\u8868\u8fbe\u662f\u5b9e\u73b0\u5f62\u5f0f\u5316\u6269\u5145\u7684\u524d\u63d0\uff0c\u5982\u4ece\u5e73\u65b9\u63a8\u5e7f\u5230\u5341\u4f4d\u6570\u5dee1\u7684\u60c5\u51b5\u3002", "to": "\u5f62\u5f0f\u5316\u6269\u5145", "width": 8.0}, {"created_at": 1752674566, "description": 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"keywords": "algebraic notation,symbolic innovation", "source_id": "chunk-f97ce783119eb62752133fb048ac4a2f", "title": "\u97e6\u8fbe\u5728\u300a\u5206\u6790\u827a\u672f\u5f15\u8bba\u300b\u4e2d\u63d0\u51fa\u7528\u5b57\u6bcd\u8868\u793a\u5df2\u77e5\u91cf\u548c\u672a\u77e5\u91cf\uff0c\u662f\u7b26\u53f7\u8868\u8fbe\u7cfb\u7edf\u5316\u7684\u8d77\u70b9\u3002", "to": "\u300a\u5206\u6790\u827a\u672f\u5f15\u8bba\u300b", "width": 10.0}, {"created_at": 1752674566, "description": "\u51e0\u4f55\u95ee\u9898\u662f\u6570\u5b66\u95ee\u9898\u7684\u5b50\u96c6\uff0c\u9700\u901a\u8fc7\u4ee3\u6570\u5de5\u5177\u89e3\u51b3\u7b26\u53f7\u8868\u8fbe\u56f0\u96be\u3002", "file_path": "unknown_source", "from": "\u6570\u5b66\u95ee\u9898", "keywords": "\u5b66\u79d1\u4ea4\u53c9,\u95ee\u9898\u5206\u7c7b", "source_id": "chunk-5d7b9b0085eaed2882fabbb5f2c18a82", "title": 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"from": "\u4e09\u89d2\u51fd\u6570", "keywords": "geometric shapes,mathematical property", "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d", "title": "\u4e09\u89d2\u51fd\u6570\u63cf\u8ff0\u76f4\u89d2\u4e09\u89d2\u5f62\u4e2d\u89d2\u4e0e\u8fb9\u7684\u5173\u7cfb\u3002", "to": "\u76f4\u89d2\u4e09\u89d2\u5f62", "width": 9.0}, {"created_at": 1752674566, "description": "\u5e73\u79fb\u548c\u65cb\u8f6c\u90fd\u662f\u521a\u4f53\u8fd0\u52a8\uff0c\u4fdd\u6301\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "file_path": "unknown_source", "from": "\u5e73\u79fb", "keywords": "\u51e0\u4f55\u53d8\u6362,\u521a\u4f53\u8fd0\u52a8", "source_id": "chunk-c1f0f38352bf492f96567cff86102e58", "title": "\u5e73\u79fb\u548c\u65cb\u8f6c\u90fd\u662f\u521a\u4f53\u8fd0\u52a8\uff0c\u4fdd\u6301\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "to": "\u65cb\u8f6c", "width": 7.0}, {"created_at": 1752674566, "description": "\u5e73\u79fb\u662f\u521a\u4f53\u8fd0\u52a8\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "file_path": "unknown_source", "from": "\u5e73\u79fb", "keywords": "\u51e0\u4f55\u53d8\u6362,\u8fd0\u52a8\u7c7b\u578b", "source_id": "chunk-c1f0f38352bf492f96567cff86102e58", "title": "\u5e73\u79fb\u662f\u521a\u4f53\u8fd0\u52a8\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "to": "\u521a\u4f53\u8fd0\u52a8", "width": 9.0}, {"created_at": 1752674566, "description": "\u5e73\u79fb\u662f\u521a\u4f53\u53d8\u6362\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u56fe\u5f62\u4e2d\u4efb\u610f\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "file_path": "unknown_source", "from": "\u5e73\u79fb", "keywords": "invariance,transformation type", "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d", "title": "\u5e73\u79fb\u662f\u521a\u4f53\u53d8\u6362\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u56fe\u5f62\u4e2d\u4efb\u610f\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "to": "\u521a\u4f53\u53d8\u6362", "width": 8.0}, {"created_at": 1752674566, "description": "\u5e73\u79fb\u662f\u56fe\u5f62\u8fd0\u52a8\u7684\u4e24\u79cd\u57fa\u672c\u5f62\u5f0f\u4e4b\u4e00\uff0c\u5c5e\u4e8e\u56fe\u5f62\u53d8\u6362\u7684\u76f4\u89c2\u4f53\u73b0\u3002", "file_path": "unknown_source", "from": "\u5e73\u79fb", "keywords": "\u56fe\u5f62\u53d8\u6362,\u6570\u5b66\u6982\u5ff5", "source_id": "chunk-d49b7902492c618044ad70c5d4b3ba7f", "title": "\u5e73\u79fb\u662f\u56fe\u5f62\u8fd0\u52a8\u7684\u4e24\u79cd\u57fa\u672c\u5f62\u5f0f\u4e4b\u4e00\uff0c\u5c5e\u4e8e\u56fe\u5f62\u53d8\u6362\u7684\u76f4\u89c2\u4f53\u73b0\u3002", "to": "\u56fe\u5f62\u8fd0\u52a8", "width": 9.0}, {"created_at": 1752674566, "description": "\u65cb\u8f6c\u662f\u521a\u4f53\u8fd0\u52a8\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "file_path": "unknown_source", "from": "\u65cb\u8f6c", "keywords": "\u51e0\u4f55\u53d8\u6362,\u8fd0\u52a8\u7c7b\u578b", "source_id": "chunk-c1f0f38352bf492f96567cff86102e58", "title": "\u65cb\u8f6c\u662f\u521a\u4f53\u8fd0\u52a8\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "to": "\u521a\u4f53\u8fd0\u52a8", "width": 9.0}, {"created_at": 1752674566, "description": "\u65cb\u8f6c\u662f\u521a\u4f53\u53d8\u6362\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u56fe\u5f62\u4e2d\u4efb\u610f\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "file_path": "unknown_source", "from": "\u65cb\u8f6c", "keywords": "invariance,transformation type", "source_id": "chunk-cceb7fc5191a1b7132a11703466c415d", "title": "\u65cb\u8f6c\u662f\u521a\u4f53\u53d8\u6362\u7684\u4e00\u79cd\uff0c\u4fdd\u6301\u56fe\u5f62\u4e2d\u4efb\u610f\u4e24\u70b9\u95f4\u8ddd\u79bb\u4e0d\u53d8\u3002", "to": "\u521a\u4f53\u53d8\u6362", "width": 8.0}, {"created_at": 1752674566, "description": "\u65cb\u8f6c\u662f\u56fe\u5f62\u8fd0\u52a8\u7684\u53e6\u4e00\u79cd\u57fa\u672c\u5f62\u5f0f\uff0c\u5e38\u4e0e\u5e73\u79fb\u7ed3\u5408\u4f7f\u7528\u6765\u63cf\u8ff0\u56fe\u5f62\u7684\u8fd0\u52a8\u3002", "file_path": "unknown_source", "from": "\u65cb\u8f6c", "keywords": "\u56fe\u5f62\u53d8\u6362,\u6570\u5b66\u6982\u5ff5", "source_id": "chunk-d49b7902492c618044ad70c5d4b3ba7f", "title": "\u65cb\u8f6c\u662f\u56fe\u5f62\u8fd0\u52a8\u7684\u53e6\u4e00\u79cd\u57fa\u672c\u5f62\u5f0f\uff0c\u5e38\u4e0e\u5e73\u79fb\u7ed3\u5408\u4f7f\u7528\u6765\u63cf\u8ff0\u56fe\u5f62\u7684\u8fd0\u52a8\u3002", "to": "\u56fe\u5f62\u8fd0\u52a8", "width": 9.0}, {"created_at": 1752674566, "description": 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