-
-
-
-
-
-
-
-
-
-
-
-
- {{item['课程']}}
- {{item['课程介绍']}}
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+ {{item['课程']}}
+ {{item['课程介绍']}}
+
+
+
+
\ No newline at end of file
diff --git a/pages/kecheng/kecheng.wxss b/前端/pages/kecheng/kecheng.wxss
similarity index 94%
rename from pages/kecheng/kecheng.wxss
rename to 前端/pages/kecheng/kecheng.wxss
index 9890932..3df464b 100644
--- a/pages/kecheng/kecheng.wxss
+++ b/前端/pages/kecheng/kecheng.wxss
@@ -1,49 +1,49 @@
-/* pages/kecheng/kecheng.wxss */
-.container {
- padding:0;
- align-items:center;
- flex-direction:column;
- justify-content:space-between;
-}
-.kecheng {
- width: 710rpx;
- height: 275rpx;
- border-radius: 20rpx;
- overflow: hidden;
- display: flex;
- justify-content: left;
- align-items: center;
- margin: 6px 0;
- box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
-}
-
-
-.kecheng_video_up {
- width: 400rpx;
- height: 225rpx;
- margin-left: 10rpx;
-}
-.kecheng_video_right {
- margin-left: 5rpx;
- width: 290rpx;
- height: 225rpx;
- overflow: hidden;
-
-}
-.kecheng_img {
- background-color: #15457b;
- width: 100%;
- height: 100%;
- border-radius: 15rpx;
-}
-.kecheng_title {
- display: block;
- font-size: 30rpx;
- text-align: center;
-}
-.kecheng_details {
- display: block;
- font-size: 25rpx;
- margin-top: 15rpx;
- margin-left: 15rpx;
+/* pages/kecheng/kecheng.wxss */
+.container {
+ padding:0;
+ align-items:center;
+ flex-direction:column;
+ justify-content:space-between;
+}
+.kecheng {
+ width: 710rpx;
+ height: 275rpx;
+ border-radius: 20rpx;
+ overflow: hidden;
+ display: flex;
+ justify-content: left;
+ align-items: center;
+ margin: 6px 0;
+ box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
+}
+
+
+.kecheng_video_up {
+ width: 400rpx;
+ height: 225rpx;
+ margin-left: 10rpx;
+}
+.kecheng_video_right {
+ margin-left: 5rpx;
+ width: 290rpx;
+ height: 225rpx;
+ overflow: hidden;
+
+}
+.kecheng_img {
+ background-color: #15457b;
+ width: 100%;
+ height: 100%;
+ border-radius: 15rpx;
+}
+.kecheng_title {
+ display: block;
+ font-size: 30rpx;
+ text-align: center;
+}
+.kecheng_details {
+ display: block;
+ font-size: 25rpx;
+ margin-top: 15rpx;
+ margin-left: 15rpx;
}
\ No newline at end of file
diff --git a/pages/wode/wode.js b/前端/pages/wode/wode.js
similarity index 95%
rename from pages/wode/wode.js
rename to 前端/pages/wode/wode.js
index 77df368..5ab1876 100644
--- a/pages/wode/wode.js
+++ b/前端/pages/wode/wode.js
@@ -1,232 +1,232 @@
-// pages/wode/wode.js
-const MyApi = require('../../Api/api.js');
-Page({
-
- /**
- * 页面的初始数据
- */
- data: {
- value: '',
- show: false,
- showCode: false,
- passwordcode: null,
- intervalId: null,
- headiconurl: '/static/image/user.png',
- urlstring: getApp().globalData.BaseURL + "/Images/ToolsIcon/toolsicon",
- LoginBtnText: '登录注册',
- UserNameText: '未登录',
- user_openid: null,
- YelayMaths_ID: null,
- IsLogin: null,
- IsAdministor: null,
- LastLoginDate: null,
- UserName: null,
- NewUser: null,
- text2: wx.getStorageSync('Config')[7].value,
- text3: wx.getStorageSync('Config')[8].value,
- text4: wx.getStorageSync('Config')[9].value,
- text5: wx.getStorageSync('Config')[10].value,
- text6: wx.getStorageSync('Config')[12].value,
- CodeURL:getApp().globalData.BaseURL + "/Images/WXCode/"+wx.getStorageSync('Config')[5].value+".png",
- },
- showdialog() {
- if (wx.getStorageSync('UserInfoCache').IsLogin ) {
- this.setData({
- show: true,
- value: ''
- });
- } else {
- wx.showToast({
- mask:true,
- icon: 'error',
- title: '请登录后操作!',
- })
- }
-
- },
-
- ChangeUserInfo() {
- wx.navigateTo({
- url: '/pages/ChangeUserInfo/ChangeUserInfo',
- });
- },
-
- onChange(e) {
- this.setData({
- passwordcode: e.detail
- });
- },
-
- onClose() {
- this.setData({
- show: false,
- passwordcode: null,
- });
- },
-
- showCode() {
- this.setData({
- showCode: true,
- });
- },
-
- OnClickCodeImage() {
- console.log("点击了二维码");
- },
-
- onCodeClose() {
- this.setData({
- showCode: false,
- });
- },
-
- GetCourse() {
- //点击确定后的课程获取
- if (this.data.passwordcode) {
- wx.showToast({
- title: '请稍后...',
- icon: 'loading',
- mask:true,
- duration: 10000
- });
-
- MyApi.ApiPasswordCodeValidate(wx.getStorageSync('UserInfoCache').user_openid, this.data.passwordcode)
- .then(result => {
- // 成功后执行的代码
- try {
- wx.hideToast();
- } catch (error) {}
-
- wx.showToast({
- title: result.Data,
- icon: 'error',
- mask:true,
- duration: 2000
- });
-
- //修改缓存已拥有课程
- if (result.ReturnData) {
- var tempcourse = wx.getStorageSync('UserInfoCache');
- tempcourse.OwnedCourse=result.ReturnData;
- wx.setStorageSync('UserInfoCache', tempcourse);
- };
- })
- .catch(err => {
- // 失败后执行的代码
- console.error("请求失败:" + err);
- });
- } else {
- wx.showToast({
- title: '请输入课程码!',
- icon: 'error',
- mask:true,
- duration: 2000
- });
- }
-
- this.setData({
- show: false,
- });
- },
-
- redirectToTargetLoginout: function () {
- wx.navigateTo({
- url: '/pages/LoginOrOutDetails/LoginOrOutDetails?IsLogin=' + this.IsLogin,
- })
- },
-
- NGT_SetPasswordCode: function () {
- this.setData({
- show: false,
- });
- wx.navigateTo({
- url: '/pages/SetPasswordCode/SetPasswordCode',
- });
-
- },
-
- /**
- * 生命周期函数--监听页面加载
- */
- onLoad(options) {
-
- var Storage = wx.getStorageSync('UserInfoCache');
- if (Storage.IsLogin === "1") {
- this.setData({
- IsLogin: Storage.IsLogin,
- headiconurl: getApp().globalData.BaseURL + "/Images/Headicons/Headicon" + Storage.HeadiconID + ".png",
- LoginBtnText: "退出登录",
- UserNameText: Storage.UserName,
- IsAdministor: Storage.IsAdministor,
- });
- } else {
- this.setData({
- IsLogin: null,
- headiconurl: "/static/image/user.png",
- LoginBtnText: "登录注册",
- UserNameText: "未登录",
- IsAdministor: null,
- });
- }
- },
-
-
- /**
- * 生命周期函数--监听页面初次渲染完成
- */
- onReady() {
-
- },
-
- /**
- * 生命周期函数--监听页面显示
- */
- onShow() {
- this.onLoad();
- },
-
- /**
- * 生命周期函数--监听页面隐藏
- */
- onHide() {
-
- },
-
- /**
- * 生命周期函数--监听页面卸载
- */
- onUnload() {
-
- },
-
- /**
- * 页面相关事件处理函数--监听用户下拉动作
- */
- onPullDownRefresh() {
-
- },
-
- /**
- * 页面上拉触底事件的处理函数
- */
- onReachBottom() {
-
- },
-
- function2_01() {
- wx.showModal({
- title: '提示',
- content: '功能安排中。。。',
- success(res) {
- if (res.confirm) {
- console.log('用户点击确定');
- } else if (res.cancel) {
- console.log('用户点击取消');
- }
- }
- });
- },
-
-
-
+// pages/wode/wode.js
+const MyApi = require('../../Api/api.js');
+Page({
+
+ /**
+ * 页面的初始数据
+ */
+ data: {
+ value: '',
+ show: false,
+ showCode: false,
+ passwordcode: null,
+ intervalId: null,
+ headiconurl: '/static/image/user.png',
+ urlstring: getApp().globalData.BaseURL + "/Images/ToolsIcon/toolsicon",
+ LoginBtnText: '登录注册',
+ UserNameText: '未登录',
+ user_openid: null,
+ YelayMaths_ID: null,
+ IsLogin: null,
+ IsAdministor: null,
+ LastLoginDate: null,
+ UserName: null,
+ NewUser: null,
+ text2: wx.getStorageSync('Config')[7].value,
+ text3: wx.getStorageSync('Config')[8].value,
+ text4: wx.getStorageSync('Config')[9].value,
+ text5: wx.getStorageSync('Config')[10].value,
+ text6: wx.getStorageSync('Config')[12].value,
+ CodeURL:getApp().globalData.BaseURL + "/Images/WXCode/"+wx.getStorageSync('Config')[5].value+".png",
+ },
+ showdialog() {
+ if (wx.getStorageSync('UserInfoCache').IsLogin ) {
+ this.setData({
+ show: true,
+ value: ''
+ });
+ } else {
+ wx.showToast({
+ mask:true,
+ icon: 'error',
+ title: '请登录后操作!',
+ })
+ }
+
+ },
+
+ ChangeUserInfo() {
+ wx.navigateTo({
+ url: '/pages/ChangeUserInfo/ChangeUserInfo',
+ });
+ },
+
+ onChange(e) {
+ this.setData({
+ passwordcode: e.detail
+ });
+ },
+
+ onClose() {
+ this.setData({
+ show: false,
+ passwordcode: null,
+ });
+ },
+
+ showCode() {
+ this.setData({
+ showCode: true,
+ });
+ },
+
+ OnClickCodeImage() {
+ console.log("点击了二维码");
+ },
+
+ onCodeClose() {
+ this.setData({
+ showCode: false,
+ });
+ },
+
+ GetCourse() {
+ //点击确定后的课程获取
+ if (this.data.passwordcode) {
+ wx.showToast({
+ title: '请稍后...',
+ icon: 'loading',
+ mask:true,
+ duration: 10000
+ });
+
+ MyApi.ApiPasswordCodeValidate(wx.getStorageSync('UserInfoCache').user_openid, this.data.passwordcode)
+ .then(result => {
+ // 成功后执行的代码
+ try {
+ wx.hideToast();
+ } catch (error) {}
+
+ wx.showToast({
+ title: result.Data,
+ icon: 'error',
+ mask:true,
+ duration: 2000
+ });
+
+ //修改缓存已拥有课程
+ if (result.ReturnData) {
+ var tempcourse = wx.getStorageSync('UserInfoCache');
+ tempcourse.OwnedCourse=result.ReturnData;
+ wx.setStorageSync('UserInfoCache', tempcourse);
+ };
+ })
+ .catch(err => {
+ // 失败后执行的代码
+ console.error("请求失败:" + err);
+ });
+ } else {
+ wx.showToast({
+ title: '请输入课程码!',
+ icon: 'error',
+ mask:true,
+ duration: 2000
+ });
+ }
+
+ this.setData({
+ show: false,
+ });
+ },
+
+ redirectToTargetLoginout: function () {
+ wx.navigateTo({
+ url: '/pages/LoginOrOutDetails/LoginOrOutDetails?IsLogin=' + this.IsLogin,
+ })
+ },
+
+ NGT_SetPasswordCode: function () {
+ this.setData({
+ show: false,
+ });
+ wx.navigateTo({
+ url: '/pages/SetPasswordCode/SetPasswordCode',
+ });
+
+ },
+
+ /**
+ * 生命周期函数--监听页面加载
+ */
+ onLoad(options) {
+
+ var Storage = wx.getStorageSync('UserInfoCache');
+ if (Storage.IsLogin === "1") {
+ this.setData({
+ IsLogin: Storage.IsLogin,
+ headiconurl: getApp().globalData.BaseURL + "/Images/Headicons/Headicon" + Storage.HeadiconID + ".png",
+ LoginBtnText: "退出登录",
+ UserNameText: Storage.UserName,
+ IsAdministor: Storage.IsAdministor,
+ });
+ } else {
+ this.setData({
+ IsLogin: null,
+ headiconurl: "/static/image/user.png",
+ LoginBtnText: "登录注册",
+ UserNameText: "未登录",
+ IsAdministor: null,
+ });
+ }
+ },
+
+
+ /**
+ * 生命周期函数--监听页面初次渲染完成
+ */
+ onReady() {
+
+ },
+
+ /**
+ * 生命周期函数--监听页面显示
+ */
+ onShow() {
+ this.onLoad();
+ },
+
+ /**
+ * 生命周期函数--监听页面隐藏
+ */
+ onHide() {
+
+ },
+
+ /**
+ * 生命周期函数--监听页面卸载
+ */
+ onUnload() {
+
+ },
+
+ /**
+ * 页面相关事件处理函数--监听用户下拉动作
+ */
+ onPullDownRefresh() {
+
+ },
+
+ /**
+ * 页面上拉触底事件的处理函数
+ */
+ onReachBottom() {
+
+ },
+
+ function2_01() {
+ wx.showModal({
+ title: '提示',
+ content: '功能安排中。。。',
+ success(res) {
+ if (res.confirm) {
+ console.log('用户点击确定');
+ } else if (res.cancel) {
+ console.log('用户点击取消');
+ }
+ }
+ });
+ },
+
+
+
})
\ No newline at end of file
diff --git a/pages/wode/wode.json b/前端/pages/wode/wode.json
similarity index 96%
rename from pages/wode/wode.json
rename to 前端/pages/wode/wode.json
index 9a83d88..7417701 100644
--- a/pages/wode/wode.json
+++ b/前端/pages/wode/wode.json
@@ -1,10 +1,10 @@
-{
- "usingComponents": {
- "van-dialog": "@vant/weapp/dialog/index",
- "van-field": "@vant/weapp/field/index",
- "van-button": "@vant/weapp/button/index",
- "van-popup": "@vant/weapp/popup/index",
- "van-image": "@vant/weapp/image/index"
- },
- "navigationBarTitleText": "个人中心"
+{
+ "usingComponents": {
+ "van-dialog": "@vant/weapp/dialog/index",
+ "van-field": "@vant/weapp/field/index",
+ "van-button": "@vant/weapp/button/index",
+ "van-popup": "@vant/weapp/popup/index",
+ "van-image": "@vant/weapp/image/index"
+ },
+ "navigationBarTitleText": "个人中心"
}
\ No newline at end of file
diff --git a/pages/wode/wode.wxml b/前端/pages/wode/wode.wxml
similarity index 96%
rename from pages/wode/wode.wxml
rename to 前端/pages/wode/wode.wxml
index 7bd610e..16a39ce 100644
--- a/pages/wode/wode.wxml
+++ b/前端/pages/wode/wode.wxml
@@ -1,98 +1,98 @@
-
-
-
- {{text6}}
-
-
-
-
-
-点击此处设置课程码
-课程码
-
-
-
-
-
-
-
-
-
- {{UserNameText}}
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- Version 0.0.9
- Copyright© 2026 YelayMaths. All Rights Reserved.
-
-
-
+
+
+
+ {{text6}}
+
+
+
+
+
+点击此处设置课程码
+课程码
+
+
+
+
+
+
+
+
+
+ {{UserNameText}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ Version 0.0.9
+ Copyright© 2026 YelayMaths. All Rights Reserved.
+
+
+
diff --git a/pages/wode/wode.wxss b/前端/pages/wode/wode.wxss
similarity index 94%
rename from pages/wode/wode.wxss
rename to 前端/pages/wode/wode.wxss
index 8143f39..479658b 100644
--- a/pages/wode/wode.wxss
+++ b/前端/pages/wode/wode.wxss
@@ -1,135 +1,135 @@
-/* pages/wode/wode.wxss */
-.container {
- padding:0;
- align-items:center;
- flex-direction:column;
- justify-content:space-between;
-}
-.code-container {
- display: flex;
- flex-direction: column;
- align-items: center; /* 水平居中 */
- padding: 20rpx;
- box-sizing: border-box;
- background-color: rgb(0, 201, 77);
-}
-
-/* 文字样式:可选加粗、颜色、间距等 */
-.tip-text {
- font-size: 34rpx;
- color: #ffffff;
- margin-bottom: 20rpx; /* 与图片保持间距 */
- text-align: center; /* 冗余保险,虽然 flex 已居中 */
-}
-
-.login {
- display: flex;
- justify-content: center;
- align-items: center;
- /* border: 3rpx solid rgb(110, 110, 110); */
- width: 710rpx;
- height: 150rpx;
- border-radius: 20rpx;
- margin: 20rpx;
- box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
-}
-.touxiangimg {
- display: flex;
- width: 100rpx;
- height: 100rpx;
- border-radius: 50rpx;
-}
-.userinfo_name {
- width: 500rpx;
- height: 150rpx;
- margin-left: 30rpx;
- border-radius: 75rpx;
- display: flex;
- justify-content: center;
- align-items: center;
-}
-.loginbtn {
- display: block;
- text-align: center;
-}
-.loginbtn[plain] {
- color: #353535;
- background-color: transparent;
- border: 0px solid #353535;
- box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
- padding: 0rpx 20rpx;
-}
-.loginuserimg {
- width: 100%;
- height: 100%;
- border-radius: 20rpx;
-}
-
-.functions1 {
- border: 3rpx solid rgb(110, 110, 110);
- width: 670rpx;
- border-radius: 20rpx;
- margin: 20rpx;
- border-radius: 20rpx;
- box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
- padding: 20rpx;
-}
-.functions2 {
- display: flex;
- justify-content: space-between;
- align-items: center;
- /* border: 3rpx solid rgb(110, 110, 110); */
- background-color: white;
- width: 710rpx;
- height: 100rpx;
- border-radius: 20rpx;
- margin: 8rpx 20rpx;
- border-radius: 20rpx;
- box-shadow: 0 2px 3px rgba(0, 0, 0, 0.5);
-}
-.jiantou {
- font-size: 40rpx;
-}
-.zuobianwenzi {
- font-size: 30rpx;
-}
-
-.function001 {
- height: 150rpx;
- width: 75rpx;
-
- padding: 10rpx 47rpx;
- text-align: center;
- align-items: center;
-}
-.functionimg001 {
- width: 75rpx;
- height: 75rpx;
-}
-
-.functionimg002 {
- width: 60rpx;
- height: 60rpx;
-}
-.functiontext001 {
- font-size: 25rpx;
-}
-.functionstools {
- position: relative;
- top: 20rpx;
- padding-bottom: 20rpx;
- display: flex;
- flex-wrap: wrap;
- justify-content: flex-start;
- align-items: center;
-}
-.version {
- position: fixed;
- bottom: 10rpx;
- color: rgb(197, 197, 197);
- font-size: 25rpx;
- text-align: center;
-}
-
-
-
+/* pages/wode/wode.wxss */
+.container {
+ padding:0;
+ align-items:center;
+ flex-direction:column;
+ justify-content:space-between;
+}
+.code-container {
+ display: flex;
+ flex-direction: column;
+ align-items: center; /* 水平居中 */
+ padding: 20rpx;
+ box-sizing: border-box;
+ background-color: rgb(0, 201, 77);
+}
+
+/* 文字样式:可选加粗、颜色、间距等 */
+.tip-text {
+ font-size: 34rpx;
+ color: #ffffff;
+ margin-bottom: 20rpx; /* 与图片保持间距 */
+ text-align: center; /* 冗余保险,虽然 flex 已居中 */
+}
+
+.login {
+ display: flex;
+ justify-content: center;
+ align-items: center;
+ /* border: 3rpx solid rgb(110, 110, 110); */
+ width: 710rpx;
+ height: 150rpx;
+ border-radius: 20rpx;
+ margin: 20rpx;
+ box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
+}
+.touxiangimg {
+ display: flex;
+ width: 100rpx;
+ height: 100rpx;
+ border-radius: 50rpx;
+}
+.userinfo_name {
+ width: 500rpx;
+ height: 150rpx;
+ margin-left: 30rpx;
+ border-radius: 75rpx;
+ display: flex;
+ justify-content: center;
+ align-items: center;
+}
+.loginbtn {
+ display: block;
+ text-align: center;
+}
+.loginbtn[plain] {
+ color: #353535;
+ background-color: transparent;
+ border: 0px solid #353535;
+ box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
+ padding: 0rpx 20rpx;
+}
+.loginuserimg {
+ width: 100%;
+ height: 100%;
+ border-radius: 20rpx;
+}
+
+.functions1 {
+ border: 3rpx solid rgb(110, 110, 110);
+ width: 670rpx;
+ border-radius: 20rpx;
+ margin: 20rpx;
+ border-radius: 20rpx;
+ box-shadow: 0 2px 5px rgba(0, 0, 0, 0.5);
+ padding: 20rpx;
+}
+.functions2 {
+ display: flex;
+ justify-content: space-between;
+ align-items: center;
+ /* border: 3rpx solid rgb(110, 110, 110); */
+ background-color: white;
+ width: 710rpx;
+ height: 100rpx;
+ border-radius: 20rpx;
+ margin: 8rpx 20rpx;
+ border-radius: 20rpx;
+ box-shadow: 0 2px 3px rgba(0, 0, 0, 0.5);
+}
+.jiantou {
+ font-size: 40rpx;
+}
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+ font-size: 30rpx;
+}
+
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+ height: 150rpx;
+ width: 75rpx;
+
+ padding: 10rpx 47rpx;
+ text-align: center;
+ align-items: center;
+}
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+ width: 75rpx;
+ height: 75rpx;
+}
+
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+ width: 60rpx;
+ height: 60rpx;
+}
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+ font-size: 25rpx;
+}
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+ position: relative;
+ top: 20rpx;
+ padding-bottom: 20rpx;
+ display: flex;
+ flex-wrap: wrap;
+ justify-content: flex-start;
+ align-items: center;
+}
+.version {
+ position: fixed;
+ bottom: 10rpx;
+ color: rgb(197, 197, 197);
+ font-size: 25rpx;
+ text-align: center;
+}
+
+
+
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similarity index 100%
rename from project.config.json
rename to 前端/project.config.json
diff --git a/project.private.config.json b/前端/project.private.config.json
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rename from project.private.config.json
rename to 前端/project.private.config.json
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rename from sitemap.json
rename to 前端/sitemap.json
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rename from static/image/defaultAvatar.png
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similarity index 100%
rename from static/image/kecheng-selected.png
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similarity index 100%
rename from static/image/wode-selected.png
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similarity index 100%
rename from static/image/wode.png
rename to 前端/static/image/wode.png
diff --git a/utils/util.js b/前端/utils/util.js
similarity index 100%
rename from utils/util.js
rename to 前端/utils/util.js
diff --git a/数据库/api_yeomai_site_develop_mysql.sql b/数据库/api_yeomai_site_develop_mysql.sql
new file mode 100644
index 0000000..015e125
--- /dev/null
+++ b/数据库/api_yeomai_site_develop_mysql.sql
@@ -0,0 +1,407 @@
+-- MySQL dump 10.13 Distrib 5.7.44, for Linux (x86_64)
+--
+-- Host: localhost Database: api_yeomai_site_develop
+-- ------------------------------------------------------
+-- Server version 5.7.44-log
+
+/*!40101 SET @OLD_CHARACTER_SET_CLIENT=@@CHARACTER_SET_CLIENT */;
+/*!40101 SET @OLD_CHARACTER_SET_RESULTS=@@CHARACTER_SET_RESULTS */;
+/*!40101 SET @OLD_COLLATION_CONNECTION=@@COLLATION_CONNECTION */;
+/*!40101 SET NAMES utf8mb4 */;
+/*!40103 SET @OLD_TIME_ZONE=@@TIME_ZONE */;
+/*!40103 SET TIME_ZONE='+00:00' */;
+/*!40014 SET @OLD_UNIQUE_CHECKS=@@UNIQUE_CHECKS, UNIQUE_CHECKS=0 */;
+/*!40014 SET @OLD_FOREIGN_KEY_CHECKS=@@FOREIGN_KEY_CHECKS, FOREIGN_KEY_CHECKS=0 */;
+/*!40101 SET @OLD_SQL_MODE=@@SQL_MODE, SQL_MODE='NO_AUTO_VALUE_ON_ZERO' */;
+/*!40111 SET @OLD_SQL_NOTES=@@SQL_NOTES, SQL_NOTES=0 */;
+
+--
+-- Table structure for table `免费视频课程数据`
+--
+
+DROP TABLE IF EXISTS `免费视频课程数据`;
+/*!40101 SET @saved_cs_client = @@character_set_client */;
+/*!40101 SET character_set_client = utf8 */;
+CREATE TABLE `免费视频课程数据` (
+ `排序` int(11) DEFAULT NULL,
+ `序号` int(11) DEFAULT NULL,
+ `课程名称` varchar(255) DEFAULT NULL,
+ `视频标题` varchar(255) DEFAULT NULL,
+ `详细介绍` varchar(255) DEFAULT NULL,
+ `发布人` varchar(255) DEFAULT NULL,
+ `发布日期` varchar(255) DEFAULT NULL,
+ `时长` varchar(255) DEFAULT NULL,
+ `视频m3u8` varchar(255) DEFAULT NULL,
+ `封面图片` varchar(255) DEFAULT NULL
+) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 ROW_FORMAT=DYNAMIC;
+/*!40101 SET character_set_client = @saved_cs_client */;
+
+--
+-- Dumping data for table `免费视频课程数据`
+--
+
+LOCK TABLES `免费视频课程数据` WRITE;
+/*!40000 ALTER TABLE `免费视频课程数据` DISABLE KEYS */;
+INSERT INTO `免费视频课程数据` VALUES (1,1,'8年级同步课程','最简二次根式','8.1.1','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/812/812.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(2,1,'8年级同步课程','二次根式的乘法','8.1.2','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/813/813.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(3,1,'8年级同步课程','二次根式的除法','8.1.3','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/814/814.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(4,1,'8年级同步课程','二次根式的加减','8.1.4','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/816/816.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(5,1,'8年级同步课程','二次根式及其性质','8.1.5','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/811/811.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(6,1,'8年级同步课程','二次根式比较大小','8.1.6','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/817/817.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(7,1,'8年级同步课程','二次根式的化简求值','8.1.7','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/818/818.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(8,1,'8年级同步课程','双重根号的化简','8.1.8','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/819/819.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(9,1,'8年级同步课程','二次根式的新运算','8.1.9','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/8110/8110.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(10,1,'8年级同步课程','同类二次根式','8.1.10','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/815/815.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(11,1,'8年级同步课程','立方根','8.1.11','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/8111/8111.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(12,1,'8年级同步课程','实数','8.1.12','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/8112/8112.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(13,1,'8年级同步课程','实数比较大小','8.2.13','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/8213/8213.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(14,1,'8年级同步课程','实数运算','8.2.14','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/8214/8214.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(15,1,'8年级同步课程','无理数的整数部分与小数部分','8.2.15','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/8215/8215.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(16,1,'8年级同步课程','笔算开平方技巧','8.2.16','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/8216/8216.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(17,1,'8年级同步课程','勾股定理的简单计算','8.2.1','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/821/821.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(18,1,'8年级同步课程','特殊角的勾股定理','8.2.2','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/822/822.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(19,1,'8年级同步课程','勾股定理的验证','8.2.3','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/823/823.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(20,1,'8年级同步课程','勾股定理与图形面积','8.2.4','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/824/824.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(21,1,'8年级同步课程','勾股定理与折叠','8.2.5','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/825/825.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(22,1,'8年级同步课程','勾股定理逆定理及应用','8.2.6','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/826/826.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(23,1,'8年级同步课程','勾股数','8.2.7','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/827/827.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(24,1,'8年级同步课程','勾股定理与实际问题','8.2.8','叶雷讲数学','2024-06-12 07:01:00','14','FreeVideos/828/828.m3u8','Images/FreeCourse/8年级上册线上同步基础课.png'),(2,2,'9年级同步课程','直接开方法解一元二次方程','9.1.2','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9102/9102.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(3,2,'9年级同步课程','配方法解一元二次方程','9.1.3','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9103/9103.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(4,2,'9年级同步课程','利用配方法求最值','9.1.4','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9104/9104.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(1,2,'9年级同步课程','一元二次方程定义及相关基础问题','9.1.1','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9101/9101.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(5,2,'9年级同步课程','公式法解一元二次方程','9.1.5','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9105/9105.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(6,2,'9年级同步课程','利用根的判别式判断根的情况','9.1.6','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9106/9106.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(7,2,'9年级同步课程','利用一元二次方程的根求参','9.1.7','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9107/9107.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(8,2,'9年级同步课程','提取公因式法解一元二次方程','9.1.8','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9108/9108.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(9,2,'9年级同步课程','利用乘法公式解一元二次方程','9.1.9','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9109/9109.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(10,2,'9年级同步课程','十字相乘法解一元二次方程','9.1.10','叶雷讲数学','2024-07-27 00:00:00','14','FreeVideos/9110/9110.m3u8','Images/FreeCourse/9年级上册线上同步基础课.png'),(11,2,'9年级同步课程','换元法解一元二次方程','9.1.11','叶雷讲数学','2024-07-27 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+/*!40000 ALTER TABLE `免费视频课程数据` ENABLE KEYS */;
+UNLOCK TABLES;
+
+--
+-- Table structure for table `用户信息`
+--
+
+DROP TABLE IF EXISTS `用户信息`;
+/*!40101 SET @saved_cs_client = @@character_set_client */;
+/*!40101 SET character_set_client = utf8 */;
+CREATE TABLE `用户信息` (
+ `OpenID` varchar(255) DEFAULT NULL,
+ `YelayMaths_ID` varchar(255) DEFAULT NULL,
+ `用户名` varchar(255) DEFAULT NULL,
+ `是否登录` varchar(255) DEFAULT NULL,
+ `是否管理员` varchar(255) DEFAULT NULL,
+ `注册日期` varchar(255) DEFAULT NULL,
+ `最后登录日期` varchar(255) DEFAULT NULL,
+ `手机号` varchar(255) DEFAULT NULL,
+ `HeadiconID` int(11) DEFAULT NULL
+) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 ROW_FORMAT=DYNAMIC;
+/*!40101 SET character_set_client = @saved_cs_client */;
+
+--
+-- Dumping data for table `用户信息`
+--
+
+LOCK TABLES `用户信息` WRITE;
+/*!40000 ALTER TABLE `用户信息` DISABLE KEYS */;
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22:45:25','2024-06-11 22:45:25',NULL,0),('omYms6_CKn0jJqAYCxiXyS0AwMhw\" union select 1,2-- ',NULL,NULL,NULL,NULL,'2024-06-11 22:45:25','2024-06-11 22:45:25',NULL,0),('omYms6_CKn0jJqAYCxiXyS0AwMhw\" union select md5(3141592657),2-- ',NULL,NULL,NULL,NULL,'2024-06-11 22:45:26','2024-06-11 22:45:26',NULL,0),('omYms6_CKn0jJqAYCxiXyS0AwMhw\" union select 1,md5(3141592657)-- ',NULL,NULL,NULL,NULL,'2024-06-11 22:45:26','2024-06-11 22:45:26',NULL,0),('omYms6_CKn0jJqAYCxiXyS0AwMhw\')) AND (SELECT*FROM(SELECT(SLEEP(3)))hyzl) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:26','2024-06-11 22:45:26',NULL,0),('omYms6_CKn0jJqAYCxiXyS0AwMhw\" AND (SELECT*FROM(SELECT(SLEEP(4)))gaqe) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:27','2024-06-11 22:45:27',NULL,0),('\")) OR (SELECT*FROM(SELECT(SLEEP(2)))hfxn) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:27','2024-06-11 22:45:27',NULL,0),('\" union select 1-- ',NULL,NULL,NULL,NULL,'2024-06-11 22:45:27','2024-06-11 22:45:27',NULL,0),('\" union select 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22:45:29','2024-06-11 22:45:29',NULL,0),('omYms6_CKn0jJqAYCxiXyS0AwMhw\') AND (SELECT*FROM(SELECT(SLEEP(2)))lmro) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:29','2024-06-11 22:45:29',NULL,0),('\" AND (SELECT*FROM(SELECT(SLEEP(3)))vrcg) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:30','2024-06-11 22:45:30',NULL,0),('\')) AND (SELECT*FROM(SELECT(SLEEP(2)))yfif) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:30','2024-06-11 22:45:30',NULL,0),('\')) AND (SELECT*FROM(SELECT(SLEEP(3)))lmkc) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:30','2024-06-11 22:45:30',NULL,0),(' AND (SELECT*FROM(SELECT(SLEEP(4)))leay) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:31','2024-06-11 22:45:31',NULL,0),('\') OR (SELECT*FROM(SELECT(SLEEP(2)))zorb) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:31','2024-06-11 22:45:31',NULL,0),('\') OR (SELECT*FROM(SELECT(SLEEP(3)))nwkl) limit 1#',NULL,NULL,NULL,NULL,'2024-06-11 22:45:31','2024-06-11 22:45:31',NULL,0),('omYms6_CKn0jJqAYCxiXyS0AwMhw\")) AND 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+INSERT INTO `网页列表` VALUES ('1','数学常识','2025-04-20 07:01:00','数学小知识','数学中的小知识',' 1.数独游戏是一种非常受欢迎的数学谜题,它需要运用逻辑推理和数字填充技巧来完成。
\r\n 2.魔方的变化数量非常巨大,通过数学计算可以得出魔方有大约 4.3×10¹⁹ 种不同的状态。
\r\n 3.汉诺塔问题是一个经典的数学难题,通过移动盘子的过程可以锻炼逻辑思维能力。
\r\n 4. 古希腊数学家阿基米德在洗澡时发现了浮力定律,兴奋地跳出浴缸,大喊 “尤里卡!”(我找到了!)。
\r\n 5.费马大定理历经三百多年才被证明,吸引了无数数学家为之努力。
\r\n 6.哥德巴赫猜想至今仍未被完全证明,这个关于偶数可以表示为两个质数之和的猜想吸引了众多数学家的关注。
\r\n 7. 黄金分割比例(约为1.618)在自然界、艺术和建筑中广泛存在,被认为具有美学上的和谐性。
\r\n 8. 圆周率 π 是一个无理数,其小数部分无限不循环,至今已计算出超过100万亿位数字。
\r\n 9. 欧拉公式 $ e^{i\\pi} + 1 = 0 $ 被誉为“数学中最美的公式”,它将五个最重要的数学常数联系在一起。
\r\n 10. 四色定理指出,任何地图只需四种颜色就能确保相邻区域颜色不同,该定理是首个主要依靠计算机辅助证明的著名数学定理。
\r\n 11. 斐波那契数列中,每一项都是前两项之和,其相邻两项的比值会逐渐趋近于黄金分割比例。
\r\n 12. 康托尔通过“对角线论证法”证明了实数集是不可数的,开创了集合论与无穷数学的新纪元。
\r\n 13. 毕达哥拉斯定理(勾股定理)指出:在直角三角形中,斜边的平方等于两直角边的平方和,即 $ a^2 + b^2 = c^2 $ 。
\r\n 14. 莫比乌斯带是一种只有一个面和一条边的拓扑结构,将纸条一端扭转180度后粘合即可制成。
\r\n 15. 零的概念最早由古印度数学家引入,它不仅是占位符,更是一个独立的数字,对现代数学发展至关重要。
\r\n 16. 纳皮尔发明了对数,极大简化了天文和航海中的复杂乘除运算,被誉为“缩短计算时间的奇迹”。
\r\n 17. 七桥问题由欧拉解决,他证明无法不重复地走遍柯尼斯堡的七座桥,由此开创了图论这一数学分支。
\r\n 18. 帕斯卡三角形不仅展示了二项式系数的规律,还隐藏着斐波那契数列、组合数等多种数学模式。
\r\n 19. 非欧几何打破了欧几里得第五公设的限制,为爱因斯坦的广义相对论提供了重要的数学基础。
\r\n 20. 拉马努金是一位自学成才的印度数学天才,留下了数千个深奥而优美的公式,许多至今仍在被研究和证明。
\r\n 21. “生日悖论”指出:在一个23人的群体中,有超过50%的概率至少两人同一天生日,这与直觉相悖但数学上成立。
\r\n 22. 卡塔兰数在括号匹配、二叉树计数、多边形三角剖分等问题中频繁出现,是组合数学中的重要数列。
'),('1','数学趣事','2025-04-20 07:01:00','数学小知识','有趣的数学',' 1.数独游戏是一种非常受欢迎的数学谜题,它需要运用逻辑推理和数字填充技巧来完成。
\r\n 魔方的变化数量非常巨大,通过数学计算可以得出魔方有大约 4.3×10¹⁹ 种不同的状态。
\r\n 3.汉诺塔问题是一个经典的数学难题,通过移动盘子的过程可以锻炼逻辑思维能力。
\r\n 4. 古希腊数学家阿基米德在洗澡时发现了浮力定律,兴奋地跳出浴缸,大喊 “尤里卡!”(我找到了!)。
\r\n 5.费马大定理历经三百多年才被证明,吸引了无数数学家为之努力。
\r\n 6.哥德巴赫猜想至今仍未被完全证明,这个关于偶数可以表示为两个质数之和的猜想吸引了众多数学家的关注。
\r\n 7. 黄金分割比例 φ(约为1.618)在自然界、艺术和建筑中广泛存在,被认为具有美学上的和谐性。
\r\n 8. 四色定理指出,任何地图只需四种颜色即可确保相邻区域颜色不同,该定理是首个借助计算机辅助证明的重要数学成果。
\r\n 9. 欧拉公式 e^(iπ) + 1 = 0 被誉为“数学中最美的公式”,巧妙地将五个最重要的数学常数联系在一起。
\r\n 10. 康托尔创立了集合论,并首次严格定义了“无穷”的不同等级,引发了数学基础的深刻变革。
\r\n 11. 斐波那契数列中相邻两项的比值会逐渐趋近于黄金分割比例,这一现象在植物叶序和螺旋壳体中屡见不鲜。
\r\n 12. 非欧几何的出现打破了欧几里得几何的唯一性,为爱因斯坦的广义相对论提供了重要的数学工具。
\r\n 13. 图灵机是一种抽象计算模型,奠定了现代计算机科学的理论基础。
\r\n 14. 蒙提霍尔问题是一个反直觉的概率谜题,正确策略能将胜率从1/3提升至2/3。
\r\n 15. 勾股定理早在古巴比伦时期就已被使用,但毕达哥拉斯学派首次给出了严格的数学证明。
\r\n 16. 纳什均衡是博弈论中的核心概念,描述了在多人决策中无人愿意单方面改变策略的稳定状态。
\r\n 17. 圆周率 π 是一个无理数,其小数部分无限不循环,至今已计算出超过100万亿位。
\r\n 18. 拓扑学研究的是在连续变形下保持不变的性质,例如“咖啡杯与甜甜圈在拓扑上是等价的”。
\r\n 19. 高斯在少年时代就快速算出1到100的和,展现了他非凡的数学天赋。
\r\n 20. 黎曼猜想是关于黎曼ζ函数非平凡零点分布的著名未解难题,被列为千禧年七大数学难题之一。
\r\n 21. 分形几何通过自相似结构描述复杂自然形态,如海岸线、云朵和山脉的轮廓。
\r\n 22. 二进制系统仅用0和1两个数字,却构成了现代所有数字设备运算的基础。
\r\n 23. 概率论起源于17世纪对赌博问题的研究,如今广泛应用于金融、人工智能和天气预报等领域。
\r\n 24. 希尔伯特在1900年提出了23个数学问题,深刻影响了20世纪数学的发展方向。
\r\n 25. 卡塔兰数在组合数学中频繁出现,可用于计算合法括号序列、二叉树结构等问题的数量。
\r\n 26. 群论最初用于研究代数方程的可解性,后来成为描述对称性的强大数学语言。
\r\n 27. 费马素性检验是一种基于费马小定理的快速判断质数的方法,虽有误判可能但效率极高。
\r\n 28. 拉格朗日中值定理是微积分中的基本结果,揭示了函数变化率与平均变化率之间的关系。
\r\n 29. 帕斯卡三角形不仅包含二项式系数,还隐藏着斐波那契数列、平方数等多种数学规律。
\r\n 30. 布尔代数由乔治·布尔创立,为逻辑运算和数字电路设计提供了数学基础。
\r\n 31. 柯尼斯堡七桥问题是图论的起源,欧拉证明了无法不重复地走遍所有桥梁。
\r\n 32. 熵在信息论中衡量不确定性,香农将其引入通信领域,奠定了现代信息科学的基石。
\r\n 33. 牛顿与莱布尼茨各自独立发明了微积分,尽管曾因优先权发生激烈争议。
\r\n 34. 质数在密码学中至关重要,RSA加密算法的安全性依赖于大整数分解的困难性。
\r\n 35. 皮亚诺公理系统为自然数提供了严谨的公理化基础,推动了数学逻辑的发展。
\r\n 36. 傅里叶变换能将复杂信号分解为简单正弦波的叠加,广泛应用于音频处理、图像压缩等领域。
'),('1','九章算数的来历','2026-01-16 07:00:00','数学小传记','关于九章算数你都知道吗','《九章算术》是中国古代一部极为重要的数学经典,成书时间大约在西汉时期(公元前2世纪至公元1世纪之间),其内容是在长期积累和整理先秦至汉代数学知识的基础上形成的。虽然作者已不可考,但传统上认为该书由张苍、耿寿昌等人整理编订。
\r\n《九章算术》共分为九章,分别是:方田、粟米、衰分、少广、商功、均输、盈不足、方程和勾股。全书以问题集的形式呈现,共收录了246个数学问题及其解法,涵盖了分数运算、比例、面积与体积计算、线性方程组求解、开方以及勾股定理等丰富内容,体现了中国古代数学的高度实用性和系统性。
\r\n该书不仅在中国数学史上具有奠基性地位,还对东亚其他国家(如日本、朝鲜)的数学发展产生了深远影响。魏晋时期的数学家刘徽曾为《九章算术》作注,进一步完善了其中的理论基础,使其成为中国传统数学体系的核心著作之一。
\r\n《九章算术》作为中国古代最具代表性的数学著作,其数学方法具有鲜明的实用性和算法化特征,体现了中国传统数学的独特风格。主要特点包括以下几个方面:
\r\n全书采用“问题—答案—术”(即问题陈述、数值答案、解题算法)的结构,共收录246个实际应用问题,内容涉及土地测量、赋税分配、工程计算、市场交易等社会生活各个方面。数学被视为解决现实问题的工具,而非抽象推理的对象。
\r\n《九章算术》中的“术”相当于现代意义上的算法,通常以简洁、通用的步骤描述如何求解一类问题。例如,“方程术”通过“直除法”(类似高斯消元)解线性方程组,“开方术”系统地处理平方根和立方根的计算。这些方法具有高度的可操作性和重复使用性。
\r\n书中对分数的四则运算、通分、约分等有完整而成熟的处理,是世界上最早系统论述分数运算的文献之一。此外,大量问题依赖精确的数值计算,体现出对算术技巧的高度掌握。
\r\n《九章算术》中的几何问题(如面积、体积计算)往往通过代数方法求解,强调“以算代证”。它不像古希腊数学那样追求逻辑公理体系和演绎证明,而是通过构造性算法和实例验证来确立方法的有效性。
\r\n书中将具体问题抽象为典型模型(如“均输”“盈不足”),每类模型配有一套通用解法。这种“模型—算法”对应的思想,使数学知识具有良好的组织性和可推广性,是中国古代数学高度系统化的体现。
\r\n在“方程”章中,《九章算术》使用算筹布列系数,通过行变换消元求解多元一次方程组,并明确引入了负数概念及正负数加减法则,这在世界数学史上具有领先意义。
\r\n综上所述,《九章算术》的数学方法以实用为导向、以算法为核心、以数值计算为基础,形成了与中国古代社会需求高度契合的数学传统,对后世中国乃至东亚数学的发展产生了深远影响。
\r\n《九章算术》作为中国古代数学的集大成之作,不仅奠定了中国传统数学的基础,还通过文化交流深刻影响了整个东亚地区的数学发展。
\r\n自汉代以后,《九章算术》及其注释本(尤其是刘徽、祖冲之等人的注解)传入朝鲜半岛和日本。在古代朝鲜(如高丽、李朝时期)和日本(如奈良、平安时代),该书长期被用作官方教育体系中的数学教材,是士人学习算学的必读经典。其以实用问题为导向的结构和清晰的算法步骤,非常适合教学与行政应用。
\r\n《九章算术》强调“经世致用”,将数学与土地丈量、赋税计算、工程营造、粮食兑换等实际政务紧密结合。这种实用主义数学观被朝鲜、日本等国广泛接受,并融入其本土的数学体系中,形成了以算法为中心、注重问题解决而非纯理论证明的东亚数学传统。
\r\n受《九章算术》启发,朝鲜和日本的学者不仅研习其内容,还在此基础上进行注释、改编甚至创新。例如,日本江户时代的“和算”虽发展出独特风格,但其早期基础仍深受《九章算术》中“方程”“盈不足”等章节的影响;朝鲜王朝的《算学启蒙》《九章详义》等著作也直接承袭了《九章》的体例与方法。
\r\n随着汉文典籍在东亚的流通,《九章算术》成为汉字文化圈共享的知识资源。它不仅传递了具体的计算技术(如分数运算、开方、线性方程组解法),还传播了中国特有的数学思维方式,如“以题带法”“寓理于算”等,对整个区域的科学文化产生了持久影响。
\r\n综上所述,《九章算术》不仅是中华文明的数学瑰宝,更是东亚数学共同体形成的重要纽带,其影响跨越国界、延续千年,至今仍被视为东方数学智慧的象征。
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+INSERT INTO `网页数据` VALUES ('1','1','1','全等三角形是初中数学一个重难点内容,相信很多同学和家长对下面这几条判定并不陌生,它们被大家俗称为“4+1”,4条是适用于所有三角形,1条是适用直角三角形。'),('1','2','2','1.jpg'),('1','3','1','很多同学,或者家长小时候在学这部分知识时,一定都背过,但是当问到“为什么这4条能判定三角形全等”时,却很少能回答出来。如果只靠死记硬背这些判定,就只能停留在“是什么”上。'),('1','6','1','对于“为什么”该怎么学?掌握这两个字——梳理'),('1','5','1','我们来一一梳理这些判定规则,帮助大家理解和记忆。'),('1','7','1','①“边边边”最好理解,因为三角形具有稳定性。也就是三角形的三条边长度确定以后,这个三角形就固定了。'),('1','4','1','②“边角边”的本质是一种动态变化。比如我们把两条边想象成是时针和分针在走,它们在开开合合;当喊停的时候,时针和分针不动了,其实也就是它们组成的角也固定住了。两条边长度不变,角也固定了,接下来只需要把第三条边连上。'),('1','8','2','2.jpg'),('1','9','1','③ 接下来是“角角边”和“角边角”。我们先来思考,三角形的角度决定了三角形的什么?'),('1','10','1','比如我们画两个三角形,它们的角度都是100°、50°、30°,我们会发现它们的形状是一样的。'),('1','11','2','3.jpg'),('1','12','1','所以角度决定了三角形的形状,而且只要有两个角相等了,它们的形状就是一样的。因为三角形三个角加起来是180°,有两个角相等了,另外一个也是相等的。'),('1','13','1','接下来再来思考,当两个三角形形状一样,边决定了三角形的什么?边的长度决定了三角形的大小。'),('1','14','1','\r\n所以当两个三角形,两个角相等决定了它们形状一样,再来任意一条边相等,这两个三角形就是全等三角形。'),('1','15','1','知道“为什么”之后,再来看“怎么做”,还是用一道题目来举例。'),('1','16','2','4.jpg'),('1','17','1','根据判定规则,我们记住这六个字,就可以解决这道题:共顶点,红配蓝(也可以其他两个颜色)'),('1','18','1','把公共顶点A的这两个三角形里面,相等的线段用红蓝两种颜色表示'),('1','19','2','5.jpg'),('1','20','1','画完之后,就变得很明显了,我们再把第三条边用相应的颜色连上,答案就出来了。'),('1','21','2','6.jpg'),('1','22','1','从“是什么”、“为什么”到“怎么做”是初中几何学习的核心方法,在学习几何的各个板块时,都要让孩子问自己,这三点是否真正学明白了。'),('2','1','1','在初中数学的考试当中,解所有题型都需要用到公式,公式的重要性不言而喻!为了更好的帮助孩子们记住初中数学常用的公式、定理,在此老师给大家总结了初中三年所有的公式及定理!这份资料十分珍贵,每次考试都能用的上!'),('2','2','2','1.jpg'),('2','3','2','2.jpg'),('2','4','2','3.jpg'),('2','5','2','4.jpg'),('2','6','2','5.jpg'),('2','7','2','6.jpg'),('2','8','2','7.jpg'),('2','9','2','8.jpg'),('2','10','2','9.jpg'),('2','11','2','10.jpg'),('2','12','2','11.jpg'),('2','13','2','12.jpg'),('2','14','2','13.jpg'),('2','15','2','14.jpg'),('2','16','2','15.jpg'),('2','17','2','16.jpg'),('2','18','2','17.jpg'),('2','19','2','18.jpg'),('2','20','2','19.jpg'),('2','21','2','20.jpg'),('2','22','2','21.jpg'),('3','1','1','家长们先来个#不生气挑战#,点击观看下方视频,看看你能忍到第几个?'),('3','2','3','2.mp4'),('3','3','1','怎么样?有感触吧!平时孩子作业时,咱们辅导孩子是不是也和视频里一样鸡飞狗跳?情绪经常崩溃,做题全靠生活经验,辅导孩子他还听不懂......'),('3','4','1','很多家长百思不得其解:明明该教的也都教了,孩子也很努力,为什么孩子在数学上就是不“开窍”呢?'),('3','5','1','这里就涉及到我们常说的一个概念——“数学思维”。这个概念听起来高大上,理解起来却很模糊。但它往往是影响孩子数学成绩的重要因素,今天就给大家解读一下这个玄乎的“数学思维”!'),('3','6','1','数学思维是何方神圣?'),('3','7','1','简单来说,「能够用数学的观点去思考问题和解决问题的能力」就是数学思维。'),('3','8','1','孩子数学跟不上,多数情况是因为数学思维不好!打个比方,假如每道数学题都是一片沙漠,孩子是迷路的旅人需要找到正确的方向,那么数学思维就是带领旅人走出沙漠的导航。'),('3','9','1','如果没有数学思维,孩子就没有一条明确的解题路线,四处试探的结果只能能是解题失败,还非常容易陷入“死记硬背、做题不会”的困境。'),('3','10','2','3.jpg'),('3','11','1','数学思维如何培养?'),('3','12','1','1、让孩子试着“讲题”'),('3','13','1','会做不一定会讲,孩子做完题后,家长不妨鼓励孩子开口讲解一下某道题,当一当家长的小老师。'),('3','14','1','孩子要想讲给家长听,就要思考、分析、整理这道题的解题思路,然后有条理地讲出来,这个知识就牢牢地成为他自己的知识,并且能够举一反三、触类旁通。'),('3','15','1','家长激励孩子多讲题,也为他建立了学习的信心和成就感。孩子能开口讲题、讲思路,这就是最好的数学思维训练模式。'),('3','16','2','4.jpg'),('3','17','1','2、抽象算式“具象化”'),('3','18','1','把抽象的数学实践到具体生活中,可以帮助孩子更容易理解。比如去超市买东西,通过标签价格、数量让孩子帮忙算算有没有超出预算等。'),('3','19','1','解题也一样,比如锯木头问题,锯四段需要6分钟,6不是4的倍数怎么算呢?这时候可以让孩子把这根木头画出来,分成4段就会发现其实只锯了3次,那问题就迎刃而解了。'),('3','20','1','转化思路,通过具象思维帮助孩子理解,善于发现这些隐藏在生活中的数学思维,学数学就会变成一件轻松有趣的事。'),('3','21','2','5.jpg'),('3','22','1','因此,学好数学需要抓住思维培养的关键期3-12岁,这个时期孩子的大脑快速发育,思维活跃,可塑性强,最有利于数学思维的培养!而数学常用的16种思维方法:分类思想、转化思想、假设思想、类比思想、极限思想、可逆思想等。'),('3','23','1','想要在小学这个关键时期,培养好这些数学思维方法,最直接有效的方法,就是选择一位资深老师,用系统的教材,通过科学的系统学习,才能构建完整的知识体系。'),('4','1','1','追本溯源化繁为简'),('4','2','1','目有千万而纲为一,枝叶繁多而本为一。纲举则目张,执本而末从。如果只在细枝末节上下功夫,费了力气却讨不了好。学习就是不断地归一,最终以一心一理贯通万事万物,则达自由无碍之化境矣(呵呵,这境界有点高,慢慢来)。'),('4','3','1','关于几何最值问题研究的老师很多,本人以前也有文章论述,本文在此基础上再次进行归纳总结,把各种知识、方法、思想、策略进行融合提炼、追本溯源、认祖归宗,以使解决此类问题时更加简单明晰。'),('4','5','1','所有问题的老祖宗只有两个:①[定点到定点]:两点之间,线段最短;②[定点到定线]:点线之间,垂线段最短。'),('4','5','2','1.gif'),('4','7','2','2.gif'),('4','8','1','由此派生:
③[定点到定点]:三角形两边之和大于第三边;
④[定线到定线]:平行线之间,垂线段最短;
⑤[定点到定圆]:点圆之间,点心线截距最短(长);
⑥[定线到定圆]:线圆之间,心垂线截距最短;
⑦[定圆到定圆]:圆圆之间,连心线截距最短(长)。'),('4','9','1','余不赘述,下面仅举一例证明:'),('4','10','1','[定点到定圆]:点圆之间,点心线截距最短(长)。'),('4','11','1','已知⊙O半径为r,AO=d,P是⊙O上一点,求AP的最大值和最小值。'),('4','12','2','3.png'),('4','13','1','证明:'),('4','14','1','由“两点之间,线段最短”得AP≤AO+PO,AO≤AP+PO,'),('4','15','1','得d-r≤AP≤d+r,AP最小时点P在B处,最大时点P在C处。'),('4','16','1','即过圆心和定点的直线截得的线段AB、AC分别最小、最大值。'),('4','17','1','(可用“三角形两边之和大于第三边”,其实质也是由“两点之间,线段最短”推得)。'),('4','18','1','上面几种是解决相关问题的基本图形,所有的几何最值问题都是转化成上述基本图形解决的。'),('4','19','1','二、考试中出现的问题都是在基本图形的基础上进行变式,如圆与线这些图形不是直接给出,而是以符合一定条件的动点的形式确定的;再如过定点的直线与动点所在路径不相交而需要进行变换的。'),('4','20','1','类型分三种情况:'),('4','21','1','(1)直接包含基本图形;'),('4','22','1','(2)动点路径待确定;'),('4','23','1','(3)动线(定点)位置需变换。'),('4','24','1','(一)直接包含基本图形。'),('4','25','1','例1:'),('4','26','1','在⊙O中,圆的半径为6,∠B=30°,AC是⊙O的切线,则CD的最小值是 。'),('4','27','2','4.gif'),('4','28','1','简析:由∠B=30°知弧AD一定,所以D是定点,C是直线AC上的动点,即为求定点D到定线AC的最短路径,求得当CD⊥AC时最短为3。'),('4','29','1','(二)动点路径待确定。'),('4','30','1','例2:'),('4','31','1','如图,在△ABC中,∠ACB=90°,AB=5,BC=3,P是AB边上的动点(不与点B重合),将△BCP沿CP所在的直线翻折,得到△B′CP,连接B′A,则B′A长度的最小值是 。'),('4','32','2','5.gif'),('4','33','1','简析:A是定点,B\'是动点,但题中未明确告知B\'点的运动路径,所以需先确定B\'点运动路径是什么图形,一般有直线与圆两类。此题中B\'的路径是以C为圆心,BC为半径的圆弧,从而转化为定点到定圆的最短路径为AC-B\'C=1。'),('4','34','1','例3:'),('4','35','1','在△ABC中,AB=AC=5,cos∠ABC=3/5,将△ABC绕点C顺时针旋转,得到△A\'B\'C,点E是BC上的中点,点F为线段AB上的动点,在△A\'B\'C绕点C顺时针旋转过程中,点F的对应点是F\',求线段EF\'长度的最大值与最小值的差。'),('4','36','2','6.png'),('4','37','1','简析:E是定点,F\'是动点,要确定F\'点的运动路径。先确定线段A\'B\'的运动轨迹是圆环,外圆半径为BC,内圆半径为AB边上的高,F\'是A\'B\'上任意一点,因此F\'的运动轨迹是圆环内的任意一点,由此转化为点E到圆环的最短和最长路径。'),('4','38','2','7.gif'),('4','39','2','8.png'),('4','40','2','9.png'),('4','41','1','E到圆环的最短距离为EF2=CF2-CE=4.8-3=1.8,'),('4','42','1','E到圆环的最长距离为EF1=EC+CF1=3+6=9,其差为7.2。'),('4','43','1','(三)动线(定点)位置需变换。'),('4','44','1','线段变换的方法:'),('4','45','1','(1)等值变换:翻折、平移;'),('4','46','1','(2)比例变换:三角、相似。'),('4','47','1','【翻折变换类】典型问题:“将军饮马”。'),('4','48','1','例4:'),('4','49','1','如图,∠AOB=30°,点M、N分别是射线OA、OB上的动点,OP平分∠AOB,且OP=6,当△PMN的周长最小值为 。'),('4','50','2','10.gif'),('4','51','1','简析:动线段(或定点)应居于动点轨迹的两侧,本题的三条动线段PM、MN、PN在OA、OB的内侧。所以本题的关键是把定线段变换到动点轨迹的两侧,从而把三条动线段PM、MN、PN转化为连接两点之间的路径。如图,把点P分别沿OA、OB翻折得P1、P2,△PMN的周长转化为P1M+MN+P2N,这三条线段的和正是连接两个定点P1、P2之间的路径,从而转化为求P1、P2两点之间最短路径,得△PMN的周长最小值为线段P1P2=OP=6。'),('4','52','1','例5:'),('4','53','1','如图,在锐角△ABC中,AB=4,∠BAC=45°,∠BAC的平分线交BC于点D,M、N分别是AD和AB上的动点,则BM+MN的最小值是 。'),('4','54','2','11.gif'),('4','55','1','简析:本题的问题也在于动线段BM、MN居于动点轨迹AD的同侧,同样把点N沿AD翻折至AC上,BM+MN=BM+MN\',转化为求点B到直线AC的最短路径,即BN\'⊥AC时,最小值为2√2。'),('4','56','1','【平移变换类】典型问题:“造桥选址”。'),('4','57','1','例6:'),('4','58','1','如图,m、n是小河两岸,河宽20米,A、B是河旁两个村庄,要在河上造一座桥,要使A、B之间的路径最短应该如何选址(桥须与河岸垂直)?'),('4','59','2','12.png'),('4','60','1','简析:桥长为定值,可以想像把河岸m向下平移与n重合,同时把点A向下平移河宽,此时转化成n上的一点到A、B的路径之和最短,即转化为定点A\'到定点B的最短路径。如下图:'),('4','61','2','13.gif'),('4','62','1','思路是把动线AM平移至A\'M,A\'N+BN即转化为求定点A\'与定点B之间的最路径。本题的关键是定长线段MN把动线段分隔,此时须通过平移把动线段A\'N、BN变为连续路径,也可以把点B向上平移20米与点A连接。'),('4','63','1','例7:'),('4','64','1','如图,CD是直线y=x上的一条定长的动线段,且CD=2,点A(4,0),连接AC、AD,设C点横坐标为m,求m为何值时,△ACD的周长最小,并求出这个最小值。'),('4','65','1','解析:两条动线段AC、AD居于动点所在直线的两侧,不符合基本图形中定形(点线圆)应在动点轨迹的两侧。首先把AC沿直线CD翻折至另一侧,如下图:'),('4','66','2','14.png'),('4','67','1','现在把周长转化为A\'C+CD+AD,还需解决一个问题:动线段A\'C与AD之间被定长线段CD阻断,动线段必须转化成连续的路径。同上题的道理,把A\'C沿CD方向平移CD的长度即可,如下图。'),('4','68','2','15.png'),('4','69','1','现在已经转化为A\'\'D+AD的最短路径问题,属定点到定点,当A\'\'D与AD共线时A\'\'D+AD最短,即为线段AA\'\'的长。'),('4','70','2','16.png'),('4','71','1','【三角变换类】典型问题:“胡不归”。'),('4','72','1','例8:'),('4','73','1','如图,A地在公路BC旁的沙漠里,A到BC的距离AH=2√3,AB=2√19,在公路BC上行进的速度是在沙漠里行驶速度的2倍。某人在B地工作,A地家中父亲病危,他急着沿直线BA赶路,谁知最终没能见到父亲最后一面,其父离世之时思念儿子,连连问:“胡不归,胡不归……!”(怎么还不回来),这真是一个悲伤的故事,也是因为不懂数学而导致的。那么,从B至A怎样行进才能最快到达?'),('4','74','2','17.png'),('4','75','1','简析:BP段行驶速度是AP段的2倍,要求时间最短即求BP/2+AP最小,从而考虑BP/2如何转化,可以构造含30°角利用三角函数关系把BP/2转化为另一条线段。如下图,作∠CBD=30°,PQ⊥BD,得PQ=1/2BP,由“垂线段最短”知当A、P、Q共线时AP+PQ=AQ\'最小。'),('4','76','2','18.png'),('4','77','1','【相似变换类】典型问题:“阿氏圆”。'),('4','78','1','“阿氏圆”:知平面上两点A、B,则所有满足PA/PB=k且不等于1的点P的轨迹是一个圆,这个轨迹最先由古希腊数学家阿波罗尼斯发现,故称阿氏圆,如下图所示,其中PO:BO=AO:PO=PA:PB=k。'),('4','79','2','19.gif'),('4','80','1','例9:已知A(-4,-4)、B(0, 4)、C(0, -6)、 D(0, -1),AB与x轴交于点E,以点E为圆心,ED长为半径作圆,点M为⊙E上一动点,求 1/2AM+CM 的最小值。'),('4','81','2','20.png'),('4','82','1','简析:本题的主要问题在于如何转化1/2AM,注意到由条件知,在M的运动过程中,EM:AE=1:2保持不变,从而想到构造相似三角形,使之与△AEM的相似比为1:2,这样便可实现1/2AM的转化,如下图取EN:EM=1:2,即可得△EMN∽△EAM,再得MN=1/2AM,显然,MN+CM的最小值就是定点N、C之间的最短路径。'),('4','83','2','21.png'),('4','84','1','之后便是常规方法先求N点坐标,再求CN的长。'),('4','85','2','22.png'),('4','86','1','【解法大一统】'),('4','87','1','万法归宗:路径成最短,折线到直线。'),('4','88','1','(所求路径在一般情况下是若干折线的组合,这些折线在同一直线上时即为最短路径)'),('4','89','1','基本图形:动点有轨迹,动线居两边。'),('4','90','1','(动点轨迹可以是线或圆,动线指动点与定点或定线、定圆的连线,动线与折线同指)'),('4','91','1','核心方法:同侧变异侧,分散化连续。'),('4','92','1','(动线在同侧,要变为异侧,一般用翻折、三角、相似的方法构造;动折线被定长线段分散时需化为连续折线,一般用平移的方法构造,如造桥选址问题)'),('4','93','1','下图是构造完成的目标图形:'),('4','94','2','23.png'),('4','95','1','再举2例说明上述规律的运用方法:'),('4','96','1','1.'),('4','97','1','如图,菱形ABCD中,∠A=60°,AB=3,⊙A、⊙B的半径为2和1,P、E、F分别是CD、⊙A、⊙B上的动点,则PE+PF的最小值为 。'),('4','98','2','24.png'),('4','99','1','思考方法如下图所示:'),('4','100','2','25.png'),('4','101','1','2.'),('4','102','1','菱形ABCD中,∠BAC=60°,P是AC上的动点,求BP+1/2AP的最小值。'),('4','103','2','26.png'),('4','104','1','思考方法如下图所示:'),('4','105','2','27.png'),('4','4','1','一、基本图形'),('5','1','2','1.png'),('6','1','2','1.png');
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\r\n 1. 圆周率π的小数部分无限不循环,到目前为止已经计算到62.8万亿位,但至今仍未找到规律。
\r\n 2. 斐波那契数列中的第144项是75025,这个数与黄金比例0.618非常接近,因此它在自然界中频频出现,比如植物叶片的排列。
\r\n 3. 自然数中的完全数很罕见,比如6、28、496等,已知的完全数至今还不到50个,但数字却可以达到数十位之多。
\r\n 4. 18世纪的数学家欧拉推算出一个奇妙的公式,即欧拉恒等式,包含5个最重要的数学常数:0、1、π、e、i。
\r\n 5. 世界上有无数个素数,但其中较大的素数非常稀少,2018年发现的最大素数长度高达2480万位。
\r\n 6. 自然数可以被分为阿姆斯特朗数与非阿姆斯特朗数,三位数中153就是一个阿姆斯特朗数,因为1^3 + 5^3 + 3^3 = 153。
\r\n 7. 数学家花费了300多年时间才证明了费马大定理,该定理表示没有整数解使得a^n + b^n = c^n (n > 2)。
\r\n 8. 在一个20层楼的正方体魔方中,所有格子的数量达到8000个,打乱后完全还原需要数百万步。
\r\n 9. 以数字7开头的奇数在任何自然数集合中占比大约14%,这是因为7被认为是最神秘的数字之一。
\r\n 10. 数字“9”在乘法运算中非常特别,9的任何倍数的各位数字和最终都会得到9,比如18、27、36等。
\r\n 11. “友爱数”很罕见,比如220和284,每个数的因子之和等于对方(220的因子和是284,284的因子和是220)。
\r\n 12. 平方和倒数的和等于1的数对叫做互质数,如3和6、4和9,这类数有数百万对。
\r\n 13. 黄金分割0.618的神奇之处在于,自然界中的许多花瓣、松果和贝壳呈现出这个比例。
\r\n 14. 数学家发现,圆的内接正n边形的面积随着n无限增大时会无限接近π,这成为了圆面积的经典证明之一。
\r\n 15. 在数论中,只有3个平方数的相加和能等于同一数的平方,比如3^2 + 4^2 + 5^2 = 12^2,这叫“勾股数”。
\r\n 16. 以e为底的自然对数是最神奇的常数之一,它等于2.718,且在金融、物理等多领域应用广泛。
\r\n 17. 在一个随机数序列中,每500个数内约有7个素数,这种分布符合“素数定理”。
\r\n 18. 数学的双射性原理指出,如果两个集合元素一一对应,就称为双射,比如自然数集和整数集的双射关系。
\r\n 19. “柏拉图正多面体”只有5种,即四面体、六面体、八面体、十二面体和二十面体。
\r\n 20. 数学家高斯在7岁时就发现自然数1至100的和是5050,得出公式n(n+1)/2
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+ `课程介绍` tinytext,
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+ `最后观看日期` varchar(255) DEFAULT NULL,
+ `观看时长` varchar(255) DEFAULT NULL
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+INSERT INTO `网页数据` VALUES ('1','1','1','全等三角形是初中数学一个重难点内容,相信很多同学和家长对下面这几条判定并不陌生,它们被大家俗称为“4+1”,4条是适用于所有三角形,1条是适用直角三角形。'),('1','2','2','1.jpg'),('1','3','1','很多同学,或者家长小时候在学这部分知识时,一定都背过,但是当问到“为什么这4条能判定三角形全等”时,却很少能回答出来。如果只靠死记硬背这些判定,就只能停留在“是什么”上。'),('1','6','1','对于“为什么”该怎么学?掌握这两个字——梳理'),('1','5','1','我们来一一梳理这些判定规则,帮助大家理解和记忆。'),('1','7','1','①“边边边”最好理解,因为三角形具有稳定性。也就是三角形的三条边长度确定以后,这个三角形就固定了。'),('1','4','1','②“边角边”的本质是一种动态变化。比如我们把两条边想象成是时针和分针在走,它们在开开合合;当喊停的时候,时针和分针不动了,其实也就是它们组成的角也固定住了。两条边长度不变,角也固定了,接下来只需要把第三条边连上。'),('1','8','2','2.jpg'),('1','9','1','③ 接下来是“角角边”和“角边角”。我们先来思考,三角形的角度决定了三角形的什么?'),('1','10','1','比如我们画两个三角形,它们的角度都是100°、50°、30°,我们会发现它们的形状是一样的。'),('1','11','2','3.jpg'),('1','12','1','所以角度决定了三角形的形状,而且只要有两个角相等了,它们的形状就是一样的。因为三角形三个角加起来是180°,有两个角相等了,另外一个也是相等的。'),('1','13','1','接下来再来思考,当两个三角形形状一样,边决定了三角形的什么?边的长度决定了三角形的大小。'),('1','14','1','\r\n所以当两个三角形,两个角相等决定了它们形状一样,再来任意一条边相等,这两个三角形就是全等三角形。'),('1','15','1','知道“为什么”之后,再来看“怎么做”,还是用一道题目来举例。'),('1','16','2','4.jpg'),('1','17','1','根据判定规则,我们记住这六个字,就可以解决这道题:共顶点,红配蓝(也可以其他两个颜色)'),('1','18','1','把公共顶点A的这两个三角形里面,相等的线段用红蓝两种颜色表示'),('1','19','2','5.jpg'),('1','20','1','画完之后,就变得很明显了,我们再把第三条边用相应的颜色连上,答案就出来了。'),('1','21','2','6.jpg'),('1','22','1','从“是什么”、“为什么”到“怎么做”是初中几何学习的核心方法,在学习几何的各个板块时,都要让孩子问自己,这三点是否真正学明白了。'),('2','1','1','在初中数学的考试当中,解所有题型都需要用到公式,公式的重要性不言而喻!为了更好的帮助孩子们记住初中数学常用的公式、定理,在此老师给大家总结了初中三年所有的公式及定理!这份资料十分珍贵,每次考试都能用的上!'),('2','2','2','1.jpg'),('2','3','2','2.jpg'),('2','4','2','3.jpg'),('2','5','2','4.jpg'),('2','6','2','5.jpg'),('2','7','2','6.jpg'),('2','8','2','7.jpg'),('2','9','2','8.jpg'),('2','10','2','9.jpg'),('2','11','2','10.jpg'),('2','12','2','11.jpg'),('2','13','2','12.jpg'),('2','14','2','13.jpg'),('2','15','2','14.jpg'),('2','16','2','15.jpg'),('2','17','2','16.jpg'),('2','18','2','17.jpg'),('2','19','2','18.jpg'),('2','20','2','19.jpg'),('2','21','2','20.jpg'),('2','22','2','21.jpg'),('3','1','1','家长们先来个#不生气挑战#,点击观看下方视频,看看你能忍到第几个?'),('3','2','3','2.mp4'),('3','3','1','怎么样?有感触吧!平时孩子作业时,咱们辅导孩子是不是也和视频里一样鸡飞狗跳?情绪经常崩溃,做题全靠生活经验,辅导孩子他还听不懂......'),('3','4','1','很多家长百思不得其解:明明该教的也都教了,孩子也很努力,为什么孩子在数学上就是不“开窍”呢?'),('3','5','1','这里就涉及到我们常说的一个概念——“数学思维”。这个概念听起来高大上,理解起来却很模糊。但它往往是影响孩子数学成绩的重要因素,今天就给大家解读一下这个玄乎的“数学思维”!'),('3','6','1','数学思维是何方神圣?'),('3','7','1','简单来说,「能够用数学的观点去思考问题和解决问题的能力」就是数学思维。'),('3','8','1','孩子数学跟不上,多数情况是因为数学思维不好!打个比方,假如每道数学题都是一片沙漠,孩子是迷路的旅人需要找到正确的方向,那么数学思维就是带领旅人走出沙漠的导航。'),('3','9','1','如果没有数学思维,孩子就没有一条明确的解题路线,四处试探的结果只能能是解题失败,还非常容易陷入“死记硬背、做题不会”的困境。'),('3','10','2','3.jpg'),('3','11','1','数学思维如何培养?'),('3','12','1','1、让孩子试着“讲题”'),('3','13','1','会做不一定会讲,孩子做完题后,家长不妨鼓励孩子开口讲解一下某道题,当一当家长的小老师。'),('3','14','1','孩子要想讲给家长听,就要思考、分析、整理这道题的解题思路,然后有条理地讲出来,这个知识就牢牢地成为他自己的知识,并且能够举一反三、触类旁通。'),('3','15','1','家长激励孩子多讲题,也为他建立了学习的信心和成就感。孩子能开口讲题、讲思路,这就是最好的数学思维训练模式。'),('3','16','2','4.jpg'),('3','17','1','2、抽象算式“具象化”'),('3','18','1','把抽象的数学实践到具体生活中,可以帮助孩子更容易理解。比如去超市买东西,通过标签价格、数量让孩子帮忙算算有没有超出预算等。'),('3','19','1','解题也一样,比如锯木头问题,锯四段需要6分钟,6不是4的倍数怎么算呢?这时候可以让孩子把这根木头画出来,分成4段就会发现其实只锯了3次,那问题就迎刃而解了。'),('3','20','1','转化思路,通过具象思维帮助孩子理解,善于发现这些隐藏在生活中的数学思维,学数学就会变成一件轻松有趣的事。'),('3','21','2','5.jpg'),('3','22','1','因此,学好数学需要抓住思维培养的关键期3-12岁,这个时期孩子的大脑快速发育,思维活跃,可塑性强,最有利于数学思维的培养!而数学常用的16种思维方法:分类思想、转化思想、假设思想、类比思想、极限思想、可逆思想等。'),('3','23','1','想要在小学这个关键时期,培养好这些数学思维方法,最直接有效的方法,就是选择一位资深老师,用系统的教材,通过科学的系统学习,才能构建完整的知识体系。'),('4','1','1','追本溯源化繁为简'),('4','2','1','目有千万而纲为一,枝叶繁多而本为一。纲举则目张,执本而末从。如果只在细枝末节上下功夫,费了力气却讨不了好。学习就是不断地归一,最终以一心一理贯通万事万物,则达自由无碍之化境矣(呵呵,这境界有点高,慢慢来)。'),('4','3','1','关于几何最值问题研究的老师很多,本人以前也有文章论述,本文在此基础上再次进行归纳总结,把各种知识、方法、思想、策略进行融合提炼、追本溯源、认祖归宗,以使解决此类问题时更加简单明晰。'),('4','5','1','所有问题的老祖宗只有两个:①[定点到定点]:两点之间,线段最短;②[定点到定线]:点线之间,垂线段最短。'),('4','5','2','1.gif'),('4','7','2','2.gif'),('4','8','1','由此派生:
③[定点到定点]:三角形两边之和大于第三边;
④[定线到定线]:平行线之间,垂线段最短;
⑤[定点到定圆]:点圆之间,点心线截距最短(长);
⑥[定线到定圆]:线圆之间,心垂线截距最短;
⑦[定圆到定圆]:圆圆之间,连心线截距最短(长)。'),('4','9','1','余不赘述,下面仅举一例证明:'),('4','10','1','[定点到定圆]:点圆之间,点心线截距最短(长)。'),('4','11','1','已知⊙O半径为r,AO=d,P是⊙O上一点,求AP的最大值和最小值。'),('4','12','2','3.png'),('4','13','1','证明:'),('4','14','1','由“两点之间,线段最短”得AP≤AO+PO,AO≤AP+PO,'),('4','15','1','得d-r≤AP≤d+r,AP最小时点P在B处,最大时点P在C处。'),('4','16','1','即过圆心和定点的直线截得的线段AB、AC分别最小、最大值。'),('4','17','1','(可用“三角形两边之和大于第三边”,其实质也是由“两点之间,线段最短”推得)。'),('4','18','1','上面几种是解决相关问题的基本图形,所有的几何最值问题都是转化成上述基本图形解决的。'),('4','19','1','二、考试中出现的问题都是在基本图形的基础上进行变式,如圆与线这些图形不是直接给出,而是以符合一定条件的动点的形式确定的;再如过定点的直线与动点所在路径不相交而需要进行变换的。'),('4','20','1','类型分三种情况:'),('4','21','1','(1)直接包含基本图形;'),('4','22','1','(2)动点路径待确定;'),('4','23','1','(3)动线(定点)位置需变换。'),('4','24','1','(一)直接包含基本图形。'),('4','25','1','例1:'),('4','26','1','在⊙O中,圆的半径为6,∠B=30°,AC是⊙O的切线,则CD的最小值是 。'),('4','27','2','4.gif'),('4','28','1','简析:由∠B=30°知弧AD一定,所以D是定点,C是直线AC上的动点,即为求定点D到定线AC的最短路径,求得当CD⊥AC时最短为3。'),('4','29','1','(二)动点路径待确定。'),('4','30','1','例2:'),('4','31','1','如图,在△ABC中,∠ACB=90°,AB=5,BC=3,P是AB边上的动点(不与点B重合),将△BCP沿CP所在的直线翻折,得到△B′CP,连接B′A,则B′A长度的最小值是 。'),('4','32','2','5.gif'),('4','33','1','简析:A是定点,B\'是动点,但题中未明确告知B\'点的运动路径,所以需先确定B\'点运动路径是什么图形,一般有直线与圆两类。此题中B\'的路径是以C为圆心,BC为半径的圆弧,从而转化为定点到定圆的最短路径为AC-B\'C=1。'),('4','34','1','例3:'),('4','35','1','在△ABC中,AB=AC=5,cos∠ABC=3/5,将△ABC绕点C顺时针旋转,得到△A\'B\'C,点E是BC上的中点,点F为线段AB上的动点,在△A\'B\'C绕点C顺时针旋转过程中,点F的对应点是F\',求线段EF\'长度的最大值与最小值的差。'),('4','36','2','6.png'),('4','37','1','简析:E是定点,F\'是动点,要确定F\'点的运动路径。先确定线段A\'B\'的运动轨迹是圆环,外圆半径为BC,内圆半径为AB边上的高,F\'是A\'B\'上任意一点,因此F\'的运动轨迹是圆环内的任意一点,由此转化为点E到圆环的最短和最长路径。'),('4','38','2','7.gif'),('4','39','2','8.png'),('4','40','2','9.png'),('4','41','1','E到圆环的最短距离为EF2=CF2-CE=4.8-3=1.8,'),('4','42','1','E到圆环的最长距离为EF1=EC+CF1=3+6=9,其差为7.2。'),('4','43','1','(三)动线(定点)位置需变换。'),('4','44','1','线段变换的方法:'),('4','45','1','(1)等值变换:翻折、平移;'),('4','46','1','(2)比例变换:三角、相似。'),('4','47','1','【翻折变换类】典型问题:“将军饮马”。'),('4','48','1','例4:'),('4','49','1','如图,∠AOB=30°,点M、N分别是射线OA、OB上的动点,OP平分∠AOB,且OP=6,当△PMN的周长最小值为 。'),('4','50','2','10.gif'),('4','51','1','简析:动线段(或定点)应居于动点轨迹的两侧,本题的三条动线段PM、MN、PN在OA、OB的内侧。所以本题的关键是把定线段变换到动点轨迹的两侧,从而把三条动线段PM、MN、PN转化为连接两点之间的路径。如图,把点P分别沿OA、OB翻折得P1、P2,△PMN的周长转化为P1M+MN+P2N,这三条线段的和正是连接两个定点P1、P2之间的路径,从而转化为求P1、P2两点之间最短路径,得△PMN的周长最小值为线段P1P2=OP=6。'),('4','52','1','例5:'),('4','53','1','如图,在锐角△ABC中,AB=4,∠BAC=45°,∠BAC的平分线交BC于点D,M、N分别是AD和AB上的动点,则BM+MN的最小值是 。'),('4','54','2','11.gif'),('4','55','1','简析:本题的问题也在于动线段BM、MN居于动点轨迹AD的同侧,同样把点N沿AD翻折至AC上,BM+MN=BM+MN\',转化为求点B到直线AC的最短路径,即BN\'⊥AC时,最小值为2√2。'),('4','56','1','【平移变换类】典型问题:“造桥选址”。'),('4','57','1','例6:'),('4','58','1','如图,m、n是小河两岸,河宽20米,A、B是河旁两个村庄,要在河上造一座桥,要使A、B之间的路径最短应该如何选址(桥须与河岸垂直)?'),('4','59','2','12.png'),('4','60','1','简析:桥长为定值,可以想像把河岸m向下平移与n重合,同时把点A向下平移河宽,此时转化成n上的一点到A、B的路径之和最短,即转化为定点A\'到定点B的最短路径。如下图:'),('4','61','2','13.gif'),('4','62','1','思路是把动线AM平移至A\'M,A\'N+BN即转化为求定点A\'与定点B之间的最路径。本题的关键是定长线段MN把动线段分隔,此时须通过平移把动线段A\'N、BN变为连续路径,也可以把点B向上平移20米与点A连接。'),('4','63','1','例7:'),('4','64','1','如图,CD是直线y=x上的一条定长的动线段,且CD=2,点A(4,0),连接AC、AD,设C点横坐标为m,求m为何值时,△ACD的周长最小,并求出这个最小值。'),('4','65','1','解析:两条动线段AC、AD居于动点所在直线的两侧,不符合基本图形中定形(点线圆)应在动点轨迹的两侧。首先把AC沿直线CD翻折至另一侧,如下图:'),('4','66','2','14.png'),('4','67','1','现在把周长转化为A\'C+CD+AD,还需解决一个问题:动线段A\'C与AD之间被定长线段CD阻断,动线段必须转化成连续的路径。同上题的道理,把A\'C沿CD方向平移CD的长度即可,如下图。'),('4','68','2','15.png'),('4','69','1','现在已经转化为A\'\'D+AD的最短路径问题,属定点到定点,当A\'\'D与AD共线时A\'\'D+AD最短,即为线段AA\'\'的长。'),('4','70','2','16.png'),('4','71','1','【三角变换类】典型问题:“胡不归”。'),('4','72','1','例8:'),('4','73','1','如图,A地在公路BC旁的沙漠里,A到BC的距离AH=2√3,AB=2√19,在公路BC上行进的速度是在沙漠里行驶速度的2倍。某人在B地工作,A地家中父亲病危,他急着沿直线BA赶路,谁知最终没能见到父亲最后一面,其父离世之时思念儿子,连连问:“胡不归,胡不归……!”(怎么还不回来),这真是一个悲伤的故事,也是因为不懂数学而导致的。那么,从B至A怎样行进才能最快到达?'),('4','74','2','17.png'),('4','75','1','简析:BP段行驶速度是AP段的2倍,要求时间最短即求BP/2+AP最小,从而考虑BP/2如何转化,可以构造含30°角利用三角函数关系把BP/2转化为另一条线段。如下图,作∠CBD=30°,PQ⊥BD,得PQ=1/2BP,由“垂线段最短”知当A、P、Q共线时AP+PQ=AQ\'最小。'),('4','76','2','18.png'),('4','77','1','【相似变换类】典型问题:“阿氏圆”。'),('4','78','1','“阿氏圆”:知平面上两点A、B,则所有满足PA/PB=k且不等于1的点P的轨迹是一个圆,这个轨迹最先由古希腊数学家阿波罗尼斯发现,故称阿氏圆,如下图所示,其中PO:BO=AO:PO=PA:PB=k。'),('4','79','2','19.gif'),('4','80','1','例9:已知A(-4,-4)、B(0, 4)、C(0, -6)、 D(0, -1),AB与x轴交于点E,以点E为圆心,ED长为半径作圆,点M为⊙E上一动点,求 1/2AM+CM 的最小值。'),('4','81','2','20.png'),('4','82','1','简析:本题的主要问题在于如何转化1/2AM,注意到由条件知,在M的运动过程中,EM:AE=1:2保持不变,从而想到构造相似三角形,使之与△AEM的相似比为1:2,这样便可实现1/2AM的转化,如下图取EN:EM=1:2,即可得△EMN∽△EAM,再得MN=1/2AM,显然,MN+CM的最小值就是定点N、C之间的最短路径。'),('4','83','2','21.png'),('4','84','1','之后便是常规方法先求N点坐标,再求CN的长。'),('4','85','2','22.png'),('4','86','1','【解法大一统】'),('4','87','1','万法归宗:路径成最短,折线到直线。'),('4','88','1','(所求路径在一般情况下是若干折线的组合,这些折线在同一直线上时即为最短路径)'),('4','89','1','基本图形:动点有轨迹,动线居两边。'),('4','90','1','(动点轨迹可以是线或圆,动线指动点与定点或定线、定圆的连线,动线与折线同指)'),('4','91','1','核心方法:同侧变异侧,分散化连续。'),('4','92','1','(动线在同侧,要变为异侧,一般用翻折、三角、相似的方法构造;动折线被定长线段分散时需化为连续折线,一般用平移的方法构造,如造桥选址问题)'),('4','93','1','下图是构造完成的目标图形:'),('4','94','2','23.png'),('4','95','1','再举2例说明上述规律的运用方法:'),('4','96','1','1.'),('4','97','1','如图,菱形ABCD中,∠A=60°,AB=3,⊙A、⊙B的半径为2和1,P、E、F分别是CD、⊙A、⊙B上的动点,则PE+PF的最小值为 。'),('4','98','2','24.png'),('4','99','1','思考方法如下图所示:'),('4','100','2','25.png'),('4','101','1','2.'),('4','102','1','菱形ABCD中,∠BAC=60°,P是AC上的动点,求BP+1/2AP的最小值。'),('4','103','2','26.png'),('4','104','1','思考方法如下图所示:'),('4','105','2','27.png'),('4','4','1','一、基本图形'),('5','1','2','1.png'),('6','1','2','1.png');
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\r\n 2. 斐波那契数列中的第144项是75025,这个数与黄金比例0.618非常接近,因此它在自然界中频频出现,比如植物叶片的排列。
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\r\n 4. 18世纪的数学家欧拉推算出一个奇妙的公式,即欧拉恒等式,包含5个最重要的数学常数:0、1、π、e、i。
\r\n 5. 世界上有无数个素数,但其中较大的素数非常稀少,2018年发现的最大素数长度高达2480万位。
\r\n 6. 自然数可以被分为阿姆斯特朗数与非阿姆斯特朗数,三位数中153就是一个阿姆斯特朗数,因为1^3 + 5^3 + 3^3 = 153。
\r\n 7. 数学家花费了300多年时间才证明了费马大定理,该定理表示没有整数解使得a^n + b^n = c^n (n > 2)。
\r\n 8. 在一个20层楼的正方体魔方中,所有格子的数量达到8000个,打乱后完全还原需要数百万步。
\r\n 9. 以数字7开头的奇数在任何自然数集合中占比大约14%,这是因为7被认为是最神秘的数字之一。
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\r\n 11. “友爱数”很罕见,比如220和284,每个数的因子之和等于对方(220的因子和是284,284的因子和是220)。
\r\n 12. 平方和倒数的和等于1的数对叫做互质数,如3和6、4和9,这类数有数百万对。
\r\n 13. 黄金分割0.618的神奇之处在于,自然界中的许多花瓣、松果和贝壳呈现出这个比例。
\r\n 14. 数学家发现,圆的内接正n边形的面积随着n无限增大时会无限接近π,这成为了圆面积的经典证明之一。
\r\n 15. 在数论中,只有3个平方数的相加和能等于同一数的平方,比如3^2 + 4^2 + 5^2 = 12^2,这叫“勾股数”。
\r\n 16. 以e为底的自然对数是最神奇的常数之一,它等于2.718,且在金融、物理等多领域应用广泛。
\r\n 17. 在一个随机数序列中,每500个数内约有7个素数,这种分布符合“素数定理”。
\r\n 18. 数学的双射性原理指出,如果两个集合元素一一对应,就称为双射,比如自然数集和整数集的双射关系。
\r\n 19. “柏拉图正多面体”只有5种,即四面体、六面体、八面体、十二面体和二十面体。
\r\n 20. 数学家高斯在7岁时就发现自然数1至100的和是5050,得出公式n(n+1)/2
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