2026新高一数学训练

2026—2027学年新高一数学集合、函数与不等式21天衔接训练第页ORG·PRACTICALDOCUMENT2026—2027学年新高一数学集合、函数与不等式21天衔接训练每日练＋3次阶段测＋开学摸底｜答案解析｜错因复盘▌初▌高▌中▌衔▌接▌▌▌2▌1▌天▌路▌径▌▌▌过▌程▌可▌追▌踪下载后可直接使用用每天30分钟完成语言、运算、图像和建模的衔接，降低开学第一月掉队风险。ORG-EDU-HSMATH-05

从这里开始：10分钟完成使用部署你将得到什么｜21天训练、63道日练、3次阶段测、1套开学摸底、答案解析、错因台账。判断项本资料对应答案适合谁2026年秋季升入高一的学生；需要暑期衔接课程的教师、机构和家长。什么时候用7月中旬至8月底；开学前21天；高一第一次月考前回补基础。最短使用路径每日30分钟→每7天一次阶段测→错因归类→最终摸底。使用提醒请先阅读“使用边界”，再按实际对象、项目或班级情况调整。一页部署清单□复制一份原文件作为工作副本□按对象填写封面/基本信息□选择本次要使用的模块□打印或复制到线上协作工具□使用后完成结果记录与复盘□将改进项纳入下一轮版本目录与使用地图01衔接说明与学习纪律02Week1：集合与代数语言03阶段测104Week2：函数概念与图像05阶段测206Week3：不等式与综合应用07阶段测308开学摸底卷09答案与错因复盘提示：本文件为可编辑版本。可删除不需要的模块，也可复制表格形成班级版、部门版或项目版。

初高中数学的四个变化变化初中常见方式高中需要补上的能力训练动作语言具体数值与图形较多集合、符号、条件与量词每题先翻译条件运算步骤相对直接代数变形链更长每步写等价依据图像画图辅助数形结合成为核心工具先定定义域再画特征学习题型模仿概念迁移与反例意识每错题写“错在条件还是方法”错因代码｜L语言理解｜A代数运算｜D定义域/条件｜G图像｜M方法选择｜E表达与检验。每次只追一个主错因。

Day1｜集合的表示本模块目标｜集合要说明对象确定性；常用列举法、描述法；元素具有互异性。知识与方法卡集合要说明对象确定性常用列举法、描述法元素具有互异性。当日训练1.用列举法表示不大于5的正整数集合。________________________________________________________________________________________________________________________________________________________________2.判断：{1,1,2}与{1,2}是否为同一集合？________________________________________________________________________________________________________________________________________________________________3.用描述法表示所有偶整数。________________________________________________________________________________________________________________________________________________________________核对与订正｜{1,2,3,4,5}；是；{x|x=2k,k∈Z}。用时正确数错因代码一句话复盘

Day2｜元素与集合关系本模块目标｜元素用∈/∉，集合之间用⊆；不要混淆对象层级。知识与方法卡元素用∈/∉，集合之间用⊆不要混淆对象层级。当日训练1.判断：1∈{1,2}；{1}∈{1,2}；{1}⊆{1,2}。________________________________________________________________________________________________________________________________________________________________2.若A={x|x²=1}，写出A。________________________________________________________________________________________________________________________________________________________________3.空集是否是任意集合的子集？________________________________________________________________________________________________________________________________________________________________核对与订正｜真/假/真；{-1,1}；是。用时正确数错因代码一句话复盘

Day3｜子集与真子集本模块目标｜n个元素的集合有2^n个子集；真子集少1个。知识与方法卡n个元素的集合有2^n个子集真子集少1个。当日训练1.三元素集合有多少子集？________________________________________________________________________________________________________________________________________________________________2.A={1,2}的所有真子集。________________________________________________________________________________________________________________________________________________________________3.若A⊆B且B⊆A，能得出什么？________________________________________________________________________________________________________________________________________________________________核对与订正｜8；∅,{1},{2}；A=B。用时正确数错因代码一句话复盘

Day4｜交集并集补集本模块目标｜交集取共同，并集取全部；补集必须先明确全集。知识与方法卡交集取共同，并集取全部补集必须先明确全集。当日训练1.A={1,2,3},B={2,3,4}，求A∩B和A∪B。________________________________________________________________________________________________________________________________________________________________2.全集U={1,2,3,4,5}，求A的补集。________________________________________________________________________________________________________________________________________________________________3.验证A∩B⊆A。________________________________________________________________________________________________________________________________________________________________核对与订正｜{2,3}；{1,2,3,4}；{4,5}；成立。用时正确数错因代码一句话复盘

Day5｜区间与数轴本模块目标｜区间端点是否包含决定括号；无穷端永远用圆括号。知识与方法卡区间端点是否包含决定括号无穷端永远用圆括号。当日训练1.把-2≤x<3写成区间。________________________________________________________________________________________________________________________________________________________________2.(-∞,1)∪[4,+∞)在数轴上表示。________________________________________________________________________________________________________________________________________________________________3.x>2与x≤5的交集。________________________________________________________________________________________________________________________________________________________________核对与订正｜[-2,3)；图略；(2,5]。用时正确数错因代码一句话复盘

Day6｜绝对值与距离本模块目标｜|x-a|表示数轴上x到a的距离。知识与方法卡|x-a|表示数轴上x到a的距离。当日训练1.解|x-2|<3。________________________________________________________________________________________________________________________________________________________________2.解|2x+1|=5。________________________________________________________________________________________________________________________________________________________________3.|x|≥4的解集。________________________________________________________________________________________________________________________________________________________________核对与订正｜(-1,5)；x=2或-3；(-∞,-4]∪[4,+∞)。用时正确数错因代码一句话复盘

Day7｜集合综合本模块目标｜把方程、不等式解集转为集合，再做运算。知识与方法卡把方程、不等式解集转为集合，再做运算。当日训练1.A={x|x²-5x+6=0}，B={2,4}，求A∩B。________________________________________________________________________________________________________________________________________________________________2.A=[-1,3],B=(1,5)，求A∪B。________________________________________________________________________________________________________________________________________________________________3.若A={1,a},且1∈A、A只有一个元素，a可能为何？________________________________________________________________________________________________________________________________________________________________核对与订正｜{2}；[-1,5)；a=1。用时正确数错因代码一句话复盘阶段测1｜集合与代数语言姓名班级日期用时得分满分：60分建议用时：45分钟1.用列举法表示{x∈Z|-2<x≤3}。（6分）________________________________________________________________________________________________________________________________________________________________2.判断并改错：{0}是空集。（6分）________________________________________________________________________________________________________________________________________________________________3.A={1,2,3}，写出所有含1的子集。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4.A=[-2,4)，B=(1,6]，求交集和并集。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________5.解|2x-3|≤5并用区间表示。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________6.全集U=R，A=(-∞,2]，求补集。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________7.已知A={x|x²-(a+1)x+a=0}只有一个元素，求a。（12分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

阶段测1答案参考答案{-1,0,1,2,3}；错误，{0}含元素0；{1},{1,2},{1,3},{1,2,3}；(1,4)和[-2,6]；[-1,4]；(2,+∞)；判别式(a+1)²-4a=(a-1)²=0，所以a=1。

Day8｜函数概念本模块目标｜函数是两个非空数集间的对应；同一个x只能对应一个y。知识与方法卡函数是两个非空数集间的对应同一个x只能对应一个y。当日训练1.判断关系y²=x是否能把y看作x的函数。________________________________________________________________________________________________________________________________________________________________2.f(x)=2x+1，求f(3)。________________________________________________________________________________________________________________________________________________________________3.同一函数由哪三要素决定？________________________________________________________________________________________________________________________________________________________________核对与订正｜不能（一个正x可对应±√x）；7；定义域、对应关系、值域。用时正确数错因代码一句话复盘

Day9｜定义域本模块目标｜优先检查分母不为0、偶次根号内≥0、实际情境限制。知识与方法卡优先检查分母不为0、偶次根号内≥0、实际情境限制。当日训练1.求f(x)=1/(x-2)定义域。________________________________________________________________________________________________________________________________________________________________2.求√(3-x)定义域。________________________________________________________________________________________________________________________________________________________________3.求√(x+1)/(x-2)定义域。________________________________________________________________________________________________________________________________________________________________核对与订正｜x≠2；x≤3；x≥-1且x≠2。用时正确数错因代码一句话复盘

Day10｜函数值与解析式本模块目标｜代入时把自变量整体替换，注意括号。知识与方法卡代入时把自变量整体替换，注意括号。当日训练1.f(x)=x²-2x，求f(-1)。________________________________________________________________________________________________________________________________________________________________2.f(x+1)=2x+3，求f(x)。________________________________________________________________________________________________________________________________________________________________3.f(2x)=x+1，求f(x)。________________________________________________________________________________________________________________________________________________________________核对与订正｜3；2x+1；x/2+1。用时正确数错因代码一句话复盘

Day11｜图像与坐标本模块目标｜图像上的点满足解析式；截距、对称、增减可从图像读取。知识与方法卡图像上的点满足解析式截距、对称、增减可从图像读取。当日训练1.点(2,5)是否在y=2x+1上？________________________________________________________________________________________________________________________________________________________________2.y=-x+3与坐标轴交点。________________________________________________________________________________________________________________________________________________________________3.画y=|x|基本图像。________________________________________________________________________________________________________________________________________________________________核对与订正｜是；(0,3),(3,0)；V形。用时正确数错因代码一句话复盘

Day12｜单调性直观本模块目标｜在区间内自变量增大时函数值整体上升/下降。知识与方法卡在区间内自变量增大时函数值整体上升/下降。当日训练1.y=2x-1在R上单调如何？________________________________________________________________________________________________________________________________________________________________2.y=-x²在[0,+∞)上如何？________________________________________________________________________________________________________________________________________________________________3.从图像判断单调区间需先做什么？________________________________________________________________________________________________________________________________________________________________核对与订正｜递增；递减；明确定义域和观察方向。用时正确数错因代码一句话复盘

Day13｜奇偶性直观本模块目标｜定义域关于原点对称后，再比较f(-x)与f(x)。知识与方法卡定义域关于原点对称后，再比较f(-x)与f(x)。当日训练1.f(x)=x²是否为偶函数？________________________________________________________________________________________________________________________________________________________________2.f(x)=x³是否为奇函数？________________________________________________________________________________________________________________________________________________________________3.f(x)=x+1是否为奇/偶？________________________________________________________________________________________________________________________________________________________________核对与订正｜偶；奇；非奇非偶。用时正确数错因代码一句话复盘

Day14｜一次、二次函数衔接本模块目标｜用斜率理解一次函数，用顶点、开口、对称轴理解二次函数。知识与方法卡用斜率理解一次函数，用顶点、开口、对称轴理解二次函数。当日训练1.过(1,3),(4,9)的一次函数解析式。________________________________________________________________________________________________________________________________________________________________2.y=x²-4x+3的对称轴与顶点。________________________________________________________________________________________________________________________________________________________________3.二次函数开口由什么决定？________________________________________________________________________________________________________________________________________________________________核对与订正｜y=2x+1；x=2,(2,-1)；二次项系数符号。用时正确数错因代码一句话复盘阶段测2｜函数概念与图像姓名班级日期用时得分满分：60分建议用时：45分钟1.判断关系y=±√x是否定义了y关于x的函数。（6分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2.求f(x)=√(2x-1)/(x+3)定义域。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________3.f(x)=x²+2x，求f(a+1)。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4.若f(x+2)=3x-1，求f(x)。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________5.一次函数过(-1,2)和(2,8)，求解析式。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________6.求y=x²-6x+5的顶点、对称轴、最小值。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________7.判断f(x)=x³+x奇偶性并说明。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

阶段测2答案参考答案不是；[1/2,+∞)且x≠-3（-3本就不在区间，可写[1/2,+∞)）；a²+4a+3；f(x)=3x-7；y=2x+4；顶点(3,-4)、轴x=3、最小值-4；奇函数。

Day15｜不等式性质本模块目标｜两边同加减不变号；同乘除负数必须变号。知识与方法卡两边同加减不变号同乘除负数必须变号。当日训练1.解-3x+2≥11。________________________________________________________________________________________________________________________________________________________________2.若a>b，比较-a与-b。________________________________________________________________________________________________________________________________________________________________3.2<x<5同时乘-2。________________________________________________________________________________________________________________________________________________________________核对与订正｜x≤-3；-a<-b；-10<-2x<-4。用时正确数错因代码一句话复盘

Day16｜一元二次不等式直观本模块目标｜先找零点，再结合开口判断符号区间。知识与方法卡先找零点，再结合开口判断符号区间。当日训练1.解x²-5x+6>0。________________________________________________________________________________________________________________________________________________________________2.解x²-4≤0。________________________________________________________________________________________________________________________________________________________________3.无实根且开口向上的二次式符号如何？________________________________________________________________________________________________________________________________________________________________核对与订正｜(-∞,2)∪(3,+∞)；[-2,2]；恒正。用时正确数错因代码一句话复盘

Day17｜基本不等式意识本模块目标｜正数条件下a+b≥2√ab；使用前检查正数和定值。知识与方法卡正数条件下a+b≥2√ab使用前检查正数和定值。当日训练1.x>0，求x+4/x最小值。________________________________________________________________________________________________________________________________________________________________2.a,b>0且ab=9，求a+b最小值。________________________________________________________________________________________________________________________________________________________________3.为何a+b≥2√ab不能直接用于负数？________________________________________________________________________________________________________________________________________________________________核对与订正｜4；6；根式及条件不满足。用时正确数错因代码一句话复盘

Day18｜分式与根式运算本模块目标｜先因式分解、通分约分；根式先化简同类根式。知识与方法卡先因式分解、通分约分根式先化简同类根式。当日训练1.化简(a²b^-1)²÷(ab^-2)。________________________________________________________________________________________________________________________________________________________________2.化简√50-2√8。________________________________________________________________________________________________________________________________________________________________3.化简(x²-1)/(x-1)，并写条件。________________________________________________________________________________________________________________________________________________________________核对与订正｜a³；√2；x+1，x≠1。用时正确数错因代码一句话复盘

Day19｜方程与函数结合本模块目标｜交点问题可转化为方程，解的个数对应交点个数。知识与方法卡交点问题可转化为方程，解的个数对应交点个数。当日训练1.y=2x+1与y=x+4交点。________________________________________________________________________________________________________________________________________________________________2.y=x²与y=4有几个交点？________________________________________________________________________________________________________________________________________________________________3.方程x²-2x+5=0对应图像与x轴关系？________________________________________________________________________________________________________________________________________________________________核对与订正｜(3,7)；2个；无交点。用时正确数错因代码一句话复盘

Day20｜实际情境建模本模块目标｜明确变量、范围、关系式、目标和结果解释。知识与方法卡明确变量、范围、关系式、目标和结果解释。当日训练1.出租车起步价12元含3km，后每公里2.5元，写x>3时费用函数。________________________________________________________________________________________________________________________________________________________________2.长方形周长20，设长x，写面积函数及x范围。________________________________________________________________________________________________________________________________________________________________3.利润=收入-成本，写出检查单位的步骤。________________________________________________________________________________________________________________________________________________________________核对与订正｜y=12+2.5(x-3)；S=x(10-x),0<x<10；开放。用时正确数错因代码一句话复盘

Day21｜综合复盘本模块目标｜用“读题—翻译—选择—运算—检验—表达”完成闭环。知识与方法卡用“读题—翻译—选择—运算—检验—表达”完成闭环。当日训练1.列出你最常见的2个错因代码。________________________________________________________________________________________________________________________________________________________________2.选一道错题写“原思路/断点/新方法”。________________________________________________________________________________________________________________________________________________________________3.为开学摸底设定分数与过程目标。________________________________________________________________________________________________________________________________________________________________核对与订正｜个性化；个性化；目标应具体可测。用时正确数错因代码一句话复盘阶段测3｜不等式与综合应用姓名班级日期用时得分满分：60分建议用时：45分钟1.解3-2x<7。（6分）________________________________________________________________________________________________________________________________________________________________2.解x²-x-6≤0。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________3.x>0，求2x+8/x最小值。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4.化简(x²-4)/(x²-x-2)，写定义条件。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________5.求y=x²与y=2x+3交点坐标。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________6.某商品成本40元，售价x元，预计销量为(200-2x)件。写利润函数并给出合理的x范围。（14分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

阶段测3答案参考答案x>-2；[-2,3]；最小值8（x=2时）；(x+2)/(x+1)，x≠2,-1；交点(-1,1),(3,9)；P=(x-40)(200-2x)，合理范围需结合销量非负和售价高于成本，可取40<x≤100。开学摸底卷｜综合诊断姓名班级日期用时得分满分：100分建议用时：75分钟1.A={1,2,3}，B={2,4}，求A∩B、A∪B。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2.解|x-1|≥3。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________3.求f(x)=√(x+2)/(x-1)定义域。（10分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________4.f(x)=x²-3x+2，求f(-1)+f(2)。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________5.过(1,3),(4,9)的直线解析式。（8分）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________6.求y=x²-4x-5的零点、对称轴和最小值。（12分）_________________________________________________________________________________________________________________________________________________