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二项式的分解因式 Factoring binomials

二项式的分解因式 Factoring binomials. Mathematics 八年级 Grade eight 云南省施甸二中 沈正零 Zhengling Shen. 教学目标 Objectives. 1 、掌握二项式分解因式的概念 the concept of binomial Factoring . 2 、了解分解因式的意义，以及它与整式乘法的关系 the meaning of factorization and its relationship with the integral expression . 3 、经历从分解因数到分解因式的类比过程

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二项式的分解因式 Factoring binomials

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  1. 二项式的分解因式Factoring binomials Mathematics八年级Grade eight 云南省施甸二中 沈正零 Zhengling Shen

  2. 教学目标 Objectives • 1、掌握二项式分解因式的概念 the concept of binomial Factoring. • 2、了解分解因式的意义，以及它与整式乘法的关系 the meaning of factorization and its relationship with the integral expression. • 3、经历从分解因数到分解因式的类比过程 • the analogy from the decomposition to Factored. • 4、感受分解因式在解决相关问题中的作用 the role of factorization in resolving the related issues. • 5、学会观察、分析，提高分析问题和解决问题的能力 Learnto observe, analyze, improve the skills of analysis and problem-solving skills.

  3. 教学内容Teaching content • 内容Content：二项式因式分解binomial factorization • 教学重点Teaching points： 1、如何提公因式to find a common factor. 2、平方差公式Squares formula. 3、在教学中要使学生掌握逆向思维和类比的数学思想方法 Analogy and reverse thinking • 教学难点Teaching Difficulties : 1、如何找公因式to find common factor. 2、如何把二项式变形为平方差公式的结构形式 How to transfer the deformation of binomials to the squares formula structure

  4. 教学方法Teaching Methods • 以学生探究发现和自主学习为主，教师讲授为辅 let students learn by themselves with assistance from the teacher • 教具准备: 需要多媒体设备辅助教学 computer-assisted 教学时数： 一课时 Teaching time:45minutes

  5. 教学过程Teaching Process • （一）教学准备 preparation • 1、复习公因式的概念review the concept of Common Factor 想一想：下列各题中两个因式的公因式是多少？ Think: what is the common factor of them? （1）993与-99 （2）3x2与-3x （3）-6a3与-2a (1) 993 and -99 (2) 3x2 and-3x (3)-6a3 and-2a 请同学们归纳一下：什么叫公因式？ summarize : What is common factor? • 2、复习乘法公式中的平方差公式 Review formula for the difference of squares (a+b)(a－b)= • 请同学们归纳一下平方差公式： ask a student to summarize the fomula

  6. （二）讲授新课 • 1、对于“993－99能被100整除吗？”小明是这样想的： Can"993-99 be divided by 100 ?" • Since 993－99=99×(992－1)=99×9800=99×98×100 • 所以小明认为它能被100整除。 the answer is yes • 师问:那么993－99还能被哪些正整数整除呢？ • Teacher asked：Can 993 －99 be divided by other positive integers? • 学生答：还能被98和99整除。 • Student A: 98 and 99 • 师: 回答正确，那么要解决这类问题的关键是什么呢？Teacher: correct, what is the key to solve such a problem ? • 生: 是要把一个数式化成几个数的积的形式 • Student ：The key is to change a number into the form of multiplication of several numbers. 。 • 师：对，这就是今天我们要学的二项式的分解因式 • Yes, that is decomposition of binomials.

  7. 2、做一做：<先把题目展示给学生，然后请同学起来回答2、做一做：<先把题目展示给学生，然后请同学起来回答 filling the blanks • 试计算下列各式： 3x(x－1)=________ (m+4)(m－4)= ________ a(a+1)(a－1)= ________ • 根据上面的算式填空： 3x2－3x=( )( ) m2－16=( )( ) a3－a=( )( )( )

  8. 3、议一议 further discussion：<根据练习情况进一步讨论> 由a(a+1)(a－1)得到a3－a的变形是什么运算？由a3－a得到a(a+1)(a－1)的变形与这种运算有什么不同？你还能再举一些类似的例子加以说明吗？How do you get a (a +1) (a-1) from a3－a ? What is the difference between it and the opposite calculation, from a3－a to a (a +1) (a-1)? 师生共同总结：（经过议论之后） 由a(a+1)(a－1)得到a3－a是整式的乘法；由a3－a得到a(a+1)(a－1)的变形是整式乘法的逆运算。 We get a(a+1)(a－1) from a3－a through multiplication, but we get a (a +1) (a-1) from a3－a is just the opposite • 师：把一个二项式化成几个整式的积的形式，这种变形叫做把这个二项式分解因式。 Teacher: the way we change binomials into a multiplication of several integral expression is factoring binomials.

  9. 4、想一想： 师问：分解因式的一般步骤是什么？（教师引导学生共同总结） What are the general steps? 生答：（１）如果二项式的各项有公因式，那么先提公因式 If there are common binomials, they should be taken out first e.g.：993 －99， x－xy 等,etc （2）如果各项没有公因式，那么可以尝试运用平方差公式进行分解 If the there is no common factor, you can try to use the formula for the difference of squares e.g.：a2－b2， 25－36x2，x2－4y2 ,等 • （3）分解因式，必须进行到每一个多项式都不能再分解为止 • Do not stop until you pick out the last common factor e.g.：x3－x=x（x2－1）= x(x +1)(x－1)

  10. （三）、课堂练习Practice • 用线连一连：matching x2－y2 (3－5x)(3+5x) 9－25x2 y(x－y) xy－y2 (2a－1)(2a+1) 4a2－1 (x+y)(x－y)

  11. 课堂小结 Summary • 1.这节课你有哪些收获？ What have you learned from this lesson ? • 2.你学到了哪些数学方法和数学思想？ What mathematical methods have you learned from this lesson ？

  12. 结果与分析 这节课充分调动了学生的积极性，主动性，以学生学习、探究为主，教师讲授为辅，在一问一答，互动讨论的情境下循序渐进地开展教学，既培养了学生的自主探究能力，也培养了学生学习数学的兴趣,体现了再创造，再发现的过程；在活动过程中学到了知识，提高了能力。 It makes the students active. With the assistance from their teachers, those students learn in interactive discussion or answering questions. It helps to cultivate their ability of independent learning.

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