Multiplication formula (chapter 15 of the first volume of eighth grade mathematics of People's Education Press)

2. Analysis of class situation and students' characteristics

Analysis of learning situation: Students have learned number operation, letter representation, merging similar items, removing brackets and so on. Through analogy, they will have questions such as "whether the formula has corresponding operation, and if so, how to proceed". Therefore, this course pays attention to students' exploration process of formulas, consciously cultivates students' reasoning ability, and allows students to experience the knowledge generation process of "special case → induction → guess → symbol representation".

3. Analysis of teaching content

This course pays attention to students' exploration process of formulas, consciously cultivates students' reasoning ability, encourages students to make induction, conjecture and symbolic representation according to special cases, expresses their thinking process in an orderly manner, cultivates students' sense of numbers and symbols, truly understands the source, essence and application of formulas, and lays a solid foundation for future study.

4. Teaching objectives

⑴. Go through the process of exploring the square difference formula, and further develop the sense of symbol and reasoning ability.

2. Can deduce the square difference formula, and can use the formula for simple calculation.

(3) Understand the square difference and its geometric background, so that students can understand the idea of combining numbers with shapes.

(4) Explore knowledge and experience the fun of learning in cooperation, exchange and discussion.

5. Cultivate students' awareness of using knowledge flexibly and their courage to explore scientific laws.

5. Analysis of teaching emphases and difficulties

Teaching emphasis: experience the discovery and derivation process of the formula, understand the essence of the formula, and use the formula for simple calculation.

Teaching difficulties: understand the meaning of letters in the formula in a broad sense, analyze specific problems in detail, and use the formula to calculate.

6. Teaching hours: 1 hour.

7. Teaching process

First, create problem situations and guide students to observe and imagine.

The teacher gives each student a square piece of paper (side length 15cm), and shows the square with multimedia courseware and square cardboard.

Teacher: On a piece of 45cm square cardboard, a square with a side length of 15cm is cut out from the middle (as shown in the figure). How many square centimeters is the remaining area?

Teacher: Is there any way to calculate the area of the remaining part?

Panel discussion:

1. can be obtained by subtracting the small square area from the large square area.

You can cut the rest into several rectangles to calculate.

Teacher: Judging from today's question, which method is better? Can your group list the formulas?

Maybe some students can list the formula quickly and get the answer of 1800 square centimeters.

Teacher: For the sake of understanding, I now put the small square in the corner of the big square (as shown in the picture).

Teacher: As we said just now, there is more than one way to calculate the area. Now we try to calculate the area by division. Please refer to the teacher's practice, first draw a dotted line on your paper, then cut (or tear off) the small square you just drew, as if to cut this part, then cut the small rectangle along the dotted line and put it on one side of the big rectangle, which just becomes a new rectangle (as shown in the figure).

Teacher: according to the requirements of our original topic, what is the length and width of the new big rectangle now? What is its area?

Health: The length of a big rectangle is (45+ 15)cm and the width is (45- 15)cm.

The area of the rectangle = (45+15) × (45-15) = 60× 30 =1800 (square centimeter).

Teacher: Do you remember these two formulas?

Health: The formula of the first method is 452- 152.

The formula of the second method is (45+ 15)×(45- 15).

Teacher: Both formulas can calculate the remaining area. What is the relationship between them?

Health: equality.

Second, exchange dialogues and explore new knowledge.

See who's quick:

( 1)(x+2)(x-2)

(2)( 1+3a)( 1-3a)

(3)(x+5y)(x-5y)

(4)(-m+n)(-m-n)

Teacher: What patterns can you find?

Teacher: Think again. If today's topic is changed to: "On a square cardboard with a side length of A cm, a small square with a side length of B cm is dug in the middle because of work needs. How much area is left? " How to express it by algebraic expression?

Health: We can use a2-b2 to represent the remaining area.

Teacher: Is there any other way?

Health: You can also use (a+b)(a-b) to represent the remaining area.

Teacher: Today, besides finding a more convenient method to find the area, it is more important that we can learn the property of (a+b)(a-b)= a2-b2 from the graph. We have learned polynomial multiplication last class. Can you work out the answer to (a+b)(a-b) by calculating polynomial multiplication?

Teacher: In order to save calculation time, we use (a+b)(a-b)= a2-b2 as the formula, and call this formula "square difference formula".

Square difference formula: (a+b)(a-b)= a2-b2.

Teacher: Which student can describe the square difference formula in words?

Health: the product of the sum of two numbers and the difference between these two numbers is equal to their square difference.

Third, apply new knowledge and experience success.

1. Example 1 Calculation:

( 1)(a+3)(a-3)

(2)(2a+3b)(2a-3b)

(3)( 1+2c)( 1-2c)

(4)

Solution: (1) Original formula =a2-32=a2-9.

(2) The original formula =(2a)2-(3b)2=4a2-9b2.

(3) The original formula = 12-(2c)2= 1-4c2.

(4) Original formula =

2. Consolidate and deepen, and expand thinking.

Calculation:

( 1)(2x+3)(2x-3)

(2)(-2x+y)(2x+y)

(3)(-x+2)(-x-2)

(4)(y-x)(-x-y)

Note: Pay special attention to the variant training of the formula when practicing. When explaining, we should closely follow the characteristics of the formula, find out the equal "term" and the opposite "term", and then use the formula.

3. Calculation of Example 2: 1998×2002.

Analysis: This is a numerical calculation problem. Ask the students to discuss in groups how to use the square difference formula to calculate.

In the teaching of this example, we should not only pay attention to the simplification and calculation of the applied formula, but also let the students feel the fun of building a mathematical "model".

4. Practice, simple calculation:

( 1)498×502 (2)999× 100 1

There is a square lawn with a length of one meter in the street garden. After unified planning, the north-south direction is lengthened by 2 meters and the east-west direction is shortened by 2 meters. What is the area of the rectangular lawn after transformation?

(First, list the algebraic expressions representing the area. )

Solution: (a+2)(a-2)= a2-4

A: The reconstructed rectangular lawn covers an area of (a2-4) square meters.

practise

Enclose a rectangular area with a fence of a certain length. Xiao Ming thinks that a square has the largest area, but Xiao Liang doesn't think so. what do you think?

Fourth, class summary.

1. What do you know through the learning activities in this class? Is there anything you don't understand?

2. What kind of formula can use the square difference formula? Remember the characteristics of the formula.

8. Operation arrangement

Required: Exercise 15.2 1 (1), (2) and (3)

Selected as: Exercise 15.2 No.65438 +0 (4), (5) and (6)

9. Ask yourself and answer yourself

By guiding students to participate in activities, students' ability to solve practical problems is cultivated. Junior high school students mainly think in images, trying to achieve the combination of numbers and shapes. Hands-on operation is a process of using both hands and brains, and it is an effective method to solve the contradiction between the abstraction of mathematical knowledge and the visualization of junior high school students' thinking. At the same time, the rich emotional experience in the process of exploration can make students change from passive "I want to learn" to active "I want to learn". Through the experimental operation, we can promote students' abstraction into concrete, cultivate students' consciousness of "using mathematics", realize the teaching goal through the design of this course, and cultivate students' mathematical quality of creation, induction, deduction and mathematical modeling.