Discover the meaning of the perfect square formula

What does the perfect square formula mean

Original publisher: Zhang Yan

Grade |Eighth Grade|Topic|Perfect Square Formula|Course Type|New Teaching|

Teaching media|Multimedia|

Teaching objectives: knowledge and skills 1. Through the process of exploring the perfect square formula, students can develop research methods from general to specific, and further develop their sense of symbols and reasoning abilities. 2. Be able to derive the perfect square formula, understand the structural characteristics of the formula, and be able to use the formula to perform simple calculations.

Process|Method|Further cultivate students’ ability to solve problems using a combination of numbers and shapes.|

Cultivate students' interest in mathematics, understand the history of mathematics, and stimulate innovation ability.

Teaching Focus|The derivation and application of (a±b)2=a2±2ab b2.|

Teaching Difficulties|The derivation of the perfect square formula, the structural characteristics of the formula and its application.|

Teaching process design|

Teaching procedures and teaching content|Teacher and student behavior|Design intention|

Explore, calculate the following formulas, what rules can you find? |(1)(p 1)2 =(p 1)(p 1)=_________;|(2)(m 2)2=(m 2)(m 2)=_________;|(3)(p-1 )2 =(p-1)(p-1)=_________;|(4)(m-2)2=(m-2)(m-2)=_________. |Answer: (1) p2 2p 1; (2)m2 4m 4; (3)p2-2p 1; (4)m2-4m 4.|1. Calculate: (a b) 2 and (a-b) 2; and explain the rules found. |（a b）2=(a b)(a b)= a(a b) b(a b)=a2 ab ab b2|=a2 2ab b2.ر|（a-b）2=(a-b)(a-b)=a(a-b) -b(a-b)=a2-ab-ab b2=a2-2ab b2.ر|2. Induction of perfect square formula|The square of the sum (or difference) of two numbers is equal to the sum of their squares, add (or subtract) it

Concept of perfect square numbers in mathematics

Perfect Square Number

(1) Properties of perfect square numbers

If a number is the perfect square of another integer, then we call this number a perfect square number, also called a square number. For example:

0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,…

Observing these perfect square numbers, we can gain an understanding of the regularity of their single digits, tens digits, sums, etc. Let's study some common properties of perfect square numbers:

Property 1: The last digit of a perfect square number can only be 0,1,4,5,6,9.

Property 2: The single-digit number of the square of an odd number is an odd number, and the tens-digit number is an even number.

Property 3: If the tens digit of a perfect square number is an odd number, then its ones digit must be 6; conversely, if the ones digit of a perfect square number is 6, then its tens digit must be an odd number.

Corollary 1: If the tens digit of a number is an odd number and the ones digit is not 6, then the number must not be a perfect square.

Corollary 2: If the ones digit of a perfect square number is not 6, then its tens digit is an even number.

Property 4: The square of an even number is a multiple of 4; the square of an odd number is a multiple of 4 plus 1.

Property 5: The square of an odd number is of type 8n 1; the square of an even number is of type 8n or 8n 4.

Property 6: The form of a square number must be one of the following two: 3k, 3k 1.

Property 7: The square of a number that is not divisible by 5 is of type 5k±1, and the square of a number that is divisible by 5 is of type 5k.

Property 8: The form of a square number has one of the following forms: 16m,16m 1, 16m 4,16m 9.

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