Topicmc9c53adc76c40e7b_1528449000663_0Topic

Formulas of reduced multiplication of the second degree

Levelmc9c53adc76c40e7b_1528449084556_0Level

Third

Core curriculummc9c53adc76c40e7b_1528449076687_0Core curriculum

II. Algebraic expressions. The student:
1) uses formulas of reduced multiplication for: ${(a+b)}^2,{(a-b)}^2,a^2-b^2,{(a+b)}^3,{(a-b)}^3,a^3-b^3,a^n-b^n.$

Timingmc9c53adc76c40e7b_1528449068082_0Timing

45 minutes

General objectivemc9c53adc76c40e7b_1528449523725_0General objective

Interpreting and manipulating information presented in both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesmc9c53adc76c40e7b_1528449552113_0Specific objectives

1. Communicating in English, developing mathematics skills, scientific, technical and IT competences; developing learning skills.

2. Transforming algebraic expressions.

3. Use of formulas for reduced multiplication of the second degree.

Learning outcomesmc9c53adc76c40e7b_1528450430307_0Learning outcomes

The studnet:

- transforms algebraic expressions,

- uses formulas for reduced multiplication of the second degree.

Methodsmc9c53adc76c40e7b_1528449534267_0Methods

1. Incomplete sentences.

2. Case study.

Forms of workmc9c53adc76c40e7b_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmc9c53adc76c40e7b_1528450127855_0Introduction

Students, working in groups with the method of incomplete sentences, review what they learned so far about algebraic expressions and methods of performing operations on algebraic expressions.

Task
Sentences that should be completed: 
- An algebraic expression is ...
- A monomial is an algebraic expression made up of ... 
- The sum of monomials is ...
- The name of the algebraic expression comes from the name of the operation we perform as ...
- Monomials are called similar terms when ...

The teacher verifies the students' statements and explains the doubts.

Proceduremc9c53adc76c40e7b_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to learn the formulas for reduced multiplication of the second degree.

Task Students, working in teams of two, analyze the material presented in the Interactive illustration. They make hypotheses and formulate the conclusion.

[Interactive illustration 1]

Conclusion:

- The square of the sum of two numbers is equal to the square of the first number plus twice the product of the first number by the second, plus the square of the second number.mc9c53adc76c40e7b_1527752263647_0- The square of the sum of two numbers is equal to the square of the first number plus twice the product of the first number by the second, plus the square of the second number.

Discussion - Can you use the formula for the square of the sum to calculate the squared difference of two expressions? Students make hypotheses, check them and formulate a conclusion.

A conclusion that students should come up with:

- If, in the formula $(x+y{)}^2$ we put in the place of y number (-y ) then we get a formula for the square of the difference of two expressionsmc9c53adc76c40e7b_1527752256679_0- If, in the formula $(x+y{)}^2$ we put in the place of y number (-y ) then we get a formula for the square of the difference of two expressions
${\left[x+(-y)\right]}^2=x^2+2x(-y)+(-y{)}^2=x^2-2xy+y^2.$

Students, working independently, solve tasks using the learned formulas.

Task
Calculate with the formula for reduced multiplication.

a) 98Indeks górny 2²

b) 102Indeks górny 2²

c) 199Indeks górny 2²

Task
Calculate.

a) $(\sqrt{3}+3{)}^2$

b) ${\left(3\sqrt{2}+2\sqrt{3}\right)}^2$

c) $(\sqrt{3}-3{)}^2$

Task
Students working in groups analyze the material presented in the Interactive illustration. They make hypotheses and formulate the conclusion.

[Interactive illustration 2]

A conclusion that students should come up with:

- The product of the sum of two expressions by their difference is equal to the difference in squares of these expressions.mc9c53adc76c40e7b_1527712094602_0- The product of the sum of two expressions by their difference is equal to the difference in squares of these expressions.

Students, working independently, solve tasks.

Task
Calculate using the squared difference formula.

a) $101·99$

b) $1002·998$

c) $590·610$

Task
Calculate.

a) $(\sqrt{2}-3)(\sqrt{2}+3)$

b) $(4+\sqrt{7})(4-\sqrt{7})$

c) $(5\sqrt{3}+6)(5\sqrt{3}-6)$

Task
Simplify the expression and then calculate its value for the given variable value.

a) $(2-3y)(2+3y)-(2-y{)}^2$ for $y = 0,5$

b) $3(2-3x{)}^2-\left(\sqrt{2}-\sqrt{3x}\right)\left(\sqrt{2}+\sqrt{3x}\right)(2+3x)$ for $y=2\sqrt{2}$

Task
Prove that the value of the expression $\left(\sqrt{5}-x\right)\left(\sqrt{5}+x\right)-\frac{1}{2}\left(5-2x^2\right)$ does not depend on the value of the variablex.

After solving all the tasks, the students present the results obtained. The teacher evaluates their work and explains any doubts.

An extra task:
Justify that the following equality is true

a) ${(5a+2b)}^2-{(5a-2b)}^2=100ab$

b) $(a+2b{)}^2+(a-2b{)}^2=2(a^2+4b^2)$

Lesson summarymc9c53adc76c40e7b_1528450119332_0Lesson summary

Students do the revision exercises.

Together, they formulate conclusions to remember.

- The square of the sum of two expressions is equal to the sum of the squares of these expressions, increased by their doubled productdoubled productdoubled product.

- The difference between two expressions equals the sum of the squares of these expressions, reduced by their doubled productdoubled productdoubled product.

- The product of the sum of two expressions by their difference is equal to the difference in squares of these expressions.

Selected words and expressions used in the lesson plan

difference squareddifference squareddifference squared

doubled productdoubled productdoubled product

reduced multiplication formulasreduced multiplication formulasreduced multiplication formulas

square of differencesquare of differencesquare of difference

square of sumsquare of sumsquare of sum