Topicmd97662399b58c2ab_1528449000663_0Topic

Short multiplication formulas of the third degreeshort multiplication formulas of the third degreeShort multiplication formulas of the third degree

Levelmd97662399b58c2ab_1528449084556_0Level

Third

Core curriculummd97662399b58c2ab_1528449076687_0Core curriculum

II. Algebraic expressions. The student:

1) applies shor multiplication formulas for:
${(a+b)}^2,{(a-b)}^2,a^2-b^2,{(a+b)}^3,{(a-b)}^3,a^3-b^3,a^n-b^n$.

Timingmd97662399b58c2ab_1528449068082_0Timing

45 minutes

General objectivemd97662399b58c2ab_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesmd97662399b58c2ab_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Conversion of algebraic expressions.

3. Applying short multiplication formulas of the third degree.

Learning outcomesmd97662399b58c2ab_1528450430307_0Learning outcomes

The student:

- converts algebraic expressions,

- applies short multiplication formulas of the third degreeshort multiplication formulas of the third degreeshort multiplication formulas of the third degree.

Methodsmd97662399b58c2ab_1528449534267_0Methods

1. Associogram.

2. Situational analysis.

Forms of workmd97662399b58c2ab_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmd97662399b58c2ab_1528450127855_0Introduction

The teacher divides the class into two groups. The task of each of the groups is to prepare an associogram.

The operations on algebraic expressions is the topic for Group 1. Group 2gathers information about exponentiation. The groups place their observations on posters specially prepared by the teacher. After finishing their work, the teams exchange their posters and add their suggestions. The groups present their posters and cooperate to choose the best expressions connected with the given concepts.

Proceduremd97662399b58c2ab_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know and applying short multiplication formulas in calculations.

The teacher divides the class into two groups. The task of each of the groups is to work out the formula for:

- Group 1 – the cube of the sum of two expressionscube of the sum of two expressionscube of the sum of two expressions,

- Group 2 – the cube of the difference of two expressionscube of the difference of two expressionscube of the difference of two expressions.

Having finished, the groups present their results. The teacher verifies the students’ answers and explains the doubts. They cooperate to formulate the conclusion.

Conclusion:

- For any expressions a, b the formulas are true:

$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$ - the formula for the cube of the sum of a and b,

$(a-b{)}^3=a^3-3a^{2}b+3a{b}^2-b^3$ - the formula for the cube of the difference of a and b.

The teacher informs the students that the relationships that they have got are the short multiplication formulas of the third degreeshort multiplication formulas of the third degreeshort multiplication formulas of the third degree.

Task
The students work individually analyzing the material presented in the applet. It is a geometric illustration of the formula for the cube of the sum . Having analysed the applet they formulate the conclusion.

[Geogebra applet]

Conclusion:

- You can illustrate graphically the cube of the sum of two expressionscube of the sum of two expressionscube of the sum of two expressions as the volume of a cube with the edge length a + b, where a > 0 and b > 0.

Task
Discussion – are the worked out relationships the only short multiplication formulas of the third degree? The students formulate hypotheses, verify them by analysing the material presented in the interactive illustration. They formulate an appropriate conclusion.

[Interactive illustration]

Conclusion:

- There are two more short multiplication formulas of the third degreeshort multiplication formulas of the third degreeshort multiplication formulas of the third degree. These are the formulas for the sum and the difference of cubes.

$a^3+b^3=(a+b)(a^2-ab+b^2)$ - formula for the sum of cubes of a and b,

$a^3-b^3=(a-b)(a^2+ab+b^2)$ - formula for the difference of cubes of a and b.

The students use the formulas to solve the tasks individually.

Task
Write down the expressions with algebraic sums.md97662399b58c2ab_1527752256679_0Write down the expressions with algebraic sums.

a) $(4-b{)}^3$

b) $(3x-5{)}^3$

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

Task
Perform the indicated operations.

a) $(x+2)(x^2-2x+4)$

b) $(a^2+2ab+4b^2)(a-2b)$

c) $(3-x)(9+3x+x^2)$

Task
Perform the operations and the reduction of like terms.

a) $(2+a{)}^3+3(a-1{)}^3$

b) $(b+5{)}^3-(5-b\left)\right(25+5b+b^2)$

Task
Factorize the expressions to the terms of maximum second degree.md97662399b58c2ab_1527752263647_0Factorize the expressions to the terms of maximum second degree.

a) $x^4+729x$

b) $64x-27x^4$

c) $1-x^6$

Having solved all the tasks, the students present their results. They assess their work. The teacher explains the doubts.

An extra task:
Prove, without the use of a calculator, that number:

a) ${11}^{12}-7^{12}$ - can be divided by 17,

b) ${17}^{18}-{16}^{18}$ - can be divided by 11.

Lesson summarymd97662399b58c2ab_1528450119332_0Lesson summary

The students do the consolidation tasks.

They formulate the conclusions to memorize.

- For any expressions a, b the formulas are true:

$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$ - the formula for the cube of the sum of a and b,

$(a-b{)}^3=a^3-3a^{2}b+3a{b}^2-b^3$ - the formula for the cube of the difference of a and b.

$a^3+b^3=(a+b)(a^2-ab+b^2)$ - formula for the sum of cubes of a and b,

$a^3-b^3=(a-b)(a^2+ab+b^2)$ - formula for the difference of cubes of a and b.

Selected words and expressions used in the lesson plan

cube of the difference of two expressionscube of the difference of two expressionscube of the difference of two expressions

cube of the sum of two expressionscube of the sum of two expressionscube of the sum of two expressions

difference of cubes two expressionsdifference of cubes two expressionsdifference of cubes two expressions

short multiplication formulas of the third degreeshort multiplication formulas of the third degreeshort multiplication formulas of the third degree

sum of cubes of two expressionssum of cubes of two expressionssum of cubes of two expressions