Topicm717b5e6363a59aec_1528449000663_0Topic

Exponentiation of powers

Levelm717b5e6363a59aec_1528449084556_0Level

Second

Core curriculumm717b5e6363a59aec_1528449076687_0Core curriculum

I. Powers of rational bases. The student:

4) exponentiates a power.

Timingm717b5e6363a59aec_1528449068082_0Timing

45 minutes

General objectivem717b5e6363a59aec_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm717b5e6363a59aec_1528449552113_0Specific objectives

1. Applying the theorem about exponentiation of powers.

2. Calculating values of powers while applying the theorem about exponentiation of powers.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm717b5e6363a59aec_1528450430307_0Learning outcomes

The student:

- applies the theorem about exponentiation of powers,

- calculates values of powers while applying the theorem about exponentiation of powers.

Methodsm717b5e6363a59aec_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm717b5e6363a59aec_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm717b5e6363a59aec_1528450127855_0Introduction

The class starts with revision of rules of operations on powers introduced during previous lessons.

Procedurem717b5e6363a59aec_1528446435040_0Procedure

Students together analyse examples.

${\left(2^2\right)}^3=2^2\cdot 2^2\cdot 2^2=2^{2+2+2}=2^6$

${\left(4^3\right)}^5=4^3\cdot 4^3\cdot 4^3\cdot 4^3\cdot 4^3=4^{3+3+3+3+3}=4^{15}$

Together they think about the relations between the exponent of the powerexponent of the powerexponent of the power that is raised to a power and the exponent of the result of exponentiation. They consider the problem in groups, analysing own examples.

Task
They check their assumptions using computers and draw a conclusion

[Slideshow]

Conclusion:

- A power of a power is equal to the power of the same base and the exponent equal to the product of exponents ${\left(a^n\right)}^m=a^{n\cdot m}$
where:
$n$ and $m$ - are natural number and $a\ne 0$.m717b5e6363a59aec_1527752263647_0- A power of a power is equal to the power of the same base and the exponent equal to the product of exponents ${\left(a^n\right)}^m=a^{n\cdot m}$
where:
$n$ and $m$ - are natural number and $a\ne 0$.

Students work in 4 persons group using the JIGSAW method.

The teacher divides students into 4 persons groups. Each member of the group gets different task from the tasks below. After solving the tasks, students gather in groups that were doing the same task. They discuss the solutions and clarify any doubts. Then, they return to the initial groups and present the solutions to other members.

Task
Write as one power.

a) $(2^3{)}^5$

b) $(3^4{)}^2$

c) $\left(\right(\frac{2}{5}{)}^3{)}^3$

d) $\left(\right(-4{)}^4{)}^8$

e) $\left(\right(7^3{)}^2{)}^0$

Task
Fill the exponents of powers.

a) $2^{12}={\left(2^6\right)}^{...}$

b) ${\left(\frac{1}{6}\right)}^{28}={\left({\left(\frac{1}{6}\right)}^7\right)}^{...}$

c) ${(0,1)}^{40}={\left({(0,1)}^{...}\right)}^{10}$

d) ${(-5)}^{81}={\left({(-5)}^3\right)}^{...}$

e) ${\left(1\frac{1}{7}\right)}^{128}={\left({\left(1\frac{1}{7}\right)}^{...}\right)}^{64}$

Task
Convert into powers of the base 2.

a) $8^3=...$

b) ${128}^3=...$

c) ${64}^4=...$

d) ${256}^3=...$

Task
Write in the form power of the base 3.

a) $9^2=...$

b) ${81}^4=...$

c) ${27}^5=...$

d) ${243}^3=...$

An extra task:
Calculate $c$ if $c^6=4^3$.

Lesson summarym717b5e6363a59aec_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

- A power of a power is equal to the power of the same base and the exponent equal to the product of exponents ${\left(a^n\right)}^m=a^{n\cdot m}$
where:
$n$ and $m$ - are natural number and $a\ne 0$.

Selected words and expressions used in the lesson plan

base of the powerbase of the powerbase of the power

exponent of the powerexponent of the powerexponent of the power

exponentiation of powerexponentiation of powerexponentiation of power

power with the natural exponentpower with the natural exponentpower with the natural exponent