Topicm4c1c95c0c42200c9_1528449000663_0Topic

The product and the quotient of powers of the same basesquotient of powers of the same basesquotient of powers of the same bases

Levelm4c1c95c0c42200c9_1528449084556_0Level

Second

Core curriculumm4c1c95c0c42200c9_1528449076687_0Core curriculum

I. Powers with rational bases. The student:

2) multiplies and divides powers of integer and positive exponents.

Timingm4c1c95c0c42200c9_1528449068082_0Timing

45 minutes

General objectivem4c1c95c0c42200c9_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm4c1c95c0c42200c9_1528449552113_0Specific objectives

1. Applying the theorems about the product and the quotient of powers of the same basesquotient of powers of the same bases.

2. Calculating values of powers, using the theorems about the product and the .

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

Learning outcomesm4c1c95c0c42200c9_1528450430307_0Learning outcomes

The student:

- applies the theorems about the product and the quotient of powers of the same basesquotient of powers of the same bases,

- calculates values of powers, using the theorems about product and quotient of powers of the same bases.

Methodsm4c1c95c0c42200c9_1528449534267_0Methods

1. Discussion.

2. JIGSAW.

Forms of workm4c1c95c0c42200c9_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm4c1c95c0c42200c9_1528450127855_0Introduction

Students recall that exponentiation is a shortened version of multiplication. They give appropriate examples.

Procedurem4c1c95c0c42200c9_1528446435040_0Procedure

Students analyse the following examples.

$2^3·2^2=(2·2·2)·(2·2)=2^5$

$3^2·3^1·3^4 = (3·3)·3·(3·3·3·3) = 3·3·3·3·3·3·3 = 3^7$

Together they wonder what the relation between the exponents of the multiplies powers of the same basessame basessame bases and the exponent of the product is. They analyse the problem based on their own examples.

Task
They check their assumptions using computers and write down the conclusion.

[Slideshow]

Conclusion:

- The product of powers of the same bases is equal to the power of the same base and the exponent equal to the sum of the exponents of the multiplied powers
$a^n\cdot a^m=a^{(n+m)}$
where:
n and m - are natural numbers.m4c1c95c0c42200c9_1527752263647_0- The product of powers of the same bases is equal to the power of the same base and the exponent equal to the sum of the exponents of the multiplied powers
$a^n\cdot a^m=a^{(n+m)}$
where:
n and m - are natural numbers.

Discussion - what formula will we obtain when we divide the powers of the same base?

Students give appropriate examples. Using the division of fraction, they write down the division as the inversion of multiplication, where one of the elements is the inversion of the divisor.

Students should draw the following conclusion:

- The quotient of powers of the same bases is equal to the power of the same base and the exponent equal to the difference of the exponents of the dividend and divisor
$\frac{a^n}{a^m}=a^{(n-m)}$ for $a\ne 0$ and natural numbers such that $n>m$.m4c1c95c0c42200c9_1527752256679_0- The quotient of powers of the same bases is equal to the power of the same base and the exponent equal to the difference of the exponents of the dividend and divisor
$\frac{a^n}{a^m}=a^{(n-m)}$ for $a\ne 0$ and natural numbers such that $n>m$.

Students work using the JIGSAW method in groups of 4.

Each member of the group gets a different task from the tasks below. After doing the tasks, the students gather in groups that were doing the same task. They discuss the solutions and clarify any doubts. Then, they return to theiroriginal groups and present the solutions to other members.

Task
Write the expression using one power.

a) $x^2·x^4$

b) $a^3·a^8$

c) $y^4·y^0$

d) $z^2·z^3·z^4$

e) $s^0·s·s^3$

Task
Write the expression using one power.

a) $x^5:x^2$

b) $a^9:a^8$

c) $y^5:y^0$

d) $z^{16}:z^8:z^4$

e) $s^{15}:s:s$

Task
Fill in the exponent of the powerexponent of the powerexponent of the power.

a) ${(-\frac{1}{2})}^7\cdot {(-\frac{1}{2})}^3:(-\frac{1}{2})={(-\frac{1}{2})}^{▫}$

b)  $3^{14}:3\cdot 3^2=3^{▫}$

c) $(-2{)}^{18}:(-2{)}^6\cdot (-2{)}^8=(-2{)}^{▫}$

d) ${(-\frac{1}{3})}^7:{(-\frac{1}{3})}^3\cdot (-\frac{1}{3})={(-\frac{1}{3})}^{▫}$

Task
Convert the units using the power of number 10.

a) 1000 km for cm,

b) 10000 m for mm,

c) 100 kg for dag,

d) 10 t for g.

An extra task:
Using one power write.

a) One eighths of the number $2^7$.

b) Sixteen times the number $2^4$.

Lesson summarym4c1c95c0c42200c9_1528450119332_0Lesson summary

Students do the revision exercises.

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

- The product of powers of the same basesproduct of powers of the same basesproduct of powers of the same bases is equal to the power of the same base and the exponent equal to the sum of the exponents of the multiplied powers:
$a^n\cdot a^m=a^{(n+m)}$
where: 
n and m - are natural numbers.

- The quotient of powers of the same basesquotient of powers of the same basesquotient of powers of the same bases is equal to the power of the same base and the exponent equal to the difference of the exponents of the dividend and divisor:
$\frac{a^n}{a^m}=a^{(n-m)}$ for $a\ne 0$ and natural numbers such that $n>m$.

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

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

product of powers of the same basesproduct of powers of the same basesproduct of powers of the same bases

quotient of powers of the same basesquotient of powers of the same basesquotient of powers of the same bases

same basessame basessame bases

shortened form of multiplicationshortened form of multiplicationshortened form of multiplication