Topicm271e3440dd4b7e88_1528449000663_0Topic

Powers with natural exponents

Levelm271e3440dd4b7e88_1528449084556_0Level

Second

Core curriculumm271e3440dd4b7e88_1528449076687_0Core curriculum

I. Powers with rational exponents. The student:

1) writes down the product of two identical elements in the form of power with the integer and the positive exponent.

Timingm271e3440dd4b7e88_1528449068082_0Timing

45 minutes

General objectivem271e3440dd4b7e88_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm271e3440dd4b7e88_1528449552113_0Specific objectives

1. Calculating powers with natural exponents, on the basis the definition.

2. Identifying the sign of the power.

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

Learning outcomesm271e3440dd4b7e88_1528450430307_0Learning outcomes

The student:

- calculates powers with natural exponents, basing on the definition,

- identifies the sign of the power.

Methodsm271e3440dd4b7e88_1528449534267_0Methods

1. Discussion.

2. Chain of associations.

Forms of workm271e3440dd4b7e88_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm271e3440dd4b7e88_1528450127855_0Introduction

The teacher introduces the subject of the lesson – calculating the value of powers with natural exponents.

Students give examples of squares and cubes of one digit numbers and calculate their values. They interpret the square of the given numbers as a product of two identical elements and a cube as the product of three identical elements.

Procedurem271e3440dd4b7e88_1528446435040_0Procedure

Example:

Students analyse the examples below. They notice that in each case we need to multiply a few identical elements.

$12·12·12·12=20736$

$\frac{1}{2}·\frac{{\displaystyle 1}}{{\displaystyle 2}}·\frac{{\displaystyle 1}}{{\displaystyle 2}}·\frac{{\displaystyle 1}}{{\displaystyle 2}}·\frac{{\displaystyle 1}}{{\displaystyle 2}}·\frac{{\displaystyle 1}}{{\displaystyle 2}}=\frac{1}{64}$

$\frac{2}{3}·\frac{{\displaystyle 2}}{{\displaystyle 3}}·\frac{{\displaystyle 2}}{{\displaystyle 3}}·\frac{{\displaystyle 2}}{{\displaystyle 3}}·\frac{{\displaystyle 2}}{{\displaystyle 3}}·\frac{{\displaystyle 2}}{{\displaystyle 3}}·\frac{{\displaystyle 2}}{{\displaystyle 3}}=\frac{128}{2187}$

The teacher informs students that multiplying many identical elements, not just two or three, can be written down in the form of power.

Definition of the power:
[The power of number a, with a natural exponent (n > 1) is called a product of n elements, each being equal to a.
We write it down as:

Number a is called the base of the power, number n – the exponent of the power. Moreover, we assume that: $a^0=1,$ for $a\ne 0$.

Using the definition, students calculate values of powers.

Task 1

Calculate:

a) $4^2,3^3,0^4,{10}^4,{135}^0,$

b) $(-5{)}^3,(-3{)}^0,-3^3,(-1{)}^{12},(-4{)}^2,$

c) $(\frac{1}{3}{)}^3,\frac{2}{5^2},(-\frac{1}{7}{)}^1,(-\frac{1}{2}{)}^4,\frac{-1}{-4^2}.$

Task 2

Calculate the values of the following powers. Think what determines the sign of the result of the exponentiation.

a) $2^3,4^2,6^3,$

b) $(-5{)}^2,(-3{)}^4,(-2{)}^6,$

c) $(-3{)}^3,(-5{)}^3,(-2{)}^5.$

Students should notice that:

1) If we raise a positive number to the power with a natural exponent, we obtain a positive number.
2) If we raise a negative number to the power with a natural, even exponent, we obtain a positive number.
3) If we raise a negative number to the power with a natural, odd exponent, we obtain a negative number.m271e3440dd4b7e88_1527752263647_01) If we raise a positive number to the power with a natural exponent, we obtain a positive number.
2) If we raise a negative number to the power with a natural, even exponent, we obtain a positive number.
3) If we raise a negative number to the power with a natural, odd exponent, we obtain a negative number.

Students use the information to do the exercises.

Task 3

Replace the dots with a proper sign <, > or = .

a) $3^2...2^3$

b) $1^2...(-1{)}^2$

c) $-3^3...3^3$

d) $4^3...(-4{)}^3$

e) $-1^5...(-1{)}^5$

f) $-3^2...-(-3{)}^2$

Task 4

Arrange the numbers from the smallest one to the greatest one.

$2^3,{24}^0,-2^1,(\frac{1}{2}{)}^4,(-2{)}^2,(-16{)}^1,-2,4^2,(-2,4{)}^2.$

Task 5

[Geogebra applet]

Students work individually, using computers. Their task is to match the powers and their properties in pairs.

An extra task:

a) $(-\frac{2}{7}{)}^4·(-0,8{)}^5$

b) $-(-3,4{)}^7·{\frac{(-1)}{-3}}^3$

c) $\frac{-1^2·(-(-6{)}^3)}{-(-0,4{)}^3·(-2{)}^5}$

Lesson summarym271e3440dd4b7e88_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

The power of number a, with a natural exponent (n>1) is called a product of n elements, each being equal to a.

We write it down as:

Number a is called the base of the powerbase of the powerbase of the power, number n – the exponent of the powerexponent of the powerexponent of the power. Moreover, we assume that: $a^0=1,$ for $a\ne 0$.

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

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

square of the numbersquare of the numbersquare of the number

cube of the numbercube of the numbercube of the number