Lesson plan

Topicm63790a1bec7524ea_1528449000663_0Topic

Operating with Exponents

Levelm63790a1bec7524ea_1528449084556_0Level

Third

Core curriculumm63790a1bec7524ea_1528449076687_0Core curriculum

I. Real numbers. The student:

1) student performs operations (addition, subtraction, multiplication, division, exponentiation, square root, logarithm) in a set of real numbers;

2) student applies the properties of monotonicity of powerpower / exponentationpower, in particular properties: if x < y and a > 1, thus aIndeks górny x < aIndeks górny y, whereas if x < y and 0 < a , 1, thus aIndeks górny x > aIndeks górny y.

Timingm63790a1bec7524ea_1528449068082_0Timing

45 minutes

General objectivem63790a1bec7524ea_1528449523725_0General objective

Performing calculations on real numbers, also using a calculator, applying the laws of mathematical operations in transforming algebraic expressions and using these skills to solve problems in real and theoretical contexts.

Specific objectivesm63790a1bec7524ea_1528449552113_0Specific objectives

1. Communicating in English, developing mathematical and basic scientific and technical and IT competences, developing learning skills.

2. Developing accounting efficiency in operations on powersoperations on powersoperations on powers.

3. Consolidation and systematization of information about powers.

Learning outcomesm63790a1bec7524ea_1528450430307_0Learning outcomes

The student:

- performs operations on powers,

- uses the rules of operations on powersoperations on powersoperations on powers in practical situations.

Methodsm63790a1bec7524ea_1528449534267_0Methods

1. Mind map.

2. Case study.

Forms of workm63790a1bec7524ea_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm63790a1bec7524ea_1528450127855_0Introduction

Prior to the lesson, students review the most important properties of operations on powersoperations on powersoperations on powers.

Students create a mind map. After finishing the task, they present their work and place the results on the board.

Procedurem63790a1bec7524ea_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to consolidate and systematize information about powers and the properties of operations on powersoperations on powersoperations on powers.

Task
Analyse the material shown below and use your knowledge to solve the tasks.

Properties of operations on powers:m63790a1bec7524ea_1527752263647_0Properties of operations on powers:

1. Multiplication of powers with the same base:

a^{n} \cdot a^{m} = a^{n + m}, w h e r e a \in R ∖ {0}, n \in C, m \in C

2. Division of powers with the same base:

a^{n} : a^{m} = a^{n - m}, w h e r e a \in R ∖ {0}, n \in C, m \in C

3. Multiplication of powers with the same exponent:

a^{n} \cdot b^{n} = {(a \cdot b)}^{n}, w h e r e a \in R ∖ {0}, b \in R ∖ {0}, n \in C

4. Division of powers with the same exponent:

a^{n} : b^{n} = {(\frac{a}{b})}^{n}, w h e r e a \in R ∖ {0}, b \in R ∖ {0}, n \in C

5. The powerpower / exponentationpower of the powerpower / exponentationpower:

{(a^{n})}^{m} = a^{n \cdot m}, w h e r e a \in R ∖ {0}, n \in C, m \in C

Task
Match in pairs.


$5^{- 5}$	${(\frac{1}{5})}^{5}$
${(\frac{1}{5})}^{- 5}$	$5^{5}$
$3^{- 3} \cdot {(\frac{1}{9})}^{- 3}$	27
${(\frac{2}{3})}^{- 2} \cdot {(\frac{3}{4})}^{- 2}$	4
${(\frac{16}{3})}^{- 3} \cdot {(0, 75)}^{- 3}$	$\frac{1}{64}$
${(2, 6)}^{- 4} \cdot {(2, 6)}^{3}$	$\frac{5}{13}$
${(3^{- 2})}^{- 3} : 3^{9}$	$\frac{1}{27}$
${(4^{- 5})}^{2} : 4^{9}$	${(0, 5)}^{38}$
${({(\frac{3}{2})}^{4})}^{- 2} \cdot {({(\frac{2}{3})}^{- 2})}^{4}$	1
${({(0, 25)}^{5})}^{- 2} : {(4^{- 2})}^{5}$	$2^{40}$

Task
Calculate the value of the expression.

a) $10^{4} \cdot 12^{- 2} \cdot 30^{- 1} \cdot 15^{2} =$

b) $\frac{4^{15} \cdot 100^{- 2} \cdot 25^{- 6}}{10^{4} \cdot 50^{- 3}} =$

c) $(1 - x) \cdot {(1 - x^{2})}^{- 1} \cdot {(1 - x^{3})}^{- 1} \cdot (1 - x^{6}), f o r x = 2^{- 1}$

Task
Students work in groups of 3 and solve the tasks.

A. Which number is greater?

a) ${(- 2)}^{100} o r 2^{- 100}$

b) $5^{- 3} + 7^{- 3} o r {(5 + 7)}^{- 3}$

c) $\frac{7^{- 2}}{5^{2}} o r \frac{5^{2}}{7^{- 2}}$

d) ${(\frac{1}{2})}^{- 10} o r {(\frac{- 1}{2})}^{- 10}$

e) ${(- 1)}^{{(- 1)}^{(- 1)}} o r {(- 2)}^{(- 2)}$

f) ${({(- 1)}^{(- 2)})}^{- 3} o r {({(- 3)}^{(- 2)})}^{(- 1)}$

B. Express the following in the form of a powerpower / exponentationpower.

a) $\frac{x^{4} \cdot x^{- 5}}{x^{- 6}} : \frac{x^{5}}{{(x^{- 4})}^{3}}$

b) $\frac{{(x^{4})}^{5} \cdot x^{6}}{{(x^{2})}^{6}} \cdot x^{- 3}$

c) $\frac{x^{- 24} \cdot x^{14}}{{(x^{- 4})}^{5}}$

Can these operations be always performed?

Students, working individually, analyse the material presented in Geogebar applet - presenting small and large numbers in exponential notationexponential notationexponential notation. They use the information learnt to solve the tasks.

Task
Analyse the material presented in the applet. When can you use exponential notationexponential notationexponential notation? Formulate the appropriate conclusions.

[Geogebra applet]

Conclusion:

- We use exponential notation to present very small and very large numbers.m63790a1bec7524ea_1527752256679_0- We use exponential notation to present very small and very large numbers.

Task
Calculate the value of the expression, write the result in exponential notationexponential notationexponential notation.

a) $3, 5 \cdot 10^{24} \cdot 4, 2 \cdot 10^{15}$

b) $3, 9 \cdot 10^{- 21} + 12, 3 \cdot 10^{- 21}$

c) $2, 3 \cdot 10^{- 25} : 9, 2 \cdot 10^{- 28}$

d) $5, 7 \cdot 10^{- 23} - 0, 35 \cdot 10^{- 23}$

Task
Calculate.

a) $\frac{2^{- 2} + 8^{0}}{{(0, 25)}^{- 1} + 5 \cdot {(- 4)}^{- 1} + {(\frac{4}{9})}^{- 1}} + 6, 75$

b) $\frac{{[{[\frac{1}{4}]}^{- 1} - {[0, 5]}^{- 1}]}^{- 2}}{{(\frac{3}{8})}^{- 1} - {(\frac{3}{5})}^{- 1}}$

After solving all the tasks, the students present the results obtained. The teacher assesses the students' work, explains all the doubts.

An extra task
Calculate $\frac{{(a^{2} \cdot b^{- 3})}^{- 2}}{a^{3} \cdot b^{4}}$ , calculate the value of the expression for $a = \frac{3^{- 2}}{3^{- 3}}, b = \frac{{(- 7)}^{0}}{9^{4}}$ .

Lesson summarym63790a1bec7524ea_1528450119332_0Lesson summary

Students do revision exercises.

Then they summarize the lesson together, formulating conclusions to remember.

- The base of power and the exponent cannot be zeros at the same time.
- We write large and small numbers in exponential notation, i.e. in the form of the product of a number between 1 and 10 by the power of the number 10 with the integer exponent.
- Exponentiation is a two‑argument operation that is a generalization of multiple element multiplication by itself. The multiplied element is called the base, and the number of multiplications (written in the upper index on the right side of the base) is called the exponent or the power.m63790a1bec7524ea_1527712094602_0- The base of power and the exponent cannot be zeros at the same time.
- We write large and small numbers in exponential notation, i.e. in the form of the product of a number between 1 and 10 by the power of the number 10 with the integer exponent.
- Exponentiation is a two‑argument operation that is a generalization of multiple element multiplication by itself. The multiplied element is called the base, and the number of multiplications (written in the upper index on the right side of the base) is called the exponent or the power.

Selected words and expressions used in the lesson plan

base of powerbase of powerbase of power

exponent of powerexponent of powerexponent of power

exponential notationexponential notationexponential notation

operations on powersoperations on powersoperations on powers

power / exponentationpower / exponentationpower / exponentation

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power / exponentation1

operations on powers1

exponential notation1

base of power1

exponent of power1

Scenariusz

Topicm63790a1bec7524ea_1528449000663_0Topic

Lesson stages

Selected words and expressions used in the lesson plan