GSSE(ENGLISH)-STD VII-Mathematics-Exponents and Powers Page 1 of 28. Master Lesson Plan. For. Exponents and Powers

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Master Lesson Plan

For

Exponents and Powers

Board Standard Subject Chapter Language Reference Link Creation date

GSSE(ENGLISH) STD VII Mathematics Exponents and_Powers English Exponents and Powers2021-06-03 19:51:56

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Exponents and Powers 1. MS_Objectives_Exponents and Powers

Notes to the teacher: This asset lays down the proposed plan for transacting this chapter.

It states the asset objectives of the MLP. This asset is for the teacher's reference and need not be taught to the students.

Students will be able to

relate the use of exponents in day to day life

identify the expanded form of numbers in exponential form explain the application of exponents in real world

illustrate exponents with a negative base

identify exponents with the same power and the same base solve the problems on exponents

write the laws of exponents to solve the problems analyse and solve the problems

identify the laws of exponents in multiplication practice love towards all

identify the product of a decimal with multiples of 10

demonstrate the laws of exponents through paper folding activity solve the given problems

Time to teach Asset Type Theme SubTheme

3 Minutes Main Script Exponents and Powers

Uses of Exponents:, To express small number to standard form, Standard form of Large numbers, Powers with negative Exponents, Meaning of Exponents , Laws of exponents, Decimal number system, Comparing very large and very small numbers, Uses of Exponents:, To express small number to standard form, Standard form of Large numbers, Powers with negative Exponents, Meaning of Exponents , Laws of exponents, Decimal number system, Comparing very large and very small numbers

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In Computer science, file sizes are expressed as an exponent.

A computer file size is measured using megabytes.

Actually, megabytes are an expression of an exponent

A megabyte is equivalent to 106 bytes.

Knowledge of exponents helps in writing computer programs efficiently.

Image Source:

https://pixabay.com/photos/technology-computer-motherboard-1396677/ https://pixabay.com/photos/programming-computer-environment-1857236/

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3 Minutes Interesting Asides Exponents and Powers Uses of Exponents:, Uses of Exponents:

3. MS_Introduction to Exponents

Introduction to Exponents

Consider the numbers 8, 16, 32, 64, and 128

Let's express these numbers as products of their prime factors i) 8

8 can be written as 2 X 2 X 2 i.e 2 multiplied thrice by itself. ii) 16

16 can be written as 2 X 2 X 2 X 2 i.e 2 multiplied by itself four times. iii) 32

32 can be written as 2 X 2 X 2 X 2 X 2 i.e 2 multiplied by itself five times. iv) 64

64 can be written as 2 X 2 X 2 X 2 X 2 X 2 i.e 2 multiplied by itself 6 times. v) 128

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128 can be written as 2 X 2 X 2 X 2 X 2 X 2 X 2 i.e 2 multiplied by itself 7 times.

In the above examples we expressed 8, 16, 32, 64, 128 as products of their prime factors. We can also write these numbers in another method.

For example: Let's consider the number 8. 8 can be written as 2 X 2 X 2 as 23

23 is called as the exponential form of 8.

Definition:

The representation of a number in the form, where and n are integers, is called

exponential representation. In is called as base and n is called as power. The expression is read as raised to the power n or nth power of .

The meaning of is,

It means is multiplied n times with itself.

Hence, exponents are used as short cuts to show that a number is to be multiplied by itself a given number of times.

The exponent is the number of times the base is multiplied by itself.

Thus, a simple way to express “10 is multiplied with itself 3 times” is 103. Similarly, 82 means 8 is multiplied with itself 2 times. i.e. 8 X 8.

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Let us consider the number 81; there are three ways in which we can represent this number. That is,

When the number 81 is represented as 81 itself the form is known as the standard

form.

When the number 81 is represented as 34 (three raised to the power of four) the

form is known as the exponential form.

When the number 81 is represented as 3 X 3 X 3 X 3, the form is known as expanded form.

Consider the following table where we will use 10 as a base and raise it to the power of 1, 2, 3, 4 (the teacher can also give another number and let the children work on it to provide the solution).

Image Source: Original Contribution by [email protected]

15 Minutes Main Script Exponents and Powers Standard form of Large numbers,_{Standard form of Large numbers}

4. DD_Exponents in Real World

Exponents - in Real World

Let us see examples of usage of exponents in our daily life, and how exponents help in representing data.

1) When we do problems on area, for example, If the area of a room is 10 ft x 10 ft, we write it as 10² sq. ft.

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3) In bank when we deposit money we get compound interest, i.e., after one year your deposit gets interest and then the interest also gets an interest.

4) The speed of light is 30,00,00,000 m/sec. These are very big numbers to handle. So they can be written in exponential form for convenient usage.

5) In computers, the capacity of the memory of a computer is in megabyte and gigabyte. 1 megabyte =1024 bytes. We can simply write it as 1 megabyte = 210 bytes.

6) Writing large numbers in exponential form i.e., the distance between planets. Simplification of chemical concentrations Ex: 0.000056 grams per litre.

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7) Calculating the volume of air in a room in cubic meters.

8)Approximate calculation of the effect of virus or bacteria in a computer.

Image Source: 1) https://pixabay.com/vectors/compound-interest-finance-interest-1296451/ 2) https://pixabay.com/illustrations/speed-of-light-space-star-universe-726251/ 3) https://pixabay.com/illustrations/binary-one-null-monitor-social-503598/ 4) https://pixabay.com/illustrations/solar-system-planet-planetary-system-11111/ 5) https://pixabay.com/photos/chemistry-lab-experiment-3005692/ 6) https://pixabay.com/vectors/warning-alert-detected-malware-2168379/

5 Minutes Day-to-day Relevance Exponents and Powers Uses of Exponents:, Uses of Exponents:

5. MS_Exponents with Negative Base

Exponents with negative base

What happens if we have a negative number as a base? The rule is the same. It is the number of times the negative number is multiplied by itself.

1. Suppose we have -2 as a base. Say for example. It is -2 multiplied by itself 4 times. Therefore,

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In terms of exponents, -2 is the base and raised to the power 4. 2. What is ?

It is -3 multiplied by itself 3 times. Therefore,

= (Odd number of negatives is negative)

=

3. Consider 25. This number can be represented as; and

Thus,

Here, the values are equal and powers are same but the bases are different.

8 Minutes Main Script Exponents and Powers To express small number to standardform, To express small number to standard form

6. MS_Comparison of Exponents

Comparing exponents with the same base (powers and bases are whole numbers)

Which is greater, or ?

= and

= =16 Therefore,

The rule or the law here is,

Which is greater, or ? Since,

by the rule,

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Which is greater, or ? and

Therefore, The rule is,

If > , and n is a natural number then > Which is greater, or ?

The bases are 2 and 3. Since, ,

Comparing very large and very small numbers, Comparing very large and very small numbers

7. QA_Finding solutions on Exponents

Answer the following:

1. Fill up the table

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3. The teacher gives one problem as homework on Monday; two problems on Tuesday and 4 problems on Wednesday and so on. How many problems will the students get on Saturday? Hint: Let the students draw a table as follows

Let them look at 1, 2, 4. Then ask them to find a pattern. Encourage them to express in a different way (exponential form).

Answers: 1.

2 .

3.

10 Minutes Assessments Exponents and Powers Meaning of Exponents , Meaning of_Exponents

8. MS_Laws of Exponents - I

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Multiplying Powers with the Same Base and Dividing Expressions with the Same Base

We know,

This is the same as,

where is the sum of the powers of the bases.

We have a rule which says: To multiply numbers with the same base, add the exponents and keep the base the same.

Multiplying Powers with the Same Base

According to the rule,

a

m

x a

n

= a

m+n

We can hence add the exponents when multiplying numbers with the same base.

Here is another example;

Dividing Expressions with the Same Base

When we divide expressions with the same base, we need to subtract the exponent of the

number we are dividing by, from the exponent of the numerator. In general, we can write is as follows.

Definition: (Of course, ), and

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=

15 Minutes Main Script Exponents and Powers Laws of exponents, Laws of exponents

9. IQ_Think Hard and Solve

The numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, ... form a pattern. What is the rule for this pattern?

This list of numbers results from finding powers of 2 in sequence. Look at the table below and you will see several patterns.

Answer the following:

1. Can you predict the next two numbers in the list after 256?

Answer: 512, 1024

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Solution: 74 = 73 times 7 74 = 343 x 7

74 = 2,401

3. The numbers 1, 5, 25, 125, 625, ... are each powers of what number?

Answer : 5º, 5¹, 5², 5³...

4. If 14 is equal to 1, then what is 1100?

Answer : 1

5 Minutes Inquisitive Questions Exponents and Powers Meaning of Exponents , Meaning of_Exponents

10. MS_Laws of Exponents - II

Laws of exponents

For example, we may start with and raise it to the power 2. This means is multiplied with itself 2 times.

Therefore, it is written as Suppose we have

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This demonstrates the exponent rule: Whenever you have an exponent expression that is raised to a power,

you can multiply the exponent and power: Definition:

(n times)

Example: Find

By the given formula, it is

Multiplying powers with the same exponent

Consider the number

Now is the product of the bases The rule is

Dividing powers with the same exponent

How to simplify ?

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Warning: This rule does NOT work if you have a sum or difference within the brackets. Exponents, unlike multiplication, do NOT "distribute" over addition.

For instance, given , do NOT give in to the temptation to say "This equals,

because this is wrong. Actually, , not 25.

HINT: When in doubt, write out the expression according to the definition of the power.

10 Minutes Main Script Exponents and Powers Laws of exponents, Laws of exponents

11. VC_Love and trust each other

We just learnt that a big number can be written in a shorter form with a base and a positive exponent/Power. it increases the value of the number. For example 22 = 2x2 =4. Similarly 25 = 32. Here 2 is the base, whereas 2 and 5 are its exponents.

In our lives, if 1 (20) person has love & kindness in heart and if he influences another, it will be now 2 (21 ) persons. Now if these two persons influence one more person there will be 4 (22) persons. If this chain continues, the number of persons with Love & kindness in their hears will be 8 (23), 16 (24) and 32 (25) and so on and will increase exponentially.

Children! if a large majority of the people in this world have love & kindness in their hearts and if they pass it on, the world will be transformed totally. Peace & happiness will prevail everywhere in this world.

Let us learn a song today:

Have you had a kindness shown? Pass it on, pass it on Was it given for you alone? pass it on, pass it on

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Let it travel down the years, let it dry another’s tears,

Let it calm another’s fears, pass it on, pass it on, pass it on

Have you had a mother’s love? Pass it on, pass it on Have you had a father’s trust? Pass it on, pass it on

You can love and trust another, as you love and trust your mother We can love and trust each other, Pass it on, pass it on, pass it on

Audio link: http://sssbpt.org/audio/EHV/Track5.mp3

Time to teach Value Type Value Sub Type Value Attribute

5 Minutes Love Kindness Songs

12. MS_Decimal Number System

Decimal Number System

1. What is ? It is 11.2.

2. What is ? It is 112.

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3. What is ? It is 1120

Here, the number 1.12 is the same. In each case, the power of 10 is different. In all cases, we have expressed the numbers as,

,

In all these expressions, we have only one digit before the decimal point. Such kind of expressions are called the standard form of the numbers.

is the standard form of 11.2 is the standard form of 112 is the standard form of 1120

Now let us understand the different ways in which 112 can be expressed.

The standard form is unique. is the standard form of 112.

The power increases as the digits after the decimal point increases.

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13. SA_Verify the law of exponents through activity

Aim: To verify the laws of exponents when the base is same.

Materials required: Paper, ruler, scissors, glue and notebook.

Instruction to the teacher: Please ask the students to bring materials on the day of activity. Type of activity: Individual activity

Setting: Indoor

Questions to recall: What is a base and an exponent of a number?

Expected answer: The representation of a number in the form, where and n are integers,

is called exponential representation.

In is called as base and n is called as power. The expression is read as raised to the power n or nth power of .

Procedure:

Step 1: Take a paper and fold it once to obtain 2 units

Step 2: Represent the first sheet as 2 raised to 0, and second after folding as 2 raised to

1

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Observation Thus 2² = 4, 2³ = 8 and ...so on. We verify the law of exponent.

Conclusion: Law of the exponent of multiplication is verified. Image Source: Original by [email protected]

20 Minutes Suggested Activity Exponents and Powers Meaning of Exponents , Meaning of_Exponents

14. QA_Questions on Laws of exponents

I. Simplify using am X an = a m+n a) 82 X 85 b) (-2)5 X (-2)3 c) (5/2)2 X (5/2)3 d) 106 X 104 X 102 e) b6 X b4 X b10 f) (3.5)5 X (3.5)8 Solution: a) 87 b) (-2)8 c) (5/2)5 d) 1012 e) b20 f) (3.5)13

II. Convert the following into exponential form and apply first law of exponents:

a) 49 X 7 b) 27 X 81 c) 243 X 81 d) 1024 X 16

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Solution: a) 49 X 7 = 72 X 71 = 73 b) 27 X 81 = 33 X 34 = 37 c) 243 X 81 = 35 X 34 = 39 d) 1024 X 16 = 210 x 24 = 214

III. Fill up the space provided with a suitable answer:

a) 105 X 108 = ---. b) (2/5)13 X (2/5)6 = ---. c) a13 = ----3 X a10.

d) (25)6 = (25)--- X (25)5.

Solution: a) 1013 b) (2/5)19 c) a d) (25)1

IV. Simplify using am/an = am-n and choose the right answer.

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i) 77 ii) 710 iii) 73 iv) 7-3 b) 615 /63 = ---.

i) 1/612 ii) 1/ 618 iii) 65 iv) 612 c) Which one shows 5-3

i) 1/125 ii) 1/-125 iii) 1/15 iv) 125 d) x11/x3 =

---i) x14 ii) x8 iii) 1/x8 iv) x

Solution:

a) 73 b) 612 c) 1/125 d) x8

V. Match the following with appropriate answers:

Solution: (i) c (ii) a (iii) e (iv) b

VI. Convert the following into exponential form using am/an = am-n

i) 125/25 ii) 81/9 iii) 256/8

Solution:

i) 125/25 = 53/52 = 51

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ii) 81/9 = 92/91 = 91 iii) 256/8 = 44/42 =42

VII. Express the number appearing in the following statements in the standard form.

For e.g. Question:The universe is estimated to be about 12,000,000,000 years old.

Answer: 1.2 X 1010 Years.

1. The distance between Earth and Moon in 384,000,000 m 2. Speed of light in vacuum is 300,000,000 m/s

3. The diameter of the earth is 1, 27, 56, 000 m. 4. The diameter of the Sun is 1, 400, 000, 00 m.

Answer:

The distance between Earth and Moon in 384,000,000 m = 3.84 X 108 m

Speed of light in vacuum is 300,000,000 m/s = 3 X 108 m/s

Diameter of the earth is 12,756,000 m = 1.2756 X 107 m

Diameter of Sun is 140,000,000 m = 1.4 X 108 m

20 Minutes Assessments Exponents and Powers Laws of exponents, Laws of exponents

15. MS_Summary_Exponents and Powers

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1 . Very large numbers are difficult to read, understand, compare and operate upon. To make all these easier, we use exponents, converting many of the large numbers in a shorter form.

2. The following are exponential forms of some numbersᴾ

10,000 = 10⁴ (read as 10 raised to 4) 243 = 3⁵,128 = 2⁷.

Here, 10, 3, and 2 are the bases, whereas 4, 5, and 7 are their respective exponents. We also say, 10,000 is the 4th power of 10, 243 is the 5th power of 3, etc.

3. Numbers in exponential form obey certain laws, which are:

For any non-zero integers, a and b and whole numbers m and n,

(a) aᵐ × aᵑ = aᵐ + ᵑ

(b) aᵐ ÷aᵑ = aᵐ - ᵑ, m > n

(c) (aᵐ)ᵑ = aᵐ ᵑ

(d) aᵐ × bᵐ = (ab) ᵐ

(e) aᵐ÷bᵑ=

aᵐaᵑ\frac{aᵐ}{aᵑ}aᵑaᵐ

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(g) (-1) even number= 1

(h) (–1) odd number = – 1

Uses of Exponents:, To express small number to standard form, Standard form of Large numbers, Powers with negative Exponents, Meaning of Exponents , Laws of exponents, Decimal number system, Comparing very large and very small numbers, Uses of Exponents:, To express small number to standard form, Standard form of Large numbers, Powers with negative Exponents, Meaning of Exponents , Laws of exponents, Decimal number system, Comparing very large and very small numbers

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GSSE(ENGLISH)-STD VII-Mathematics-Exponents and Powers Page 1 of 28. Master Lesson Plan. For. Exponents and Powers

a

x a

= a

​aᵐaᵑ\frac{aᵐ}{aᵑ}aᵑaᵐ​​

aᵐaᵑ\frac{aᵐ}{aᵑ}aᵑaᵐ