Download POWERS AND ROOTS

Bartomeu Ramon
CC San Alfonso
POWERS AND ROOTS
Age of students: 12-13
Language Aims

To present a lexical set of words for students
to understand and work with powers and
roots

To practise pronunciation of the words with
the class

To reinforce the language by providing
students with tasks which invite them to use
the language
Content Aims

To introduce powers, index laws
and roots, and the operations with
powers working with practical
activities in order to engage the
students
1. REMEMBER WHAT YOU ALREADY KNOW
Modeling new vocabulary
Power and roots concepts
2. OPERATIONS AND PROBLEMS WITH POWERS
LISTENING FOR THE RIGHT ANSWER
NOUGHTS AND CROSSES
INDEX FORM
HANGMAN
3. TRY YOURSELF WITH A TEST
1
Bartomeu Ramon
CC San Alfonso
MODELING NEW VOCABULARY
WORK IN PAIRS
Vocabulary :
1. We are going to read some words and to repeat them aloud chorally and individually:
Power
/paʊər/
Square
/skweər/
Cube
/kjuːb/
Root
/ruːt/
Base
/beɪs/
Exponent
/ɪkˈspəʊ.nənt/
Index
/ˈɪn.deks/
2. Now write down the words in the activity one in the blank spaces:
_________
5
3
_________
________
________
3
√8
__________
________
2
Bartomeu Ramon
CC San Alfonso
POWER AND ROOTS CONCEPTS
Powers
9 is a square number.
8 is a cube number.
3×3=9
2×2×2=8
3 × 3 can also be written as 32.
2 × 2 × 2 can also be written as 23
This is pronounced "3 squared".
which is pronounced "2 cubed".
Index form
The notation 32 and 23 is known as index form. The small digit is called the index
number or power.
You have already seen that 32 = 3 × 3 = 9, and that 23 = 2 × 2 × 2 = 8.
Similarly, 54 (five to the power of 4) = 5 × 5 × 5 × 5 = 625
and 35 (three to the power of 5) = 3 × 3 × 3 × 3 × 3 = 243.
The index number tells you how many times to multiply the numbers together.

When the index number is two, the number has been 'squared'.

When the index number is three, the number has been 'cubed'.

When the index number is greater than three you say that it is has been
multiplied 'to the power of'.
For example:
72 is 'seven squared',
33 is 'three cubed',
37 is 'three to the power of seven',
45 is 'four to the power of five'.
3
Bartomeu Ramon
CC San Alfonso
LISTENING FOR THE RIGHT ANSWER
1. Activity
Working in pairs you have to read aloud to your partner the powers you have in your
worksheet, how many times they are multiplied for itself and the result. Then, he or she
has to write it correctly in her/his worksheet. Take in account you have different powers
on your worksheets
STUDENT A
Index
form
Multiplied form
Result
43
27
72
53
132
65
2. Activity:
Some of you will have to listen powers readed aloud from your partners and write them
at the board.
4
Bartomeu Ramon
CC San Alfonso
LISTENING FOR THE RIGHT ANSWER
1. Activity
Working in pairs you have to read aloud to your partner the powers you have in your
worksheet, how many times they are multiplied for itself and the result. Then, he or she
has to write it correctly in her/his worksheet. Take in account you have different powers
on your worksheets
STUDENT B
Index
form
Multiplied form
Result
83
34
26
112
24
104
2. Activity:
Some of you will have to listen powers read aloud from your partners and write them at
the board.
5
Bartomeu Ramon
CC San Alfonso
POWER AND ROOTS CONCEPTS
Square root
The opposite of squaring a number is called finding the square root.
Example
The square root of 16 is 4 (because 42 = 4 × 4 = 16)
The square root of 25 is 5 (because 52 = 5 × 5 = 25)
The symbol '√ ' means square root, so √ 36 means 'the square root of 36'.
Cube root
The opposite of cubing a number is called finding the cube root.
Example
The cube root of 27 is 3 (because 3 × 3 × 3 = 27)
The cube root of 1000 is 10 (because 10 × 10 × 10 = 1000)
3
The symbol √
means cube root, so √
‘the cube root of 64’.
6
Bartomeu Ramon
CC San Alfonso
NOUGHTS AND CROSSES
1. You will split the classroom into two groups (Team A and team B) and choose a
spokesperson for each team. In turns, the spokesperson of team A will have to read
aloud a root and the other team will have to write the correct answer at the
whiteboard so that ONLY if they answer right, they will be allowed to write a
“nought or a cross”. After that will be the turn of team B.
Important: each time has only ONE minut to guess the correct answer.
Roots for
Team A
Write the
correct answer
Roots for
Team B
√
√
√
√
√
√
3
3
√
3
√8
3
√
Write the
correct answer
√
3
√
3
√
7
Bartomeu Ramon
CC San Alfonso
INDEX LAWS
Multiplying numbers with the same base
We often need to multiply something like the following:
4 3 × 45
We note the numbers have the same base (which is 4) and we think of it as follows:
43 × 45 = (4 × 4 × 4) × (4 × 4 × 4 × 4 × 4)
We get 3 fours from the first bracket and 5 fours from the second bracket, so altogether
we will have 3 + 5 = 8 fours multiplied together.
43 × 45 = 43+5= 48
In general, we can say for any number a and indices m and n:
Dividing numbers with the same base
As an example, let's divide 36 by 32:
So we have 36÷32=36-2=34
We cancelled out 2 of the threes on top and the 2 threes on the bottom of the fraction,
leaving 4 threes on the top (and the number 1 on the bottom).
In general, for any number a (except 0) and indices m and n:
8
Bartomeu Ramon
CC San Alfonso
Raising an index expression to an index
As an example, let's raise the number 42 to the power 3:
(42)3 = 42 × 42 × 42
From the multiplication example above, we can see that this is going to give us 46. We
could have done this as:
(42)3 = 42×3 = 46
In general, we have for any base a and indices m and n:
(am)n = amn
Raising a product to a power
Number example:
(5 × 2)3 = 53 × 23
In general:
(a·b)n = anbn
Raising a quotient to a power
Number example:
In general:
9
Bartomeu Ramon
CC San Alfonso
HANGMAN
1. Split the classrom into two teams. Each team choose a representative and team
A start saying a letter to find out the unknown word. Each team can only fail 6
times. Then it is the turn of team B.
Roots for
Team A
Write the
correct answer
Roots for
Team B
Write the
correct answer
82 ·52
4
26 · 56
8
5
Team A
___ ___ ___ ___ ___ ___ ___ ___
Team B
___ ___ ___ ___ ___
10
Bartomeu Ramon
CC San Alfonso
NOUGHTS AND CROSSES
1. You will split the classroom into two groups (Team A and team B) and choose a
spokesperson for each team. In turns, the spokesperson of team A will have to read
aloud a root and the other team will have to write the correct answer at the
whiteboard so that ONLY if they answer right, they will be allowed to write a
“nought or a cross”. After that will be the turn of team B.
Important: each time has only ONE minut to guess the correct answer.
Roots for
Team A
√
Write the
correct answer
Roots for
Team B
√
3
√
√
√
√
3
3
√
3
√8
3
√
Write the
correct answer
√
3
√
3
√
11