Square & Square Roots
Objective: Students will be able to square a number and find the square root of a number.
Standards: 7NS 2.4
Squaring a Number
2• 2
2
2
4
=4
=2
9
= 32
= 3• 3
=9
=3
4•4
=4
16
2
= 16
5
5
2• 2
Draw a rectangle (we are going to
define a square and why its called
square & square root).
This rectangle has an area of 4.
What would be the dimensions?
[2 & 2]
Show multiplying 2 times 2,
writing it using exponents and
getting 4.
This is called “squaring a
number”.
4
4
=
3• 3
9
4
=2
2
3
3
Taking a Square Root
25
5•5
16
= 4•4
=4
The inverse of squaring a number
is taking the square root. We
write like this 4 . We need to
think: “what number multiplied
by it self equals 4? “ We know
that 2 times 2 is 4. Because 2
times 2 equals 4, then the square
root of 4 is 2.
25
= 52
= 5•5
= 25
=5
Continue for the rest of the
squares (9, 16, 25 & 36).
Think-Pair-Share:
* What shape did we draw in all
the cases? [a square]
6
6
36
6•6
36
= 62
= 6•6
= 36
=6
* Why do you think raising a
number to the second power is
called squaring a number?
*When we use the
we
only want the positive square
root. This is called the
Principal Square Root
MCC@WCCUSD 11/04/2011
Simplifying Expression with Square Roots
*Remember: the
represents the Principal Square Root (a positive square root).
When we have x 2 , the square root must be x , since we don’t know if x is a positive or negative
Example 1
Simplify.
Generic Square
Decomposition
?
?
m10
m•m•m•m•m•!
m•m•m•m•m
Traditional
m10
m10
= m•m•m•m•m•m•m•m•m•m
=
= m5 • m5
(m • m • m • m • m)• (m • m • m • m • m)
= m5 • m5
=
(m )
5 2
= m5
= m5
You Try 1
Simplify.
b8
?
?
b !
m
8
10
b•b•b•b• !
b•b•b•b
= b•b•b•b•b•b•b•b
=
(b • b • b • b)• (b • b • b • b)
= b4 • b4
b8
= b4 • b4
=
(b )
4 2
= b4
= b4
MCC@WCCUSD 11/04/2011
Example 2
Simplify.
Generic Square
Area Model
x + x + x!
?
?
Decomposition
x x2 !
+
2
x! x !
+
x2 !
x
2
!9x
3• 3• x • x
x2 !
x2 !
x2 !
x2 !
Traditional
9x 2
9x 2
=
3• 3• x • x
=
32 • x 2
x2 !
=
3x • 3x
(3x)
x2 !
= 3x
=
2
= 3x
!
You Try 2
Simplify.
Generic Square
Area Model
?
y
2
y
!
y4
!
+
y4
y2
?
!
= 22 g•33•
g2 g32g•y g3•
y gy gy
y• y• y• y
!
+
36 y 4
!
2
+ y
2
y4
!
y4
!
+ y
2
y4
!
y4
!
+ y
2
y4
!
y4
!
+ y
2
y4
!
y4
!
Decomposition
+ y
!
y4
!
y2
y4
y4
y4
y4
y4
y4
!4
!4
!4
!4
!4
y4
y
!
+
y2
+
y
2
!
y
!
y
!
y
!
y
!
!
36y 4
36y 4
y4
+
y2
2
Traditional
!
!
y4
y4
y4
y4
y4
y4
!4
y
!4
y
!4
y
!4
y
!4
y
!4
y
!
!
!
!
!
!
= 2 • 2 • 3• 3• y • y • y • y
=
(2 • 3• y • y )• (2 • 3• y • y )
= 6y 2 • 6y 2
= 6y 2 • 6y 2
=
(6y )
2 2
= 6y 2
= 6y 2
MCC@WCCUSD 11/04/2011
Example 3
Simplify.
Generic Square
Decomposition
100a 30b 24
?
?
10 •10 • a15 • a15 •
1512
15
12
b g•ab ga gb gb
10 g10
100a 30b 24
= 10 • 10 • a15 • a15 • b12 • b12
100a 30b 24
12
Traditional
!
=
12
(10 • a
15
• b12 )(10 • a15 • b12 )
= 10a15b12
!
=
(10a
b
=
(10a
b
15 12
)(10a
15 12
b
)!
)
15 12 2
= 10a15b12
You Try 3
Simplify.
?
169a 64b18
?
!
13•13• a 32 • a 32 •
13g13 ga
b 9 •b 9
32
ga
32
169 x 64 y18
169 x 64 y18
9
gb gb
9
= 13 • 13 • x32 • x32 • y 9 • y 9
=
(13x
32
(13x
32
=
(13 • x32 • y9 )(13 • x32 • y9 ) !
=
=
(13x
= 13x 32 y 9
32
y 9 )(13x32 y 9 )
y 9 )(13x 32 y 9 )
y
)
9 2
= 13x32 y 9
MCC@WCCUSD 11/04/2011
!
Squares and Square Roots
Number
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
Squaring
a Number
Expanded
Notation
Perfect
Square
22
= 2•2
=4
= 9!
3
2
4
2
= 3• 3
= 4 • 4!
!
6 !
7 !
8 !
9 !
5
2
2
2
2
2
!
!
12 !
13 !
14 !
15 !
16 !
17 !
18 !
10
2
112
2
2
2
2
2
2
2
= 5 • 5!
= 6 • 6!
= 7 • 7!
= 8 • 8!
= 9 • 9!
= 10 •10 !
= 11•11!
= 12 •12 !
= 13•13 !
= 14 •14 !
= 15 •15 !
= 16 •16 !
= 17 •17 !
= 18 •18 !
20 2
= 19 •19
= 20 • 20 !
25 2
= 25 • 25 !
19 2
= 16 !
= 25 !
= 36 !
= 49 !
= 64 !
= 81!
= 100 !
= 121!
= 144 !
= 169 !
= 196 !
= 225 !
= 256 !
= 289 !
= 324 !
= 361!
= 400 !
= 625 !
Taking the
Square
Root
4
9
!
16 !
25 !
36 !
49 !
64 !
81 !
100 !
121 !
144 !
169 !
196 !
225 !
256 !
289 !
324 !
361 !
400 !
625 !
Identify
Factor
Pairs
=
2•2
=
3• 3
=
4•4
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Square
Root
(Principle)
=2
= 3!
!
5•5!
6•6!
7• 7!
8•8!
9•9!
10 •10
11•11
!
!
12 •12
13 •13
= 4!
!
!
!
15 •15 !
16 •16 !
17 •17 !
18 •18 !
19 •19 !
20 • 20 !
25 • 25 !
14 •14
= 5!
= 6!
= 7!
= 8!
= 9!
= 10 !
= 11!
= 12 !
= 13 !
= 14 !
= 15 !
= 16 !
= 17 !
= 18 !
= 19 !
= 20!
= 25!
MCC@WCCUSD 11/04/2011
Squares and Square Roots
Number
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
Squaring
a Number
Expanded
Notation
Perfect
Square
22
= 2•2
=4
32
= 3• 3
= 4 • 4!
42
5
2
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Taking the
Square
Root
Identify
Factor
Pairs
4
=
2•2
9!
=
3• 3
!
25 !
!
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16
!
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Square
Root
(Principle)
=2
!
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MCC@WCCUSD 11/04/2011
CAHSEE Released Test Questions:
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MCC@WCCUSD 11/04/2011
CST Math 7 Released Test Questions:
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MCC@WCCUSD 11/04/2011
Warm-Up
!
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!
CST:%Grade%7NS1.2%%%%%%%%%%%%%%%%%%%%%%%%
%
Review:%Grade%7%%%%%%%%%%%%%%%%%%%%%%%
Classify!each!expression!as!rational!or!
irrational.!!
!!
a.!!! 96
!
!
!
b.!! 70
!
!
!
c.!! 40
!
%
!
%
!
x"
!
!
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Current:%Grade%7%%%%%%%%%%%%%%%%%%%%%%%%
Other:%Grade%4MG%1.4%%%%%%%%%%%%%%%%%%%%%%%%
Simplify.!!
Find%the%area%of%the%square.%%
!
%
!!
(3x )
3 2
%
!
%
!
6% in
!
%
!
%
! %
6 in
%
MCC@WCCUSD 11/04/2011