1 standards square units and second power, then square roots cubic units and cube numbers a pattern...

1

STANDARDS

Square Units and Second Power, then Square Roots

Cubic Units and Cube Numbers

A pattern of Powers of 10’s

In between what whole numbersis the square root?

Scientific Notation

Exponent Properties

END SHOW

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

2

2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base.

2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why.

2.1 Entender exponentes enteros negativos. Multiplicar y dividir expresiones que involucran exponentes.

2.4 Usar la relación inversa entre elevar una potencia y sacar su raíz cuadrada perfecta; para un entero que no es cuadrado, determinar sin calculadora los dos enteros entre los cuales se encuentra dicha raíz y explicar porqué.

GRADE 7: Number Sense

GRADE 8: Algebra2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, and taking a root. They understand and use the rules of exponents.2.1 Los estudiantes entienden y usan operaciones como tomar el opuesto, encontrar el reciproco, y sacar la raíz. Ellos entienden y usan las reglas de los exponentes.



3

STANDARDS

1

1

1x1 =12

1

= 1

What is the area of the square?

1 = 1

What is the length of the side?



4

STANDARDS

2x2 = 22

2

243

21

= 4

4 = 2





5

STANDARDS

3x3 = 32

3

3

987

654

321

=9

9 = 3





6

STANDARDS

4x4 = 42

4

4

16151413

1211109

8765

4321

= 16

16 = 4





7

STANDARDS

5x5 = 52

5

5

2524232221

2019181716

1514131211

109876

54321= 25

25 = 5



The SQUARE OF A NUMBER is the total of square units used to form a larger square.

The SQUARE ROOT OF A NUMBER is the opposite of the square. It is when you find the lenght of the side in a square with a given number of square units.



8

STANDARDS

2x2 = 22

2

243

21

4

= 4

= 2

3x3 = 323

3

987

654

321

9

=9

= 3

4x4 = 424

4

16151413

1211109

8765

4321

16

= 16

= 4

1x1 = 52

5

5

2524232221

2019181716

1514131211

109876

54321

25

= 25

= 5

THE SQUARE OF A NUMBER

1

1

1x1 =12

1

1

= 1

= 1

THE SQUARE ROOT OF A NUMBER



9

STANDARDS

3x3 = 32

3

3

=9

SUMMARIZING:

We say 3 SQUARE or THREE TO THE SECOND POWER.



10

STANDARDSSUMMARIZING:

We say 5 SQUARE or FIVE TO THE SECOND POWER.

5x5 = 52

5

5

2524232221

2019181716

1514131211

109876

54321

= 25



11

STANDARDSWhat means 7 square?

7

7

7

7

7x7 = 72 Why?

= 49

49 square units.



12

STANDARDSWhat means 13 square?

13x13 = 132 Why?

13

13

13

13

169 square units.

= 169



13

STANDARDS

11

1

2

2

23

3

3

4

4

4

1x1x1 = 13

= 1

1 CUBED

2x2x2 = 23

= 8

2 CUBED3x3x3 = 3

3= 27

3 CUBED

4x4x4 = 43

=64

4 CUBED

What is the volume for these cubes?



14

STANDARDS

3

3

3

3x3x3 = 33

= 27

Three to the THIRD POWER

OR

What is 3 cubed?

That is 27 cubic units!



15

STANDARDS

4

4

4

4x4x4 = 43

=64

What is 4 cubed?

Four to the THIRD POWER

OR

That is 64 cubic units!



16

STANDARDS

2x2 = 22

4

= 4

= 2

1x1 =12

1

11

1

= 1

= 1

Which whole numbers is between?3

What is the largest perfect square that can be made with 3 square units?

There is no possible perfect square with 3 square units.

3

21

We either take out 2 or add 1.

Taking out 2:

Adding 1 more:

43210

The square root of 3 is between 1 and 2.

3 1.73

2

243

21

1 4

3PRESENTATION CREATED BY SIMON PEREZ. All rights reserved


17

STANDARDS

2x2 = 22

2

2

4

= 4

= 23x3 = 3

2

9

=9

= 3





43210


8 2.83

4 9

8

Taking out 3:

3

3

Adding 1 more:



18

STANDARDS

3x3 = 32

3

3

9

=9

= 3

4x4 = 42

16

= 16

= 4





Adding 4 more:

43210


12 3.46

9 16

12

Taking out 3:

4

4



19

STANDARDS

10x10 = 102

100

100 square units.

10

10

10

10

10x10x10 = 103

1000

10x10x10x10 = 104

10000

10x10x10x10x10 = 105

100000

10x10x10x10x10x10 = 106

1000000

10x100

10x1000

10x10000

10x100000

10

10

10

1000 cubic units.Can you continue the pattern?

Finding a pattern:



20

STANDARDS

10x10

102

100

10x10x10

103

1000

10x10x10x10

104

10000

10x10x10x10x10

105

100000

10x10x10x10x10x10

106

1000000

10x10010x100010x1000010x100000 10x1

101

10

ONES

TENSHUNDREDS

ONE

THOUSANDS

TEN

THOUSANDS

HUNDRED

THOUSANDS

Then: 500 = 5 x100

=5x102

100

1

ONE

MIL

LIONS

7000 = 7 x1000

=7x103

and 8x106

=8x1000000

=8000000

3x105 =3x100000

=300000

9x100=9x1

=9



21

Write in scientific notation: 1,750,000

10x10

102

100

10x10x10

103

1000

10x10x10x10

104

10000

10x10x10x10x10

105

100000

10x10x10x10x10x10

106

1000000

10x10010x100010x1000010x100000 10x1

101

10

ONES

TENSHUNDREDS

ONE

THOUSANDS

TEN

THOUSANDS

HUNDRED

THOUSANDS

100

1

ONE

MIL

LIONS

1 7 5 0 0 0 0

Then:

1,750,000 = 1.750 millions = 1.750 106

x This is the number in scientific notation!

OR 1 ,7 5 0 , 0 0 0.

6 places to the left

= 1.750 106

x STANDARDS



22

Write in Scientific Notation the following numbers:

= 1.32 103

x1,320


1,320

= 3.79 104

x37,900


37,900

= 5.591 101

x55.91

1 place to the left

55.91

11.45 103

x

11.45 103

x

1 place to the left

= 1.145 104

x

237.6 105

x

237.6 105

x


= 2.376 107

x

STANDARDS



23

Write in Standard Notation the following numbers:

= 2.85

STANDARDS

2.85 103

x 1000

= 2,851.

= 2,851

5.71 106

x = 5.71 1000000

= 5,710,000.

= 5,710,000

8.093 101

x = 8.093 10

= 80.93

27.9 102

x = 27.9 100

= 2790.

= 2,790



24STANDARDS

ONES

TENS

HUNDREDS

ONE

THOUSANDS

HUNDREDTHS

THOUSANDTHS

TENTHS

10x10

102

100

103

1000

10x100 10x1

101

10

100

1 110

10x 110 10x 1

10010x 1

100010x 1

10000

1100

11000

110000

10x 1 100000

TEN

THOUSANDTHS

10-1

10-2 10

-310

-4

What pattern do you see emerging in the exponents?

They decrease from left to right!

.1 .01 .001 .0001

What about the decimals?

Observe the following pattern:



25

100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

10987654321

=

How many hundredths does the unit have?

STANDARD 1.2



26

10

9

8

7

6

5

4

3

2

1

=

How many parts does the tenth have?

STANDARD 1.2



27

STANDARD 1.2

Penny

=$ .01

1 cent

1¢

$1100

A hundredth of a dollar.



28

STANDARD 1.2

Dime

=$ .1

10¢

10 cents

$1 10

A tenth of a dollar



29

STANDARD 1.2

• Name = 1 dollar• Worth = $1.00• Worth = 10 dimes• Word = 100 cents• Worth = 100 ¢

Dollar

=



30

HundredthsTenthsUnits or Ones

DECIMAL POINT

STANDARD 1.2


31

Write in Scientific Notation the following numbers:

4 places to the right

= 3.45 10-4

x

STANDARDS

.000345 .000345= 3.45 110000

x

= 3.45 10-4

x

OR

3 places to the right

= 6.75 10-3

x.00675 .00675= 6.75 1 1000

x

= 6.75 10-3

x

OR


32

Write in Standard Notation the following numbers:

STANDARDS

= 4.35 110000

x4.35 10-4

x

= 4.35 10000

=.000435

OR 4.35 10-4

x


000 =.000435

= 7.26 1 10000000x7.26 10

-7x

= 7.26 10000000

=.000000726

OR 7.26 10-7

x


000000 =.000000726

= 40.1 1 1000000x40.1 10

-6x

= 40.1 1000000

=. 0000401

OR 40.1 10-6

x


0000 =.000000726


33

STANDARDS

Exponential FormStandard Numbers

10x10

10x100

10x1000

10x10000

10x100000

=100

=1000

=10000

=100000

=1000000

101 10

1x

101 10

2x

101

103x

101 10

4x

101 10

5x

102

=

103

=

104

=

105

=

106=

What is then?

7 1 x 71

72=

7 1 x 72

73=

7 1 x 73

74=

7 1 x 74

75=

7 1 x 75

76=

Z1 x Z1 Z2=

Z1 x Z2 Z3=

Z1 x Z3 Z4=

Z1 x Z4 Z5=

Z1 x Z5 Z6=

am an = a m+n

Product of Powers:

For any real number a and integers m and n

= x 2+5

= x 7

x x 2 5 y y y 2 5 7 = y 2+5+7

= y 14

Write the expressions as a single power of the base:

Do you remember?

10x10x10x10x10x10 10x100000= 1000000= 106=

Let’s look for a pattern:


34

STANDARDS

100000010

10000010

1000010

100010100101010

=100000

=10000

=1000

=100

=10

=1


104

=

103

=

102

=

101

=

100=

105

=106 10

1

105 10

1

104 10

1

103

101

102 10

1

101 10

1

10x10x10x10x10x10101

11x10x10x10x10x10

1= =100000If

Let’s look for a pattern:then

am

an = am-n

Quotient of powers:

For any real number a, except a=0, and integers m and n

106 10

1 106–1 = 105

=

What is happening?

= x 9–3

= x 6

=y7–6 xx

9

3yy

7

6

= y

Simplify the quotients:


35

STANDARDS

1010

=1 100=10

1 101


Let’s concentrate in this part:

Power to the zero:

a0 = 1

(4y) 0

(-3kp)0 = 1

= 1 00 UNDEFINED!

101–1 =If

then


36

STANDARDS

100000010

10000010

1000010

100010

10010


10-4

=

10-3

=

10-2

=

10-1

=

10-5

=101 10

6

101 10

5

101 10

4

101

103

101 10

2

10x10x10x10x10x1010

1

1

1x10x10x10x10x10 1

=If

Let’s look for a pattern:then

100000 1

=

10000 1=

1000 1

=

1001=

101

=

101–6 =

101–5 =

101–4 =

101–3 =

101–2 =

100000 1

=

a =-nn

1

a

Power with Negative Exponents:

For any real number a, and any integer n, where a = 0

31

a 51

xa =-3 x =-5

y =-99

1

y


1 standards square units and second power, then square roots cubic units and cube numbers a pattern...

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