1 STANDARDS Square Units and Second Power, then Square Roots Cubic Units and Cube Numbers A pattern...
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Transcript of 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
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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.
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3
STANDARDS
1
1
1x1 =12
1
= 1
What is the area of the square?
1 = 1
What is the length of the side?
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4
STANDARDS
2x2 = 22
2
243
21
= 4
4 = 2
What is the area of the square?
What is the length of the side?
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5
STANDARDS
3x3 = 32
3
3
987
654
321
=9
9 = 3
What is the area of the square?
What is the length of the side?
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6
STANDARDS
4x4 = 42
4
4
16151413
1211109
8765
4321
= 16
16 = 4
What is the area of the square?
What is the length of the side?
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7
STANDARDS
5x5 = 52
5
5
2524232221
2019181716
1514131211
109876
54321= 25
25 = 5
What is the area of the square?
What is the length of the side?
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.
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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
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9
STANDARDS
3x3 = 32
3
3
=9
SUMMARIZING:
We say 3 SQUARE or THREE TO THE SECOND POWER.
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10
STANDARDSSUMMARIZING:
We say 5 SQUARE or FIVE TO THE SECOND POWER.
5x5 = 52
5
5
2524232221
2019181716
1514131211
109876
54321
= 25
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11
STANDARDSWhat means 7 square?
7
7
7
7
7x7 = 72 Why?
= 49
49 square units.
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12
STANDARDSWhat means 13 square?
13x13 = 132 Why?
13
13
13
13
169 square units.
= 169
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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?
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14
STANDARDS
3
3
3
3x3x3 = 33
= 27
Three to the THIRD POWER
OR
What is 3 cubed?
That is 27 cubic units!
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15
STANDARDS
4
4
4
4x4x4 = 43
=64
What is 4 cubed?
Four to the THIRD POWER
OR
That is 64 cubic units!
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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
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17
STANDARDS
2x2 = 22
2
2
4
= 4
= 23x3 = 3
2
9
=9
= 3
Which whole numbers is between?8
What is the largest perfect square that can be made with 8 square units?
There is no possible perfect square with 8 square units.
We either take out 3 or add 1.
43210
The square root of 8 is between 3 and 4.
8 2.83
4 9
8
Taking out 3:
3
3
Adding 1 more:
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18
STANDARDS
3x3 = 32
3
3
9
=9
= 3
4x4 = 42
16
= 16
= 4
Which whole numbers is between?12
What is the largest perfect square that can be made with 12 square units?
There is no possible perfect square with 12 square units.
We either take out 3 or add 4.
Adding 4 more:
43210
The square root of 12 is between 3 and 4.
12 3.46
9 16
12
Taking out 3:
4
4
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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:
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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
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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
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22
Write in Scientific Notation the following numbers:
= 1.32 103
x1,320
3 places to the left
1,320
= 3.79 104
x37,900
4 places to the left
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 places to the left
= 2.376 107
x
STANDARDS
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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
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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:
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25
100999897969594939291
90898887868584838281
80797877767574737271
70696867666564636261
60595857565554535251
50494847464544434241
40393837363534333231
30292827262524232221
20191817161514131211
10987654321
=
How many hundredths does the unit have?
STANDARD 1.2
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26
10
9
8
7
6
5
4
3
2
1
=
How many parts does the tenth have?
STANDARD 1.2
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27
STANDARD 1.2
Penny
=$ .01
1 cent
1¢
$1100
A hundredth of a dollar.
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28
STANDARD 1.2
Dime
=$ .1
10¢
10 cents
$1 10
A tenth of a dollar
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29
STANDARD 1.2
• Name = 1 dollar• Worth = $1.00• Worth = 10 dimes• Word = 100 cents• Worth = 100 ¢
Dollar
=
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30
HundredthsTenthsUnits or Ones
DECIMAL POINT
STANDARD 1.2
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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
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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
4 places to the left
000 =.000435
= 7.26 1 10000000x7.26 10
-7x
= 7.26 10000000
=.000000726
OR 7.26 10-7
x
7 places to the left
000000 =.000000726
= 40.1 1 1000000x40.1 10
-6x
= 40.1 1000000
=. 0000401
OR 40.1 10-6
x
6 places to the left
0000 =.000000726
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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:
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34
STANDARDS
100000010
10000010
1000010
100010100101010
=100000
=10000
=1000
=100
=10
=1
Exponential FormStandard Numbers
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:
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35
STANDARDS
1010
=1 100=10
1 101
Exponential FormStandard Numbers
Let’s concentrate in this part:
Power to the zero:
a0 = 1
(4y) 0
(-3kp)0 = 1
= 1 00 UNDEFINED!
101–1 =If
then
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36
STANDARDS
100000010
10000010
1000010
100010
10010
Exponential FormStandard Numbers
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
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