Transcript File

8-19-15
Mini Math Facts Lesson
1
MATH MUST KNOWS:
•Whenever you operate on numbers (add, subtract,
multiply, etc.) the first number is what you have and the
second number is what you will do to it. For example,
even though 2 + 3 = 5 and 3 + 2 = 5, the first one means
you started with 2 and added 3. The second means you
started with 3 and added 2.
•The definition of subtraction is to add the inverse.
Example:
3 – 7 = 3 + (- 7) = - 4
(The first
is what you have and the second is what you do to it!)
•Parentheses are to be treated as a (“one”) number.
Ex: (x + 5)2 = (x + 5) (x + 5).
What is 2 + 3(x – 8) ? 2 + 3x – 24 = 3x – 22
2
MATH MUST KNOWS (cont’d):
•Numbers are generally divided into three sections, those
greater than zero, those less than zero, and at zero.
When considering possible solutions, absolute values, or
domain restrictions, think in those terms.
•To express positive values, we use > 0 To express
negative values, we use < 0 . Anything but zero is: ≠ 0
•Anytime you deal with RATE, you are really working
with SLOPE.
•We might not know an answer, but we can represent it!
•If you know the whole (Ex: 10), but not each part, you
can use x & 10 – x for each part.
3
MATH MUST KNOWS (cont’d):
• When solving, you are “un-doing” the order of
operations. In other words, you add or subtract, then
multiply or divide, then address exponents—the opposite
of the order of operations!
•Important: – a = – 1a In other words, the opposite of a
number is the same thing as the number times negative
one.
•Also remember: ( – a)2 ≠ – a2
2
•The bar in fractions 3 , the division sign bar
, or the
top of a radical sign
is a vinculum! It is a
medieval symbol for grouping, gaining popularity today!

4
Divisibility of Numbers:
•How do we know if a number is divisible by 2?
•It ends in 0, 2, 4, 6, or 8
•How do we know if a number is divisible by 3?
•Add the digits. If the sum is divisible by 3, then the
main number is. Ex: 1,278? Ex: 5,773?
•How do we know if a number is divisible by 4?
•If it ends a multiple of 4 (and we know most of them
from 0 to 96), then it is divisible by 4. Any numbers to
the left do not matter as any multiple of 100 (700, 1,000,
10,000 etc.) is already divisible by 4. Ex: 781,348?
5
Divisibility of Numbers (cont’d):
•How do we know if a number is divisible by 5?
•It ends in 0 or 5
•How do we know if a number is divisible by 6?
•If it is even and divisible by 3.
•How do we know if a number is divisible by 9?
•Add the digits. If the sum is divisible by 9, then the
main number is. Ex: 2,187? Ex: 6,884?
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Classification of Numbers:
Natural No.’s:
1,2,3,…
Whole No.’s:
0,1,2,3,…
Integers:
…-3,-2,-1,0,1,2,3,…
Rational:
a ,b≠0
b
Real:
Rational + Irrational
Complex: a + bi
Real + Imaginary
7
Irrational Numbers:
•Irrational number are numbers that cannot be
expressed as a ratio or rational number.
•It goes on infinitely and has no exact repeating
pattern. Some examples are pi and square roots.
Number
Rational
3

5.9

0.353535… = 0.35

9.545445444…
Irrational

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Irrational Numbers (cont’d):
How is 0.353535… a rational number? It is also 0.35
So, if we have:
Then:
100 N =
35.35
1N=
0.35
99 N =
35
99
99
N=
35
99
Wow! No more repeating numbers!
Now, check it in your calculator—it’s 0.353535…!
9
Irrational Numbers (cont’d):
9? Fascinating number!
1
9
= 0.111… = 0.1
2
= 0.2
9
3
= 0.3
9
So , what would 8 = 0.8
9
Then , what would 9 = 0.9 or 1?
9
How is that possible? Mathematical Paradox!
10
Exact Values:
The nature of an EXACT VALUE is one that can be
expressed as an integer, a rational number, or a
decimal value that is finite or has an infinite repeating
pattern (think 1/3). For others, there are exact value
representations such as pi and square roots. Can all
irrational numbers be shown as an exact value? No,
only those that have another representation.
8
Examples: 5,
, 6.7,  , 3 , 2 7, etc.
3
An example that is not: 2.5894217598557432…
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Radical Information:
•Powers and roots are just specialized versions of
multiplication and division. 23 is really the
multiplication of 2 times itself 3 times. Roots are really
just division problems where the number that goes into
it is the number that comes out. For instance the square
root of 9 is 3 because if we divide by 3, the answer is
also 3.
3
6
3
Square
Area is
9 units
6
Square
Area is
6 units
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Radical Information continued:
Since  2  times  3 has the same
result as  2  3 , numbers under the
radical can multiply (and divide)
across radicals.
Remember, what happens in Vegas
stays in Vegas, and what happens
under the radical stay under the
radical!
13
Radical Rules for Reals:
1. No fractions under the radical.
2. No radicals under the fraction.
3. No perfect powers under the radical.
4. No negative values under the even radical.
How do you see the solution to the following problem?
If I have 3  3 ,
then,do you see it as or as?


9  3
 3
2
 3
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Estimating Radicals:
43 ?
Can you estimate the value of
6
7
43
36
Try
49
18 :
4
16
Try
5
18
25
77 :
8
64
9
77
81
15
Building Squares:
Squares:
Geometry:
Difference:
1*1 = 1
1
2*2 = 4
3
3*3 = 9
5
4*4 = 16
7
5*5 = 25
9
What is the pattern? What can you do to see if it
is true?
6*6 = 36
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