Math 6A (Periods 2 & 4)

Decimals 3-3

Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.

Note that
1/10 = 1/10¹
1/100 = 1/10²
1/1000 = 1/10³

We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.


SO 1/10 = 1/10¹= 0.01 and it is equal to 10^-1
Notice that 10^-1 is NOT a negative number-- it is a small number
and 10^-21 is not a negative number it is a VERY TINY number

AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example

zero and six thousand three hundred ninety-four ten-thousandths.

Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?

Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.

So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."

Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so

zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400

Carefully see the distinction!!

Getting back to our 0.6394

to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004

0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)

0(10⁰) + 6(10^-1)+ 3(10^-2)+ 9(10^-3)+ 4(10^-4)

14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point

Algebra Honors (Periods 5 & 6)

Multiplying Binomials Mental 5-4

Look at
(3x - 4)(2x+5)

remember FOIL
First terms (3x)(2x)
Outer terms (3x)(5)
Inner Terms (-4)(2x)
Last Terms (-4)(5)

6x² + 15x -8x -20
6x² + 7x - 20
Or use the box method as we have done in class
This is a quadratic polynomial
The quadratic term is a term of degree two

Remember a linear term has a term of degree 1 such as y = 3x + 5

6x² + 7x - 20

The 6x² is the quadratic term
the +7x is the linear term
and the - 20 is the constant term

(x +1)(x +3) = x² + 4x + 3

(y + 2)( y + 5) = y² + 7y + 10
( t -2)( t -3) = t² -5t + 6

( u -4)(u -1) = u² - 5u + 4

What about
( u-4)(u +1) = u ² -3u -4

See the difference between the two?

(7 - k)(4 -k)

28 - 11k + k²


r + 3)(5 - 5)
r² - 25 - 15



(3x - 5y)(4x + y)
12x² - 17xy - 5y²

a + 2b)(a-b)

careful....
a² + ab - 2b²

n(n-3)(2n+1)
first distribute the n
(n² -3n)(2n +1)
2n³ - 5n² - 3n

Solve for
(x-4)(x +9) = (x +5)(x -3)
x² + 5x - 36 = x² + 2x -15
5x - 36 = 2x - 15
3x = 21

x = 7
or in solution set notation {7}

Math 6High (Period 3)

Mixed Numbers & Improper  Fractions 2.7 

What are Mixed Numbers?  --> sum of a whole number and a fraction
6 3/5 = 6 + 3/5

improper fraction -->  numerator is greater than or equal to the denominator.
4/3 and 8/8 are both improper

Write 2 2/5 as an improper faction
2 + 2/5  10/5 + 2/5 = 12/5

3 1/4 = 13/4

2 5/6 = 17/6


Write an improper fraction as a mixed number

7/3 =  2 1/3

5/4 = 1 1/4

11/6 = 1  5/6

16/5 = 3  1/5

order the following from least to greatest

4/3,  5/6,   1,  2/3,   1 1/6

first notice two are smaller than 1 and two are larger
Then all you have to do is compare the two smaller than 1
2/3 < 5/6

compare the two that are larger than 1

1 1/6 < 4/3
so
2/3 < 5/6 < 1 < 1 1/6 < 4/3

Refer to your notes on the various ways to compare two fractions.

Then we ordered numbers on a number line.
We graphed 7/5 on a number line  changing 7/5 to 14/10 to make it easier to plot.

Math 6A ( Periods 2 & 4)

The Decimal System 3-2

Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9

Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)

Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?

Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
Think of the comma as a 'signal' for you to mention a place name category!!

In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."

The expanded notation for 420,346 is given by

4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)

Using exponents the expanded notation may be given as
4(10⁵) + 2(10⁴)+0(10³)+3(10²)+4(10¹)+6(10⁰)

What hmmm.. how is (10⁰) = 1
any number to the zero power is 1.


From yesterday we noticed that
10² ⋅ 10³ = 10⁵
and
10⁶ ⋅ 10³ =10⁹
so we wrote a rule for any exponent values a and b
10^a ⋅ 10^b = 10^a+b
Then I asked what would be a rule for

10⁸
10⁵
and
10⁶
10⁴
Students decided that since we added the exponents when multiplying we could subtract the exponents when dividing!!

so could we write a rule for any exponent values a and b?
YES
10^a
10^b
= 10^a-b
but then
10⁵
10⁵
would be
10 ^5-5 which is 10⁰
BUT WAIT... isn't any number divided by itself equal to ONE. That is,
10⁵
10⁵
= 1
So
10⁰ = 1

279,043
is read as two hundred seventy-nine thousand, forty-three.
and in expanded notation becomes:
2(100,000) + 7(10,000) + 9(1000) + 0(100) + 4(10) + 3(1)

or with exponents:
2(10⁵) + 7(10⁴) + 9(10³) + 0(10²) + 4(10¹) + 3(10⁰)
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???

Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.

so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value

10t + 2 becomes the correct expression

What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 in front of the x for the ones' digit? No it is... invisible!!

Some of you asked that I put the Magic Number Trick that we did online... try this one on your parents...

Have your 'victim' follow these directions:
Choose any four-digit number
Write the thousand's digit
Write the thousands' and hundreds' digits
Write the thousands' hundred's and tens' digits.
Add these numbers

Multiply by 9

Find the sum of the digits of the original number
add this sum to the previous results.

Your 'victim's' answers should always be the same as their original number chosen.
Example
I used in class 1492
Write the thousand's digit-------> 1
Write the thousands' and hundreds' digits ---> 14
Write the thousands' hundred's and tens' digits--->149
Add these numbers 1 + 14 + 149 = 164
Multiply by 9 or 164(9)= 1476

Find the sum of the digits of the original number 1 + 4 + 9 + 2 = 16
add this sum to the previous results. 1476 + 16 = 1492

Voila!! It worked!!
Email me your parent's comments or post them here!!

Math 6A (Periods 2 & 4)

Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.

A product in which each factor is the SAME is called a power of that factor.

2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
2⁴ = 16

The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.

When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
4³ = 4 X 4 X 4 = 16 X 4 = 64

The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.

We read 12² as "twelve squared" and to evaluate it
12² = 12 X 12 = 144

Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 10¹ but the exponent is invisible = 10
Second Power: 10² = 10 X 10 = 100
Third Power 10³ = 10 X 10 X 10 = 1000
Fourth Power 10⁴ =10 X 10 X 10 X 10 = 10,000
Fifth Power 10⁵ = 10 X 10 X 10 X 10 X 10 = 100,000

Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:

The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.

The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.

For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000

When you need to multiply other bases:

first multiply each
For example

3⁴ X 2 ³ would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648

What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3 or 3 ⁶
We just add the exponents if the bases are the same!!

Well then, what about (3⁴)² ?
Wait.. look carefully isn't that saying 3⁴ Squared?
That would be (3⁴)(3⁴), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 3⁸.
OR
we could have made each (3⁴) = (3⋅3⋅3⋅3)
so (3⁴)² would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 3⁸
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(3⁴)² = 3⁸

1 to any more is still just 1
1⁵ = 1

0 to any power is still 0!!

Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a³ + b²
would be (3)³ + (5) ²
= 27 + 25 = 52




Check out this great Video on the Powers of Ten
POWERS OF TEN

Algebra Honors ( Periods 5 & 6)

Monomial Factors of Polynomials 5-3

In the first chapters we reflected that if a, b, and c, were real numbers and c was not equal to 0,
then
(a + b)/c = a/c + b/c
It also applies to monomials
(5m + 35)/ 5 = 5m/5 + 35/5... which simplifies to  m + 7

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and then add the results.
From this point on, our textbook has us assume that not divisor equals 0

Divide the following
26uv - 39v
    13v

You can separate each part so that you have
(26uv)/13v  - 39v/13v
which simplifies to
2n -3

Divide

3x⁴ - 9x³y + 6x²y² 
-3x²

This time you notice that -3x² is a factor of all three terms of this polynomial
and you can separate the polynomial into three separate terms or... as taught in class you can easily do each ( carefully)
so that you simplify the fraction to
-x² + 3xy -2y²

Divide:
x³y - 4y + 6x
     xy

Here you might want to show the three terms to see what is happening to each individual term...

x³y 
  xy

-4y
xy

+6x
xy

which simplifies to

x² -4/y + 6/x

You definitely could simplify by crossing out the common factors  each term shares with the divisor.

One polynomial is evenly divisible or just divisible by another polynomial if the quotient is also a polynomial.
So the first two examples show divisibility but this last one does NOT.

You factor a polynomial by expressing it as a product of other polynomials. The factor set for a polynomial have integral coefficients.
You can use division to test for factors!

The greatest monomial factor of a polynomial is the GCF of its terms!

Factor
5x² + 10x
The greatest monomial factor is 5x
You don't want to change the value of your polynomial-- you just want to factor it!
Pull out the 5x and divide by 5x as well.. because you are NOT CHANGING the VALUE
Its simply using the Distributive Property

5x(5x² + 10x)
     5x

5x(x + 2)
To check-- just multiply out using your knowledge of the distributive property!!

Factor
4x - 6x³ + 14x

The greatest monomial factor is 2x

2x(4x - 6x³ + 14x)
   2x
2x(2x - 3x² + 7)

Factor
8a²bc² - 12ab²c²
The greatest monomial factor is 4abc²


4abc²(8a²bc² - 12ab²c²)
  4abc²

4abc²(2a-3b)

Practice these and you will be able too do the division steps mentally.
Check your factorization by multiplying the resulting factors!
Make sure you end up with where you started-- when you check!!