07.12.13
6:58

I
am reviewing some basic algebra through the Udacity site. Some notes:
the commutative property means that order of any two variables will
not change the outcome. Addition and Multiplication both have and
commutative property. Associative property means that the order of
any three or more variables will not change the outcome, and again
both addition and multiplication but not subtraction or division
share this property. For example

5+2
= 7 and 2+5 = 7, also 2*5 = 10 and 5*2 = 10, this is an example of
the commutative property. 5-2= 3 and 2-5 = -3, 3 is not = to -3 to
subtraction does not have the commutative property. Just like 10/2 =
5 and 2/10 = 0.2 so division also does not share this property.

The
next section is about powers, any thing to the first power is itself
and anything to the zero power is one. So 3

^{1 }is 3 and 3^{0}is 1.
Order
of operations: ( ) and [ ] first, simplify exponents, multiply and
divide from left to right, add and subtract from left to right.

Area
of a square is l

^{2 }or l*l (length^{2}).
Area
of a rectangle is l*w or Length times Width

Area
of a triangle is 1/2 base times height.

07.14.13
3:08

This
section is about fractions. An improper fraction is one in which the
numerator is larger than the denominator. For example 4/3, this can
be simplified to 1 1/3. 1 and 1/3 is a mixed number, which is a
whole number and a fraction.

The
denominator must be the same to add or subtract fractions, this is
done by finding the lowest common denominator (the smallest number
that both denominators go into). For example if I'm going to add 1/3
and 1/2 I have to find a number that both 3 and 2 go into. 12 is one
but it's high and that would make the fraction more difficult so the
lowest common denominator is 6. So to change both fractions I look
first at what I have to multiply 3 by to get 6, that’s 2 so I
multiply both the numerator and denominator by 2 to get 2/6. Going
through the same process for 1/2 give me 3/6 then I just add the
numerators, leaving me with 5/6.

Multiplying
fractions is pretty straight forward just multiply the numerators
together and the denominators together. Dividing fractions is the
same as multiplying by the reciprocal. so if I want multiply 1/3 by
1/2 I get 1/6 (1*1 is 1 and 3*2 is 6) but if I divide 1/3 by 1/2 I
get 2/3. This is because the reciprocal or 1/2 is 2/1 and then I
multiply, so 2*1 is 2 and 1*3 is 3 so the answer is 2/3.

07.15.13
09:36

This
section is on decimals and rounding. Starting with a decimal such as
3.462 and rounding up to the nearest whole number gives us 3,
rounding to the first place gives us 3.5, this is because if a number
to the right of the decimal is less than 5 it is rounded down and if
it is greater than five it is rounded up.

The
second part is about circles, Circumference is the distance around a
circle or the perimeter of a circle. Diameter is the length of a line
that passes through the center and whose end points lie on the
circle. Radius is the distance of a line with one end point on the
center of the circle or origin and the other on the circle.

Pi
equals circumference/diameter. Circumference = 2*pi*r . Area equals
pi*r

^{2}.
07.19.13
07:52

This
part of section 2 is about scientific notation. Scientific notation
takes a number and simplifies is down to a decimal times 10

^{x}. For example 535,000,000 can be written as 5.35 * 10^{8}. The key to remember is that only one number should come before the decimal in scientific notation. Small numbers can also be written in scientific notation. For example .000632 can be written as 6.32 * 10^{-4}, and again remember only one number comes before the decimal.
Section
3 starts out with rates. Rates are things like miles per hour or
diapers per package, they are used to compare two quantities with
different units. So I can go 67 mph which would be the same as saying
I went 67 miles in one hour. 16 diapers per 4 packages is the same as
saying there are 4 diapers in each package or 4 diapers per package.
Both of these are examples of a rate because miles, hours, diapers
and packages are all different units.

A
"Unit Rate" is any rate where the value of the denominator
is equal to one.

A
ratio is used when comparing two quantities with the the same units.
For example 2 out 3 of my girls wear diapers, this is a ratio because
there is only one unit, in this case it is "my girls".
Ratios can also be written with colons so 2/3 of my girls can be
written as 2:3.

07.19.13
04:56

This
part of section 3 is about Conversion Factors, a conversion factor is
a number that relates the quantity of one thing to the quantity of
another thing and it always equals one. For example 12 eggs /1 carton
is a conversion factor because 12 eggs is equal to 1 carton.

The
last part of section three is about converting decimals and percents.
To change a decimal to a percent move the decimal 2 places to the
right, and to change a percent to a decimal move the decimal to
places to the left. So if I want to say that 2 of my 3 children wear
diapers that equals .667 of my children which is 66.7%. If my oldest
daughter finishes 75% of her homework she has finished .75 of it and
so on.

Unit
4 starts out with variables. A variable is a symbol or letter that
can be used to represent an unknown value.

07.23.13

Unit
4 section one is about algebraic expressions, an algebraic expression
has terms that include numbers and variables that are connected by
addition, subtraction, multiplication and division. 3x-4 is an
algebraic expression. To evaluate an expression the value of the
variable is plugged in. For example if x=3 evaluate 3x-4 it would be
3(3)-4 or 9-4 which is 5. 4x+3(x-2) to simplify this I need to
combine like terms, like terms are variables that are raised to the
same power so x and 2x are like terms but not x^2 or 2y. So to
simplify combine the coefficients of like terms (coefficients are
numbers that variables are multiplied by) but leave the variable
alone. Constants are numbers with out variables so in our example
4x+3(x-2), 4 is a coefficient and 2 is a constant. To simplify we
follow the order of operations. So 4x+3x-6 turns into 7x-6, and our
original expression is simplified.

Unit
4 Section two is about linear equations, linear equations have an
equals sign, an expression on both sides of the equals sign and no
variable has an exponent higher than one. So 5x+3=40 is a linear
equation. These can be solved for the variable. There are some that
don't have a real answer and others where x (the variable) can be all
real numbers and still make the equation true. For example
2x-3=-3+2x, simplified down this is all real numbers because no
matter what number is x is substituted for the equation remains true.
Now 2x=2x +2 doesn't have a solution because if we simplify it we get
0=2 which is not true.

The
same principles that apply to linear equations also apply to
inequalities. Such as, if you do something to one side then you have
to do the same thing to the other side. The only difference is that
if you divide or multiply by a negative in an inequality the sign
flips. So -3x<9 simplifies/ solves to x>-3.

Unit
4 Section three is about proportions and similar triangles.
Proportions start with a rate and then can be used to find other data
by cross multiplying and dividing. For example if I work out and lose
3 pounds a week how many days will it take me to lose 23 pounds. So 3
pounds/7 days = 23 pounds/x days. This problem can be made into a
proportion. So now I multiply 23 pounds by 7 days to get 161 and then
divide by 3 pounds, so because I have the variable pounds on the top
and bottom of the equation I can cancel it out and I get 53 2/3 days.

07.25.13

Continuing
with Unit 4 section three, Similar triangles are triangles that have
the same angles but different sides. The angles will all have the
same measure and the sides will be in proportion to each other. This
is important because similar triangles can be used to find heights in
the real world among other things. Lets say that we hold up a yard
stick and it casts a 10 foot long shadow, we can then find the height
of anything else by measuring its shadow because the sun is at the
same angle to the yardstick as it is to every other vertical object.
So lets say that we want to know how tall the tree outside is. We
measure the shadow and it is 60 feet long, so now we can set up a
proportion, 3/10 = x/60. Now we cross multiply and divide, 60 * 3 is
180, divided by 10 is 18. So the tree is 18 feet tall.

Unit
4 Section 4 is about the Pythagorean Theorem which states that in a
right triangle a²
+ b²
= c², where c is the length of the
hypotenuse. The Hypotenuse is the longest side of a triangle
and in a right triangle it is opposite the 90 degree angle.

## No comments:

## Post a Comment