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 31 is 3 and 30 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 l2 or l*l (length2).
Area of a rectangle is l*w or Length times Width
Area of a triangle is 1/2 base times height.
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.
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*r2.
This part of section 2 is about scientific notation. Scientific notation takes a number and simplifies is down to a decimal times 10x . For example 535,000,000 can be written as 5.35 * 108. 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.
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.
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.
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.