Algebra 11.21.13

This past week I reviewed graphing basics and how to find the distance between two points. Let's start with some graphing basics. Points on a graph are denoted (x,y) where x is the distance a point is from the origin (0,0) on the x or horizontal axis and y is the distance from the origin on the y or vertical axis. A graph has 4 quadrants. Points in quadrant 1 will have both a positive x and y value so (x,y). Points in quadrant 2 will have a negative x and positive y (-x,y). In quadrant 3 both the x and y values are negative (-x,-y) and in quadrant 4 the y value is negative while the x value is positive (x,-y). It is important to note that while a graph will not show all the numbers the x and y axis extend indefinitely in both the negative and positive directions.

We can find the distance between two points using the Pythagorean Theorem a squared plus b squared = c squared where c is the hypotenuse of a right triangle. No matter which two points we look at they all can be looked at as being a part of a right triangle. So if we were going to find the distance between the point (1,2) and (2,4) we need to look at the big picture. The distance between the two x coordinates 1 and 2 is one and the distance between the two y coordinates 2 and 4 is 2. 1 squared is 1 and 2 squared is 4, one plus 4 is 5. Then we take the square root of 5 to get the distance between those two points. So the distance formula is      
                               (x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−−        Distance =      √

Algebra 08.04.13

An equation is two expressions that are set equal to each other. Conditional equations, identities, and not equations. A Conditional equation is such that its truth is dependent on which number x stands for. For example 2x+1=5 is only true when x = 2. An Identity is when both sides are exactly the same so 3n+1=3n+1 because when we simplify we get 0=0. x-3 and 2/x +1= 2/x +2 are not equations simply because they do not equal anything or are false.

Algebra 08.02.13

The last part of lesson two focuses on exponents. When a dealing with several variables multiplied together that have exponents the exponents can be added together to simplify the expression. For example if I have x^2*y^4*x^9*3 (where ^ is used to show there is an exponent) I can simplify to x^11*y^4*3.

Lesson three is about polynomials. The distributive property allows us to take the expression a(b+c) and simplify it to ab+ac. For example say we have the expression 5-(6-x) we can think of the negative sign as a -1 and distribute it to get 5-6+x or -1+x. When multiplying polynomials together make sure to multiply every term in the first polynomial by every term in the second polynomial. For example (x+2)(x+5)= x^2+5x+2x+10 which simplified is x^2+7x+10.

Algebra 08.01.13

Lesson 2 in Udacity's College Algebra course is about expressions. It starts by going back over the commutative property of both addition and multiplication. As a reminder the commutative property just means that the order of terms or factors doesn't matter. So a+b+c is the same thing as b+a+c and a*b*c is the same as c*a*b and so on. When combining like terms it is important to remember that they need to have the same variable and power (x cannot be added to x squared). So 2x+3x+4-2y can be simplified to 5x-2y+4 but no further.

A polynomial is an expression with constants and/or variables that are combined using addition, subtraction and multiplication, where all exponents are non-negative integers. The degree of a term is the sum of exponents on the variables in that term. So x squared times y cubed would have a degree of 5 because the exponents 2 and 3 add up to 5. The degree of a polynomial is equal to the highest degree of any of its terms. the standard form of a polynomial is to put the terms in order from highest degree to lowest degree.

Algebra 07.28.13

Yesterday I finished up the Introductory Algebra Review on Udacity ( I know the certificate says 07.25.13 but that was for getting 80% of the questions right, I finished out the last unit and test on the 27th), I'm starting the College Algebra course today. Lesson one is about numbers, numbers are a construct in that they do not exist in the real world, we use the concept of numbers to do everything from balancing our checkbooks to launching satellites into orbit. Natural numbers are the numbers 1 to infinity, increasing by 1 at each step so 1, 34 and 375 are natural numbers but -3, 3.6, and 0 are not. Whole numbers are all natural numbers and the number 0. Integers are all whole numbers and negative numbers that are not fractions or decimals so they range from -infinity to +infinity. These are all sets of numbers, of course a set can be made up of any combination of numbers. Another way to express the relationship of of one set of numbers to another is to say x is a sub-set of y, for example whole numbers are a subset of integers because all whole numbers are integers. Rational numbers are any number that can be written as a/b where a and b are integers and b is not 0. Irrational numbers cannot be written as ratios of integers, so any square root that is not a perfect square is an irrational number, pi is also an irrational number.

There is also a review of variables, constants, expressions and equations. I am not going to go back over them because that information is in my previous algebra posts.

Algebra 07.27.13

Yesterday we learned a bit about slope and how horizontal lines will always have a slope of zero because if you divide zero by any number it is still zero. This brings us to vertical lines, these lines have a change in x of zero, because we can't divide by zero the slope of any vertical line is undefined. Given two points (x₁, y₁) (x₂,y₂) we can then find the slope using this equation where "m" stands for slope, m=(y₁-y₂)/(x₁-x₂). So how do we find the graph or visual representation of a line from this information? We use the slope of a line and a point on the line, we can also represent a line using slope-intercept form. This uses the slope and the y intercept to represent a line or y=mx+b where again m is slope and b is the value of y at the y intercept. We can also use point-slope form y-y₁=m(x-x₁) where (x₁, y₁) is any point on the line. If we solve for y we can convert point-slope form to slope-intercept form. 

Algebra 07.26.13

Unit 5 Section 1 starts out by introducing t-charts and ordered pairs for displaying data points. A t-chart looks like a t with the two variables (x and y) on the top and numbers on the bottom. This t-chart  shows some data points for the equation 2x = y.







This can also be shown with ordered pairs, for example (0,0) (1,2) (2,4). Both represent the same data and this data can then be plotted on a graph as shown above. An intercept is where the line crosses the x and y axis, in this case both the x and y intercepts are at (0,0). In any linear equation the x and y intercepts can be found by setting x or y to zero. If you set x to 0 and solve the result will be the y intercept, and if you set y to 0 and solve the result will be the x intercept of the line. Not all lines will have an x and y intercept, vertical lines will only cross the x axis and horizontal lines will only cross the y axis. Vertical lines will have equations that look like x=2 while horizontal lines will have equations that look like y=5.

Unit 5 Section 2 is about slope. Slope is how steep a line is and is calculated by rise/run, or the change in y divided by the change in x. Horizontal lines will always have a slope of 0 because the y value does not change and 0 divided by any number is still 0.

Algebra learning journal to date.

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 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.

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*r².

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 * 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.

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.