M8N1: i.Simplify expressions containing integer exponents

M8D1. c.Use set notation to denote elements of a set

• Give a solution using the roster method:A = { 1, 2, 3, 4, 5, 6, 7 }, B is a subset of A, the elements of B are even. The numbers in A that are even are 2, 4, and 6, so B = {2, 4, 6}.

• What is the intersection of A = { x is odd } and B = { x is between –4 and 6 }, where the elements of the two sets are integers? Since "intersection" means "only things that are in both sets", the intersection will be all the numbers which are both odd and between –4 and 6.
  
  {–3, –1, 1, 3, 5}

• What is the union of A = { x is a natural number between 4 and 8 inclusive }and B = { x is a single-digit negative integer }? Since "union" means "anything that is in either set", the union will be everything from A plus everything in B. Since A = { 4, 5, 6, 7, 8 } and B = { –9, –8, –7, –6, –5, –4, –3, –2, –1 }, then their union is:
  
  { –9, –8, –7, –6, –5, –4, –3, –2, –1, 4, 5, 6, 7, 8 }

M8D1. a.Demonstrate relationships among sets through use of Venn diagrams

A typical Venn diagram uses overlapping circles to represent groups of items or ideas that share common properties. In a venn diagram, all elements of a set are contained within a given circle and elements which are shared between two sets are contained within the overlapping regions of the circles.

Example #1

The intersection of two sets A and B, written as , consists of those elements
that are common to both set A and set B.
For example, if A = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} and B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20},
then = {6, 12, 18}

Example #2

The union of two sets A and B, written as , consists of those elements
that are common to set A or set B or both.                          For example, if A = {cat, dog, hamster, goldfish} and B = {rat, mouse, beaver, hamster},
              then = {cat, dog, hamster, golfish, rat, mouse, beaver}

M8A2. c.Graph the solution of an inequality on a number line

Example #1

Graph of the Point 2

 The following graph represents the inequality x≤2 . The dark line represents all the numbers that satisfy x≤2 . If i pick any number on the dark line and plug it in for x , the inequality will be true.

Example #2

Graph of the Inequality x≥2

An inequality with a " ≠ " sign has a solution set which is all the real numbers except a single point (or a number of single points). Thus, to graph an inequality with a " ≠ " sign, graph the entire line with one point removed. For example, the graph of x≠2 looks like:

M8A5: b.Solve systems of equations algebraically and graphically

X2 + Y2 =100
(y+2)2 +Y2 =100
Y2+4Y+4+Y2=100
2Y2+4Y+4=100
2Y2+4Y-96=0
2(Y2+2Y-48)=0
2(y-6)(y+8)=0

y-6=0                    y+8=0
                or
y=6                        y=-8                    


 

M8A4: c.Graph the solution set of a system of linear inequalities in two variables

A "system" of linear inequalities is a set of linear inequalities that you deal with all at once. Usually you start off with two or three linear inequalities. The technique for solving these systems is fairly simple.

Example #1

Solve the following system:

2x – 3y < 12 x + 5y < 20 x > 0 Just as with solving single linear inequalities, it is usually best to solve as many of the inequalities as possible for "y" on one side. Solving the first two inequalities, I get the rearranged system:       Example #2

The line for the first inequality in the above system,  y > ( 2/3 )x – 4, looks like this:

y > ( ²/₃ )x – 4 y < ( – ¹/₅ )x + 4 x > 0  Elizabeth

M8N1.j.Express and use numbers in scientific notations

Scientific notations is simply a method for expressing, and working with, very large or very small numbers. It is a short hand methods for writing number, and as easy methods for calculations. Number in scientific notation are made of three parts : the coefficient,the base, and the exponent.

Example #1

5.67 x 105

This is scientific notation for the standard number, 567,000. Now look at the number again with the three parts lableled.

Example #2

5.67 x 10

cofficient base exponent

In order for a number to be in correct scientific notation, the following conditions must be true:

1. The coefficent must be greater than _> or equal to 1 and < less than 10.
2. The base must be 10.
3. The exponent must show the number of decimal places that the decimal needs to be moved to changed the number to standard notation. A negative exponent means that the decimal is moved to the left when changing to standard notation.