Math Notes

JANUARY 2017
HAPPY NEW YEAR!
AND WELCOME TO CHAPTER 4:  LINES AND LINEAR EQUATIONS.

Link to section 4.1 notes below:

4.1.pdf

Section 4.2 quiz coming up.   Link to notes below:

4.2.pdf






December, 2016
CHAPTER 3 NOTES:   CHAPTER 3 TEST WILL BE ON TUESDAY, DECEMBER 20TH.   BELOW ARE SOME HELPFUL NOTES AND SOME REVIEW PROBLEMS.   STUDY!!!!!

solving equations 1.pdf.

solving word problems.pdf

types of equations.pdf

chapter 3 test review.pdf


November 8, 2016

Chapter 2 Test tomorrow.  Test Prep Practice Attached.  Good Luck!

chapter 2 test prep.pdf

November 1, 2016

Chapter 2 Test coming up next Wednesday, Nov 9th.  

SCIENTIFIC NOTATION!
Copies of more specific notes are available in Room 310.

Big Ideas:
**Scientific Notation has 3 parts....
---a coefficient (number equal to or greater than 1, but less than 10)
---multplied by a base of 10
---with an exponent that is an integer
**A positive exponent makes the number BIGGER...moving the decimal that many places to the RIGHT
**A negative exponent makes the number SMALLER...moving the decimal that many places to the LEFT
**When comparing numbers in scientific notation...first compare the exponents; if same, then compare the bases
**When adding or subtracting numbers in scientific notation...make sure the powers of 10 are the same in both, then add/subtract the bases
**When multiplying/diving numbers in scientific notation....multiply/divide the coefficients;  multiply/divide the powers of 10;  multiply/divide the units;  make sure the answer is in scientific notation.




 October 12, 2016
The Chapter One TEST is coming up within the next week.  Study Guide attached here:

CH 1 math.docx
,

October 4, 2016

Chapter 1 Section 5 quiz coming this week.  Test on Chapter 1 at the end of next week.   Here is the study guide so far!

CH 1 math.docx

September 27, 2016

Section 1.5 ZERO AND NEGATIVE EXPONENTS

Anything to the ZERO power is equal to 1!!


For negative exponents..........   make the number a fraction, with one in the numerator, and the base to the POSITIVE power as a denominator

EXAMPLE:     8-2 =  1/8=  1/84
3-3 = 1/33 = 1/27
                     3m-2 = 3/m2
                     (3m)-2 = 1/(3m)2   

September 26,  2016

Section 1.4:  The Power of a Product and the Power of a Quotient

If you have different bases, but the same exponent.......you can multiply or divide the bases and keep the same exponent.

Example:  

6* 73  =  (6 * 7)3  = 423
 
18 3  /   63  = (18/6)3  = 33

And the opposite is also true:  

423   =  ( 6 * 7)3  =  6​* 7​3




September 21, 2016

Quiz on Section 1.3 coming up soon.  Here is a study guide for all of Chapter 1 so far!

MATH CH1.docx


September 19, 2016
Study guide for Section 1.2

Section1.2.docx


September 15, 2016

CALCULATING SIMPLE INTEREST AND COMPOUND INTEREST

Simple Interest = (p)(r)(t)    
p=principal....amount you started with
r= interest rate.  You must change it to a DECIMAL by dividing by 100
Example:  5% interest =  0.05
t=time

Example of  simple interest:

Shana deposits $100 in a bank account that earns 5% simple interest.   How much will be in her account at the end of 5 years?

(100)(0.05)(5) =  $25 interest

Compound Interest    A= p (1 + r )n
A = amount at the end
p = principal....amount you started with
r = interest rate  You must change it to a DECIMAL by dividing by 100
Example:  5% interest =  0.05
n = time

Example of compound interest:

Shana deposits $100 in a bank account that earns 5% interest compounded yearly.  How much will be in her account at the end of 5 years?

Formula      A (amount at end) = p ( 1 + rate as a decimal)​number of years  

A = 100( 1 + .05) ​5 = $127.63 (she made $27.63 in interest)


TRY THESE TWO FOR PRACTICE:

1.  Jason invests $2500 in an account that earns 3.5% interest compounded annually.  How much money will be in his account in 20 years?

2.  Kelly has $4000 to inivest for 10 years.  She has two choices.   Her bank will pay her 2% interest compounded annually   OR    12% simple interest.  How should Kelly invest her money?

ANSWERS:
1.  $4974.47
2.  simple interest gets her $4800;  compound interest gets her $4875.97.  She should go for the compound interest!


REMEMBER:   When you are dealing with money......put in the dollar sign  AND round to two decimal places!



September 13, 2016

Notes for Section 1.2

THE PRODUCT AND QUOTIENT OF POWERS

Example 1:  10* 109 =

10*10*10*10*10*10         *       10*10*10*10*10*10*10*10*10  =  1015

10* 109 =

10 6+9  = 1015

Example 2:      a4  * a
a4+3 =    a7



PRODUCT OF POWERS PROPERTY:

am * an = am+n

When the bases are the same, keep the base, add the exponents.

Showing work:
Write problem        x23 * x18 =
Write intermediate step     x23+18 =
Write answer        x41

QUOTIENT OF POWERS PROPERTY:

When the bases are the same, keep the same base and subtract the exponents!

55  divided by 52 = 55-2 = 53


Make sure to use parentheses around the bases that need it, like negative numbers and fractions!

When in doubt…..expand it out!!!!

EXAMPLE:

(82a5b3)
(3a4b10)
       Simplify the numerator and denominator separately, by                                         multiplying numbers

(4a3 b4)(2a3b)              (coefficients)  then like variable terms

8*3*a5*a4*b3*b10    =    24 a9 b13        Then divide the like terms   and get   2 a3 b 8

4*3*a3a3*b4*b        =    12 a6 b 5

Remember…..if you don’t see an exponent, there is that invisible 1  !!!





September 8, 2016
Math classes will be having a quiz on Chapter 1, Section 1 at the end of this week or the beginning of next week.
Below are notes to study for this section!

Section1.1prob.docx


CHAPTER 1, SECTION 1
EXPONENTIAL NOTATION

Objectives:
*Understand and use exponential notation
*Use exponents to write prime factorization of a number

VOCABULARY
 
* Exponents....show repeated multiplication
*Exponential Notation.....write using exponents, such as 23
* Base.....the number that is multiplied by itself
Examples:
43.....4 is the base
(-4)5......-4 is the base
(2x)6.....2x is the base
-43.....4 is the base
*Exponent....shows how many times the number is multiplied by itself
Examples:
43.....3 is the exponent
(-4)5......5 is the exponent
(2x)6.....6 is the exponent
-43.....3 is the exponent
*Power...the expression that has a base and an exponent
*Expanded Notation....written out in the larger format, Ex:
23= 2*2*2

23= 2*2*2=8    It DOES NOT equal 2*3 =6

To EVALUATE....means to give the answer, solve the problem

Divisibility Rules
*A number is divisible by 2 if......it ends in a 0, 2, 4, 6 or 8
*A number is divisible by 3 if .......the digits add up to a number divisible by 3 
*A number is divisible by 5 if........it ends in a 0 or 5

Prime Factorization
Prime numbers......have exactly two factors.
Examples:  2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Composite numbers.....have more than two factors  (anything that is not
 prime)

500 = 5*5*5*2*2=  22 * 53

Doing exponents on the calculator

Base-----Carat-----Exponent----Equals



***************************************************************************
***************************************************************************
***************************************************************************

END OF NOTES FOR 2015-2016 SCHOOL YEAR
May 25, 2016

CHAPTER 10 TEST ON FRIDAY!!!!!!
CHAPTER REVIEW NOTES ATTACHED!!!!!

UNIT 10Review.docx

May 11, 2016
Chapter 10 TEST on Friday, May 27th
There is a QUIZ coming up on sections 10.1 and 10.2.  Study guide attached here!

UNIT 10.docx


May 2, 2016
Welcome Back!
Chapter 9 Test on Thursday.   Study Guide attached!

Ch9StudyGuide.PDF


April 14, 2016

Quiz tomorrow on Section 9.1

9.1  Understanding and Applying Congruent Figures 

Figures with the same SIZE and SHAPE are congruent.

Corresponding means matching.

If two triangles are congruent, you can find 3 sets of congruent angles, and 3 sets of congruent sides.

If two quadrilaterals (4-sided figure) are congruent, there would be 4 sets of congruent angles and 4 sets of congruent sides


Statement of Congruence:   

  Δ ABC is congruent to ΔDEF  (remember your symbol for congruence (equal sign with a squiggle on top)

In a statement of congruence, the points have to be in correponding order.  In the above example....A corresponds to D; B corresponds to E and C corresponds to F

List your original shape in alphabetical order, call it ABC, rather than CBA

Correponding sides are marked with tick marks; corresponding angles are marked with arcs

All angles in a triangle add up to 180 degrees.

METHODS TO PROVE TRIANGLES ARE CONGRUENT:

SSS ---side side side....if the 3 sides of a triangle are congruent to 3 sides of another triangle, the triangles are congruent

SAS--if two sides and the angle in between them in one triangle, are 
congruent to two sides and the angle in between then are congruent, then the two triangles are congruent

ASA or AAS--if two angles and one side of a triangle are congruent to two angles and one side of another triangle, the triangles are congruent

METHODS THAT DO NOT WORK:

AAA---angle angle angle
SSA/'donkey theorem"---two sides and an angle



March 30, 2016

Chapter 8 Test will be next Tuesday (April 5)
Study Guide attached!

CHAPTER 8 StudyGuide.docx

Part 2 of study guide!

CH 8 STUDY GUIDE 2.docx



March 7, 2016

There will be Chapter 7 test this week.  Depending on the class and how things are progressing for students, the test will be either Thursday or Friday.
I am attaching two files.  One is Notes for the entire chapter, divided into sections.  The second is a set of Extra Practice Problems that students can do to prepare them for the test.  Feel free to download and print!  Good luck!

Chapter7NOTES.PDF
Chapter7EXTRAPRACTICE.PDF


February 18, 2016

Quiz tomorrow on Chapter 7.1 (all three notes attached as PythNotes 1 and 2, and 3 below!)


PythNotes3.PDF


February 17, 2016

Quiz on this first section of the Pythagorean Theorem this Friday.  More notes attached here.

PythNotes2.PDF

February 15, 2016
CHAPTER 7:  THE PYTHAGOREAN THEOREM
7.1  Understanding the Pythagorean Theorem and Plane Figures
(please note.....we have moved on to Course 3...Volume B!)

Notes and Examples for the Pythagorean Theorem are attached in the file below!

PythNotes1.PDF

Please note:  For homework......you must write out work in a specific order for full credit!

Formula.....Substitute......Solve.
You must write the formula, a+ b = c2    each time.

Example work format:

a2  + b2  =  c2

32  + 42  =  c2

9 + 16 = c2

25  =  c2

5 = c

If there are units given in the problem (cm, ft, m....) don't forget to include them in your answer!

February 11, 2016

Chapter 6 Test tomorrow.
Study guide attached.  At the end of the notes is a practice test.  Most students did it in class or were given it to do for homework.  I have attached an extra blank copy as a way to study.  

Chapter6REVIEW.pdf


CHAPTER 6 TEST ON FRIDAY
February 10, 2016

KEY CONCEPTS FOR CHAPTER 6:


**A RELATION pairs a set of inputs with a set of outputs
**The FOUR TYPES OF RELATIONS are:  one-to-one, one-to-many, many-to-one and many-to-many
**A FUNCTION is a type of relation that assigns one output to each input.  All functions are either one-to-one or many-to-one relations
**Functions can be LINEAR (make a line), NON-LINEAR (not a line, usually a curve), INCREASING (getting bigger or going up, having a positive slope) or DECREASING (getting smaller or going down, having a negative slope)
**A LINEAR FUNCTION can be represented as an equation in the form        Y = MX + B, and has a CONSTANT rate of change (slope)
**Functions can be represented in different ways....algebraically (an EQUATION), graphically (on a GRAPH) numerically in tables (CHART) or by verbal descriptions (WITH WORDS)



February 9, 2016

6.4  COMPARING TWO FUNCTIONS

REMEMBER:
If a function can be represented by Y = MX + B  it is a linear function

To compare functions, you need to graph or chart each one separately, then be able to compare the rate of change or the slope, to answer the question posed....such as...which one is faster?; which one sold more?  which one is less expensive?; etc

February 8, 2016

6.3  LINEAR AND NON-LINEAR FUNCTIONS

Remember:  Rate of change = Slope (m)
Starting Point = Y- intercept (b)

A linear function makes a line when graphed; has a CONSTANT slope
A non-linear function does not make a line; usually makes a curve...the rate of change (slope) VARIES

If an equation is in slope-intercept form ( Y = MX + B) it is linear.

Questions you may be asked:

1.  Is it a function?  You can tell by doing the VERTICAL LINE TEST
2. Is it linear or non-linear?   Line = linear.  Constant rate of change= linear.  If not.....non-linear
3.  Is in increasing or decreasing?  Look from left to right (like you are reading)....follow the line...is it going up (increasing) or down (decreasing)

Word problems......Look for:
Rate of change (m)
Starting point, initial amount (b)

See notes attached here:
6.3wordproblems.pdf
February 2, 2016
6.2 REPRESENTING FUNCTIONS

There are four ways to look at a function.
--mapping
--graphing
--writing an equation
--making a chart

Graphing, charting and writing an equation are shown with examples on the worksheet we did in class.  Copy (with my notes) attached:

representingfunctions.pdf 

Writing an equation.

Remember the formula.....Y = MX + B

M = slope = rate (per)
B = y-intercept = starting point, initial cost, beginning

X is always the input, the horizontal axis
Y is always the output, the vertical axis

EXAMPLE:
Janice plans to enroll in a Spanish class.  She has to pay a registration fee of $100, plus $20 per each hour-long class she attends.  

Write the equation for this problem. 

Start with Y = MX + B

M = slope, which equals the rate.  In this case, the rate is $20 per class
B = y-intercept, which equals the starting point, or initial cost.  In this case, the initial cost is the registration fee of $100

So, the equation is:   Y = 20 X  +  100

You could make a chart of this information:

# Hours0123
Cost (dollars)100120140160

You could then graph this on a coordinate graph, putting the number of hours on the X-axis and the amount of money on the Y-axis

You could also make a mapping diagram where the first oval is the # of hours (input) and the second oval is the amount of money (output) and draw arrows to make a one-to-one relationship


January 28, 2016

CHAPTER 6:  FUNCTIONS
6.1:  UNDERSTANDING RELATIONS AND FUNCTIONS

VOCABULARY:
FUNCTION--a type of relation that assigns exactly one output to each input
RELATION-- pairs a set of inputs with a set of outputs
INPUT--the independent variable...the one YOU have control of
OUTPUT--the dependent variable...the result
INDEPENDENT VARIABLE--input
DEPENDENT VARIABLE--output
MAPPING DIAGRAM---  a diagram that pairs a set of inputs with a set of outputs.  Four types:  one to one, one to many, many to one, many to many.  For examples of each of these, open the link below

Mapping Diagram.docx

VERTICAL LINE TEST--a test to determine whether a graph is a function.  If a vertical line intersects a graph at more than one point, then the graph DOES NOT represent a function


REMEMBER:  Independent Variable=Input=I control   all start with I !!!

A function is a special relation in which each input relates to exactly one output.
All functions are either 0NE-TO-ONE or MANY-TO-ONE

Linear Function:  When you graph it....it creates a line


January 25, 2016

CHAPTER 5 TEST THIS WEEK.  STUDY GUIDE ATTACHED...........

CHAPTER 5 SG.docx

January 22, 2016

CHAPTER 5 TEST NEXT WEDNESDAY OR THURSDAY!  STUDY NOTES FOR 5.1, 5.2. 5.3. 5.4 AND 5.5!!

5.5  Inconsistent and Dependent Systems of Linear Equations

Lines that have the same slope are parallel.

Lines that are parallel NEVER intersect.
 
There are NO SOLUTIONS when you try to solve a system of parallel lines.


January 21, 2016

5.4  Solving a System of Equations by Graphing

FIRST...... A REVIEW OF GRAPHING EQUATIONS!


Given the slope-intercept formula, y = mx + b  graph the equation

Step 1:  Graph the y -intercept  ( the +b part of the equation)

Step 2:  Find the slope (m), and use slope to find another point.  It's good to find at least 3 points to make sure you are counting correctly.  Remember, slope = RISE/RUN


Step 3.  Use a ruler and draw a line through the points, putting arrowheads on the ends

EXAMPLE:   y = x + 3    I looked at the equation.  The y intercept is 3, so I placed a dot on the 3 on the y-axis.  Then I looked at the slope....no number before the x (where the slope is supposed to be) so it must be 1, or 1/1.  So, from my 3 on the y, I went up 1 (rise) and right 1 (run) and put a new point.  I did a few more to be sure, then drew my line, remembering to add arrowheads on the ends.




Now do this same thing with the system of equations.

Step 1:  Graph the first equation
Step 2:  Graph the second equation on the same grid
Step 3:  Identify the point of intersection as x and y coordinates
Step 4:  Plug your x and y into both equations to see if they are true.

EXAMPLE:
I graphed these 2 equations in the same grid (sorry, no picture....you will just have to take my word for it)

Y  =  1/4 X  -  1
Y  =  5/4 X  + 3

The two lines met at point (  -4 , 2 ).  That means my X is -4 and my Y is 2. 

Plug into first equation:

Y  =  1/4 X  - 1
2  =  1/4 (-4)  - 1
2  =  -1   -  1
2  =  2

Plug into second equation

Y  =  5/4 X  + 3
2  =  5/4 (-4)  + 3
2  =  -5  +  3
2  =  2


January 15, 2016

5.3  Solving Real World Problems Using Systems of Linear Equations

Example:
The school that Abhasra goes to is selling tickets to the annual dance competition.  On the first day of ticket sales, the school sold 3 senior tickets and 8 child tickets for a total of $74.  The school took in $116 on the second day by selling 6 senior tickets and 8 child tickets.  What is the price of one senior ticket and one child ticket?

STEP 1:
Assign variables.
 Determine what you need to know and assign a letter to stand for each.  You can usually find this in the question at the end.  In the example, you are asked to find the cost of a senior ticket and the cost of a child ticket.  So, I decided to use X and Y as my variables and said:
 Let X = the price of one senior ticket
 Let Y = the price of one child ticket

STEP 2:
Write one equation that uses both variables
.  Find the info in the problem.  For my equation, I will use the second sentence:   On the first day of ticket sales, the school sold 3 senior tickets and 8 child tickets for a total of $74. 

So that means   3X  +  8 Y  =  74

STEP 3:
Write a second (different) equation.  For this equation I will use the third sentence in the problem:  The school took in $116 on the second day by selling 6 senior tickets and 8 child tickets.

So that means      6X + 8Y = 116

STEP 4:
Solve the system of equations using substitution or elimination. 

3X   +  8 Y   =   74
6X   +   8Y   =   116
,
To eliminate, I must either subtract or multiply the second equation by (-1).

My result:       -3X   =   -42

X  =  14

Substitute  14 for X back into the first equation:
3 (14 ) + 8Y  =  74
42 +  8Y  =  74
8Y  =  32
Y = 4

So.....A senior ticket costs $14 and a child ticket costs $4.


January 12, 2016

Study guide for tomorrow's quiz attached!

Elim and Sub.docx

January 11, 2016

SUBSTITUTION METHOD:

Remember:  you can only substitute things of equal value!  

Use the substitution method to solve for a system of equations:

Example:  

y  =  x + 4
y  =  3x +8

STEP 1:
Make a new equation by substituting the value of y from the first equation into the second equation:
x   +  4  =   3x  +  8
STEP 2:
You now have an equation with variables on both sides.   Move the variable term from one side to the other

x  +  4   =   3x  +  8
-x                -x
______________________________________
                                4   =  2x  +  8

STEP 3:
You now have a 2-step equation;  solve for x.  First remove the constant (the term that is added or subtracted to the side with the variable by doing the opposite)

4  =  2x  +  8
                        -8               -8
                    _________________

                       -4       =    2x

Then remove the coefficient (the number that is multiplied or dividing the variable) by doing the opposite

-4   =   2x
                         2         2

                -2  =  x

STEP 4:  Now you need to find y.  Pick one equation and plug in the value you found for x
y  =   x   +  4
y =     -2  + 4
                        y  =  2

STEP 5:  Check your work by plugging x  and  y into the other equation

y  =  3x  +  8
2  =  3 ( -2 )  +  8
2  =   -6  +  8
2  =  2    CORRECT!

January 6, 2016

More on Elimination Method:

If, when you line up the equations, you cannot automatically add to eliminate one term.....you must do something to one of the equations (multiply it by an integer) to make it so that you can.


Example:
Equation 1:   x + y = 8
Equation 2:   x + 2y = 10

Step 1:  I am going to change the second equation by multiplying it by -1

x    +     y       =    8
-x    +   -2y     =    -10
________________________________________________  

Step 2:  Add:
      -1y       =     -2

Step 3:    Solve for y
y = 2

Step 4:  Find the other variable by substituting in  the one you know into one of the equations
x  +   y   =   8
x   +   2   =   8
x = 6

Step 5:  Check your work by substituting both values you found into the other equation:
x   +   2y        =   10
6   +   2 (2)    =    10
6    +    4        =   10
10          =    10
  

January 5, 2016

5.2  SOLVING SYSTEMS OF LINEAR EQUATIONS USING ALGEBRAIC METHODS

Elimination Method:
n
Given two equations, how can you find the point of intersection?

Example:
-9x - y = 15
5x + y = -7

FIRST:  Line them up so that like terms are in the same row:

-9x          -y          =          15
5x           +y         =           -7

SECOND:   Look to see which terms you could add or subtract to get zero.  In the example above, you can add -y  and +y to cancel each other out. Do that, along with adding the other terms:

-9x          -y            =            15
        +5x           +y           =             -7
                      ____________________________   
-4x          +0           =             8
 
THIRD:   This turns into an equation you can solve!

-4x = 8
x = - 2

FOURTH:   Pick one of your equations and substitute the value you found for x:

5x + y = -7
5 (-2) + y = -7
-10  + y = -7
y = 3

LAST:  Write out your answer as:

x = -2      y =  -3             OR

                                           ( -2 , 3 )








January 4, 2016

Chapter 5:  SYSTEMS OF LINEAR EQUATIONS

5.1:  Introduction:

System of Linear Equations:  a set (2 or more) of linear equations that has one or more variables

Unique Solution:  the single pair of variables that satisfies BOTH

The solution to a system of equations is a point on BOTH lines; the point of intersection 

Example:  
Two equations: 
   2x + y = -5                            7x - y = 14

Choices:   a.  (1, -8)    b. (1, 2)    c.  (1, -7)     d.  (-8, -6)

Substitute your x and y in BOTH equations....only one set will make BOTH equations true.


The answer is c.......     (1 , -7)
First Equation:
2x + y = -5
2 (1) + -7 = -5
2 + -7 =n -5
-5 = -5

Second Equation
7x - y = 14
7 (1) - (-7) = 14
7 - (-7) = 14
7 + 7 = 14



 December 17, 2015

Study Guide for Unit 4 Test attached via link below.  Test is Tuesday, 12/22!

UNIT 4.docx


December 15, 2015
4.4  SKETCHING GRAPHS OF LINEAR EQUATIONS

Two methods:

Method #1:

Given the slope-intercept formula, y = mx + b  graph the equation

Step 1:  Graph the y -intercept  ( the +b part of the equation)

Step 2:  Find the slope (m), and use slope to find another point.  It's good to find at least 3 points to make sure you are counting correctly.  Remember, slope = RISE/RUN


Step 3.  Use a ruler and draw a line through the points, putting arrowheads on the ends

EXAMPLE:   y = x + 3    I looked at the equation.  The y intercept is 3, so I placed a dot on the 3 on the y-axis.  Then I looked at the slope....no number before the x (where the slope is supposed to be) so it must be 1, or 1/1.  So, from my 3 on the y, I went up 1 (rise) and right 1 (run) and put a new point.  I did a few more to be sure, then drew my line, remembering to add arrowheads on the ends.



Method #2

Given the slope (m) and a point on the line (x,y), graph the equation

Step 1:  Plot the point given.  Use this as your starting point.

Step 2:  Find the slope (m), and use slope to find another point.  It's good to find at least 3 points to make sure you are counting correctly.  Remember, slope = RISE/RUN

Step 3.  Use a ruler and draw a line through the points, putting arrowheads on the ends

EXAMPLE:  Given the slope is equal to 2, and the point is (1,0) graph the line.   I would first put a dot at (1,0).  Using that as my starting point, I would go up 2, then over 1 to find an additional point.  I would do that a couple more times, then draw my line.

SOME EXCEPTIONS!

When the slope is zero, the equation looks like this:   y = 2, or y = 4, or, in the graph below, y = -1 and the line is horizontal.



When the slope is undefined, the equation looks like this   x = 8 or x = -5   or in the graph below x = 3  and the line is vertical.




December 9 and 10, 2015
4.3 WRITING LINEAR EQUATIONS IN SLOPE-INTERCEPT FORM 
( y = mx + b )Four Main Parts:
A.  The slope (m) is always with the x....it is the coefficient of the x-term
The y-intercept (b) is always a constant (no variable attached)
When there is no x-term, it means that the slope is 0..... (0 times any x-term would eliminate the term)
When there is no constant, the y-intercept is 0....(to add 0 as a constant would not change the equation)

Examples:
y = -2x + 4              m = -2     b = 4
y = -x + 3                m = -1     b = 3
y = 1/3x - 2m = 1/3   b = -2
y = 4m = 0    b = 4
y = 2/5 xm = 2/5   b = 0

B.  Given the slope (m)  and the y-intercept (b), write the equation.

Example:  m = 1/4   b= 3  Equation:  y = 1/4x + 3
m = -2    b = -5   Equation:  y = -2x + -5   or  y= -2x - 5
m = 0   b = 3   Equation:  y = 3
m = 2/3   b = 0   Equation  y = 2/3x

C.  Parallel Lines:
Slopes are the same
y-intercepts are different

Example:  Find the equation of the following line:  the y-intercept is 6.  The line is parallel to y = 4x -7  (Find the slope---4---and keep it the same.  They give you the y-intercept---6....all you need to write a linear equation is the slope (m) and the y-intercept (b)

Answer y = 4x + 6

D.  Write an equation ( y = mx + b ) if you know the slope (m) and a point (x,y)

EXAMPLE:  

A line has a slope of -5 and passes through the point (1, -8).  Write the equation for that line.

STEP 1:   Find the slope  (m)

     m = -5  (it told you so in the problem)

STEP 2:  Find and label the x and the y

x = 1      y = -8   (The problem gave you the ordered pair, and all you had to do was remember that the first one was the x and the second one was the y)

STEP 3:  Substitute m, x, and y into y=mx + b
         y               m       x            we still don't know what b is
- 8     =      -5      (1)     +      b

STEP 4:  Solve for b
Simplify first.....

- 8  =  -5  +  b            Now it's just a one-step equation
+5      +5

-3   =    b

STEP 5:  Substitute m and b into    y = mx + b

y =  -5x   +  -3                 or   y = -5x  -  3


December 7, 2015   CHAPTER 4 TEST ON FRIDAY 12/18!!!

4.2  UNDERSTANDING SLOPE INTERCEPT FORM

Definitions:

Y-intercept:  the y-coordinate of the point that intersects the y-axis  (where the line crosses the Y)
X-intercept:  the x-coordinate of the point that intersects the x-axis  (where the line crosses the X)
Slope-Intercept Form:  
y = mx + b

m = slope
b = y-intercept

EXAMPLE:




Using points on the graph above (-2,1) and (0, 3)  I can see that the rise is 2 and the run is 2, making the slope 2/2 or 1.
 So m = 1

I can see that the line crosses the Y-axis at the 3, so the y-intercept is 3
 So b =3

Substituting those into the formula   y = mx + b

the equation of the line becomes:

y = 1 x  +  3

I can simplify that to 
 
y = x + 3



December 1, 2015
FINDING THE SLOPE FROM TWO ORDERED PAIRS:

SLOPE =    change in Y
                    change in X

Example:
   x   y              x    y
( 8 , 4 )         ( 5 , 3 )

change in Y  = 4 - 3   = 1
change in X =   8 - 5  =  3

slope  =  1/3


Example:  

    x     y             x   y
( -8 , 12 )        ( 6 , 8 )

change in Y = 12 - 8  =   4
change in X =  -8 - 6  =  - 14

slope  = 4/-14, which reduces to - 2/7

Slope remains in fraction form.  Don't make into a decimal or a mixed number.  It CAN be a whole number.

Lines with a POSITIVE slope.....go UP like climbing a mountain (look at it from left to right)
Lines with a NEGATIVE slope....go DOWN like skiing down the mountain (look at it from left to right)
Lines that are HORIZONTAL.....have a slope of 0
Lines that are VERTICAL....have a slope that is UNDEFINED.

Come get a picture of slope mountain if you want a reminder of what this looks like.

November 30, 3015
UNIT 4:  LINES AND LINEAR EQUATIONS
4.1:  FINDING AND INTERPRETING SLOPES OF LINES

Coordinate Graphing:  
Ordered Pairs (x, y)
x = horizontal axis
y = vertical axis

Origin = middle of the graph, point (0,0)



The four quadrants of a coordinate plane.

To graph a point, go over on X first, then up/down on Y
Example:  Point (2,3).....go to 2 on the X axis, then up 3!

Vocabulary:

Slope:  How steep a line is.  calculated as a ratio:  
  vertical change
horizontal change

or

change in y
change in x

or

rise
run

Rise:  The change  vertically (up and down)....like....you RISE from a chair

Run:  The change horizontally  (side to side)....like....you RUN across the room.

Unit Rate:  a comparison of two things, contains the word "per",  the rate for ONE, such as miles per gallon, dollars per hour, heartbeats per minute.....  

Constant of Proportionality:  abbreviated as "k", means the same as unit rate and the same as slope




November 16, 2015

Skill #9

Step 1: Simplify by combining like terms on the same side of the equation
Step 2:  Eliminate the variable term from one side (it is easier to eliminate the smaller term)
it then becomes a two step equation.....same directions as below
Step 3:  Eliminate the constant 
Step 4:  Eliminate the coefficient

November 12, 2015

 STUDY GUIDE FOR QUIZ  (SKILLS #7 AND #8)

Ms Calandriello's Tips for solving equations:

EXAMPLE:  -5(6-2b) + 4 = -66

1) Are there parentheses???  Use the distributive property to distribute the number outside throughout the parentheses
               -5*6 =-30     -5*-2b+= 10b
2) Rewrite the equations
-30 + 10b + 4 = -66
3) Combine any like terms.
-30 + 4 = -264) Rewrite the equation
10b - 26 = -66 

5) Then......where is the variable?  Which side?  Point to the variable!
6) What is in the way of the variable being by itself on that side?  
7) Remove the CONSTANT first (the number that is added or subtracted) by doing the OPPOSITE operation
6) Do the same thing to the other side of the equation.
10b -26 = - 66+26 +26
7) Rewrite new equation
10b = -40
8)  Remove the coefficient (do the opposite operation...multiplication or division, or multiply by the coefficient.
Either divide both sides by 10, or....if you have Ms. Mulqueen....multiply by the recipricol 1/10
9) Do the same thing for the other side of the equation
10 b = -40 OR (1/10) (10/1) = (-40/1) (1/-40
 ---- -----10           10

10) Rewrite the new equation.
11) Solve for the variable.

                       b  = -4
12)  Check your work.  Write original equation, substituting your solution for the variable
-5 (6 - 2*-4) +4 = -66-5 (-14) + 4 = -66-70 + 4 = -66-66 = -66 yippeeeee!



November 3, 2015

SOLVING ONE AND TWO STEP EQUATIONS

An equation is like a balance scale.
To keep things balanced, whatever you do to one side you must do to the other.
Both sides are equal (shown by the EQUAL sign).  


The goal is to get the variable on one side of the equation by itself.  This is called isolating the variable.

VOCAB REVIEW:
Constant:  a number that has no variable attached to it, never changes, Examples:  7, -12, 1/3
Coefficient:  the number you are multiplying the variable by.  In the term 2n, 2 is the coefficient, n is the variable

Ms Calandriello's Tips for solving equations:

For ONE STEP equations:
Ask yourself the following questions.....in order....
1) Where is the variable?  Which side of the equation?  Point to it!
2) What is in the way of the variable being by itself on that side of the equation?
3) What operation (addition, subtraction, multiplication, division) is involved?
4) Decide on the opposite operation to get rid of what is in the way.....if it is a constant that is added or subtracted.....do the opposite.  If it is a coefficient that is multiplied or divided, do the opposite....or multiply by the recipricol.
5) Do the SAME THING to the other side of the equation.
6) Solve for the variable.

For TWO STEP equations: 

1) If you can combine any like terms on either side of the equation, do that first.
2) Then......where is the variable?  Which side?
3) Remove the CONSTANT first (the number that is added or subtracted)
4) Do the same thing to the other side of the equation.
5) Rewrite new equation
6) Remove the coefficient (do the opposite operation...multiplication or division, or multiply by the coefficient.
7) Do the same thing for the other side of the equation
8) Rewrite the new equation.
9) Solve for the variable.

I know that for ONE step equations, I have listed actually 6 steps and for TWO step equations, I have actually listed 10 steps!  The ONE and the TWO actually refer to how many things need to be removed from the variable side and how many times the equation needs to be written.

You should all have PLENTY of examples from class in your notes to refer to!



October 26, 2015

CHAPTER3:  ALGEBRAIC LINEAR EQUATIONS


EXPRESSION VS. EQUATION:

An equation has an equal sign; an expression does not
An equation can be solved; an expression can only be simplified
Equations are made up of expressions.

It is like a sentence and a phrase....an equation is a complete thought; an expression is not

Example:  2x + 4 = 8  is an equation
2x + 4 is an expression

EVALUATING EXPRESSIONS:

Must be given the value for the variable.  

Write the expression
Substitute the variable
Solve vertically, using order of operations
Circle answer

Example:  Evaluate 8x - 2   if   x=3
Write:   8x-2
Substitute:    8 (3) - 2
Solve:  24 - 2
Answer:  22

COMBINING LIKE TERMS:

Combine means add
Like terms  must have the same variable to the same power

Constant  a number that stays the same  (7, 2 303)
Variable   a letter symbolizing a number that changes  (x, y, a)

To simplify....group and combine like terms

October 20, 2015


CHAPTER 2:  SCIENTIFIC NOTATION

Important Vocabulary:  Know these and use them in your answers!

Coefficient:  The multiplicative factor in scientific notation  (the number between 1 and 10 that gets multiplied by the power of 10)
Scientific Notation:  A way of expressing really large or really small numbers.  Used a lot by scientists
Standard Form:  The regular way of writing a number out.  

How to put standard numbers into Scientific Notation:
1) Find the coefficient....where can you put the decimal point so that your number is between 1 and 10.
2) Write    * 10
3) Decide on your exponent.....count the number of spaces from your new decimal place to your original decimal place
4) Decide if it is positive or negative......is the original number a BIG number (larger than 10?)....your exponent will be positive....Is the original number less than one?  A decimal?  your exponent will be negative

COMPARING NUMBERS IN SCIENTIFIC NOTATION:
**** First compare the exponents.  If the exponents are different, the larger exponent shows the larger number.If the exponents are the same, move to step 2. 
****When exponents are the same, compare the coefficients.  The larger coefficient shows the larger number
(negative exponents can make these a little tricky.  Try to think of a number line.....numbers to the LEFT are LESS) 

When comparing numbers in scientific notation with numbers in standard form, you need to change one so they are BOTH in standard form, or BOTH in scientific notation.


October 19, 2015

CHAPTER 2:  SCIENTIFIC NOTATION

Remember......when you multiply, you make the number BIGGER
when you divide, you make the number SMALLER
Make sure your answer MAKES SENSE!

Scientific Notation has two parts:
1) a coefficient that is equal to or greater than 1, but less than 10
2) a power, with a base of 10 and an exponent that is an integer.

Check for Scientific Notataion:
You can tell if a number is written correctly in scientific notation if you can answer YES to the following questions:
***Is the coefficient equal to or greater than 1, but less than 10?
***Is the operation MULTIPLICATION?
***Does the power have a base of 10?
***Is the exponent an integer?

Multiplying by a positive power of 10....move the decimal to the right the number of places that equals the exponent.  Add zeros when necessary

Multiplying by a negative power of 10....move the decimal to the left the number of places that equals the exponent.  Add zeros when necessary.


October 15, 2015

We had an in-class review today for the Chapter 1 test tomorrow.  Students should have it at home and be using it to study.  The review, complete with answers and some work, courtesy of Mr. Droesch, is linked below:

Chapter 1 Test Review with answers.pdf

October 13, 2015

SECTION 1.6   
REAL WORLD PROBLEMS....SQUARES AND CUBES

Square Roots:
There are two square roots of a number...the positive and the negative
You cannot take the square root of a negative number.
If you are just given the radical symbol....the answer they are looking for is the positive square root.
If you are given the radical signal with a negative sign in front of it...the answer they are looking for is the negative square root.
If it asks for the square roots of a number.....you need to write down both!
If you are solving an equation.....   x= ?   you need to write down both!

Cube roots have only one answer....might be positive or negative
You CAN take the cube root of a negative number

The opposite of squaring a number is finding the square root.

The opposite of cubing a number is finding the cube root.

Area.....problems about area involve squaring and square roots
Volume.....problems about volume involve cubing and and cube roots

October 7, 2015

CHAPTER ONE REVIEW!!!

We are almost at the end of Chapter 1 and everyone will be having a chapter one TEST at the end of next week.  I have prepared a Chapter One review that you can click on below.

CHAPTER ONE REVIEW.docx

At the end of the review, there is a Study Guide for students to fill out.  If they fill it out,it will be quite useful to them.  
If they would like help filling it out, I will be hosting a Test Review Session next Wednesday, October 14th after school, from 2:20 to 3:20ish.  Any student is welcome to attend.  Any student is welcome to use their filled out Study Guide to study.  Students with test accommodations in their IEPs may be able to use the Study Guide during the test.  Parents should check IEPs or check with their child's case manager to see if this is the case.


October 5, 2015
SECTION 1.5 ZERO AND NEGATIVE EXPONENTS

Negative exponents have NOTHING to do with negative numbers.  Negative exponents are all about FRACTIONS.

RULE:

 a-n = 1/an

EXAMPLE:

5-2 = 1/52 = 1/25



October 1, 2015
SECTION 1.5 ZERO AND NEGATIVE EXPONENTS

Any number to the zero power is equal to one, as long as the number itself is not zero.

a0= 1    70=1    1000= 1         

EXAMPLE:

7* 70    = 3+0   =  73

PEMDAS reminder........

If you have something that looks like this......

1 * 10+ 2 * 10+ 3 * 100       

you need to remember that you have different operations here....you just can't add the exponents.  PEMDAS tells us that we have to do the multiplication parts first.   So.....

1 * 10= 100

2* 10= 20

3 * 10​​​0  = ​3

Then......   100 + 20 + 3 = 123





September 30, 2015
USING PROPERTIES OF EXPONENTS TO SIMPLIFY EXPRESSIONS.

With some expressions, you may need to use more than one property to simplify.  Follow order of operations!  PEMDAS!

Examples:

[ (-4)​* (-4)3 ]6  =

[ (-4) 2+3]6   =

[(-4)5] =

(-4) 5*6 =

(-4) 30




September 28, 2015

POWER OF A POWER

(am)n = am*n = amn

Example:
 (2
43  =  

24*3 = 2 12   OR

(2*2*2*2) * (2*2*2*2) * (2*2*2*2) =  212  OR

2* 2* 24 = 2​ ​4+4+4​ = 2​ 12


September 24, 2015
Section 1.2
THE PRODUCT AND QUOTIENT OF POWERS

QUOTIENT OF POWERS PROPERTY:
When the bases are the same, keep the same base and subtract the exponents!

55  divided by 52 = 55-2 = 53

September 23, 2015
Section 1.2
THE PRODUCT AND QUOTIENT OF POWERS

Example 1:  10* 109 =

10*10*10*10*10*10         *       10*10*10*10*10*10*10*10*10  =  1015

10* 109 =

10 6+9  = 1015

Example 2:      a4  * a
a4+3 =    a7

PRODUCT OF POWERS PROPERTY:

am * an = am+n

When the bases are the same, keep the base, add the exponents.

Showing work:
Write problemx23 * x18 =
Write intermediate stepx23+18 =
Write answerx41

September 22, 2015

WORD PROBLEMS:

There are 20 bacteria in a given sample in a laboratory.  During the early phase of culture growth, the number of bacteria doubles every hour.  How many bacteria are there after 3 hours?  Write your answer in exponential form.

Make a chart:

BacteriaHour
200
401
802
1603

Answer=  160.  In Exponential Form......2* 5  (use prime factorization)


Karen ate at a restaurant.  One day later, Karen told 3 friends about the restaurant.  The day after that, each of her friends told 3 friends about the restaurant.  If the pattern continues, how many friends are told after Day 5?

Make a chart:  (finish)

DayFriends
0Karen
13
29
327
4
5

Shana deposits $100 in a bank account that earns 5% interest compounded yearly.  How much will be in her account at the end of 5 years?

Formula      A (amount at end) = ( 1 + rate as a decimal)​number of years  

A = ( 1 + .05) ​5

September 21, 2015
Study Guide for Section 1.1. Quiz:  For whatever reason, this sometimes does not read well on screen....does read better if you print it out.  Sorry...not quite sure why!

Try these questions.....check answers below to see if you are right!

1.  Identify the base and the exponent for the following:  

a.   -42b.   (-4) 5c.  (1/2) 4


HINT:  BASE IS THE NUMBER THAT IS MULTIPLIED REPEATEDLY; EXPONENT IS THE NUMBER OF TIMES IT IS MULTIPLIED

2.   Is the following correct or incorrect?  State why.

-4= -4 * -4 *-4


HINT:  DECIDE WHAT THE BASE IS AND DETERMINE IF IT IS MULTIPLIED THE CORRECT NUMBER OF TIMES.

3.  Write in exponential notation:

a.   ab*ab*ab*abb.  0.5 * 0.5 *0.5


HINTEXPONENTIAL NOTATION HAS A BASE AND AN EXPONENT


 
4.  Expand and evaluate the following:
 
a.  53b.  (1/4) 4


HINT:  EXPAND MEANS TO WRITE IT OUT AS REPEATED MULTIPLICATION;   EVALUATE MEANS TO FIND THE ACTUAL ANSWER


5.  Write the prime factorization of 36 in exponential notation.



HINT:  SHOW WORK. USE A LADDER OR A FACTOR TREE.  REMEMBER TO USE PRIME NUMBERS AS FACTORS.  THE MOST IMPORTANT PRIME NUMBERS TO REMEMBER FOR THIS ARE:  2, 3, 5, 7, 11



6.  Order the following from least to greatest.

-24       (-2)4         43


HINT: SHOW WORK.  FIRST EXPAND AND EVALUATE EACH.  THEN PUT THEM IN ORDER


ANSWERS
1.  a)  base = 4, exp = 2   b).  base = -4  exp = 5   c)  base = 1/2 exp = 4

2.  Incorrect.  The base is 4, not negative 4, because it is not in parentheses

3.   a)   (ab)4  must use parentheses    b)   0.53

4. a)     5 * 5 * 5 = 125    b)   (1/4)(1/4)(1/4) (1/4)=  1/256

5.   22 * 32

6.  -23= (-) 2 * 2* 2*2 = -16

(-2)4= (-2)(-2)(-2)(-2)= 16

43=  4 * 4 *4= 64

In order:  -16, 16, 64,  so......-2​3, ​(-2)​4​, 4​3
   

September 18, 2015
Divisibility Rules
*A number is divisible by 2 if......it ends in a 0, 2, 4, 6 or 8
*A number is divisible by 3 if .......the digits add up to a number divisible by 3 
*A number is divisible by 5 if........it ends in a 0 or 5

Prime Factorization
Prime numbers......have exactly two factors.
Examples:  2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Composite numbers.....have more than two factors  (anything that is not
 prime)

500 = 5*5*5*2*2=  22 * 53

Doing exponents on the calculator

Base-----Carat-----Exponent----Equals

Sept​ember 17, 2015
CHAPTER 1, SECTION 1
EXPONENTIAL NOTATION

Objectives:
*Understand and use exponential notation
*Use exponents to write prime factorization of a number

VOCABULARY
 
* Exponents....show repeated multiplication
*Exponential Notation.....write using exponents, such as 23
* Base.....the number that is multiplied by itself
Examples:
43.....4 is the base
(-4)5......-4 is the base
(2x)6.....2x is the base
-43.....4 is the base
*Exponent....shows how many times the number is multiplied by itself
Examples:
43.....3 is the exponent
(-4)5......5 is the exponent
(2x)6.....6 is the exponent
-43.....3 is the exponent
*Power...the expression that has a base and an exponent
*Expanded Notation....written out in the larger format, Ex:
23= 2*2*2

23= 2*2*2=8    It DOES NOT equal 2*3 =6

To EVALUATE....means to give the answer, solve the problem


THIS WEEK, MATH CLASSES ARE ALL DOING STAR MATH TESTING ON DIFFERENT DAYS.....SO THE DATE OF THE NOTES MAY BE SLIGHTLY OFF FOR YOUR MATH CLASS......WE WILL ALL CATCH UP BY THE END OF THE WEEK!



September 15, 2015

SKILL # 6 and 7
MULTIPLYING AND DIVIDING INTEGERS

The rules are the same for both multiplication and division!

If you have two integers with the SAME sign.....your answer will be POSITIVE.

If you have two integers with DIFFERENT signs....your answer will be NEGATIVE.

STEP ONE:  Decide on the sign of the product or quotient
STEP TWO:  Do the math!

Examples:

-2 x -4= +8    (both factors are negative....same sign...so answer is positive)

-2 x 4 = -8  (one is negative, one is positive....different signs....so answer is negative)

When you get to three or more numbers.....either do them in pairs, as you progress from left to right.   OR  
*if you have an ODD number of negative integers, your answer will be negative
* if you have an EVEN number of negative integers, your answer will be positive

September 14, 2015

SKILL # 5
SUBTRACTING INTEGERS


To subtract integers.....ADD THE OPPOSITE!

THE RULE:
To subtract two integers, add the opposite of the number being subtracted.

Then....you have an addition problem!  Follow the rules for adding integers (reminder on how to do those....look at skill #4

Examples:
7-4=
7+ (-4)=
3

-2 - (-6) =
-2 + 6=
4


September 11, 2015
SKILL # 4
ADDING INTEGERS


The Rules:
1)Adding integers with the same sign.....
add the absolute value of the numbers (think just digits, no signs)
keep the same sign.

Example:  4 + 3 = 7       ( -4) + (-3)= (-7)
sometimes parentheses are put around negative numbers just to keep the negative sign and the number together......that's all....

2) Adding integers with different signs.....
subtract the absolute values
use the sign of the number with the larger absolute value

Example:  5 + (-7) = -2    {Think.....7-5 = 2; 7 >5 so we will use the negative sign that went with the 7}
(-2) + 5 = 3

Different teachers used different examples to show this.  Think about Mr. Droesch's Tug'o'War or Ms. Schulten's Revolutionary War Soldiers, or Ms. Mulqueen's pairing of + and - and the pairs (kapoweeeeeee!) blowing up!


September 10, 2015
SKILL #3
LOCATE IRRATIONAL NUMBERS ON A NUMBER LINE

Remember:  An irrational number is a decimal that does not terminate, and does not repeat.  Pi is a good example:  3.14159........  Square roots of numbers that are not perfect squares are also good examples.  

TABLE OF PERFECT SQUARES

     
12   1 x 1
 22  2 x 2  4
 32  3 x 3 9
 42  4 x 4  16
 52  5 x 5  25
 62  6 x 6 36
 72  7 x 7  49
 82  8 x 8 64
 92  9 x 9 81
 102  10 x 10 100
 112  11 x 11  121
 122  12 x 12 144

TABLE OF PERFECT CUBES

     
13   1 x 1 x 1
 23  2 x 2 x 2  8
 33  3 x 3 x 3 27
 43  4 x 4 x 4  64
 53  5 x 5 x 5  125
 63  6 x 6 x 6 216
 73  7 x 7 x 7  343
 83  8 x 8 x 8 512
 93  9 x 9 x 9 729
 103  10 x 10 x 10 1000
 113  11 x 11 x 11  1331
 123  12 x 12 x 12 1728

To graph an irrational number on a number line, first decide what two integers it comes between.
PI = 3.14159.......   so it would be somewhere between 3 and 4, and it would be closer to the 3!

To graph a square root:  
Find the two perfect squares it comes between.
Find the square root of those perfect squares
Graph between those two numbers

For example:
(sorry, I can't figure out how to make the root symbol on the computer!  I will use the words!  Bear with me!)

Locate the square root of 21 on the number line.

1.  Look at table of perfect squares.
2.  Find the two perfect squares that 21 comes between......that would be 16 and 25.
3.  So......you could say......    
                16<21<25
4.  Which means that....
         the square root of 16<the square root of 21<the square root of 25
5.  Which means that:
          4<the square root of 21<5
6.  So you would graph the square root of 21 somewhere between 4 and 5, a little bit closer to 5, and estimate it to be around 4.6
7.  You could check with your calculator  and get 4.5825756........

EXAMPLE:   Graph the CUBE root of 30 on a number line
Use the same process
1.  Look at table of perfect cubes.
2.  Find the two perfect cubes that 30 comes between......that would be 27 and 64.
3.  So......you could say......    
                27<30<64
4.  Which means that....
         the cube root of 37<the cube root of 30<the cube root of 64
5.  Which means that:
          3<the cube root of 30<4
6.  So you would graph the cube root of 30 somewhere between 2 and 3, a little bit closer to 2, and estimate it to be around 2.1
7.  You could check with your calculator!




September 9, 2015
SKILL #2
WRITING RATIONAL NUMBERS AS TERMINATING OR REPEATING DECIMALS  otherwise known as WRITING FRACTIONS AS DECIMALS


Example:  3/4  means 3 divided by 4 .  If you are doing it with long division, the numerator (top number) goes inside the "house".  
If you are doing it with a calculator.....think about reading from top to bottom.....3 divided by 4, and put it into the calculator in that exact order.
3 divided by 4 equals 0.75
So......3/4 =0.75

5/6 means 5 divided by 6

19/4 means 19 divided by 4

It helps to think to yourself....is this fraction smaller than one, or larger than one?  And, then, when you do your division, you will know whether or not you put the numbers in the calculator in the right order.

3/4 is less than one.  0.75 is less than one

19/4 is more than one.  4.75 is more than one.


September 8, 2015
SKILL #1
THE REAL NUMBER SYSTEM


Students all received handouts with diagrams about the Real Number System.  One is a flow chart; one is a Venn Diagram.  Below are some definitions and examples of the terms in these diagrams.

There are REAL and IMAGINARY numbers.  In Grade 8, we will only be concerned with REAL numbers.  REAL numbers are all the numbers we have used so far in our math careers.  
There are two categories of REAL numbers:  Rational Numbers and Irrational Numbers. 
IRRATIONAL NUMBERS....Numbers that have no end and do not repeat.  Examples used include Pi, and decimal numbers such as
 34.56839583930288593.......  never ending and no repeating patterns.  
All the rest of the numbers are RATIONAL NUMBERS!
RATIONAL NUMBERS include any number that can be written in fraction form....think of the term "ratio".  These can be numbers such as 21, which can be written as 21/1, or 0.7, which can be written as 7/10
Rational Numbers have two categories:  Integers and Non-Integers.
NON-INTEGERS:  Fractions, terminating or repeating decimals...Ex. 3/4,      2   3/8, 1.567 or 3.3333333333......
INTEGERS: No fractions or decimals.  There are two categories...NEGATIVE NUMBERS and WHOLE NUMBERS.  Whole numbers also have two categories...ZERO and POSITIVE NUMBERS


September 3, 2015
Schulten--period 2
Beware if your children ask for a change in allowance!  The problem posed today.....would you rather have $10 a week for the rest of the school year, or 1 cent today, then doubled each day afterwards for the rest of the school year? Students discussed in groups.  The $10/week today was easy to figure out.  The penny deal was a bit more complicated and a table, graph and equation were used to figure it out.
DAY #
MONEY EARNED
 TOTAL
 1  1cent  1 cent
 2  2  3
 3  4  7
 4  8  15
 5  16  31
 6  32  63
 7  64  1.27
 8  128  2.55
 9  256  5.11
 10  5.12  10.23
 11  10.24  20.47
 12  20.48  40.95

We talked about the rate of "exponential growth" and how ....it starts out slow, but gets bigger really quickly!
We did the same thing with a story about a grain of rice, doubled each day and converted the numbers into exponents.  Students are reminded:

Repeated addition is multiplication!
Repeated multiplication is exponents!

Mulqueen--Period 3
We did a neat experiment with a deck of cards and putting them in a logical order to create a specific outcome.  Students worked in small groups and then as a large group.
Students also brainstorm a list of words that could be used to classify numbers.   This is what they came up with:
even                  odd                              negative                  positive
fraction              decimal                        percents                  counting
simplified           numerals                     digits                       equal

The focus was then put on converting fractions into decimals.  Students were reminded that the top number was called the numerator; the bottom number is the denominator.  The line between is called the VINCULUM (who knew?) and it means to divide.  Thus a fraction is converted to a decimal by dividing the numerator by the denominator.  Example 3/4 is the same as 3 divided by 4, which equals 0.8.  Students are allowed to use calculators.  Students were also reminded that 0 can never be used as a denominator


Droesch---Period 4
Students played a fun game reviewing math class rules and policies!


September 2, 2015
Today in math classes teachers went over behavioral and academic expectations.  Nothing too new here....Just remember the 3 Dover Middle School expectations....Be Positive, Be Respectful, Be a Learner....and everyone will do just fine.  
Teachers also talked about accessing the text book on line, and offered to sign out textbooks to anyone who did not have the ability to access the internet at home.
There will be multiple opportunities for students to be assessed.  There will be a short quiz after each section, and a test at the end of each chapter. 
Grading for all math classes is comprised of 20% homework and 80% assessments.  If students are putting in the work and still scoring low on assessments, please reach out to me or the math teacher about ways to improve on this!  We are all open and working for the success of each student.



August 27, 2015
Hello Everyone,
It's almost time to get started.I am going to use this page to jot down some 8th grade math notes and give you some examples on how to solve various problems.
  Also, just so you know, you can access your math book online!  Go to Useful Links my home page and find Math Book  A and B. You will need to log in to access these books.  Your math teacher will probably give you a log in to access this book, but in case you forget it, you can use mine:  
User name:  seight151
        Password:  e4v3a
Good luck.  I will see you at DMS soon!