A book explaining anything and verything about 8 grade math.

All about Math :) 

 A book written by: Omar H, Elias, Mohammed, and Safa 

 

All about Math

 

Elias Hajjar

 

Omar Hisham

 

Safa Sahir

 

 

Mohammad Khater

 

 

Publisher

-----------

EOSK

Copyright © 2016

Thank you ISG Dammam for allowing us to make this book.

Thank you Mrs. Urooj for giving us 

permission to publish this book.

 

And thank you EOSK for making us publish this book.

 

 

 

Acknowledgements

Grade Levels 8-9

Table of Contents

Chapter 1:

Lesson 1......................................................................................................................6

Lesson 2......................................................................................................................7

Lesson 3......................................................................................................................8

Chapter 2:

Lesson 1......................................................................................................................11

Lesson 2......................................................................................................................12

Lesson 3......................................................................................................................13

Chapter 3:

Lesson 1......................................................................................................................16

Lesson 2......................................................................................................................17

Lesson 3......................................................................................................................18

Chapter 4:

Lesson 1......................................................................................................................21

Lesson 2......................................................................................................................22

Lesson 3......................................................................................................................23

Chapter 5:

Lesson 1......................................................................................................................26

Lesson 2......................................................................................................................27

Lesson 3......................................................................................................................28

➢ 1-1 Variables and Expressions

 

 

 

 

➢ 1-2 Solving Equations with Variables on Both Sides

 

 

 

➢ 1-3 Solving for a Variable

Chapter 1: Equations

Variables and expressions:



What is a variable?

A variable is a letter or a symbol used to represent a changing value.

 

What is a constant?

A constant is a value which does not change.

 

Translating Algebraic expressions into words:

Example 1

 

X+3                                                                                           m - 7    

 

The sum of X and 3. -2                                                              The difference of m and 7

X increased by 3                                                                      7 less than m                                                         

 

2 . y                                                                                           k÷5

2 times y                                                                                    k divided by 5   

The product of 2 and y                                                              The quotient of K and 5                                                  

Lesson 1.2

Solving equations by Addition and Subtraction




An equation is a mathematical statement stating that 2 equations are equal.

A solution is the variable that makes the statement true.

   



SOLVING EQUATIONS BY USING ADDITION

Example 1

x-10=4

x-10=

   +10          +10

X            =      4

 

Check/Verify                                                                                       x + 7=9

                                                                                                           9-7 x = 2      check verify



Lesson 3

Solving equations by multiplying or Dividing

 

 

Using Inverse methods to undo operations, Remember that an equation is like a balanced scale.

 

Solving Equations by Using Multiplication

 

Example 1

-4 = k over -5

(-5)(-4) = (-5) (k over -5)

20 = k

 

Verify




Solving Equations by using Division

Example 2

7x = 56

7x = 56 over 7

X = 8

Verify     

 

1. M - 4= 318 

 

2. X - 2.4= 3.5                                                   

 

3. 7.5 = w - 3.4

 

4. K / 4 = 8

 

5. -2=w / -7

 

6. Z /3 = -9

Chapter 1 Review

Chapter 2: Inequalities 

 

➢ 2-1 Solving two- Step

 and Multistep Inequalities

 

 

 

 

 

 

➢ 2-2 Solving Compound

 Inequalities

 

 

 

 

 

 

 ➢ 2-3 Solving Absolute

 Value Inequalities

 

2-1 Solving two- Step and Multistep Inequalities

Objective: to understand how to solve inequalities and multistep inequalities

 

Vocabulary:an inequality is a relation that holds between two values when they are different.

The notation a ≠ b means that a is not equal to b hence the cross on the equal sign.

The notation a < b means that b is actually less than a because the thats the less than sign. 

The notation a > b means that b a is greater than b because if the point is to the left its more than and when its to the right its less than.

The notation a ≤ b  means that b is less than or equal to a hence the line under and the less than sign above it. The notation a ≥ b  means that hence the half equal sign and the greater than sign that a is greater than or less or equal to.

Real life situations:An 18-wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is 65,000 pounds. The cab (front) of the truck weighs 20,000 pounds, and the trailer (back) of the truck weighs 12,000 pounds when empty. In pounds, how much cargo can the truck carry and still be allowed to cross the bridge?

 

Explanation: its easy to solve  for example  :take this 6 ≤ 7 that means that  7 is greater than or equal to 6 so see its that easy. Now lets move on to the multistep inequalities. The multistep inequalities are also not hard so lets take for example:

Lesson 2: solving compound inequalities

 

Objective how to solve compound inequalities

 

Vocabulary:A compound inequality is an equation with two or more inequalities joined together with either "and" or "or" (for example, and ; or ). When two inequalities are joined with and, they are often written simply as a double inequality, like: .

 

Real life situation:Some of your friends’ bike speeds are between 5 to 6 meters per second. If they take part in a 3000m race, calculate the shortest time by writing an inequality to model this situation.

 

For example:

This means that the graph of a compound inequality with an "and" represents the intersection of the graph of the inequalities. A number is a solution to the compound inequality if the number is a solution to both inequalities. It can either be written as x > -1 and x < 2 or as -1 < x < 2.

Try solving some yourself:

Solve for d.
d+7≥14 or d+7≤5
Write your answer as a compound inequality with integers.

 

Bibliography:

https://www.ixl.com/math/algebra-1/solve-compound-inequalities

http://www.mathplanet.com/education/algebra-1/linear-inequalitites/solving-compound-inequalities

http://www.chegg.com/homework-help/definitions/compound-inequalities-27

 

Lesson:2-4

Objective: to learn how to solve absolute value inequalities

vocabulary:

The absolute value of a number like 5 for example is written like this | 5 | .

This basically means that 5 is the place away from 0 on the number line so its 5 steps away.

This lao means that for example |5| =c means that it is 5 place away from 0 so it can either be 5 positive steps or 5 negative steps.

Explanation: let’s take for example how to solve this equation

 

| X+5 | =10

 

What you need to do is separate the equations so make two separate equations and then start solving each one :

X+5   =10

     -5      -5

X=5

Or

X+5 =-10

X+5  =-10

    -5       -5

X=-15

If you ever get a question with a negative absolute value just know that there is no solution and that is because you can’t actually have a negative absolute value. So just remember if you see it write no solution. It would look for example like this:

|-5| = No solution

Now let’s do the “Real Life Situations”:

The street built in the city must be 25 feet in width with a tolerance of 0.5 feet. Streets that are not within the tolerated widths must be repaired. Which of the following inequalities can be used to assess which streets are within tolerance? (W is the width of the street) .

The answer is 0.02.

 

 

15<18:

13>7:

 |x+13|=26

 

|b-3|=1

 

|a+7|=14

 

|x+12|=24


|d-9|=18

 

d+7≥14 or d+7≤5

Chapter 2 Review

➢ 3-1 Relations and Functions

 

 

 

 

➢ 3-2 Writing Functions

 

 

 

 

 

➢ 3-3 Graphing Functions

Chapter 3: Functions

Chapter 3-1: Graphing Relationships

 

Lesson Objective: Learn how to read graphs and know how to draw a graph.

Vocabulary Words

Continuous Graph

Discrete Graph

 

A continuous graph is a graph that is made up of connected lines or curves.

A discrete graph is a graph that is made of only distinct points.

Graphs can be used in real life situations such as: when doctors use graphs to measure a patient’s heart rate.


This is an example of a discrete graph:




Continuous graphs are very easy to read and create, because there are certain keywords that are used when describing events using continuous graphs. These keywords are: remains the same, rises at a steady rate, increases rapidly, remains high, slows down and returns to the normal rate. Remains the same means that the line is straight, rises at a steady rate means the line is slanted and rising, increases rapidly means the line becomes steeper than the previous keyword (rises at a steady rate), remains high means that the line stays straight while it is high in the graph, slows down means that the line is slanting to the right and returns to the normal rate means it becomes straight and is at the same level of the start of the line.

 

3-2 Relations and Functions

 

Lesson Objective: Identify functions and find the domain and the range of both relations and functions.

Vocabulary Words: Relation,Domain, Range,and Function

Relations and functions can be used in real life situations such as: the scoring system of a track meet.

Ordered pairs that represent relationships are called relations.

The domain of a relation is the set of first coordinates of the ordered pairs(x-values).

The range of a relation is the set of second coordinates of the ordered pairs (y-values).

A function is a special type of relation that pairs each domain value with only one range value.

There are multiple representations of functions, here are a few examples of them:

This is the mapping diagram form of a function.

As you can see, not all the y-values have been paired with an x-value, which isn’t required to make a mapping diagram form of a function.

However, it is required to have each domain value paired with only one range value. Mapping diagrams are used to determine whether a group of ordered pairs is a function or not.

 

This is the graph form of a function.

To make a graph function, all you have to do is plot the ordered pairs in points on the graph.

This is the table form of a function.

All you have to do to make one is to organize the domain and range into two groups, in the form of a table. The domains of a relation are all the x-values of the ordered pairs, and the ranges of a relation are all the y-values of the ordered pairs. To identify between a function and a relation, you must use a mapping diagram. If one or more x-values are paired up with more than one y-value, the ordered pairs are not a function.

 

 

 







Chapter: 3-3 Writing Functions

 

Lesson Objective: Identify independent and dependent variables. Learn how to write an equation in function notation.

Vocabulary Words:

Independent variable

Dependent variable

Function rule

Function notation

You can use function rules to calculate the money you can earn for working certain amounts of time.

The input of a function is an independent variable. The output of a function is a dependent variable. Any algebraic expression that defines a function is a function rule. Function rules are used in input and output tables. These function rules tell you how an input and output table was made. Function notation for y is f(x) (it is read: f of x), this is used only if an equation has two variables that define/explain it.

Function notations are used because they show us the value of x.

Example of function notation: f(x)= 2x+6

If x=3

f(3)= 2(3)+6

(Those examples are exactly the same as: y= 2x+6) Here is an example of an input and output table:

 

 

 

 
 

1.Write whether this is a continuous or discrete graph:                                           

7.Find the function rule of the following input and output tables:

 

                                                        Input

Output

                                                       2

4

                                                       3

5

                                                       4

6

                                                       5

7

 

2.Write whether this is a continuous or discrete graph:

3.Write whether this is a continuous or discrete graph:

4.Give the domain and range of the following:

 

 

5.Give the domain and range of the following:

 

 

6.Give the domain and range of the following:

 

 

 

Chapter 3 Review

Unit 4: Linear Functions 

 

 

➢ 4-1 The Slope Formula

 

 

 

 

 

 

4-2 Slope Intercept Form

 

 

 

 

 

➢ 4-3 Point- Slope Form

 

Lesson 1: Slope formula

Objective: To learn about slope formulas

Vocabulary: Y-intercept : The y-coordinate where the line intersects the y-axis,the x-coordinate is always 0.

X-intercept: The x-coordinate where the line intersects the x-axis, the y-coordinate is always 0.

 

 

To find the y coordinate, replace the x with a 0.

Solve .

If you want to find the x intercept: Do the same with the x coordinate but replace the y with a 0 instead of the x.  

 

 Real world application

 

 

 

 

 

 

 

 

200x+250y=3800

 

picture bibliography:http://www.algebra-class.com/image-files/standard-wd-prob1.gif

http://www.mathexpression.com/images/c2-finding-y-intercept.png

 

 

Lesson 1: Slope formula

Objective: to learn the formula of Slope intercept form

 

Vocabulary: None

Slope intercept form = y=mx+b

b= y intercept

m= slope

Example:x= 3, y= 9

y= 2x + b

9= 2(3) +b

9=6+b

9-6=b

3=b

Application: You want to save for a car. You open an account with $500. Each month you will deposit an additional $100.

m=100,b=500

y=mx+b

y=100x+500

 

Lesson 2 Slope intercept form 

Lesson 3 Point slope form

 

 

Objective:to learn about point slope form

Vocabulary: slope formula: formula to find a slope (m=y2-y1/x2-x1)

Point slope form: the formula to find a point on a slope (y-y1=m(x-x1))

Explanation:

 

Slope formula example:

(1,2) (2,1)                         Slope formulas are used only when a slope is not given.

m=y2-y1/x2-x1

m=1-2/2-1

m= -1

Point slope form example:

Slope=-1, (2,1)

y-y1=m(x-x1)

y-1= -1(x-2)

 

 

Real application

A new website has 200 hits in the first day, and every day after that it gets 50 more hits. Write an equation in point-slope form modeling the number of hits the website gets.

We know that one point on this line would be (1, 200) because on the first day, the website gets 200 hits. After this, the hits on the website increase by 50 every day, so the slope is 50. Plugging in these points, we get y – 200 = 50(x – 1).

 

Bibliography: https://braingenie.ck12.org/skills/105437

 

Questions:

  1. Find the answer.
  2. 1.(3,2) (2,3)

  3. 2.Slope =1, (5,2)

  4. 3.(5,9), (9,5)

  5. 4.Slope= 18, (167,98)

  6. 5.(123,321), (652,867)Try it

  7. Find the y intercept, then the x intercept.
  8. 6.10x-5y + 2= 10

  9. 7. 1x + 2y=3

  10. 8. 12x+25y=76

  11. 9. 4x + 2y=3

  12. 10. 20x + 9y=56

  13. Try it: replace all y=3 and x=4

  14. 11.y=1x+b

  15. 12.y=2x+b

  16. 13.y=3x+b

  17. 14.y=10x+b

Chapter 4 Review

➢ 5-1 Solving systems by Graphing

 

 

 

 

 

➢ 5-2 Solving systems by Substitution

 

 

 

 

 

➢ 5-3 Solving Special systems

Chapter 5: Systems

 

Chapter 5.1

 

System of linear equations: a set of two or more linear equations containing two or more variables.

Objective: to learn about solving systems using graphs
Explanation: {2x+2y=6

                    {4x-6y=12

                    

                                                                      {y=2x+3

                                                                      {y=4x-2

                                             

                                                                      Graph.

To graph, remember that b is the y intercept, so graph b. And the slope is equivalent to rise/run. You rise based on the numerator and go horizontally based on the denominator. The area where the two lines meet is the answer.

 

Real life application

Two classmates, Dedra and Christina, plan to meet in the computer lab to type up their research papers. Dedra can type at a speed of 3 pages per hour, whereas Christina can type2 pages per hour. So far, Dedra already has 5 pages typed up, compared to Christina's 15 pages. Once they sit down and start typing together, the two students will reach the same page count before too long. What will the page count be? Graph.

{y=3x+5

{y=2x+15

35 pages after 10 hours

 

Chapter 5.1

System of linear equations: a set of two or more linear equations containing two or more variables.

Objective: to learn about solving systems using graphs
Explanation: {2x+2y=6

                    {4x-6y=12

                    

                                                                      {y=2x+3

                                                                      {y=4x-2

                                             

                                                                      Graph.

To graph, remember that b is the y intercept, so graph b. And the slope is equivalent to rise/run. You rise based on the numerator and go horizontally based on the denominator. The area where the two lines meet is the answer.

 

Real life application

Two classmates, Dedra and Christina, plan to meet in the computer lab to type up their research papers. Dedra can type at a speed of 3 pages per hour, whereas Christina can type2 pages per hour. So far, Dedra already has 5 pages typed up, compared to Christina's 15 pages. Once they sit down and start typing together, the two students will reach the same page count before too long. What will the page count be? Graph.

{y=3x+5

{y=2x+15

35 pages after 10 hours

Lesson 5-2: Solving systems by substitution

 

Objective: To learn how to solve systems by substitution

 

Vocabulary:A system of equations is basically when you have 2 or more equations with the same set of unknown. It could also be linear and non linear equations.

The subsitution method basically uses replacement to elimintate one of the variables nd thats when you slove a system of equations

 

Explanation: Lets take for example

7y-3x=15

y+5x=10

So start with solving one of the equations with x or y  so

y=10-5x

Then you have to

Replace the "y" value in the first equation by what "y" now equals.  Grab the "y" value and plug it into the  other equation. So like this

7(10-5x)-3x=15

 

 

Chapter: 5-3 Solving Special Systems

 

Lesson Objective: Solve special systems of linear equations and be able to classify systems of linear equations and to be able to determine the amount of solutions.

Vocabulary Words

Inconsistent System

Consistent System

Independent System

Dependent System

A linear system (System of linear equations) that has at least one solution is called a: consistent system.

Any linear system that does not have a solution is called an: inconsistent system.

Real World Application: Linear equation systems can be used to find out two things at once on a graph.

To find out whether a system of linear equations is consistent or inconsistent, you need to write the system in slope-intercept form, check whether there are exactly similar slopes or different y-intercepts.

An independent system is any linear equation system with only one solution. A dependent system is a linear equation that has infinitely many solutions.

If a linear equation system has exactly one solution, different slopes, and intersecting lines on a graph it is consistent and independent. If a linear equation system has infinitely many solutions, same slopes, same y-intercepts, and coincident lines on a graph (lines exactly on the same place) it is consistent and dependent. If a linear equation system has no solutions, same slope, different y-intercepts, and parallel lines on a graph it is inconsistent.

Example of system of linear equations:

This is an example on how to know how many solutions a linear system of equations has:

y=10-x

-y=8-x

We first have to change the system into slope-intercept form and check if the slopes and y-intercepts are the same or different. If the slopes are the same but the y-intercepts are different then the system does not have a solution and if the slopes and y-intercepts are the same, the system has infinitely many solutions.

 

1.y = 6x − 11

 

−2x − 3y = −7

 

 

2. 2x − 3y = −1

    y = x − 1

 

 

3. y = −3x + 5

   

   5x − 4y = −3

 

 

 

4.−3x − 3y = 3

 

    y = −5x − 17

 

 

  1.  

 

 

 

 

 

Chapter 5 Review

Chapter 1:

1. 322

3. 10.9

5. 14

Chapter 2:

Q1:  True

Q3: -29

Q5. -21, 7

Q7. -39, 13

Q9. -4, 22

 

 Chapter 3: 

Q 1. Discrete

Q 3. Discrete

Q 5. Domain: -9, -2, 4, 11; Range: -5, 2, 8, 15

 

Chapter 4: 

 

Q1. y-3= -1 (x-2)

Q3. y-5=-4(x-9)

Q5. y-321=1.0321(x-123)

Q7. x- 3, y- 3/2

Q9. x- 3/4, y-3/2

Q11. 3/4=b

Q13. 1/4=b

 

Index

Chapter 5: 

 

Q1. (2,3) 

Q2. (1,2)

Chapter 1: Equations  

Chapter 2: Inequalties 

 

Chapter 3: Functions  

 

 

 

Chapter 4: Linear Equations 

 

 

Chapter 5 systems of equations and inequalities 

 

All about Math 

mnx

 

 

 

ISBN 123-023-12567-3240

25 $