Constructing Math

By German Moreno

1.2Introduction to signed numbers

Integers are the set of whole numbers and their opposites. The number line can be used to represent the set of integers. Look carefully at the number line below and the definitions that follow.

Definitions

The number line goes on forever in both directions. This is indicated by the arrows.
Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.
Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line.
The integer zero is neutral. It is neither positive nor negative.
The sign of an integer is either positive (+) or negative (-), except zero, which has no sign.

Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, +3 and -3 are labeled as opposites.

Example 1

The highest elevation in North America is Mt.McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt.McKinley to the bottom of Death Valley?

Understanding the problem

The distance from the top of Mt.McKinley to sea level is 20,320 feet and the distance from sea level to the bottom of Death Valley is 282 feet.

Devising a Plan

The total distance is the sum of 20,320 and 282,

Carrying out the Plan

20,320+ 282 = 20,602 feet.

Looking Back

The distance from the top of Mt.McKinley to the bottom of Death Valley is the same as the distance from +20,320 to -282 on the number line. We add the distance from +20,320 to 0, and the distance from 0 to -282, for a total of 20,602 feet.

Elevation / Integer
20,320 feet above sea level / +20,320
sea level / 0
282 feet below sea level / -282

The problem above uses the notion of opposites: Above sea level is the opposite of below sea level. Here are some more examples of opposites:

top, bottomincrease, decreaseforward, backwardpositive, negative

Practice 1

Write an integer to represent each situation then graph each number on the number line:

10 degrees above zero /
a loss of 16 dollars /
a gain of 5 points /
8 steps backward /

Example 2

Name the opposite of each integer then graph each integer.

-5 -4 2 3 -10

Practice 2

Name 4 real life situations in which integers can be used.

Comparing and Ordering Integers

The set of integers is composed of the counting (natural ) numbers, their opposites, and zero.
Begining with zero, numbers increase in value to the right (0, 1, 2, 3, …) and decrease in value to the left (…-3, -2, -1, 0)
When comparing numbers the order in which they are placed on the number line will determine if it is greater than or less than another number.
If a number is to the left of a number on the number line, it is less than the other number. If it is to the right then it is greater than that number.

Example 3: If the lowest score wins, order the following golf scores from best to worst: Tigre Madera –4, Jack Nickles +1, Nick Cost –2, Freddy Pairs –5, John Weekly +3

Practice 3

The Picksburg running back, nicknamed "the trolley," had five carries with the following results: +18, -3, +4, 0, -1. List these yardage figures from best to worst.

Using the < and > symbols

If there are two numbers we can compare them. One number is either greater than, less than or equal to the other number.

If the first number has a higher count than the second number, it is greater than the second number. The symbol ">" is used to mean greater than. In this example, we could say either "15 is greater than 9" or "15 > 9". The greater than symbol can be remembered because the larger open end is near the larger number and the smaller pointed end is near the smaller number.

If one number is larger than another, then the second number is smaller than the first. In this example, 9 is less than 15. We would have to count up from 9 to reach 15. We could either write "9 is less than 15" or "9 < 15". Once again the smaller end goes toward the smaller number and the larger end toward the larger number.

If both numbers are the same size we say they are equal to each other. We would not need to count up or down from one number to arrive at the second number. We could write "15 is equal to 15" or use the equal symbol "=" and write " 15 = 15".

Example 4

6 -6

Practice 4

-11 1

9 -10

Absolute Value

Absolute value is the distance a number is from zero on a number line. We use |n | to indicate absolute value.

|-n| = |n| =n; where n is an integer.
Example 5

|-7|

Practice 5

|-1000|

|-4|

|21|

Example 6

Practice 6

Example 7

Jack Nickles is –4 after 36 holes of golf. How many strokes away from even par is he? (Par for a course is a score of zero strokes above or below.)

1.3 Adding Integers

Adding Integers That Have The Same Sign

When adding integers of the same sign, we add their absolute values, and give the result the same sign.

Example 1:

2+5=

(-7)+(-2)=

(-80)+(-34)=

Adding Integers That Have Different Signs

When adding integers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the integer with the larger absolute value.

Example 2:

8+(-3)=

Example 3:

8+(-17)=

Example 4:

-22+11=

Example 5:

53+ (-53)=

Properties of Addition

Addition Property of 0

A + 0 = A

Commutative Property of Addition

A + B = B + A

Associative Property of Addition

(A + B) + C = A + (B + C)

Example 6

Application Problems

Example 7

Cynthia had $ in her checking account. She wrote a check for $83 and was charged $17 for overdrawing her account last month. What was her account balance?

Example 8

Which sum is farther from zero, the sum of 101 and 85, or the sum of -98 and -104?

Example 9

Ms. Wilburson's candy store is selling lots of Super Chompers (a kind of candy bar). The numbers of Super Chompers she sold per hour for the first 5 hours of the day are 100, 70, 77, 59 and 34. How many did she sell in those first five hours?

Example 10

If a dense plastic block is dropped into a tank of water, it experiences a change in velocity of -3 m/s. If the original velocity was 20 m/s, what was the velocity immediately after it hit the water?

1.4 Subtracting Integers

Subtracting Integers

Subtracting an integer is the same as adding its opposite.

Examples:

In the following examples, we convert the subtracted integer to its opposite, and add the two integers.
7-4=7+(-4)=3
12-(-5)= 12+(5)=17
-8-7=-8+(-7)=-15
-22-(-40)=-22+(40)=18

Note that the result of subtracting two integers could be positive or negative.

Practice

1. 5 – 7 =

2. 8 – 6 =

3. 4 – (-7)=

4. -5 – 8=

5. -8 – (-8)=

6. -3 – (-5)=

Application Problem

Example 2

In Buffalo, New York, the temperature was -14 ° F in the morning. If the temperature dropped 7° F, what is the temperature now?

Example 3

Roman Civilization began in 509 B.C. and ended in 476 A.D. How long did Roman Civilization last?

Practice 1

In the SaharaDesert one day it was 136° F. In the GobiDesert a temperature of -50° F was recorded. What is the difference between these two temperatures?

Practice 2

The Punic Wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last?

Practice 3

Metal mercury at room termperature is a liquid. Its melting point is -39°C. The freezing point of alcohol is -144°C. How much warmer is the melting point of mercury than the freezing point of alcohol?

Practice 4

The water potential in one plant cell was calculated to be -3 bar. The water potential in another cell was found to be -11 bar. What was the difference in water potentials? Note: the bar is a pressure unit equal to about 14.5 PSI.

1.6 Multiplying Integers

To multiply a pair of integers if both numbers have the same sign, their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If one or both of the integers is 0, the product is 0.

Example 1:

In the product below, both numbers are positive, so we just take their product.
4×3=12

In the product below, both numbers are negative, so we take the product of their absolute values.
(-4)×(-5)=|-4|×|-5|=4×5=20

In the product of (-7)×6, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-7|×|6|=7×6=42, and give this result a negative sign: -42, so (-7)×6=-42.

In the product of 12×(-2), the first number is positive and the second is negative, so we take the product of their absolute values, which is |12|×|-2|=12×2=24, and give this result a negative sign: -24, so 12×(-2)=-24.

Practice 1

To multiply any number of integers:

1. Count the number of negative numbers in the product.
2. Take the product of their absolute values.
3. If the number of negative integers counted in step 1 is even, the product is just the product from step 2, if the number of negative integers is odd, the product is the opposite of the product in step 2 (give the product in step 2 a negative sign). If any of the integers in the product is 0, the product is 0.

Example 2

4×(-2)×3×(-11)×(-5)=?

Counting the number of negative integers in the product, we see that there are 3 negative integers: -2, -11, and -5. Next, we take the product of the absolute values of each number:
4×|-2|×3×|-11|×|-5|=1320.
Since there were an odd number of integers, the product is the opposite of 1320, which is -1320, so
4×(-2)×3×(-11)×(-5)=-1320.

Practice 2

Application Problems

Example 3

Which product is closer to zero, the product of 85 and -102, or the product of -63 and -126?

Example 4

Stephanie is saving for a trip to her cousin’s house in another state. She figures she needs $216 to have a comprtable trip. To earn money she mows lawns. Each mowing earns her $16. She already has mowed seven lawns. How many more lawns must she mow to get at least $216?

Practice 3

Chantele has three children. Her older daughter had a throat culture taken at the clinic today. Her baby received three immunization shots and her son received two shots. THE co-pay amounts we e $8 for each shot, an $18 office charge for each child an a $12 charge for the throat culture. How much did Chantele pay?

Practice 4

There is a 3 degree drop in temperature for every thousand feet that an airplane clibs into the sky. If the temperature on the ground is 50 degrees, what will be the termperature when th plane reaches an altitude of 24,000 feet?

1.7 Dividing Integers

Dividing Integers

To divide a pair of integers if both integers have the same sign, divide the absolute value of the first integer by the absolute value of the second integer.
To divide a pair of integers if both integers have different signs, divide the absolute value of the first integer by the absolute value of the second integer, and give this result a negative sign.

Example 1

In the division below, both numbers are positive, so we just divide as usual.
4÷2=2.

In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second.
(-24)÷(-3)=|-24|÷|-3|=24÷3=8.

In the division (-100)÷25, both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-100|÷|25|=100÷25=4, and give this result a negative sign: -4, so (-100)÷25=-4.

In the division 98÷(-7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |98|÷|-7|=98÷7=14, and give this result a negative sign: -14, so 98÷(-7)=-14.

Practice 1

Application Problems

Example 2

Because of sea floor spreading, the Atlantic Ocean is getting wider at a rate of about one cm per year. At that rate of expansion, how much wider will the Atlantic be at the end of one-fifth of a century?

Example 3

If there were 1,500,220,586 hydrogen atoms in a collection of H2O molecules, how many oxygen atoms were there?

Example 4

The melting point of trifluoroacetophenone is -40C. The melting point of 4–nitroacetophenone is 81C. Another similar compound melts at -46C. What is the average melting point of the compounds?

Practice 2

Three cars came to a sudden stop on the highway. The acceleration of one was -28 m/s2. The acceleration of another was -32 m/s2. The acceleration of the other car was exactly halfway between the other two accelerations. What was the acceleration of the third car?

Practice 3

It was a close race for the pennant (division championship) in the baseball league this year. The Rattlers won. Each team is awarded a point for every game won, and the team with the most points is the winner of the pennant. The Rattlers won 66 games. The Tigers were -2 (two games behind), the lizards were -6, and the rollers were -12. What was the average number of games won by the teams that did not win the pennant?

Practice 4

Mr. Bloop likes to hit fly balls to his nephew on Saturday afternoons. His nephew catches about two out of every five fly balls hit to him. If Mr. Bloop hits 70 fly balls, how many will he probably catch?

1.5Rounding and Estimating

Example 1

1. 96,547

Tens

2. 771,978

hundred thousands

3. -842

tens

4. -63,217

ten thousands

5. -6,024,047

hundreds

Practice

1. 705,072

thousands
2. 6,665

tens

3. -9,897

hundreds

4. -4,838,348

millions

5. -33,892

thousands

Application Problems

Example 2

Sebastian recorded his observations at the bird feeder for a Year. He counted: 226 cardinals, 532 house sparrows, 213 goldfinches, 119 blue jays, 86 mourning doves, 64 downy woodpeckers, and 416 grackles. Estimate the total number of birds he counted by rounding to the nearest tens place?

Practice

Amanda performs an endothermic reaction in a flask on the bench. At the beginning of the reaction the temperature in the flask is 129C. The temperature decreases by 213C. Estimate the new temperature?

1.8 Order of Operations

The Order of Operations is very important when simplifying expressions and equations. The Order of Operations is a standard that defines the order in which you should simplify different operations such as addition, subtraction, multiplication and division.

This standard is critical to simplifying and solving different algebra problems. Without it, two different people may interpret an equation or expression in different ways and come up with different answers. The Order of Operations is shown below.

Example 1

64÷8×5
(9-1)+2×2
82×12-4
46÷2+1
84÷4×44
(9-1)-(35÷7)

Practice 1

5 x 8 + 6 ÷ 6 - 12 x 2

150 ÷ (6 + 3 x 8) – 5

5 - 2×2 = ?

(8 - 3)×4 = ?

(8 - 7)×6 - 10/5 + 4 = ?

7 - (11 - 8) + 14 = ?

(2 + 8)/(6 - 1) + 7×2 = ?

(1 + 2×3) - 7/(4 - 3) + 2 = ?

(12/(3×2) + 4)/(13 - (8 + 2)) = ?

2.1 Introduction to Variable Expressions

Variable

A variable is a letter that represents a number.

Don't let the fact that it is a letter throw you. Since it represents a number, you treat it just like you do a number when you do various mathematical operations involving variables.

x is a very common variable that is used in algebra, but you can use any letter (a, b, c, d, ....) to be a variable.

Algebraic Expressions

An algebraic expression is a number, variable or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots.

2x + y, a/5, and 10 - r are all examples of algebraic expressions.

Evaluating an Expression

You evaluate an expression by replacing the variable with the given number and performing the indicated operation.

Value of an Expression

When you are asked to find the value of an expression, that means you are looking for the result that you get when you evaluate the expression.

Example 1

Write a variable expression that represents the perimeter of the following rectangle and fill in the table below.

Side / Perimeter
8
10
12
14
16

Example 2

r / C

Example 3

Write a variable expression for a cell phone plan and fill in the table below?

Minutes (m) / Cost
(C)
500
1000
1500
2000
2500
3000

Example 4

Write an expression for the volume of object below and fill in the table.

s / t / V

2.2 Combining Like Terms

A term is a constant, a variable or the product of a constant and variable(s)in an expression. In the equation 12+3x+2x2=5x-1, the terms on the left are 12, 3x and 2x2, while the terms on the right are 5x, and -1.

Combining Like Terms is a process used to simplify an expression or an equation using addition and subtraction of the coefficients of terms. Consider the expression below

5 + 7

By adding 5 and 7, you can easily find that the expression is equivalent to 12

What Does Combining Like Terms Do?

Algebraic expressions can be simplified like the example above by Combining Like Terms. Consider the algebraic expression below:

12x + 7 + 5x

As you will soon learn, 12x and 5x are like terms. Therefore the coefficients, 12 and 5, can be added. This is a simple example of Combining Like Terms. You get 17x + 7.

What are Like Terms?

The key to using and understanding the method of Combining Like Terms is to understand like terms and be able to identify when a pair of terms is a pair of like terms. Some examples of like terms are presented below.

Example 1

The following are like terms because each term consists of a single variable, x, and a numeric coefficient.

2x,

5x,

0x,

-2x,

-x

Each of the following are like terms because they are all constants.

-2,

27,

9043,

0.6

Each of the following are like terms because they are all y2 with a coefficient.

3y2,

y2,

-y2,

-6y2

For comparison, below are a few examples of unlike terms.

The following two terms both have a single variable with an exponent of 1, but the terms are not alike since different variables are used.

17x,

17z

Each y variable in the terms below has a different exponent, therefore these are unlike terms.

15y,

19y2,

31y5

Although both terms below have an x variable, only one term has the y variable, thus these are not like terms either.

19x,

14xy