NEW Number Concepts and Operations

UNDEFINED Any non-zero number divided by itself (EX: 3/3) is ONE. However when x=0 in x/x; the quotient is UNDEFINED, because the denominator is 0.

0/0 is

UNDEFINED Any division problem with a 0 in the denominator is UNDEFINED there is no division occuring if you are trying to divide by 0

3/0 is

C 22 residents per square mile

A county with an area of 425 square miles has a population of 9,350 residents. Which rate best represents the relationship between the population of the county and the area of the county? A 22 square miles per resident B 9,350 residents per square mile C 22 residents per square mile D 425 square miles per resident

24 2 8 4

A group of 24 friends are making teams to play tag. Every player must be on a team, and each team must have the same number of players. If there is 111 team, there will be _____players on the team. If there are _______ teams, there will be 121212 players on each team. If there are 333 teams, there will be ______ players on each team. If there are _____ teams, there will be 666 players on each team.

A. 4%

As part of a survey, 300 girls were asked to name their favorite sport. The results showed that 12 of the girls named bowling as their favorite sport. What percentage of the girls in the survey named bowling as their favorite sport? A. 4% B. 12% C. 25% D. 0.04%

The order of the addends will not change the sum. EX: (2 + 3) + 5 = 10 2 + (3 + 5) = 10

Associative Property of Addition

The order of the factors will not change the product. EX: (2 x 3) x 5 = 30 2 x (3 x 5) = 30 Associative property of multiplication: Changing the grouping of factors does not change the product. For example, (2 \times 3) \times 4 = 2 \times (3 \times 4)(2×3)×4=2×(3×4) left parenthesis, 2, times, 3, right parenthesis, times, 4, equals, 2, times, left parenthesis, 3, times, 4, right parenthesis.

Associative Property of Multiplication

(either by dividing by a positive power of ten, like 10, 100, 1000, etc. or by multiplying by a negative power of ten, like 1/10, 1/100, 1/1000, etc.) - move the decimal point to the left - So, to divide 54 by 100 (or to multiply 54 by 100 (or to multiply 54 by 1/100 or its equivalents, 0.01 or 10⁻² ) makes 54 into 0.54 after two moves of the decimal point to the left.

Base Ten: To make a number smaller

Algorithm- is an established and well-defined step-by-step problem-solving method used to achieve a desired mathematical result

Basic operations and their Algorithims

- divide the numerator by the denominator and represent the remainder as a fraction EX: 5/2 = 4 + 1 /2 = 2 X 2 +1 / 2 = 4/2 + 1/2 = 2 1/2

Changing improper fractions to Mixed numbers

A. All whole numbers are integers.

Choose the statement that best describes the relationship between whole numbers and integers. Choose 1 answer: A. All whole numbers are integers. B. All integers are whole numbers.

The coefficients are the numbers that multiply the variables or letters.

Coefficient

4p + 3

Combing like terms to create an equivalent fraction for 4p + 6 - 3

4x

Combing like terms to create an equivalent fraction for 5x-x

algebra tiles base-ten blocks counting colored squares

Concrete representation examples

Putting your keys in your front pocket and your wallet in your back pocket. - For the commutative property to hold, the end result must be identical in spite of the order the actions were preformed.

Determine which activity can be used to illustrate the commutative property to a fifth-grade class? A.Washing clothes and drying clothes. B.Put your keys in your front pocket and your wallet in back pocket. C. After dark, turning your lights on your car and driving your car. D. Washing dishes and putting them in the cabi

Associative The associative property of multiplication is (a*b)*c = a*(b*c). Terms stay in the same order on each side of the equals sign, but are associated differently

Determine which property describes: (a*b)*c=a*(b*c)

in mathematics a digit is a number symbol (e.g., 1,2,3) used in numerals to represent numbers (these are real numbers or integers) in positional numeral systems

Digit

says that everything within the parentheses needs to be multiplied by the number on the outside - You can add and then multiply and then add a(b + c) = a X (b + c) 8(5 + 2) = 8 X 7 = 56 OR a(b + c) = (a X b) + (a X c) 8(5 + 2) = (8 X 5) + (8 X 2) = 56 -ALSO WORKS WITH subtraction 4(8 - 5) = 4(8) - 4(5) = 32 -20 = 12 AGAIN, if the quantity in the parentheses is evaluated first 4(8-5) = 4(3) = 12

Distributive Property

the expanded form of numbers shows the place value of each digit EX: 263 = 200 + 60 + 3

Expanded Form

1. Change the percent to a common fraction or a decimal fraction 2. Multiply the fraction times the quantity 3. The percentage is expressed in the same units as the known quantity. EX: To find 25% of 360 books, change 25% to 0.25 and multiply times 360 0.25 X 360 = = 90 books

Find the Percentage of a Known Quantity *remember "percent" means "for every one hundred"

B. ab/2² Of the choices given, ab/2 is the LCM. Since both numbers are even, both have factors of 2. So, the product can be reduced by 2. If the numbers have only one factor of 2 in either of them such as in 6 and 16, then 96 is the product and a multiple. However, if 96 is reduced by 2, then 48 is the LCM.

Given two integers, (a) and (b) what could be the least common multiple (LCM)? A. ab B. ab/2 C. Same as the least common multiple for two odd integers D. Greatest Common Factor

- can be multiplied as if they were whole numbers

Multiplying Decimals

x9 cannot write the variable in FRONT OF CONSTANT, the constant must always be written in FRONT of variable ( 9x )

NOT a proper way to denote multiplication

8²- 60 < 11¹ < 2² x 50

Order the expressions from least to greatest 11¹, 8²- 60, 2² x 50

-the basic foundation for understanding mathematics computation

Place Value

$54.00 90*.20=18, 90-18=72 , 72*.25= 18, 72-18=54

Samantha works at a clothing store where she receives an employee discount of 20% on all clothing items. She wants to buy a sweater that usually sells for $90, but is on sale for 25% off. How much will the sweater cost Samantha if she applies both the 25% off sale price and her 20% employee discount

is a form of writing a number as the product of a power of 10 and a decimal number greater than or equal to 1 and less than 10 EX: 2,400, 000 = 2.4 X 10⁶ ; 240.2 = 2.402 X 10² ; 0.0024 = 2.4 X 10-³

Scientific Notation

S¹¹ . t¹⁰

Simplify S³ . t⁴ . S⁸ . t⁶

Algebraic expressions, like 5x or 6y² are considered "like" each other if they have the same variable parts raised to the same power (sometimes known as the degree) EX's: "5x" and "8x" are "like" terms, while "5x" and "5y" are not. neither are "5x" and "5x³ Similarly, "6y²" and "-1.4y²" are like terms, but "6y²" and "6x²" are not. Neither are "6y²" or "6y⁵" - All constant terms (ordinary Real numbers with no variable parts) are like terms with each other b/c they all have NO variable at all (and therefore do have "the same variable parts raised to the same power")

The concept of "Like Terms"

they have no factors (besides 1) in common EX: 34 and 15 are relatively prime b/c 34 = 2 X 17 and 15 = 3 X 5 and there are no factors in common NOTE: that the least common multiple of two numbers that are relatively prime can be found by multiplying the two numbers together: LCM of 34 and 15 is 34 X 15 or 510

Two numbers are relatively PRIME if

1. we can factor each number completely as a product of prime numbers 42= 2 x 3 x 7 28 = 2 x 2 x 7 70 = 2 x 5 x 7 2. Find the factors that are COMMON to EACH number. ( 2 and 7) 3. Multiply 2 x 7 = 14

What is the GCF of 42, 28, and 70?

13

What is the coefficient of the term 13a in the expression 13a+6b ?

Think of your checking account being overdrawn by $7 and you write another check for $12. How much would be in your checking account?

Which choice best describes an explanation a teacher could give a student to help him/her judge the reasonableness of the solution in the following scenario. Evaluate: -7 - 12 The student gives an answer of 5.

0¹ The number raised to any non zero number is one

Which choice is a real number?

rational

Which number system is a repeating decimal a member of?

D. 6 = 3 x 2

Which equation shows that 3 is a factor of 6? A. 9 = 3 + 6 B. 6 = 3 + 3 C. 18 = 3 x 6 D. 6 = 3 x 2

D. 18 = 6 x 3

Which equation shows that 6 is a factor of 18? A. 18 = 24 -6 B. 18 = 6 + 12 C. 18 = 108 ÷ 6 D. 18 = 6 x 3

C. 7 and 7 E. 1 and 49

Which of the following are factor pairs for 49? Choose all answers that apply: A. 2 and 23 B. 3 and 13 C. 7 and 7 D. 4 and 11 E. 1 and 49

A. 3 and 22 B. 2 and 33 C. 6 and 11 D. 1 and 66

Which of the following are factor pairs for 66? Choose all that apply A. 3 and 22 B. 2 and 33 C. 6 and 11 D. 1 and 66 E. 4 and 16

D. 5

Which of the following numbers is a factor of 55? A. 3 B. 7 C. 6 D. 5

D. 72

Which of the following numbers is a multiple of 8? A. 60 B. 35 C. 18 D. 72

- "Like Terms" are added and subtracted with each other. - they are added or subtracted by adding or subtracting their coefficients and replacing the variable parts unchanged. - One can use the Distributive Property and Commutative Property of Multiplication to justify this process. 5x + 8x = x(5 + 8) = x(13) = 13x OR one can think of coefficient as a system to track how many of a particular quantity exist, and so "5x + 8x" means that there are 5 of something and then 8 more of that same type of thing, which combine to yield a total of 13 of that thing.

"Like Terms" can be added and subtracted with each other

UNDEFINED/INDETERMINATE You can raise nothing to nothing This differs than if any other number is raised to the 0 power

0⁰ is

12p

9p + 3p=

40%× 35% .40× .35= .14, .14×N = 14N

A recent survey reveals that 40% of young people between 15 and 25 years of age will buy an iPod this year. Of those who buy iPods 35% enjoy listening to country music. If N is the number of young people between 15 and 25 how many people between 15 and 25 this year will enjoy listening to country music on their iPods?

1800×.33= 594, 360×.25=90, 100×.20=20 1800-594=1206 , 360-90=270 ,80 1206+270+80= 1556

A store is having an end-of-the-year computer sale. What is the total cost of a discounted computer system (i.e., computer, monitor, and printer)? Item Price Discount Computer $1,800. 33.3% Monitor. $360. 25% Printer. $100. 30%

Have the class research products online such as stereos, game stations, and etc, and choose an item they would like to purchase.

A teacher decides to assign students a project where they will develop an amortization schedule of a purchase so they can apply a lesson using the future value formula and observe the details line by line as payments are made. What would be the best approach in assigning a purchase?

is the distance of a number from zero on the number line - this action ignores the + or - sign of a number - the notation for absolute value is 2 vertical lines , so |x| is the graphic used to describe the action of absolute value. EX: |-5| = 5 OR |5| = 5

Absolute Value

128

Amy cut 32 feet of chain into pieces that were each 1/4 ft long. How many of these pieces did Amy have after cutting the chain? Be sure to use the correct place value

2y + 6

Combing like terms to create an equivalent fraction for 3y + 6 - y

2 terms (abc, def) 3 factors

Complete the statement to describe the expression abc + def The expression consists of _____ terms, and each term contains _____ factors.

6.5

Convert 13/2 to a decimal.

1.8628 x 10¹³

The United States debt is over 18,628,000,000,000. How would you most likely see this number correctly represented?

7.14

Estimate √51

are fractional numbers that are written using base ten . - A mixed decimal number: has a whole number part as well EX: 0.28 is a DECIMAL number EX: 3.9 is a MIXED decimal number - Decimal numbers are fractions whose denominators are powers of 10 (e.g. 10, 100, 1000, and so forth) EX: 0.098 is equivalent to 98/1000

Decimal Numbers

10^3 ~10,000,000/8,000=1,250 ~10^2=100 ~10^3=1000

Determine the best estimate of how many months it will take to spend 10 million dollars if you spend $8,000 a month?

-is a mathematical operation involving 2 numbers that tell how many groups there are or how many are in each group. -May be taught as THE INVERSE OF MULTIPLICATION -DIVISOR- (number doing the dividing/outside bracket) -DIVIDEND-(number being divided/inside bracket)

Division

3 + (-3) = 0 An additive inverse is a number added to the original number that results in the addictive identity element 0. The numbers 3 and -3 are opposites thus their sum is

Example of the additive inverse element:

just as expanded form, show place value by multiplying each digit in a number by the appropriate power of 10. 523 = 5 X 10² + 2 X 10 + 3 X 1 OR 5 X 10² + 2 X 10¹ + 3 X 10⁰

Expanded Notation

a symbolic way of showing how many times a number or variable is used as a factor -In the notation 5³, the exponent shows that 5 is a factor used three times 5³ = 5 X 5 X 5 = 125 - a NEGATIVE EXPONENT indicates a reciprocal EX: 5-³ = 1/5³ = 1 / 5x5x5 = 1/125

Exponential Notation

17 -- 24

Express 0.7083 as a fraction.

4.375 - 3÷8

Express 4 3⁄8 as a decimal number.

The GCD of two or more nonzero integers is the largest positive integer that divides into the numbers without producing a remainder - sometimes the term used is "divides evenly into" - This is useful for simplifying fractions into their lowest terms

Greatest Common Divisor

they cannot be simpliied throuhg addition or subtraction. For EX: "2x + 9y" is considered FULLY simplified. So. is "y + 3y⁵" NOTE: while unlike terms can't be added or subtracted, such terms CAN be multiplied and divided. That material, however is not relevant to a standard elementary school curriculum so won't be explored)

If terms are NOT like each other

-is a whole number that includes all positive and negative numbers, including 0 -This may be representesd on a number line that extends in both directions from 1. -You might have -45, -450,000, 0, 234, or 78,306 -integers do not include decmials or fractions -many real life situations are represented in integers

Integers

1. Find GCF 2. GCF is 11 3. 11 inches wide

Jazmin is completing an art project. She has two pieces of construction paper. The first piece is 44 inches wide and the second piece is 33 inches wide. Jazmin wants to cut the paper into strips that are equal in width and are as wide as possible. How wide should Jazmin cut each strip?

Her discourse cut off opportunities for sense-making by hurrying students through the solution of the task.

Not knowing where to begin, the students began to urge her to give them some help. Wanting them to feel successful and stay engaged, Ms. Jones pointed out to the students that the problem involved subtracting from the time Pat and her sister got home the minutes and hours they were out. She told her students that they needed to convert hour to minutes when subtracting minutes, and that there are 60 minutes in an hour.

1. Find LCM for 7 and 9 2.To find the number of times Miranda went to the store, we need to divide by how many dollars she spends each time: 63 ÷ 7 = 9 3.The least possible number of times that Miranda has been to the store is 9 times.

Miranda and Savannah are excited that a new store just opened in town! They go together the first day it opens! Each time Miranda goes to the store she plans to spend $7, and each time Savannah goes to the store she plans to spend $9. A few weeks from now, Miranda and Savannah are surprised to find out that they have spent the exact same total amount of money at the store. What is the least possible number of times that Miranda has been to the store? _______ times

J. p = (185-25) ÷ 40

Mr. Anderson had 185 pieces of wood. He sold 25 pieces of wood to his neighbor and stacked the rest of the wood into piles around his house. Each pile of wood contained 40 pieces of wood. Which equation can be used to find p, the number of piles of wood Mr. Anderson made? F. p = (185+25) + 40 G. p = (185 25) 40 − 40 H. p = (185+25) X 40 J. p = (185-25) ÷ 40

It will not engage students in high level forms of thinking

Mr. Grant wants his fourth-grade math students to learn to work collaboratively, to discuss alternative approaches to solving tasks, and to justify their solutions. Students are grouped in pairs and Mr. Grant gives his students the following task: Express the ratios below in the lowest forms: 15/25, 18/6, 9/36, 18/15 Which of the following best describes the level of student engagement expected with the given task?

B. instruction in math should build on existing knowledge of math.

Mr. Jansen's second-grade students have been working on addition and are now ready to begin learning two-digit subtraction. To introduce this new unit of study he first has students work on addition problems they are familiar with and then follows up by demonstrating to them how those problems are related to subtraction. This approach best demonstrates Mr. Jansen's understanding that — A. students with learning disabilities learn much slower than other students. B. instruction in math should build on existing knowledge of math. C. he must first review students to make sure they understand addition. D. he must build student confidence before beginning new instruction.

Commutative and associative Jamie used the commutative property to change the order of the numbers to be added when he moved the 45 up with the 15. Then he used the associative property to group the 15 and 45 for a sum of 60 and the 12 and 28 together for a sum of 40.

Mrs. Johnson teaches fourth graders with learning disabilities. She asks them to add the following numbers: 15 12 28 +45 Jamie responds very quickly with 100. When Mrs. Johnson asks him how he solved it he replies: "I added 15 and 45 and got 60 and then I added 12 and 28 and got 40. Next, I added 60 and 40 together and got 100." Which of the following two properties of numbers did Jamie use to solve this problem?

the order of operations

Mrs. Johnson teaches fourth graders, some with learning disabilities. Many of them are having trouble with expressions similar to this one: 12 + (7 X 2) - 2(5 + 3). Students' trouble is most likely the result of misunderstanding

she should teach math concepts from the concrete to the representational to the abstract.

Ms. Senath teaches math to a diverse group of students in her fourth-grade class, some of them with learning disabilities. It is important for her to remember when presenting math instruction to her students that —

1. Find GCF 2. Divide number of boys by GCF ( 35÷ 5 = 7) 3. 7 BOYS

Rafaela is a physical education teacher and has 25 girls and 35 boys in her class. She wants to divide the class into teams of the same size, where each team has the same number of girls and the same number of boys. If Rafaela creates the greatest number of teams possible, how many boys will be on each team?

Multiples for 6 and 9 until the same number 18 days

Ronald and Tim both did their laundry today. Ronald does laundry every 6 days and Tim does laundry every 9 days. How many days will it be until Ronald and Tim both do laundry on the same day again? ____ days

is the process/operation of removing objects from a larger group, or finding parts of a whole. -when a number (formally called a subtrahend) is subtracted from another number (formally called a minuend), the resulting number (the difference) is smaller than the minuend. -CHILD MIGHT knwo the operation, butnot understand the word "left" in problem that calls for subtraction. -TEACHERS NEED TO develop a list of key words linked to the 4 basic operations -It is not approriate to teach a procedure without teaching them how to reason

Subtraction

The number 0 raised to any non-zero number is still 0. Because, no matter how many times you multiply nothing by nothing, you still have nothing, and since 0 is considered a WHOLE NUMBER, it must also be a REAL NUMBER.

The number o¹ is a REAL NUMBER

Total amount invested - Principal

The principal is $127,000 and the monthly payments are $583.56. The students want to know how much interest over the period of the note (30 years) will accrue. They decide the first step is to calculate the total amount to be invested in the house. Payment amount x Number of payments = Total amount invested What should the next step of the algorithm be?

Rational numbers, integers, whole numbers, and natural numbers The number 1 is a member of the natural numbers. Since the natural numbers are a subset of the whole numbers {0, 1, 2, 3, 4,...} and the whole numbers are a subset of the integers, and the integers are a subset of the rational numbers.

The product of two rational numbers can belong to which of the following combinations of sets of numbers?

zero Zero is a number. Zero can be a solution. Zero has a value and is only represented

The symbol Æ is often used in mathematics. This symbol does not represent (the) ____

Least Common Multiple The smallest number that both 3 and 5 will divide into without a remainder is 15. A multiple of 3 and 5 is a number that both 3 and 5 divide into evenly. Thus, 15 is the LCM

The teacher creates the word problem: Cass and Tori are neighbors. They both have mowed their grass today. Cass mows her lawn every 3 days. Tori mows her grass every 5 days. How many days will it be before they both mow their grass on the same day? Based on the word problem, what will the lesson most likely cover?

*Cardinal The meaning of five in this example is a cardinal meaning. It is the number of triangles in the set. The ordinal meaning of five would be the 5th member in a row. Initial rot sequence occurs before children pair numbers with objects. It is just the expression more of word chunk

The teacher has her pre-kindergarten students count the number of triangles in the set: one-two-three-four-five. To reinforce the idea, she has them count the triangles again: one-two-three-four-five. She asks the students, "What is the special meaning of five here?" Which of the following meanings of the number five is the teacher asking for?

Relative magnitude -Relative magnitude includes ordering or comparing place value of real numbers on a number line.

The teacher tells the class to put the following numbers in order from smallest to largest. 8 -3.5 9⁄2 π 2.1 2.03 √ 2 Based on the exercise the teacher has given, the exercise is most likely trying to assess which of the following number concepts.

Place Value

Three sudents are playing a game using the spinner above. Each student draws three boxes on a piece of paper. The spinner is spun, and each student writes the number in any one of the three boxes. The spinner is spun two more times, and each time the students write the number in a remaining empty box. The students then compare numbers, and whoever has the largest three-digit number wins the game. This game would be particularly useful in helping students' understanding of:

To CONVERT a percent to a decimal fraction: divide the percent by 100, or move the decimal point 2 places to the left EX: 6% = 0.06 - b/c "/100" means to divide by 10² b/c of the base ten system, division by 2 powers of ten is accomplished by moving the decimal 2 places to the left. EX's: 100% = 10/10 = 1.0 25% = 25/100 = 0.25 150% = 150/100 = 1.5

To CONVERT a percent to a decimal fraction: *remember "percent" means "for every one hundred"

To CONVERT percent to a common fraction: place the percent in the numerator and use 100 as the denominator (simplify if necessary) and reduce as far as possible. 6% = 6/100 = 3/50

To CONVERT percent to a common fraction: *remember "percent" means "for every one hundred"

B. move the decimal to the left 2 places

To convert a percentage to a decimal, a student can A. move the decimal to the left 1 place B. move the decimal to the left 2 places C. move the decimal to the right 2 places D. move the decimal to the right 1 place

$988.00 $5150 - $320 - $465 - $250 = $4115 Now, multiple by 24%. $4115 x 0.24 = $988

Troy and Rosa want to purchase a house. The bank will approve a loan for a house including property tax and insurance on the house for 24% of their adjusted monthly income. Their combined gross monthly income is $5150. They have a car payment of $320 and pickup payment of $465. Also, they have a furniture payment of $250. What is the maximum payment the bank will approve?

1 7

Tyler is cleaning up his 35 toy cars. He wants to put every car in a toy bin, and he wants each bin to have the same number of cars. If there is______ bin, there will be 35 cars in the bin. If there are 5 bins, there will be _____ cars in each bin.

A. 42 is a multiple of 7 B. 6 is a factor of 42

We know 6 x 7 = 42 So, which of the following statements are also true? Choose all answers that apply: A. 42 is a multiple of 7 B. 6 is a factor of 42 C. 42 is a factor of 7

A. It increases

What happens to the value of the expression 1+3ƒ as ƒ increases? A. It increases B. It decreases C. Stays the same

It decreases

What happens to the value of the expression 10 −− as "d" increases from a small positive number d to a large positive number? A. It increases B.It decreases C. It stays the same

it decreases Let's think about 100 - x, for different values of x. x 100-x 0 100 1 99 2 98 10 90 50 50 Looking at the table, is 100 - x increasing or decreasing as x increases? 100-x decreases as x increases *IT DECREASES*

What happens to the value of the expression 100-x as x increases?

2x2x2x2x2x2

What is the prime factorization of 64?

1 raised to any power is always still 1

What is the solution, if any, to the expression: 1^n = ?

D. Greatest Common Multiple

Which of the following concepts is needed to solve the word problem: 'In planning a mosaic, an artist counts up her yellow tiles and green tiles. She counts 28 yellow, and 42 green. She wants to create areas that have the same number of rows of each color. What is the greatest number tiles each row can have and still share in common?' A. Least Common Multiple B. Least Common Factor C. Greatest Common Factor D. Greatest Common Multiple

Y2-5=0 An irrational number cannot be expressed as a fraction, in decimal form it goes to infinity. A rational number can be expressed as a fraction or a ratio. 5 is a prime number.

Which of the following has an irrational rather than a rational solution? A. Y2 - 36 = 0 B.Y2 - 49 = 0 C. Y2 - 16 = 0 D. Y2 - 5 = 0

C. 0⁰ Any variable raised to the zero power is

Which of the following is not equivalent to 3⁰? A. -2⁰ B. 1 C. 0⁰ D. a⁰

Distributive property

Which property is used in answering the following problem? Given the area of this rectangle is 56 in^2, write the equation to solve for x. (5+x)(4x)

A. Whole B. integers C. rational

Which set(s) does the number 3 belong to? A. whole numbers B. integers C. rational numbers D. irrational numbers

1, 2, and 3 whole numbers integers rational numbers

Which set(s) of numbers does 3 belong to? 1. whole numbers 2. integers 3. rational numbers 4. irrational numbers

Whole numbers are the numbers starting at 000 and counting up forever: {0, 1, 2, 3, 4, 5, 6, 7, 8,9,10, 11...} Key idea: Whole numbers don't include negative numbers, fractions, or decimals. set of natural numbers and ZERO "0" - it starts at ZERO "0" and goes up by 1's to to create the infinite set 0, 1, 2, 3, 4, 5 ,.....

Whole Numbers

36 factor numbers then do 12 x 3 = 36

Yadira's mom is buying hot dogs and hot dog buns for the family barbecue. Hot dogs come in packs of 12 and hot dog buns come in packs of 9. The store does not sell parts of a pack and Yadira's mom wants the same number of hot dogs as hot dog buns. What is the smallest total number of hot dogs that Yadira's mom can purchase? ______ hot dogs

1. Find GCF for 72 and 24 2. GCF is 24 3.The greatest number of identical packages that Zayed can make is 24 4.To find the number of pencils in each package, we need to divide the total number of pencils by the number of packages: 72 ÷ 24 = 3

Zayed is helping his classmates get ready for their math test by making them identical packages of pencils and calculators. He has 72 pencils and 24 calculators and he must use all of the pencils and calculators. If Zayed creates the greatest number of identical packages possible, how many pencils will be in each package? _____ pencils

The LCM is the smallest common multiple of two or more numbers. For example, the LCM of 2 and 5 is 10. The least common factor is the smallest common factor of two or more numbers. The smallest factor of every number is 1. So the least common factor of any two numbers will be 1.

What is the difference of LCM and least common factor

53

What is the prime factorization of 53?

3x3

What is the prime factorization of 9?

A. Whole number AND B. Integer

What type of number is 12? Choose all answers that apply: A. Whole number B. Integer C. None of the above

C. None of the above

What type of number is 5.15.15, point, 1? Choose all answers that apply: Choose all answers that apply: A. Whole number B. Integer C. None of the above

1. A composite number is a number that has more than two factors (including 1 and itself) 2. 11, 29,43, and 47 each have only 2 factors 3. The factors of 25 are 1, 5, and 25 4. 25 is the composite number

Which of these numbers is composite? A. 11 B. 25 C. 29 D. 43 E. 47

If it has no factor, then it would be a prime: a number with no factors. Examples of primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.

What if the number you have has no multiples?

Simply find the prime factorization of the numbers. Prime factorization is the representation of a number with only its prime factors. Multiply the prime numbers that're shared in each number only once, cross out these numbers, and then multiply the rest of the numbers. Example: find the least common multiple of 3, 4, and 90. Prime factorization of 3:3 Prime factorization of 4: 2*2 Prime factorization of 90: 2*3*3*5 Shared prime factors: 2,3 Therefore, the lcm is 2*3*2*3*5, which is 180.

What if you have really big numbers? Is there a trick to find LCM?

integer

What type of number is -3

- Both addition and multiplication are commutative properties - subtraction and division ARE NOT commutative (however 5-4= 1 but 4-5 = -1) changing the order of the addends (terms being added) of factors (in multiplication) do not change the result. a + b = b + a EX: 6 + 8 = 14 8 + 6 = 14

Commutative Property of Addition

31

What is the prime factorization of 31?

EX: 91 is close to 100, so it's square root will be close to 10. -So, to determine if 91 is prime or composite, the division should be used to test each prime number less than 10 (2, 3, 5, and 7) - 91 ÷ 2 yields 45.5, a decimal - 91 ÷ 3 yields 30.333...., a decimal - 91 ÷5 yields 18.2, a decimal - 91 ÷ 7 yields exactly 13 - Therefore, 91 is a composite number, composed of the two factors 7 and 13. EX: 101 is close to 100, so its square root should also be close to 10. - division should be used to test each prime number less than 10 (2, 3, 5, and 7) - 101 ÷ 2 yields 50.5, a decimal - 101 ÷ 3 yields 33.666..., a decimal - 101 ÷ 5 yields 20.2, a decimal - 101 ÷ 7 yields 14.428571428571428571..., a decimal - Therefore, 101 is a PRIME number, with no natural number factors besides

-To test whether a a given value is a PRIME or COMPOSITE

1. Area = l x w The area of the rectangle is 18 square centimeters, so we're looking fo 2 numbers that have a product of 18 2. 1 x 18 = 18 2 x 9 = 18 3 x 6 = 18 3. 4 x 5 = 20 4. The following could be the rectangles length and width: 1 cm and 18cm 2 cm and 9 cm 3 cm and 6 cm

A rectangle has an area of 18 square centimeters. Which of the following could be the rectangle's length and width? (Area = l x w x h) 1 cm1, space, c, m and 18\text{ cm}18 cm18, space, c, m A. 1 cm and 18 cm B. 4 cm and 5 cm C. 2 cm and 9 cm D. 3 cm and 6 cm

A. 2 and 10 C. 1 and 20 D. 4 and 5

A rectangle has an area of 20 square centimeters. Which of the following could be the rectangle's length and width? (Area =equals length \times×times width) Choose all that apply A. 2 and 10 B. 3 and 8 C. 1 and 20 D. 4 and 5

B. 9 cm and 9 cm C. 3 cm and 27 cm D. 1 cm and 81 cm

A rectangle has an area of 81 square centimeters. Which of the following could be the rectangle's length and width? (Area = equals length x width) Choose all answers that apply: A. 7 cm and 12 cm B. 9 cm and 9 cm C. 3 cm and 27 cm D. 1 cm and 81 cm

1. To find the number of trees, we need a number that is a factor of 54 and 27, so they both can be divided up into equal rows. 2. So if each row had 9 oak trees, there would be 54 ÷ 9 = 6 rows of trees, and 27 ÷9 = 3 rows of pine trees * this creates equal rows, but is NOT the greatest number of trees per row (GCF). 3.To find the greatest number of trees, we want to find the greatest common factor of 54 and 27start color green, 27, end color green. 4. To do so, find factors of 54 and 27 5. 54: 1,2,3,6,9,18,27,54 27: 1,3,9,27 6. The GCF is 27 the greatest number of trees he can plant in each row is 27

Abe is going to plant 54 oak trees and 27 pine trees. Abe would like to plant the trees in rows that all have the same number of trees and are made up of only one type of tree. What is the greatest number of trees Abe can have in each row? _______ trees

a number that, when added to any element x in a set, always yields x. - The idea of an additive identity is that an initial value is kept IDENTICAL to what began as, even while a quantity is added to it. - The quantity added to it, of course, must be zero, as in, 5 + 0 = 5 -ZERO itself is sometimes referred to as "The Additive Identity" -The Additive Identity is used quite often in an algebraic context, for example, in order to "cancel out" a value from one side of an equation and bring it to the other side.

Additive Identity

1. 593262 is divisible by 9 if it can be divided by 9 without leaving a remainder 2. A number is divisible by 9 if the sum of the digits is divisible by 9 3. Add the digits 5+9+3+2+6+2 If 27 is divisible by 9, then 593262 mut be divisible by 9 4. 27 is divisible by 9, therefore 593262 must also be divisible by 9

Is 593262 divisible by 9?

Inverses, in general are opposites - when an additive inverse is added to a number, it "cancels out/it yields the Additive Identity/it yields the Additive Identity/it has a sum of zero. - The additive inverse of a number always has the same ABSOLUTE VALUE (numbers distance from zero) as the original value, but with the opposite sign , like 5 and -5 equals 0. - The additive inverse of -2 is 2 b/c -2 + 2 = 0 - The additive inverse can be found by subtracting the original number from zero.

Additive Inverse

decimal representations of fractional parts rather than working with the actual fractions THIS IS NOT A WISE mathematical choice- b/c decimals must sometimes be rounded giving an APPROXIMATE answer. while EXACT answers can be found if fractions are used.

B/C the use of claculators has become commonplace in the classroom, it is often the case that students move to -

Babylonians used a base 60 (sexigesimal) system while the - Mayans used a base 20 (vigesimal) system -Computer languages use a base 10 system with Hindu-Arabic origins - The Common curret number system passed along to students now is a base 10 system with Hindu-Arabic origins

Babylonians/Mayans Hindu/Arabic

Any nonzero non-zero number raised to the zero power = one - with one exception -0 itself cannot be raised to the power of 0 at all, that operation, much like division by 0, is UNDEFINED. - the tens place is 10¹ - the hundred place is 10² - the thousandths place is 10³, etc. - In the opposeite direction of the ones place, there is a tenths place from 10⁻¹ or 1/10 - the hundreths place rom 10⁻² 1/100 - the thousandths place from 10⁻³ or 1/1000 7 X (10⁰)= 7(1) = 7

Base 10 system

(whether X by a positive power of ten, lie 10, 100, 1000, etc or b/c of dividing by a fraction/power with a negative exponent on ten, like 1/10, 1/100, 1/1000., etc.) MOVE THE DECIMAL TO THE RIGHT - move one position for each power of 10 EX: So 54 multiplied by 10 (or divided by 1/100-same effect either way) uses 10² and so requires 2 moves of the decimal point - While no actual decimal point is wrtten in "54" , it is properly understood to be "54.0," so two moves to the right makes the product of 54 x 100 = 5,400

Base Ten: To make a number larger

1 + 1/2 + 5 + 6 1/2 + 1 to get a close estimate, 14 of the actual answer 14.131666.... When asked to estimate the answer to the addition problem above, students should first determine what whole numbers or benchmark fractions the numbers are closest to, then replace the original values with the benchmarks, and finally add the easier-to-work-with values in order to find an appropriate estimate.

Benchmarking Fractions in order to obtain reasonable estimates for tedious fractions EX: 7/8 + 2/3 + 5.2 + 6 1/2 + 0.89

A set is considered closed for an operation if the operation on any 2 elements in the set always yields a value that is a member of the set. EX: the set of whole numbers is closed under the operation of addition b/c the sum of any two whole numbers is a whole number. EX: the set of whole numbers is also closed under the operation of multiplication for a similar reason. - the set of Whole numbers is NOT CLOSED under wither SUBTRACTION or DIVISION , b/c the difference -answer to a subtraction problem - of 2 whole numbers does not always result in a whole number (for ex: 9 - 12 = -3 and -3 is an integer, but not a whole number) The quotient- -of 2 whole numbers does not always result in a whole number EX: 6 divided by 5 = 1.2 -1.2 is a Rational number, but not a Whole number

Closure

a whole number that is a multiple of two or ore given numbers EX: common multiples of 2, 3, and 4 are 12, 24, 36, 48

Common Multiple

No. LCM stands for Least Common Multiple. A multiple is a number you get when you multiply a number by a whole number (greater than 0). A factor is one of the numbers that multiplies by a whole number to get that number. example: the multiples of 8 are 8, 16, 24, 32, 40, 48, 56... the factors of 8 are 1, 2, 4, 8. The term least common factor doesn't really make sense since the least common factor of any pair of numbers is 1.

Is the least common factor the same as the lcm?

Commutative property of multiplication: Changing the order of factors does not change the product. For example, 4 \times 3 = 3 \times 44×3=3×44, times, 3, equals, 3, times, 4. The commutative property of multiplication says that changing the order of factors does not change the product. Here's an example: 4 \times 3 = 3 \times 44×3=3×44, times, 3, equals, 3, times, 4 Notice how both products are 121212 even though the ordering is reversed. Here's another example with more factors: 1 \times 2 \times 3 \times 4 = 4 \times 3 \times 2 \times 11×2×3×4=4×3×2×11, times, 2, times, 3, times, 4, equals, 4, times, 3, times, 2, times, 1 Notice that both products are 242424. a X b = b X a EX:a X b = b X a

Commutative Property of Multiplication

numbers written in the form: a + bi where i = √-1 and where "a" and "b" are real numbers - the value of "i" is called the imaginary unit - the number "a" is the real part of the complex number - the value "bi" is the imaginary part - if b = 0, then a complex number has no imaginary part and is simply a real number - if a = 0 and b≠0, then the complex number is classified as a "pure imaginary number" - do NOT always retain their imaginary components when combined (this can happen b/c either the two "bi" parts of the number were additive inverses and so cancel out when the quantities are added, like (5 - 2i) + (4 + 2i), which equals simply 9) OR (the imaginary compnents can cancel throguh subtraction when two "bi" parts are identical, like (5 - 2i) - (4 - 2i), which equals simply 1. - In addition, because i =√-1, squaring both sides yields i² = ( √-1)² = -1, which is a real number. - it is also true that if two complex quantities are a pair of conjugates, then both their product and their sum will be Real values. -ALL complex numbers have CONJUGATES- the conjugate for any complex number of the form a + bi is simply a-bi (same numeric quantities, but with the opposite operation/sign between them) (Their sum will always be 2a (a real number), and their product is also always a Real number) EX: (two abi's crossed out) (a + bi) (a - bi) = a²- abi + abi -b²i² =a² + 0 -b²(-1) = a² +b²

Complex Numbers

Natural numbers GREATER THAN 0 that are divisible by at least one other number besides 1 and themselves have AT LEAST 3 factors EX: 1, 3, and 9 - EVERY composite number can be written as a product of prime numbers.

Composite Numbers

The term "denseness" means that between any 2 numbers, there is always at least one additional number. - some number sets are dense, while others are not - The set of RATIONAL numbers is DENSE b/c, for example EX: between 1/4 and 1/3, there is the rational number of 7/24 - The set of WHOLE numbers is NOT DENSE b/c there may not be a whole number between 2 whole numbers. EX: While it's true that 6 is a whole number between the whole numbers 4 and 10, NOT EVERY PAIR of whole numbers has another whole number between them - like 7 and 8 -there's not whole number between them

Denseness

The *domain of any expression is the set of all possible input values. In the case of rational expressions, we can input any value except for those that make the denominator equal to 0 (since division by 0 is undefined). In other words, the *domain of a rational expression includes all real numbers except for those that make its denominator zero. Example: Finding the domain of ×+1 −−−−−−− (x-3) (×+4) Let's find the zeros of the denominator and then restrict these values: (x-3)(x+4) = 0 x-3=0 OR x+4 = 0 (Zero Product Property) x = 3 OR x = -4 (SOLVE FOR X) So we write that the domain is all real numbers except 3 and -4 or simply x≠3, -4

Domain of rational expressions

ESTIMATION: used to make an approximation that is still close enough to be useful -When using estimation, use the LEADING or the LEFT-most digit to make an estimate. (sometimes called "Front End Estimation") EX: 532 + 385 + 57 = 500 + 400 + 60 = 960 ROUNDING: may be used to estimate a sum, difference, or product and to make mental approximations. - look at the digit to be rounded and use the following rules for rounding. - useful to have fewer digits when dividing numbers with remainders EX: To divide a 20-cm long wire into 3 equal pieces and to find the length of each piece, divide 20 by 3 to get the length of each piece. 20/3 = 6.6666666666666........ the 6 repeats an infinite number of times typically the answer is based on the required number of decimal places here decimal places refer to the number of digits to the right of the decimal usually rounding to the tenths or hundredths place is sufficient to round the answer (6.6666666...) to the nearest hundredths place, the next digit in the thousandths place is 6, so the 6 in the hundreths place is rounded up, giving 6.67 as the desired answer.

Estimation/Rounding

Find the prime factors of each: 6 = 2 * 3 15 = 3 * 5 X = X (it can't be factored, so it is prime) 1) Copy one set of numbers into LCM. Let's start with 2 * 3 from the 6 LCM = 2 * 3 *... 2) Compare the factors of 15 with what is already in the LCM. The 3 is already in the LCM, so don't put in another one. Our LCM does not yet have a 5, so include it. LCM = 2 * 3 * 5 *... 3) Compare X to what is already in the LCM. We don't have any X in our LCM, so X is also included. Our LCM = 2 * 3 * 5 * X = 30X

Find the LCM of 6,15, X

- a quick way to handle comparisons of fractions would be to round each fraction to a benchmark fraction value EX: To order the values 2/3 and 3/7, one could say that 2/3 is more tha the familiar fraction 1/2 (b/c half of 3 would be 1.5 and 2 is larger than that) - B/C it is known that 2/3 is is more than 1/2 while 3/7 is less than 1/2, it becomes clear that 3/7 is smaller than 2/3 (by virtue of 3/7 being smaller than a value known to be smaller than 2/3, the benchmark fraction used in this case, 1/2)

Finding equivalent fractions

Identity property of multiplication: Identity property of multiplication The identity property of multiplication says that the product of 111 and any number is that number. Here's an example: 7 \times 1 = 77×1=77, times, 1, equals, 7 The commutative property of multiplication tells us that it doesn't matter if the 111 comes before or after the number. Here's an example of the identity property of multiplication with the 111 before the number: 1 \times 6 =61×6=6 The product of 111 and any number is that number. For example, 7 \times 1 = 77×1=77, times, 1, equals, 7.

Identity property of multiplication:

Integers are just like whole numbers, but they also include negative numbers: {...−5,−4,−3,−2,−1,0,1,2,3,4,5...} Key idea: Like whole numbers, integers don't include fractions or decimals. positive and negative counting numbers AND zero - DO NOT INCLUDE decimals or fractions - the infinite set ....-3, -2, -1, 0, 1, 2, 3, .... - WHOLE number that includes all positive and negative numbers - may be represented on a number line that extends in both directions from 1 - Many real life situations are resprented with integers

Integers

are REAL numbers that CANNOT be represented as a ratio of two integers - In order to be expressed exactly (without rounding), irrational numbers are often written with symbols like: √2, π, e - the decimal form of the number never terminates and never repeats - when irrational numbers are written using only ordinary numbers, they must be rounded, as irrational numbers, by definition cannot be written in fraction form. -can be approximated and positioned on a number line √2 = 1.41 π = 3.14 or 22/7 e = 2.72

Irrational Numbers

1. A number is divisible by 3 if if it can be divided by 3 without leaving a remainder 2. A number is divisible by 3 if the sum of the digits is divisible by 3 3. Add the digits of 204430 2+0+4+4+3+0 If 13 is divisible by 3 then 204430 must also be divisible by 3 4. If 13 is not divisible by 3, then 204430 must not be divisible by 3

Is 204430 divisible by 3?

1. 262015 is divisible by 4 if it can be divided by 4 without leaving a remainder. 2. A number is divisible by 4 if the last two digits are divisible by 4. [Why?] We can rewrite the number as a multiple of 100 plus the last two digits: 262015=262000+15, Because 262000 is a multiple of 100, it is also a multiple of 4. So as long as the value of the last two digits,15, is divisible by 4, the original number must also be divisible by 4! 3. Is the value of the last two digits, 15, divisible by 4? 4. NO, 15 is not divisible by 4, so 262015 is not divisible by 4

Is 262015 by 4?

1. 313756 is divisible by 10 if it can be divided by 10 without leaving a remainder 2. A number is divisible by 10 if the LAST digit is 0. 3. The last digit of 313756 is 6, so NO it isn'y divisible by 10

Is 313756 divisible by 10?

333707 is divisible by 2 if it can be divided by 2 without leaving a remainder. Any even number is divisible by 2. 333707 is odd, so it is not divisible by 2.

Is 333707 divisible by 2?

1. 404985 is divisible by 5 if it can be divided by 5 without leaving a remainder. 2. A number is divisible by 5 if the last digit is a 0 or a 5. 3. The last digits of 404985 is 5, so YES it's divisible by 5

Is 404985 divisible by 5?

How to determine LCM: 1 of 2 methods can be applied 1. Generate a list of multiples and look for the smallest value on both lists. EX: Find LCM of 12 and 15 12 (12, 24,36, 48, 60, 72, 84...) 15 (15, 30, 45, 60, 75, 90...) LCM= 60

Least Common Multiple

is a number that when multiplied by any element "x" in a set , always yields "x". - the idea of the multiplicative identity is that an initial value a kept "identical" to what it began as, even while a quantity is multiplied with it. - That quantity multiplied-by, of course, must be the number 1, as in, 5 x 1 = 5. - The number 1 is sometimes referred to as " The Multiplicative Identity". - The utility of the MI comes in realizing that 1 comes in may forms, like 2/2 or 8/8 (as long as the numerator and denominator of a fraction are the same as each other, the quantities divide with a quotient of 1) - The strategy of "multiplying by a form of 1" is used quite frequently in math, especially to create denominators as fractions.

Multiplicative Identity

When a number and its multiplicative inverse are multiplied together, they "cancel each other out"/it yields the Multiplicative Identity/they multiply to equal 1. - An MI is also known as a "reciprocal", and more formally stated, is a number that when multiplied by "x" yields 1. - The MI is commonly denotated as 1/x or x⁻¹ - In fraction form, the multiplicative inverse of a/b would be b/c b/c their product will always be 1 ( the form of 1 written ab/ab) - ONLY the number ZERO has NO MI, no reciprocal (b/c division be zero is undefined/impossible)

Multiplicative Inverse

- also known as "Counting Numbers" -is a positive or nonnegative integer - quantity concepts that can be pointed to in nature-one tree, two people, three dogs - STARTS at "1" and goes up by "1's" - the same way kids first learn to count - nonnegative integers also include zero "0" - A list of positive integers would include whole numbers but not zero "0" - Include 1, 2, 3, 4, 5......(infinity) -ARE ALL WHOLE NUMBERS -DO NOT INCLUDE negatie numbers, fractions, or decimals

Natural Numbers

PEMDAS 1. simplify within grouping symbols, such as parentheses or brackets, starting with the INNERMOST 2 Apply exponents - powers and roots 3. Perform all multiplications and divisions in order from left to right 4. Perform all additions and subtractions in order from left to right.

Order of Operations

another way of expressing a fractional number - percent ALWAYS expresses a fractional number in terms of 1/100 or 0.01 EX: 40 parts of 100 is 40% - a percent is easily converted to a common fraction or decimal representation ***The fraction equivalents of a percent often can be simplified by dividing a common factor EX: 25% = 25/100 (divide both by common factor of 25) = 1/4

Percents *remember "percent" means "for every one hundred"

involves the knowledge of 2 sets of numbers - EVERY composite number can be written as a product of prime numbers - TO PERFORM prime factorization: 1. Start by factoring a known prime number into the given number 2. Then repeat this procedure on all factors until no more factorization can take place 3. When a factor is repeted in a prime factorization, express the repeated factor by using an exponent USE FACTOR TREE to help find prime factorization of a compositre number

Prime Factorization

Natural numbers greater than 1 that are divisible ONLY BY themselves and 1 - The first 11 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Prime Numbers

the process of writing a number as a product of prime factors - ALL integers can be factored into a product of prime numbers or a product of prime factors and -1. - it DOESN'T matter which pair is used first, as long as factoring continues until there are ONLY PRIME numbers EX: 48 48 = 3 X 16 (start by factoring 48 by a known prime number) 48 = 3 X 2 X 8 (Factor 16 by 2, the lowest divisible prime of 16) 48 = 3 X 2 X 2 X 4 (Factor 8 similarly) 48 = 3 X 2 X 2 X 2 X 2 (Factor 4 similarly) 48 = 3 X 2⁴ (Express 2x2x2x2 as 2⁴)

Prine Factorization/Factor Trees

What is a rational expression? A polynomial is an expression that consists of a sum of terms containing integer powers of x, like 3x²-6x-1 A rational expression is simply a quotient of two polynomials. Or in other words, it is a fraction whose numerator and denominator are polynomials. These are examples of rational expressions: 1 x+5 ×(×+1) (2×-3) −, −−− , −−−−−−−−−− × ×²-4×+4 ×-6 - Notice that the numerator can be a constant and that the polynomials can be of varying degrees and in multiple forms.

Rational Expressions

numbers that can be expressed as a RATIO (fraction or comparison) of two integers - a/b where b≠0 -simply b/c "0" can NEVER BE A DENOMINATOR -RATIONAL numbers ARE: - natural numbers - whole numbers - integers (as each of these can be expressed in fraction form with a denominator) of 1 - CAN BE expressed as: - common fractions - mixed numbers - decimals (like 0.25 which is also 1/4) - finite decimals (repeating) (0.72.72.72.. also written as 0.72(with a line over 2), which is 8/11 when written in fraction form) - ALL rational numbers can be written EXACTLY in both fractions and decimal forms and can be graphed on a number line. - Note: EXACT form meaning a value that HAS NOT been rounded or approximated)

Rational Numbers

Consider the rational expression 2x+3 −−−− ×-2 ​ We can determine the value of this expression for particular xxx-values. For example, let's evaluate the expression at ×=1 2 (1) +3 5 −−−−− = −−− 1-2 -1 = -5 From this, we see that the value of the expression at ×= 2 Now lets find the value at x=2 2(2) + 3 7 −−−−− = −− 2- 2 0 = UNDEFINED! An input of 2 makes the denominator 000. Since division by 0 is undefined, x=2 is not a possible input for this expression

Rational expressions and undefined values

describe any number that is positive, negative, or zero and can be used to measure continuous quantities - ALSO INCLUDES numbers that have decimal representations, even those with inifinite decimal sequences, such as π

Real Numbers

A rational expression is a ratio of two polynomials. The domain of a rational expression is all real numbers except those that make the denominator equal to zero. * should know rational expressions/factoring polynomials A rational expression is considered simplified if the numerator and denominator have no factors in common. We can simplify rational expressions in much the same way as we simplify numerical fractions. For example, the simplified version of 6 3 − is − 8 4 Notice how we canceled a common factor of 222 from the numerator and the denominator: Example 1: Simplifying ײ+3× −−−− ײ+5× Step 1: Factor the numerator and denominator The only way to see if the numerator and denominator share common factors is to factor them! ײ+3× ×(×+3) −−−− = −−−−− ײ+5× ×(×+5) Step 2: List restricted values At this point, it is helpful to notice any restrictions on x. These will carry over to the simplified expression. Since division by 000 is undefined, here we see that x≠0 and ×≠ -5 ×(×+3) −−−−− ×(×+5)* Step 3: Cancel common factors Now notice that the numerator and denominator share a common factor of x. This can be canceled out. ×(×+3) −−−−− = ×(×+5) ×(×+3) −−−−− (cross out both x's to cancel) = ×(×+5) ×+3 −−− ×+5 Step 4: Final answer Recall that the original expression is defined for x≠0,−5 The simplified expression must have the same restrictions. Because of this, we must note that x≠0. We do not need to note that x≠−5, since this is understood from the expression. In conclusion, the simplified form is written as follows: ×+3 −−− for x≠0 ×+5

Simplifying Rational Expressions

How can it's answer be 75 75 is the highest common factor of those measurements. If you had a 1cm tape measure you would have lay out the tape 825 times for the first dimension, but would only need to use a 75cm tape measure 11 times. Similarly a 75cm tape measure could be used 9 times for the 2nd dimension and 6 times for the 3rd dimension.

The length,breadth and height of a room are 825cm, 675cm and 450cm respectively. Find the longest tape which can measure the three dimensions of the room exactly?

To CONVERT a common fraction to a percent: 1. carry out the division of the numerator by the denominator of the fraction to get the decimal equivalent 2. then to convert the decimal to a percentage move the decimal point 2 placed to the right (adding 0's as placeholders if needed) and round as necessary. EX: 1/4 = 1÷4 = .25 = 25/100 = 25% EX: 2/7 = 2÷7 = .286 = 286/1000 = 28.6% which can be rounded to 29%

To CONVERT a common fraction to a percent: *remember "percent" means "for every one hundred"

simply move the decimal place twice to the right and put % symbol behind the number EX: 2/3 = 0.666.... = - move twice to the right - with a repeating decimal, one must round the value to 66.67%, 66.7%, or 67% depending on the context b/c while theres a bar to show repeating decimals, it's not appropriate to use in a percentage.

To Convert a Decimal to a Percent *remember "percent" means "for every one hundred"

e.g. LCM (10,8) Situations similar to this come up all the time in manufacturing where one company will only give you parts in packages of X, and another will only give you parts in packages of Y. If you need one of each to make your product and you can't afford to waste parts you will need to buy the LCM(X,Y) of each part. It's true, LCM is useful in real life.

You want to plan a barbecue since: -hot dogs often come in packs of 10 -and hot dog buns come in packs of 8 (or 12) If you don't want to waste any hot dogs or buns, you need to buy the LCM of hot dogs and buns