CHAPTER 1 - SUMMARY AND REVIEW

The place value of the digit 7 on this number: 70,322,454 is

7 ten millions.

24. P.E. CLASSES In a physical education class, the students stand in a rectangular formation of 8 rows and 12 columns when the instructor takes attendance. How many students are in the class?

8*12=96

Some of the key words and phrases that are often used to indicate subtraction are

: loss, decrease, down, backward, fell, less than, fewer, reduce, remove, debit, in the past, remains, declined, take away. --- To answer questions about how much more or how many more, we use subtraction ---- WEIGHTS OF CARS A Chevy Suburban weighs 5,607 pounds and a Smart Car weighs 1,852 pounds.How much heavier is the Suburban? -- The phrase how much heavier indicates subtraction 5,607-1,852= 3,755 The Suburban weighs 3,755 pounds more than the Smart Car.

8. Subtract 287 [from] 535. Show a check of your result.

=> 535 - 287 not 287 - 535, noticed that the word "from" in the sentence.

prime number

A counting number greater than 1 whose only two factors are the number 1 and itself.

factor tree

a diagram showing the prime factorization of a number

Associative property of addition: The way in which whole numbers are grouped does not change their sum.

(1+3)+5=1+(3+5)

27. INHERITANCE A father willed his estate, valued at $1,350,000, to his four adult children. Upon his death, the children paid legal expenses of $26,000 and then split the remainder of the inheritance equally among themselves. How much did each one receive?

(1,350,000-26,000 legal expenses)/4 children = 331,000 for each child after paid legal expenses.

associative of multiplication

(ab)c=a(bc)

To evaluate. What will we do?

(find the value of) expressions that involve addition and subtraction written in horizontal form, we perform the operations as they occur from left to right

commutative of addtition

1+3=3+1

whole numbers written in standard form are:

1,2, 530, 5030, ...

21. Find the prime factorization of 1,260.

1,260=10.126=(2.5).(6.21)=(2.5).(2.3)(7.3)=(2^2)(3^2)(5)(7)

23. TOSSING A COIN During World War II, John Kerrich, a prisoner of war, tossed a coin 10,000 times and wrote down the results. If he recorded 5,067 heads, how many tails occurred? (Source: Figure This!)

10,000-5,067=4,933

Every subtraction has a related addition statement

10-3=7 => 7+3=10

25. FLOOR SPACE The men's, women's, and children's departments in a clothing store occupy a total of 12,255 square feet. Find the square footage of each department if they each occupy the same amount of floor space.

12,255/3=4,085 square feet(ft^2)

Commutative property of addition: The order in which whole numbers are added does not change their sum.

2+5=5+2

22. TEETH Children have one set of primary (baby) teeth used in early development. These 20 teeth are generally replaced by a second set of larger permanent (adult) teeth. Determine the number of adult teeth if there are [12 more of] those than baby teeth.

20+12=32

26. MILEAGE The fuel tank of a Hummer H3 holds 23 gallons of gasoline. How far can a Hummer travel on one tank of gas if it gets 18 miles per gallon on the highway?

23*18=414 miles.

17. Use front-end rounding to estimate the difference: 49,213 - 7,198

50,000-7,000 => 43,000.

prime factorization

Breaking down a composite number until all of the factors are prime a number written as the product of its prime factors

We can use multiplication to count objects arranged in rectangular patterns of neatly arranged rows and columns called rectangular arrays. Some key words and phrases that are often used to indicate multiplication are: double, triple, twice, of, times

CLASSROOMS A large lecture hall has 16 rows of desks and there are 12 desks in each row. How many desks are in the lecture hall? The rectangular array of desks indicates multiplication: The number of desks in the lecture hall 16.12

To check: Difference, what the method we use?

Difference + subtrahend = minuend

To estimate quotients, we use a method that approximates both the dividend and the divisor so that they divide easily.

Estimate the quotient for by finding The dividend is approximately The divisor is approximately 154,908 / 46 => 150,000 / 50

To evaluate (find the value of) expressions that involve more than one operation, use the order-of-operations rule. Order of Operations 1. Perform all calculations within parentheses and other grouping symbols following the order listed in Steps 2-4 below, working from the innermost pair of grouping symbols to the outermost pair. --- 2. Evaluate all exponential expressions. --- 3. Perform all multiplications and divisions as they occur from left to right. --- 4. Perform all additions and subtractions as they occur from left to right. ***** When grouping symbols have been removed, repeat Steps 2-4 to complete the calculation. If a fraction bar is present, evaluate the expression above the bar (called the numerator) and the expression below the bar (called the denominator) separately. Then perform the division indicated by the fraction bar, if possible.

Evaluate: 10+3[2^4-3(5-2)] = 10 + 3 [16 - 3(3)] = 10 + 3 [16 - 9] = 10 + 3 [7] = 10 + 21 => 31

To find the GCF of two (or more) whole numbers using prime factorization: 1. Prime factor each number. 2. Identify the common prime factors. 3. The GCF is a product of all the common prime factors found in Step 2. ***If there are no common prime factors, the GCF is 1.

Find the GCF of 36 and 60. 36=2.2.3.3 60=2.2.3.5 =>2.2.3=12 *The GCF is the product of the circled prime factors. GCF (36, 60) = 2 . 2 . 3 = 12.

what does "to factor" mean?

It means to express it as the product of other whole numbers.

LCM

Least Common Multiple the smallest multiple of two (or more) numbers (the smallest number that is a multiple of both)

footage

Length. A portion of a film is called footage, such as daily footage or news footage. raw, unedited material as it had been originally recorded

factors

Numbers that are multiplied together to get a product

The greatest common factor (GCF) of two (or more) whole numbers is the largest common factor of the numbers. The greatest common factor of two (or more) numbers is the largest whole number that divides them exactly

The factors of 18: 1, 2, 3, 6, 9, 18 The factors of 30: 1 , 2 , 3 , 5, 6 , 10, 15, 30 The common factors of 18 and 30 are and 6. The greatest common factor of 18 and 30 is 6, which is written as: GCF (18, 30) = 6. 18/6=3; 30/6=5.

Commutative property of multiplication

The order in which whole numbers are multiplied does not change their product.

Be careful when translating the instruction to subtract one number from another number. Why?

The order of the numbers in the sentence must be reversed when we translate to symbols --- Subtract 41 from 97 => 97 - 41 = ?

Associative property of multiplication:

The way in which whole numbers are grouped does not change their product.

There are divisibility tests to help us decide whether one number is divisible by another.

They are listed on page 61

How to write this number 2,568,019 in words?

Two million, five hundred sixty-eight thousand, nineteen2,568,019

We often round numbers when....

When we don't need exact results.

STOCKING SHELVES Boxes of rice are being stacked next to boxes of instant mashed potatoes on the same bottom shelf in a supermarket display. The boxes of rice are 8 inches tall and the boxes of instant potatoes are 10 inches high. ---- a. What is the shortest height at which the two stacks will be the same height? b. How many boxes of rice and how many boxes of potatoes will be used in each stack?

a) 1st. Find factorization of 8 and 10, we get 8=2^3 and 10=2.5 => LCM(8,10)=(2^3)(5)=40 b) 40/ the height box of rice = 40/8 =5 boxes of rice. 40/ the height box of instant mashed potatoes = 40/10=4 boxes of instant mashed potatoes.

Consider the whole number 402,198. --------- a. What is the place value of the digit 1? b. What digit is in the ten thousand column?

a. 1 hundred b. 0

Round 34,759,841 to the a. nearest million b. nearest hundred thousand c. nearest thousand

a. 35,000,000 b. 34,800,000 c. 34,760,000

a.Find the factors of 12. b. Find the first six multiples of 4. c. Write as 5+5+5+5+5+5+5+5 a multiplication

a. factors of 12 are:1,2,3,4,6,12 b. 4.1=4;4.2=8;4.3=12;4.4=16;4.5=20;4.6=24. c. 8 times of number 5 => 8.5=40.

Application problems that involve forming equal-sized groups can be solved by division. Some keywords and phrases that are often used to indicate division:

split equally, distributed equally, shared equally, how many does each, how many left (remainder), per, how much extra (remainder), among --- BRACES An orthodontist offers his patients a plan to pay the $5,400 cost of braces in 36 equal payments. What is the amount of each payment? The phrase: 36 equal payments indicates division: $5400/36=150.

To write a whole number in words, start from

the left. Write the number in each period followed by the name of the period (except for the ones period, which is not used). Use commas to separate the periods.

The digit 4 on this number 7,334,376 tells

the number of thousands

(find the value of) expressions that involve addition and subtraction written in horizontal form, we perform how?

the operations as they occur from left to right ex: Evaluate: Working left to right, do the subtraction first.

The multiples of a number are

the products of that number and 1, 2, 3, 4, 5, and so on.

The least common multiple (LCM) of two whole numbers is

the smallest common multiple of the numbers. The LCM of two whole numbers is the smallest whole number that is divisible by both of those numbers. Multiples of 2: 2,4,6,8,10,12 Multiples of 3: 3,6,9,12,15... The common multiples of 2 and 3 are: 6,12,... The least common multiple of 2 and 3 is 6, which is written as : LCM 2, 3) = 6. 6/2=3; 6/3=2

tossing a coin

thrown from side to side throwing lightly

evaluate

to find the value of an expression

To write a number in expanded form (expanded notation) means

to write it as an addition of the place values of each of its digits ex: The expanded form of 32,159 is: 30,000 + 2,000 + 100 + 50 + 9

division ladder

tool used to find LCM, GCF, and distributive property

8/0

undefined never divide by 0

c. To evaluate an expression such as means to find its___-

value

a. The set of _____numbers is {0, 1, 2, 3, 4, 5, }.

whole number

The set of whole numbers is

{0, 1, 2, 3, 4, 5, p}.

composite number

A whole number greater than 1 that has more than 2 factors.

addition

An operation in which we combine two numbers to make a larger one.

Find the LCM of two (or more) whole numbers by listing: 1. Write multiples of the largest number by multiplying it by and so on. 2. Continue this process until you find the first multiple of the larger number that is divisible by each of the smaller numbers. That multiple is their LCM

Find the LCM of 3 and 5. Since 15 is the first multiple of 5 that is divisible by 3, the LCM(3, 5) = 15.

To find the LCM of two (or more) whole numbers using prime factorization: 1. Prime factor each number. 2. The LCM is a product of prime factors, where each factor is used the greatest number of times it appears in any one factorization

Find the LCM of 6 and 20. 6=2.3 20=2.2.5 =>(2.2.5).3 LCM (6, 20) = 2 . 2 . 3 . 5 = 60

The area of a rectangle is the measure of the amount of surface it encloses (covers all). Area is measured in square units, such as square inches (written in^2. ) or square centimeters (written cm^2 ). Area of a rectangle = length . width or A = l.w Letters (or symbols) that are used to represent numbers are called variables. ex: a, l, w, x, y and so on.

Find the area of the rectangle shown below. 25 in. long and 4 in. wide A = lw Replace l with 25 and w with 4. Multiply. The area of the rectangle is 100 square inches, which can be written in more compact form as 100 in^2.

The arithmetic mean, or average, of a set of numbers is a value around which the values of the numbers are grouped. ---- To find the mean (average) of a set of values, divide the sum of the values by the number of values

Find the mean (average) of the test scores and 73. = (74 + 83 + 79 + 91 + 73) / 5 => 400/5 = 80.

GCF

Greatest Common Factor the greatest factor that is common to two or more numbers

To write division, we can use

a division symbol, a long division symbol, or a fraction bar.

exponent

a mathematical notation indicating the number of times a quantity is multiplied by itself the number that indicates how many times the base is used as a factor

Another way to answer a division problem is to think in terms of multiplication and write

a related multiplication statement. 8/2=4 => 4.2=8

To write multiplication, we use

a times symbol X a raised dot . and parentheses ( ).

a. Write 7,018,641 in words. b. Write "one million, three hundred eighty-five thousand, two hundred sixty-six" in standard form. c. Write 92,561 in expanded form.

a. seven million, eighteen thousand, six hundred forty-one. b. 1,385,266. c. 90,000+2,000+500+60+1

Multiplication of whole numbers is repeated ...

addition but with different notation. Repeated addition: The sum of four 6's Multiplication 6 + 6 + 6 + 6 = 4 x 6 = 24

even number

any number divisible by 2

odd number

any number that is not divisible by 2

d. The _____of a rectangle is a measure of the amount of surface it [encloses.]

area

How to find mean (average) of something?

average = sum of terms / number of terms

To make large whole numbers easier to read, we use

commas to separate their digits into groups of three, called periods.

Inequality symbols are used to

compare whole numbers > means is greater than < means is less than ex: 7>3, 6<9 ...

if a whole number is a factor of a given number, it also ...

divides the given number exactly 6/1=6; 6/2=3; 6/3=2...

e. One number is _____by another number if, when we divide them, the remainder is 0.

divisible

16. Find [the quotient of ] 125,000 and 500.

do division.

Find the [product of ] 23,000 and 600.

do multiplication.

To divide whole numbers, think of separating a quantity into

equal-sized groups.

Rounding a Whole Number 1. To round a number to a certain place value, locate the rounding digit in that place. 2. Look at the test digit, which is directly to the right of the rounding digit. 3. If the test digit is 5 or greater, round up by adding 1 to the rounding digit and replacing all of the digits to its right with 0. ***If the test digit is less than 5, replace it and all of the digits to its right with 0

ex: Round 9,842 to the nearest ten=> 9,840 Round 63,179 to the nearest hundred => 63,200.

To estimate a sum, use

front-end rounding to approximate the addends. Then add. Estimate the sum: The estimate is 10,600. Round to the nearest thousand. Round to the nearest hundred. Round to the nearest thousand. 7,000 600 3,000 10,600 7,21

To estimate a product, use

front-end rounding to approximate the factors. Then multiply

To estimate a difference, use

front-end rounding to approximate the minuend and subtrahend. Then subtract. Estimate the difference.

Some key words and phrases that are often used to indicate addition are:

gain, increase ,up ,forward, rise ,more than ,total combined, in all ,in the future ,extra ,altogether Translate the words to numbers and symbols: VACATIONS There were 4,279,439 visitors to Grand Canyon National Park in 2006. The following year, attendance increased by 134,229. How many people visited the park in 2007? The phrase increased by indicates addition: => 4,279,439 + 134,229

Whole numbers are often used

in tables, bar graphs, and line graphs

b. The symbols > and < are _____symbols.

inequality symbol :)

The distance around a rectangle or a square is called

its perimeter. ---- Perimeter of a rectangle = length +length + width + width Perimeter of a square =side +side +side +side

The position of a digit in a whole number determines

its place value

To check the result of a division, we ...

multiply the divisor by the quotient and add the remainder. The result should be the dividend

f. The grouping symbols ( ) are called, and the symbols [ ] are called.

parentheses and brackets

g. A ____number is a whole number greater than 1 that has only 1 and itself as factors.

prime

willing

ready to do something ad. being ready or having a desire to ("They are willing to talk about the problem.")

occupy

reside or have one's place of business in (a building) to take control of a place

A place-value chart

shows the place value of each digit in the number.

A process called long division can be used to divide whole numbers. Follow a four-step process:

• Estimate • Multiply • Subtract • Bring down