3.1 Numeration Systems
Base Ten Blocks
Consists of 1's, 10's, 100's, and 1000's respectively
3.2 Addition of Whole Numbers
Additions is a binary operation because two numbers are combined to form another number
Associative Property of addition of whole numbers
If a,b, and c are any whole numbers then (a+b)+c=a+(b+c)
Basic addition words counting on and doubles
Identity property of addition of whole numbers
There is a unique whole number 0, called the additive identity, such that for anywhole number a,a +0=a=0+a.
3.3 subtraction of whole numbers
Definition of subraction of whole numbers: For any whole numbers (a) and (b) such that a> equal to b, a-b is the unique whole number (c) such that b+c=a
Missing addend Model relates subtraction and addition
3.5 Division of whole numbers
Missing factor model
For any whole numbers a and b, with b equals 0, a/b=c, if only if c, is the unique whole number such that b*c+a
The division algorithm : given any whole numbers a and b with b + 0, there exist unique whole numbers q (quotient) and r (remainder) such that
3.4 Numeration systems and whole number operations
Rectangualr- Array Model
An array is suggested when objects are arranged with the same numbers of objects in each row and column
Multiplication of whole numbers: cartesian porduct model
For finite sets A and B, if n(A)= a and n(B)=b, then a * b=n)A*B)
Partial-Products Algorithm (array model, partial products, standard algorithm)
4.1 divisibilty
Divisibility: An even whole number is a whole number that has a remainder 0 when divided by 2. An odd whole numbers is a whole number that has remainder 1 divided by 2
4.2 Prime and Composite Numbers
Factors: one method used in elementary schools to determins the positive factors of non-zero whole numbers is to use squares of paper
Prime and Composite numbers: any whole number with exactly two distincy, whole number divisors is a prime numbber, or a prime. Any whole number greater than 1 that has a whole number factor other than 1 and itself is a composite number, or a composite
Prime Factorization: A factorization containing only prime numbers is a prime factorization
Theorem: Fundamental Theorm of Arithmetic: Each composite number can be written as a product of primes in one and only one, way except for the order of the prime factors in the product
4.3 Greatest Common Divisor
The greatest common divisor (GCD) or the greatest common factor (GCF) of two whole numbers a and b not both 0 is the greatest whole number that divides both a and b.
The Prime Factorization Method:To find the GCD of two or more non-zero whole numbers, first find the prime factorizations of the given numbers and then identify each common prime factor of the given numbers. The GCD is the product of the common factors, each raised to the lowest power of that prime that occurs in any of the prime factorizations. Numbers, such as 4 and 9, whose GCD is 1 are relatively prime.
Least Common Multiple (LCM): The least common multiple (LCM) of two non-zero whole numbers a and b is the least non-zero whole number that is simultaneously a multiple of a and a multiple of b.
5.1 Addition and Subtraction of Integers
Integers: The set of numbers (-1, -2,-3,-4..) is the set of negative integers. the set (1,2,3,4..) is the set of positive integers the integers. The integer ) is neither positive nor negative.
Definiton: the union of the set of negative integers, the set of positive integers, and (0) is the set of integers, denoted by I
Number line model: always start at zero, and always face + right direction. if the number is positive walk foward, if the number line is negative walk backwards
Integer addition model (chip model)
5.2 Multiplication and division of Integers
Nummber Line Model for multiplication: The rules for walking the number line are the same as for addition or subtraction of integers
Theorum: Properties of integers multiplication: the set of integers I satifies the following properties of multiplication for all integersa,b,c
6.1 the set of rational numbers
The common core standard, when using legnth on a number line a and b are whole numbers with b =0 the reason for this is that legnth are greater than or equal to 0.
Equivalent or equal fraction
6.2 addition subtraction estimation with rational numbersn
\A mixed number, is the sum of an integer and a proper fraction. the figue shows a nail that is 2 3/4 in long the mixed number 2 3/4 means 2+3/4
6.3 multiplication, division, and estimation with rational numbers
Definition of multiplication of rational numbers: if a/b and c/d are rational numbers, then a/b* c/d = ac/bd
Identity and inverse properties of multiplication of rational numbers: Multiplicative inverse property of rational numbers a/b, the multiplication inverse (reciprocal) is the unique rational numbers b/a such that a/b * b/a=1=b/a*a/b
Properties of Exponents: for any non-zero rational numbers a and b and any integers m and nnare true, a square root of zero= 1
6.4 Studetn thinking with rational numbers
Definition of Ration: a ration , denoted as a/b, a/b or a:b where a and b are rational numbers is a comparison of two quatities. Ratios can represent part to whole or whole to part comparisons
Proportions: A proportions is a statement that two given are equal
Scale drawings: ratios and proportions are used in scale drawings. The scale is the ratio of the size of the drawing of the size of the object.
Using the bar model to solve ratio and proportion problems: in the comparison model one bar is longer or the same size as the other
7.1 Terminating Decimals
Decimal: The word decimal comes from the Latin decem, meaning "ten".
Terminating Decimal: are numbers that can be written with a finite number of places to the right of the decimal point.
Theorem: A rational number a/b in the simplest form can be written as a terminating decimal if, and only if, the prime factorization of the denominator contains no primes other than 2 or 5
Ordering Terminating Decimal: the steps used to compare terminating decimals are similar to those for comparing whole numbers. 1: line up the numbers by value and append zeros, if necessary. 2: start at the left and find the first place where the face values are different. 3: compare these digits, the digits with the greater face value in this place represents the greater of the two numbers
7.2 operations on decimals
Definition of Scientific Notations
A positive number is written as the product as the product of a number greater than or equal to 1 and less than 10 and an integer power of 10. to write a negative number and place a negative sign in front of the result
Mental computation
Some of the tools used for mental computations with whole numbers can be used to perform mental computations with decimals
Round off errors: are typically compounded thwn computations are invloved. the greatest possible error in measuring is defined as one half of the measuring unit
7.3 Repeating Decimals
Repeating decimals: when finding the decimal representation of fraction by division, if the division pattern repeats, then the deciaml is called a repeating decimal, and the repeating block digits is the repetend
In general if a/b is any rational numbers in simplest form with b = 0 and b>a and not represent a terminating decimal, the repetend has at most b-1 digits
Ordering Repeating decimal: repeating decimal might lie on a number line or we compare them using place value in manner similar to how we ordered terminating decimals
7.5 real numbers
Irrational Numbers: the ancient greek discovered numbers that are not rational, that is, numbers with decimal representaion that neither terminates nor repeats to find such decimals, consider their characteristics: there must be an infinite number of non-zero digits to the right of the decimal
Real numbers: a real number is any number that can be represented by a decimal
estimating a square root: some squares are rational numbers. other like square root of 2 are irrational numbers
fractions: set of fractions can be extened to be of the form a/b wherea and b are real numbers with b=0 such as sqare root of 3/5
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