Kategorier: Alle - equations - variables - functions - expressions

af Cindy Shin 14 år siden

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MATH 171 (Chapter Summary)

Algebraic thinking is a fundamental concept in mathematics that involves the use of equations and functions. A function relates elements of one set to another, ensuring each input has a unique output.

MATH 171 (Chapter Summary)

MATH 171 (Chapter Summary)

Chapter 3 : Whole Numbers and Their Operations

Mental Mathematics and Estimation for Whole-Number Operations
Computational estimation

Computational estimation is the process of forming an approximate answer to a numerical problem. This is especially useful when the computation is done on a calculator.

Mental mathematics

Mental mathematics is the process of producing an answer to a computation without using computational aids. It can help in our everyday estimation skills. It is essential that we have these skills even in a time when calculators are readily available.

Algorithms for Whole-Number Multiplicationi and Division
Division
Multiplication
Multiplication and Division of Whole Numbers
Division of Whole Numbers

Repeated-Subtraction Model

Multiplication of Whole Numbers

Properties of Multiplication

Closure property of multiplication : For whole numbers a and b, a*b is a unique whole number.

Commutative property: For whole numbers a and b, a*b = b*a.

Associative Property: For whole numbers a,b, and c, (a*b)*c = a*(b*c).

Identity property of multiplication : There is a unique whole number 1 such that for any whole number a, a*1 = a = 1*a.

Zero multiplication property : For any whole number a, a*0 = 0 = 0*a.

Area Model

Cartesian-Product Model

The Cartesian-Product model offers another way to discuss multiplication.

Repeated-Addition Model

When we put equal-sized groups together we can use mulipication. We write 3+3+3+3+3 as 3x5 and say "three times five" or "three multiplied by five." The repeated-addition model can be illustrated in several ways, including number lines and arrays.

Algorithms for Whole-Number Addition and subtraction

An algorithm (named for the ninth-century Arabian mathematician Abu al Khwarizmi) is a systematic procedure used to accomplish an operation.

Addition and Subtraction of Whole Numbers
Subtraction of Whole Numbers

Properties of Subtraction

Commutative property does not exist for subtraction : Say we have to compute 2 - 3 , now if we do 2-3 = -1 and if we place the number as 3-2, it equals 1. Hence Commutative property is not applicable to subtraction.

Associative property also does not exist for subtraction. Say we have 2-(3-4) = 2-(-1) = 3. Now if we change the order of subtraction, (2-3)-4 = -1-4=-5. Hence associative property also does not exist for subtraction.

Identity : Same as that of Additive identity. 0+(-N) = -N , where N is positive.

Inverse : N is the inverse of a number –N where N is positive, since N-N=0.

Solving Problems by using Model

Missing-Addend Model

Take-away Model

Addition of Whole Numbers

Properties of Addition

Closure Property of Addition : If a and b are whole numbers, then a + b is a whole number.

Commutative Property of Addition : If a and b are any whole numbers, then a + b = b + a.

Associative Property of Addition : If a, b, and c are whole numbers, then (a + b) + c = a + (b + c).

Identity Property of Addition : There is a unique whole number 0, the additive identity, such that for any whole number a, a + 0 = a = 0 + a.

Solving problems by using Model

Set Model

A set model is one way to represent addition of whole numbers.

Chapter 2 : Numeration Systems and Sets

Other Set Operations and Their Properties
Properties of set opertions

Distributive Property of Set Intersection over Union

For all sets A, B, and C, A ∩ (B ∪ C)= (A ∩ B) ∪ (A ∩ C) is the distributive property of set intersection over union.

Associative Property of Set Union

A ∪ (B ∪ C) = (A ∪ B) ∪ C is the Associative property of set union.

Associative Property of Set Intersection

The property A ∩ (B ∩ C) = (A ∩ B) ∩ C is the associative property of set intersection.

Set Operations

Set Union

The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and is read ‘A union B’

Set Interaction

The intersection of two sets X and Y is the set of elements that are common to both set X and set Y. It is denoted by X ∩ Y and is read ‘X intersection Y’.

Describing Sets
Venn Diagrams Word Problems
Venn Diagrams
Universal sets, Subsets, etc.
Set-Notation
Defintion of Sets
Numeration Systems
Other Number Base Systems

We use 100 to represent ten 10s, or one 100. In the base-five system, we need a symbol to represent five 5s. The number 100 means 1•10² + 1•10¹ + 1•1, whereas the number 100 base five means 1•5² + 1•5¹ + 1•0, or 25.

Ancient Numeration Systems

Hindu-Arabic

Today, we use Hindu-Arabic numeration system which all numerals are constructed from the 10 digits-0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, and it’s place value is based on powers of 10, the number base of the system.

Roman

Greek

Mayan

Egyptian

Babylonian

Chapter 1 : An Introduction to Problem Solving

Explorations with Patterns

A sequence is a set of numbers in a specific order. What this means is that the set of numbers can be put into a one-to-one correspondence with the Counting Numbers (1, 2, 3, 4, ... ).

Other Sequences

Figurate numbers provide examples of sequences that are neither arithmetic nor geometric. Such numbers can be represented by dots arranged in the shape of certain geometric figures. The number 1 is the beginning of most patterns involving figurate numbers. Here is a website that shows a great example of Pascal’s Triangles. It also contains references about the triangle numbers.

Not only Pascal’s triangles but also a sequence such as 3, 3, 3, 3, 3,…was interesting that it is considered both an arithmetic and geometric sequence because you can both multiply 1s and add 0s to get 3. I didn’t think this could be a sequence before I took this class, so it was a fresh shock to me.

Fibonacci Sequences

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonacci, "son of Bonaccio").

Geometric Sequences

A geometric sequence is a sequence such that each successive term is obtained from the previous term by multiplying by a fixed number called a common ratio.


The sequence 5, 10, 20, 40, 80, .... is an example of a geometric sequence. The pattern is that we can multiply by a fixed number of 2 to the previous term to get to the next term. It is geometric if you are always multiplying by the same number each time.

Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same, i.e., the difference is a constant.

In order to identify if a pattern is an arithmetic sequence you must examine consecutive terms. If all consecutive terms have a common difference you can conclude that the sequence is arithmetic.

Consider the sequence: 3, 10, 17, 24, 31, ...

Since 10 - 3 = 7, 17 - 10 = 7, 24 - 17 = 7, 31 - 24 = 7 the sequence is arithmetic. We can continue to find subsequent terms by adding 7. Thus, the sequence continues : 38, 45, 52, etc.

Strategies for Problem Solving
Four-Step Problem-Solving Process

Chapter 6 : Rational Numbers as Fractions

Munltiplication and Division of Rational Numbers
Division of Rational Numbers

If a/b and c/d are any rational numbers,

then a/b ÷ c.d = e/f if, and only if, e/f is the unique rational number such that d/c * e/f = a/b.

Multiplication of Rational Numbers

If a/b and c/d are any rational numbers,

then a/b * c/d = a*c / b*d.

Multiplication with Mixed Numbers

Properties of Multiplication of Rational Numbers

Multiplication Property of Zero for Rational Numbers

Multiplication Property of Zero for Rational Numbers:

If a/b is any rational number,

then a/b * 0 = 0 = 0 * a/b.

Multiplication Property of Inequality for Rational Numbers

Multiplication Property of Inequality for Rational Numbers:

1) If a/b > c/d and e/f, then a/b * e/f > c/d * e/f.

2) If a/b > e/c and e/f < 0, then a/b * e/f < c/d * e/f.

Mutiplication Property of Equality for Rational Numbers

Mutiplication Property of Equality for Rational Numbers:

If a/b and c/d are any rational numbers such that

a/b = c/d, and e/f is any rational number then,

a/b * e/f = c/d * e/f.

Distributive Property of Multiplication Over Addition for Rational Numbers

Distributive Property of Multiplication Over Addition for Rational Numbers:

If a/b, c/d, and e/f are any rational numbers, then

a/b(c/d + e/f) = (a/b * c/d) + (a/b * e/f)

Multiplicative Identity and Multiplicative Inverse of Rational Numbers

Multiplicative Identity and Multiplicative Inverse of Rational Numbers:

1. The number 1 is the unique number such that for every rational number 1/b, 1*(a/b) = a/b = (a/b)*1.

2. For any nonzero rational number a/b, b/a is the unique rational number such that a/b * b/a = 1 = b/a * a/b.

Addition, Subtraction, and Estimation with Rational Numbers
Estimation with Rational Numbers
Subtraction of Rational Numbers

If a/b and c/d are any rational numbers,

then a/b - d/c is the unique rational number e/f

such that a/b = d/c + e/f.

If a/b and c/d are any rational numbers,

then a/b - c/d = a/b + (-c/d)

If a/b and c/d are any rational numbers,

then a/b - c/d = ad-bc/bd.

Properties of Addition for Rational Numbers

Additive Inverse Property of Rational Numbers:

For any rational number a/b, there exists a unique rational number -a/b. the additive inverse of a/b,

such that a/b + (-a/b) = 0 = (-a/b) + a/b.

Addition Property of Equality:

If a/b and c/d are any rational numbers such that

a/b = c/d, and if e/f is any rational number,

then a/b + e/f = d/c + e/f.

Mixed Numbers
Adding Rational Numbers

If a/b and c/b are rational numbers,

then a/b + c/b = a+c/b.

If a/b and c/d are any two rationanl numbers,

then a/b + c/d = ad+bc/bd.

The Set of Rational Numbers
Denseness of Rational Numbers

Given two different rational numbers a/b and c/d, there is another rational number between these two numbers.

Ordering Rational Numbers

If a, b, and c are integers and b > 0, then a/b > c/b if, and only if, a>c.

Equality of Fractions

Two fractions a/b and c/d are equal if, and only if,

ad = bc.

Simplifying Fractions

A rational number a/b is in simplest form if b > 0 and GCD(a,b) = 1; that is, if a and b have no common factor greater than 1, and b > 0.

Ex)

80/120 = 8*10/12*10 = 8/12

then, 8/12 = 2*4/3*4 = 2/3.

Hence, 80/120 = 2/3.

Equivalent or Equal Fractions

Fundamental Law of Fractions

Let a/b be any fraction and n a nonzero integer. Then, a/b = an/bn

Chapter 5 : Intergers and Number Theory

Clock and Modular Arithmetic
Modular Aritmnetic
Clock Arithmetic
Greatest Common Divisor and Least Common Multiple
Least Common Multiple (LCM)
Greatest Common Divisor (GCD)
Prime and Composite Numbers
Sieve of Eratosthenes

Sieve of Eratosthenes is a method the Greek mathematician Eratosthenes developed for identifying prime numbers.

Prime Factorization

A factorization containing only prime numbers is a prime factorization.

Divisibility

If a and b are any integers, then b divides a, written b|a, if, and only if, there is a unique integer q such that a = bq.

Divisibility Rules

Divisibility Test for 6

Divisibility Test for 6

An integer is divisible by 6 if, and only if, the integer is divisible by both 2 and 3.

Divisibility Test for 11

Divisibility Test for 11

An integer is divisible by 11 if, and only if, the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that are odd powers of 10 is divisible by 11.

Divisibility Test for 9

Divisibility Test for 9

An integer is divisible by 0 if, and only if, the sum of the digits of the integer is divisible by 9.

Divisibility Test for 3

Divisibility Test for 3

An integer is divisible by 3 if, and only if, the sum of its digits is divisible by 3.

Divisibility Test for 8

Divisibility Test for 8

An integer is divisible by 8, if, and only if, the last three digits of the integer represent a number divisible by 8.

Divisibility Test for 4

Divisibility Test for 4

An integer is divisible by 4 if, and only if, the last two digits of the integer represent a number divisible by 4.

Divisibility Test for 10

Divisibility Test for 10

An integer is divisible by 10, if, and only if, its units digit is divisible by 10; that is, if, and only if, the units digit is 0.

Divisibility Test for 5

Divisibility Test for 5

An integer is divisible by 5, if, and only if, its units digit is divisible by 5; that is, if, and only if, the units digit is 0 or 5.

Divisibility Test for 2

Divisibility Test for 2

An integer is divisible by 2, if, and only if, its units digit is divisible by 2.

Multiplication and Division of Integers
Ordering Integers

For any integers a and b, a is less than b,

written a < b, if, and only if, there exists a positive integer k such that a + k = b.



Theorem


5-10 : a < b (or equivalently, b > a) if, and only if, b - a is equal to a positive integer; that is, b - a is greater than 0.


5-1 1: a) If x < y and n is any integer,

then x + n < y + n.

b) If w < y, then -x > -y.

c) If x < y and n < o, then nx > ny.

d) If x < y and n < 0, then nx < ny.


Integer Division

If a and b are any integers, then a/b is the unique integer c, if it exists, such that a = bc.

Integer Multiplication

Properties of Integer Multiplication

Closure property : ab is a unique integer.

Commutative property : ab = ba.

COmmutative property : (ab)c = a(bc).

Multiplicative properties of multiplication over addition for integers : a(b + c) = ab + ac and (b + c)a = ba + ca.

Multiplicative properties of multiplication over subtraction for integers : a(b - c) = ab - ac and (b - c)a = ba - ca.

Zero multiplication property : 0 is the unique integer such that for all integers a, a*0 = 0 = 0*a.

Chip Model, Number-Line Model

Integers and the Operations of Addition and Subtraction
Integer Subtraction

Integer subtraction can be modeled with a charged field. For example, consider -2 - -7. To subtract -7 from -2, we first represent -2 so that at least 7 negative charges are present. To subtract -7, remove the 7 negative charges, leaving 5 positive charges.

Hence, -2 - -7 = 5.

Properties of Integer Addition

Closure property : a + b is a unique integer.

Commutative property: a + b = b + a.

Associative property : (a + b) + c = a + (b + c).

Identity : 0 is the unique integer such that, for all integers a, 0 + a = a = a + 0.

Absolute Value

Distance is always a positive number or zero. The distance between the point corrsponding to an integer and 0 is the absolute value of the integer.

Integer Addition

Number-Line Model

Charged-Field Model

A model similar to the chip model uses positive and negative charges. A field has 0 charge if it has the same number of positive (+) and negative (-) charges. It can be represented in many ways using the charged-field model.

Chip Model

Chapter 4 : Algebraic Thinking

Functions

A function from set A to set B is a correspondence from A to B in which element of A is paired with one, and only one, element of B.

Functions as Graphs

It is one of the most widely recognized representations of a function is as a graph.

Functions as Tables and Ordered Pairs

It is another useful way to describe a function. The function could be given using ordered pairs. For example, when 0 is the input and 1 is the output, that is recorded as the ordered pair (0, 1).

Functions as Arrow Diagrams

It can be used to examine whether a correspondence represents a function.

Functions as Equations

We can write an equation to depict the rule. The output values can be obtained by substituting the values 0, 1, 3, 4, and 6 for x in f(x) = x + 7, as shown:

f(0) = 0 + 7 = 7

f(1) = 1 + 7 = 8

f(3) = 3 + 7 = 10

f(4) = 4 + 7 = 11

f(6) = 6 + 7 = 13

Functions as Machines

It is another way to prepare students for the concept of a function.What goes in the machine is referred to as input and what comes out as output.

Functions as Rules

It is like a game called guess my rule. We figure out a certain rule from the question.

Equations
Variables

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations.

For instance, in the formula x + 3 = 5, x is a variable which represents an "unknown" number.

Algebraic expressions

An algebraic expression is a mathmatical expression containing variables, numbers, and operation symbols.