MAT 157

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MAT 157 by Mind Map: MAT 157

1. Properties of addition for Rational Numbers.

1.1. Additive Inverse Property

1.1.1. For any rational number a/b, there exists a unique rational number -a/b called the additive inverse of a/b, such that a/b+(-a/b)=0=(-a/b)+a/b.

1.2. Properties of the additive inverse for rational number are analogous to those of the additive inverse for integers.

1.3. Estimation with Rational number- Many of the estimation and mental math techniques that are used with whole numbers also work with rational number. Estimations okays an important role in judging the reasonableness of computations.

2. Key Vocabualry

2.1. In the division problem A divided by B equal C. A is the DIVIDEND, B is the DIVISOR, and C is the QUOTIENT. note that a divided by b can be written as a/b=c.

3. Test 1

3.1. Chapter1

3.1.1. section1 Mathematics & Problem Solving George Polya"s 4-step problem solving 1. understanding the problem 2. Devising a Plan 3. Carrying Out the Plan 4. Looking Back Gauss's Approch n(n+1)/2

3.1.2. Section 2- Exploration with Pattern Inductive Reasoning -Method for making vernalization based on observations and patterns Conjecture- statement thought to be true, but is not yet proven to be true or false. Counterexample- Proves the conjecture false in Arithmetic Sequences- is one in which each successive term from the second term on is obtained from the previous term by adding or subtracting of a fixed number Example: 3,6,9,12.... Where"add 3" is the sequence An=a1+(n-1)d an=frist, n=term,d=difference Fibonacci Sequence- this is not arithmetic as there is not fixed term. Example:1,1,2,3,5,8,13,21,34,55,89,144 Fn=Fn-1=Fn-2 Geometric Sequences- Each successive term of a geometric sequence is obtained from its predecessor by multiplying by a fixed nonzero number, the ratio. Example: a Child has 2 biological parnets, 4 grandparents, 8 great grandparents, 16 great-great grandparents, and so on. in this case the first term and ratio are 2 An=A1x R^n-1 A1= first term R= ratio, N-1 desired term

3.2. Chapter 2

3.2.1. Section 1- Numeration Systems Numerals- the written symbols for digits 1. Hindu-Arabic Tally System Egyptian Numeration System Babylonian Numeration system Mayan Numeration System Roman Numeration System

3.2.2. Section 2&3 Vocab. Sets: is understood to be any collection of objects Elements: members of the set or individual objects in a set Ono-to-One Correspondence def.: of the element of set P and S can be paired so that each element of P there is exactly one element of S and for each element of S the is exactly one element of P, then the two sets P and S are said to be in one to one correspondence. Short cut: fundamental counting principal- 5*4*3*2*1 Equivalent Sets Two set A and B are equivalent, written A~B if an only of there exist a one-to-one correspondence between the sets. Example: A=(p,q,r,s), B=(a,b,c), C=(x,y,z)and D=(b,a,c), Set A and B are not equivalent and are not equal Set B and C are equivalent, but are not equal Cardinal Numbers The cardinal number of a set denotes that number of elements in the set Example: The Set, (a,b),(p,q), (x,y), (b,a),(*,#) are equivalent to one another and share the property of "twoness"; That is these sets have the same cardinal number, namely, 2. Sets Universal set or the universe, Denoted U, is the set that contains all elements being considered in a given discussion. look at page 85 in book Subsets B is a subset of A, if and only if, every element of B, is in the element of A Example:A=(1,2,3,4,5,6) and B=(2,4,6) All of the elements of B are contained in set A and we say that be is a subset of A. Set Intersections The intersection of two set, A and Bm is the set of all elements common to both A and B. Example: A=(1,2,3,4) and B=(3,4,5,6) Thus, A intersects with B=(3,4) Set Union The union of two sets A and B, is the set of all elements in A to in B Example: A=(1,2,3,4), B=(3,4,5,6) the Set union would be (1,2,3,4,5,6,) Set Difference Also Called the relative complement. The complement of A relative to B, is the set of a; Elements in B that are not in A. Example: A=(d,e,f), B=(a,b,c,d,e,f), C=(a,b,c) thus the set difference would be (a,b,c)- Properties of set Operations Associative Property Commutative Property Cartesian Products For any set A and B, the Cartesian produce of A and B, written A X B, (reads A cross B) is the set of all ordered pairs such that the first component of each pair is an element of B. Example: Suppose a person has three pairs of pant P=( blue, white, green) and two shirts S=(blue, red). There are 3 X 2 or 6 possible different pant-and-shirt pairs.

4. Test 2

4.1. Chapter 3

4.1.1. Section 1:Addition of Whole Number Set Model Example:Set A n(A)= {a,b,c,d} Set B n={e,f,g} n(A) + n(B)= {a,b,c,d,e,f,g}=4+3=7 n( A U B) The numbers a+b are the ADDENDS and a+b is the SUM Number Line (Measurement) Model Solve 4+3 using a number line. ALWAYS START AT ZERO WITH YOUR FIRST NUMBER!! Addition Properties Closure Property Commutative Property Associative Property: Also known as the GROUPING PROPERTY Identity Property Mastering Basic Addition Facts Counting On Doubles Making 10 (and Then add any leftovers) Counting Back

4.1.2. Section 1: Subtractions of Whole Numbers Inverse Operations- operations that undo each other. Subtractions in the inverse of addition Take-Away Model- you have 8 blocks, take away 3 blocks=8-3+5 blocks left. Missing Addend Model- Relates addition and subtractions 3+__=8 Put in 3 blocks???=8 8-3=5 Number line Fact Families Comparison Model Number line Model Properties of Subtractions Closure- {1,3,5,7,...} (3-5=-2) no answer is not WHOLE number Associative (a-b)-c=a-(b-c) Yes Commutative a-b=b-a No Identity- a-0=a, Yes However: 0-a=0 is not true.

4.1.3. Section 2: Algorithms for Whole-Number Addition and Subtractions Addition Algorithms To help students understand algorithms, we should start with manipulative. Children can touch, move around, and be led to developing their own algorithms (a procedure to accomplish an operation.) After working with manipulative, then move to paper/pencil operations. Regrouping or trade problems are then used to describe carrying Lattice Algorithm for Addition Subtraction Algorithms Use base-ten block to provide a concrete model for subtraction as we did in addition The concept of remove or take away in used Then paper/pencil algorithms are introduced Equal-Addition Algorithm Based on the fact that the difference two number does not change if we add the same amount to both numbers.

4.1.4. Section3 : Multiplication and Division of whole number Multiplication of Whole Number Repeated-Addition Model The Array and Area Model Cartesian-Product Model Properties of Whole Number Multiplications Closure property Commutative property Associative property Identity Property Zero Multiplications property Distributive property Divison of Whole Numbers Set (Partition) Model Missing-Factor Model Repeated subtraction Model The division algorithm Given any whole number a and b with b=0, there exist unique whole numbers q (quotient) and r (remainder) such as a=bq+r. When a is "divided" by b and the remainder is 0, we say that a is divisible by b or that b is a divisor of a or that b divides a Relating Multiplication and Division as Inverse Operations Division is the inverse of multiplication Divison with a remainder of 0 and multiplication are related Division by 0 or 1 n divided by 0 is undefined ( there is no answer to the equivalent multiplication problems) 0 divided by n=0 0 divided by 0 is under fined also Order of Operations When solving equations, students have difficulties involving the order of arithmetic operations

4.1.5. Section 4: Algorithms for whole-numbers multiplications and division Multiplications algorithms Single digits numbers times two digit number Multiplications with two-digit factors hundred- tens-ones Lattice Multiplications Division algorithms The typical instructions for "long division" is Using base-ten block to develop standard division algorithms Division by a two-digit divisor

4.2. Chapter 4: Functions

4.2.1. Fuctions: is a relationship that assigns exactly on output value for each input value

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

4.2.3. Functions as Rule- each element of A is pair with exactly one element of B.

4.2.4. Functions as Machines

4.2.5. Functions as Equations

4.2.6. Functions as Arrow Diagrams

4.2.7. Functions as table and Ordered Pairs

4.2.8. Relations- a relation from set A to set B is a correspondence between elements of A and element of B. but unlike functions, do not require that each element of A be paired with one, and only one element B Any set of order pairs is a relations Every function is a relation but not every relation is a function

5. Test 3

5.1. Chapter 5

5.1.1. Section 1 Integer Additon Chip Model for Addition: Black chips are used to represent positive integers and negative integers are red. Charged-Field Model: Similar to chips but use the positive(+) and (-) charges. Number Line Model: Always start at zero (0) and move line to first number, then go to second number. Pattern Model: Addition of integers can also be motived by using patterns of addition of whole numbers. (pg. 254). Absolute Value- The distance between the point corresponding to an integer and 0 is the absolute value of the integer. Thus, the absolute value of both 4 and -4, written |4| and |-4|=4. Notice that if x>=0, the |x|=x and if x<0, then -x is positive. Properties of Integer Addition Closure property of addition of integers Commutative property of addition of integers Associative property of addition of integers Identity element of addition of integers- 0 is the unique integer such that, for all integers a, 0+a=a=a+0. Uniqueness of the Addition Inverse For every integer a, there exists a unique integer -a, the additive inverse of a, such that a+ -a=0=-a +a. Also, -(-a)=a and -a+-b=-(a+b). Integer Subtraction Chip Model for Subtraction- Same as addition, but notice when you have 3-(-2) you must first put in 3 black chips, then put in 2 black and red chips, then subtract 2 chips to make the equation true. Subtractions Using the missing addend approach: 5-3=n if and only if 5=3+n Subtraction Using Adding the Opposite Approach: a-b=a+-b, Example: 2-8=2+-8=-6 -(b-c)=-(b+-c)=-b+-(-c)=-b+c. Properties of Subtraction: Cannot do commutative nor associative with subtraction.

5.1.2. Section 2 Multiplication of Integers Patterns Model for Multiplication of Integers Chip Model and Charged-Field Model for Multiplications- See pages 270-271 Number-Line Model pages 271-272 Properties of integer Multiplication Integer Division The quotient of two negative integers, is a positive integer and the quotient of a positive and negative integers, is negative. Order of Operations on integers Use the order of operation: parenthesis, exponents, multiple/divide, left to right, add/subtract left to right.

5.1.3. Section 3 Section 3 Divisibility Rules

5.1.4. Section 4 Prime Numbers Numbers in which there are only two (2) factors or positive divisors. Some example are 2- (2x1), 3-(3x1), 5-(5x1). Composite Numbers Numbers in which there are more than two factors or positive divisors. Some example are 4- (4x1 & 2x2), 12-( 1x12, 2x6, 3x4). Prime Factorization 1. Composite numbers can be expressed as products of two or more whole numbers greater tun 1. A factorization containing only prime numbers Factor tree Ladder Model Number of Divisor they are groups of divisors Sieve of Eratosthenes Method for identifying prime number-pg.309

5.1.5. Section 5 Greatest Common Divisor (Factor)-GCE: the greatest common divisor of two natural numbers a and b is the greatest natural number that divides both a and b. Colored rod Method The intersections of Sets Method Prime Factorization Method Least Common Multiple- The least common multiple (LCM) of a and b is the least natural number that is simultaneously a multiple of b Number line Method Colored Rods Method- pag. 322 Intersection of Sets Method Prime Factorization Method Ladder Method or the division by prime methods

6. Test 4

6.1. Chapter 6

6.1.1. Section 1: The set of rational numbers In the rational number a/b, a is the numerator and b is the denominator. it can be represented as a/b or as "a divided by b". 1. Uses of rational Numbers 1. division problem or solution to a multiplication problem such as 2x=3 is 3/2 2. Partiton, or part, of a whole such as joe ate 1/2 of the pizza for dinner. 3. Ratio such as the ratio of girls to boys in the class was ten to twelve. 4. Probability such as when you toss a fair coin, the probability of getting heads is 1/2. Research has found teaching with the area model (part of a whole) is preferred over the set model ( group of colored chip). Rational number can be represented on a number line. Proper fraction Improper Fraction 2. Equivalent or equal Fractions A. Equivalent fraction are numbers that represent the same point on a number line. B. The value of the fraction does not change if its numerator and denominator are multiplied by the same nonzero integer. C. Fundamental Law of Fractions. Let a/b be any fraction and n a nonzero integer. Then, a/b=an/bn. D. Is 7/-15? Yes because 7/-15x -1/-1=-7/15 E. Equivalent fractions can be found from dividing n/n into a fraction such as 12/42=2/7 x 6/6 =2/7 3. Simplifying Fractions A. 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. B. To write a fraction a/b in simplest form; that is in lowest terms, we divide both a and by the GCD. C. Example of simplifying a fraction: 60/120 divided 10/10 = 6/21 divided by 3/3 = 2/7. 4. Equality of Fractions A. Simply both fractions to the same simplest forms: 10/35=5x2/5x7=2/7 B. Rewrite both fractions with the same least common denominator. Since LCM (42, 35)=210 then, 12/42= 420/1470 and 10/35= 420/1470 hence: 12/42= 10/35 C. Rewrite both fractions with a common denominator (not necessarily the least). 5. Ordering Rational number A. start by comparing fractions with like denominators such as 7/8 >5/8. B. With unlike denominators, best start with fractions strips or bars to compare the fractions visually. C. Comparing fractions with unlike denominators can be accomplished by rewriting the fractions with the same positive denominators. Compare 3/4 and 9/16. find the common denominator (16), then change 3/4 to 12/16 and compare to 6/16, thus; 3/4 > 9/16. D. Comparing rational numbers can be accomplished by estimating on a number line for the number closest to 0, 1/2, 1. Also you can look at the denominators to estimate the order. 6. Denseness of Rational Numbers Given two different rational numbers a/b and c/d, there is another rational number between these two numbers.

6.1.2. Section 2: Addition, Subtraction, and Estimation with Rational Numbers. Mixed Numbers Numbers that are made up of an integer and a fractional part of an integer. A mixed number is a rational number, and therefore, it can always be written in the form a/b.

6.1.3. Section 3: Multiplication and division of Rational Numbers Multiplication using repeated addition, or part of an area. Multiplicative Identify- The number 1 is the unique number such that for every rational number a/b. Multiplicative inverse- For any nonzero rational number a/b, b/a is the unique rational number such that a/b, b/c=1=b/a, a/b. Distributive Property of multiplication over addition- Multiplication property of Equality Multiplication property of inequality Multiplication property of zero

6.2. Chapter7

6.2.1. Section 1 Decimals are most familiar when dealing with money Decimals can be written in expanded form; 12.618= 1X10^1 + 2X10^0 + 6X10^-1 + 1X10^-2 + 8X10^-3 Covert each to decimals: 25/10=((2X10)/10)10+5/10=2+5/10=2.5. Terminating Decimals-Secimals that can be written with only a finite number of places to the right of the decimals point. Ordering Terminating Decimals- Change decimal by adding place value such .36 and .9 change to .36 and .90 to ease in ordering decimals. Line up Decimals such as: 1. Line up the numbers by place value. 2. Start at left and find the first place where the face values are different. 3. Compare these digits. The digit with the greatest face value in this place represents the greater of two numbers

6.2.2. Section 2 1. Algorithms for addition and subtraction of terminating decimals 1. Use base-ten blocks 2. Change it to a problem we already know hoe to solve. 3. Line up the decimals points and add or subtract. 2. Algorithms for multiplying decimals if there is n digits to the right of the decimals point in one number and m digits to the right of the decimals point in a second number, multiply the two numbers, ignoring the decimals, and then place the decimal point so that there are n+m digits to the right of the decimal point in the product. Scientific Notation- Use when you have wither very small are very large numbers.

6.2.3. Section 3 Ways to convert some rational numbers to decimals: 7/8=7/3^3=7x5^3/2^3X5^3=875/1000=.875 or use as a division problem-"numerator divided by denominator point zero, zero, zero" n/d.000 7/8=7 8.000 Repeating Decimals 2/11=0.1818181818181 Writing Repeating Decimals as Fractions A short way of writing the repeating decimals 0.323232... is 0.32. you can write this repeating decimals as a fractions. Ordering Repeating Decimals write the decimals one' under the order, in their equivalent forms without the bars, and line up the decimal points ( or place values as follows: 1.34783478, 1.34782178.