Math (802)

Probability and Statistics (005) Review carefully

Counting techniques for solving problems Research on Teaching data analysis Probability: Range -- difference between greatest and least numbers in data set Mean -- average Median -- middle value of all numbers Mode -- value that appears with greatest frequency Probability -- likelihood of particular outcome

Number Concepts and Operations (002)

Integer -- whole number that includes all positive and negative numbers, including zero Natural Numbers -- positive integer Rational Numbers -- a number that can be expressed as a ratio or quotient of 2 nonzero integers Irrational Numbers -- cannot be expressed as an exact ratio of 2 integers (square root of 2, pi) Real Numbers --any number, positive or negative and zero to measure continuous quantities Place Value --numbers in value scale Basic Operations: Addition Subtraction -- subtrahend subtracted from minuend Multiplication Division -- divisor outside bracket and dividend (number being dividing) inside bracket Fractions: Finding equivalent fractions, simplifying fractions Adding and subtracting homogeneous fractions Changing improper fractions to mixed numbers Adding and subtracting mixed numbers Changing mixed numbers to improper fractions Adding fractions with different denominators Multiplying fractions -- multiple numerators and denominators Dividing fractions -- reciprocal of second fraction; 1/5 divided by 3/8 becomes 1/5 times 8/3 Decimal numbers Fractional numbers written using base 10. Mixed decimal number has a whole number part and decimal number part. Adding and subtracting decimals Multiplying decimals Dividing decimals -- number of digits to right of decimal point in the divisor (number doing the dividing) is how far the decimal point in the dividend is moved; example: 1.44 divided by .3 becomes 14.4 divided by 3 Percents -- expressed as fractional number in terms of 1/100 Mathematical Terms and Operations: Exponential Notation -- how many times a number is used as a factor Absolute Value -- distance from zero, ignoring + and - Expanded Form -- place value of each digit; example 263 = 200+60+3 = 2 hundreds, 6 tens, 3 ones Expanded Notation --shows place value by multiplying each digit by the power of 10; example: 523 = 5 x 10 to the 2nd + 2 x 10 + 3 x 1; note 10 to the zero power = 1 Scientific Notation -- number as product of power of 10 and a decimal; example: 2,400,000 = 2.4 x 10 to the 6th power Estimating and rounding Number Theory Prime factorization Composite numbers have at least 3 factors; example: 9 is a composite because it has 3 factors: 1, 3, 9 Greatest Common Divisor Common Multiple Numbers and their Properties: Commutative Property Associative Property of Multiplication and Addition: (a+b)+c= a+(b+c) Property of zero Distributive property Order of Operations 1. Simplify within grouping symbols such as brackets 2. Apply exponents -- powers and roots 3. Perform all multiplications and divisions in order from left to right 4. Perform all additions and subtraction in order from left to right PEMDAS -- Please excuse my dear Aunt Sally -- Parentheses, Exponentiation, Multiplication, Division, Addition, Subtraction

Mathematical Process (006) Review carefully

Logical reasoning Problem-solving strategies Developmental considerations Historical Development in Mathematics: Egyptians (wrote numbers )and Babylonians (made greatest advances in science, math, and astronomy) Greeks (built on accomplishments of Egyptians and Babylonians; formal study of math) Hinds (5th century developed system that allowed for astronomical calculations) Arabs (first to solve sophisticated algebraic equations) Concept of Zero (India around 650 ce) Number System (Hindu-Arabic)

Mathematics Instruction (001)

Principles of Math: Equity -- high expectations and strong support for all students Curriculum -- coherent, focused on concepts Teaching -- understanding what students know and need to learn while challenging and supporting Learning -- students must learn with understanding Assessment --support learning Technology -- essential in teaching and learning math Math for ELLs: Math is not a universal language Nomenclature of Math: Special vocabulary Developmentally Appropriate Instruction: Children develop basic understanding of numbers at 2 years old. Infants understand concepts of "none" and "more". Rote counting begins around 2 or 3 Cognitive Development and Math: Requires children to create and recreate math relationsips in their own minds. Piaget's developmental stages: Sensorimotor stage (birth-2) Preoperational (2-7) Concrete Operational (7-11) Formal Operational (11-adult) Math in real-life situations -- encourage students to relate math to their daily lives. Standard and Metric systems of measurement Manipulatives in math Use of technology Learning environment -- 4 categories: 1. tasks -- projects, questions, problems, exercises 2. discourse -- manner of representing, thinking, talking 3. environment -- setting for learning 4. analysis -- systematic reflection TEKS by grade: Pre-K -- explore concrete models and materials, counts to 10, begins to recognize concept of 0 K -- whole number concepts through 20, concepts of "part of" and "whole", sorting 1st -- creates setsof 10s and 1s using concrete objects, reads and writes up to 99, separates whole into 2, 3, 4 equal parts, models and creates addition and subtraction problems, identifies coins by name and value 2nd -- concrete models of 100s, 10s, and 1s; begins to use place value; fractional parts; addition and subtraction of 2-digit numbers; determines value of collection of coins up to $1; decribes money symbols 3rd -- place value up to 9,999; fraction names and symbols; addition and subtraction for whole numbers through 999; rounding and estimating solutions; multiplication facts through 12 4th -- place value up to 999,999999 and decimals for 10ths and 100ths, including money; generate equivalent fractions; multiplcation to solve 2 digits times 2 digits; division to solve 1 digit divisors and 3 digit dividends; rounding, compatible numbers to estimate 5th -- place value to 999,999,999,999; decimals to 1000s place; common factors; multiply 3 digits times 3 digits; division with 2 digit divisors and 3 digit dividends; solutions with a remainder 6th -- nonnegative rational numbers; equivalent forms of rational numbers; integers; prime factorization using exponents; factors of positive integer; common factors; greatest common factor Analysis of teaching and learning Making connections with the real world Thematic instruction Planning and organizing thematic units -- selecting theme, designing integrated curricular, gathering materials, arranging thematic activities Types of questioning strategies Formal and informal assessments -- STAAR

Patterns and Algebra (003)

Problem-Solving Situations Linear and Nonlinear Functional Relationships -- linear function is a straight line and always satisfies the following: f(x+y)=f(x) + f(y). Nonlinear function does not satisfy constraints of linear function; example: f(x)=x to the 2nd is nonlinear Algebraic Pattern Algebraic Expression -- note when algebraic terms are divided, exponents are subtracted Multiplication of Binomials -- FOIL Algebraic Relationship Algebraic Solution Performing Operations with Negative and Positive Numbers Graphs and Symbolic Representations -- pictorial graphs, bar graphs, line graphs, pie charts Identifying Patterns using Concrete Models -- begin in Kinder Proportional reasoning Variables, Equations, and Inequalities

Geometry and Measurement (004) (more details -- need to review)

Van Hiele Theory of Geometric Thinking - visualization, analysis, informal deduction, formal deduction, rigor Points, lines, angles, lengths, and distances: 1. Line segment -- any portion of a line between 2 end points. 2. Ray -- line segment that extends forever in one direction. 3. Angle -- 2 rays that share an endpoint. Right angle, acute angle, obtuse angle, congruent (If 2 angles have the same size regardless of how long their rays are, they are congruent), supplementary angle (2 angles that add up to 180), complementary angles (2 angles that add up to 90) 4. Parallel Lines and perpendicular lines. Transverse is a line that crosses them. Polygons -- many sided plan. Angle in a polygon is a vertex. Triangle = 3 sides Quadrilateral = 4 sides Pentagon = 5 sides Hexagon = 6 sides Octagons = 8 sides Triangles: Isosceles -- 2 equal sides Equilateral -- all 3 side are equal Scalene -- 3 unequal sides Acute -- all 3 angles are acute Right triangle -- one angle is a right angle Obtuse -- one angle is obtuse Reflex -- angle is greater than 180 but less than 360 Pythagorean Theory -- for any right triangle with legs a and b and hypotenuse (longest side) c: a squared + b squared = c squared Congruent: side-angle-side, side-side-side, angle-side-angle, angle-angle-side Quadrilaterals: Parallelograms Rectangles Squares Circles: diameter, radius (half of diameter) Formulas for finding perimeter and area: Triangle P=S1+S2+S3; A=1/2(bxh) Rectangle and Square: P=S1+S2+S3+S4; A=law Circle: circumference=2 pi r or pi diameter; pi r squared Similarity and Congruence Similar if they have the exact same shape, even if not the same size Congruent if they have the same shape and size 3-D Figures: Rectangular solid -- volume = l x w x h Cube --volume = l x w x h or side cubed Cylinder -- volume = pi x r squared x h Prism -- volume = area of the prism's base by its height Sphere Vertex -- union of 2 points Face -- each plane region Edge -- union of 2 faces Symmetry -- same if folded in half Tessellations Ordered pairs Coordinate Plane -- x axis and y axis, origin in the point intersection of the 2 axes Concepts of measurement Time -- digital and analog Temperature Money Transformations: Translation -- moving Rotation -- turning Reflection -- flipping Conversions within and between measurement systems: Linear -- inches, feet, yards, miles, millimeters, centimeters, meters, kilometers; 2.5 cm = 1 inch, more than 1.5 km = 1 mile Mass -- ounces, pounds, tons, grams, kilograms; 28 grams = 1 ounce Volume -- teaspoon, tablespoon, cup, pint, quart, gallon, milliliter, liter; a liter is slightly larger than a quart 1760 yards = 1 mile 5280 feet = 1 mile 1 gallon = 4 quarts = 128 fluid ounces 1 quart = 2 pints 1 pint = 2 cups = 16 fluid ounces 1 ton = 2000 pounds Logical Reasoning: Deductive reasoning -- move from assumption to conclusion Inductive reasoning -- examine particular instance to make a general assumption Axiomatic Structure: Axiom is a math rule