Including, but not limited to: Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are whole numbers and b ≠ 0, which includes the subsets of whole numbers and counting (natural) numbers (e.g., 0, 2, etc.). Various forms of positive rational numbers Whole numbers Decimals (less than or equal to one) Fractions (proper or equal to one) Percents (less than or equal to 100%) Event – a probable situation or condition Outcome – the result of an action or event Mutually exclusive events – events that cannot happen at the same time Ex: Sample space – a set of all possible outcomes of one or more events Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p ≤1 Probability = Notation for probability P(event) The closer a probability of an outcome is to 1, the more likely the outcome will occur; whereas, the closer a probability of an outcome is to 0, the less likely the outcome will occur. Ex: Theoretical probability – the likelihood of an event occurring without conducting an experiment Various types of theoretical experiments Ex: Coins, drawing objects out of box without looking, spinners with even and uneven sections, choosing a random card, marbles, number cubes, etc. Representation of theoretical probability as a fraction, decimal, or percent Ex: Three out of the ten sections are blue: , 0.3, 30% Simple event – an event that consists of a single outcome Compound events – events that consists of two or more simple events and consists of more than one outcome Compound independent events – events with more than one outcome, and one event does not affect the outcome of the other Compound dependent events – events with more than one outcome, and the outcome of one event affects the outcome of the subsequent event or events Proportional reasoning to make predictions using theoretical probability Ex:
Tree diagrams Tables Fundamental Counting Principle – if one event has a possible outcomes and a second independent event has b possible outcomes, then there are a • b total possible outcomes for the two events together Probability – a ratio between the number of desired outcomes to the total possible outcomes, 0 ≤ p≤1 Probability = Notation for probability P(event) The closer a probability of an outcome is to 1, the more likely the outcome will occur; whereas, the closer a probability of an outcome is to 0, the less likely the outcome will occur. Ex: Theoretical probability – the likelihood of an event occurring without conducting an experiment Experimental probability – the likelihood of an event occurring from the outcomes of an experiment Various types of experiments Ex: Coins, drawing objects out of box without looking, spinners with even and uneven sections, choosing a random card, marbles, number cubes, etc. Representation of probability as a fraction, decimal, or percent Ex: Three out of the ten sections are blue: , 0.3, 30% Complement of an event – the probability of the non-occurrence of a desired outcome Ex: The probability of selecting a face card from a deck of cards is or . The complement of selecting a face card from a deck of cards is the probability of selecting any card but a face card from a deck of cards which is or . The outcomes of an event and its complement complete the sample space. Relationship between an event and its complement expressed as a ratio or numerical expression The sum of the probability of an event and its complement will always be 1. Ex: If, P(A) = , then P(not A) = or . Therefore, P(A) + P(not A) = = 1 Relationship between theoretical and experimental probability Law of large numbers – as the number of trials increases the difference between the experimental and theoretical probability will be closer to zero Simple event – an event that consists of a single outcome Ex: Compound events – events that consists of two or more simple events and consists of more than one outcome Compound independent events – events with more than one outcome, and one event does not affect the outcome of the other Ex: Compound dependent events – events with more than one outcome, and the outcome of one event affects the outcome of the subsequent event or events
Determine THE CIRCUMFERENCE AND AREA OF CIRCLES Including, but not limited to: Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are whole numbers and b ≠ 0, which includes the subsets of whole numbers and counting (natural) numbers (e.g., 0, 2, etc.). Various forms of positive rational numbers Whole numbers Decimals (less than or greater than one) Fractions (proper, improper, and mixed numbers) Circle A figure formed by a closed curve with all points equal distance from the center No straight sides No vertices No parallel or, perpendicular sides Diameter – a line segment whose endpoints are on the circle and passes through the center of the circle Radius – a line segment drawn from the center of a circle to any point on the circle and is half the length of diameter of the circle Semicircle – half of a circle Quarter circle – one-fourth of a circle Circumference – a linear measurement of the distance around a circle Positive rational number dimensions Pi (π) – the ratio of the circumference to the diameter of a circle π ≈ 3.14 or Formulas for circumference from STAAR Grade 7 Mathematics Reference Materials Circumference using the radius of a circle C = 2πr, where C represents the circumference of the circle and r represents the radius of the circle, and π is approximately 3.14 or Ex: Circumference using the diameter of a circle C = πd, where C represents the circumference of the circle, d represents the diameter of the circle, and π is approximately 3.14 or Ex: Area – the measurement attribute that describes the number of square units a figure or region covers Area is a two-dimensional square unit measure. Positive rational number dimensions Formula for area of a circle from STAAR Grade 7 Mathematics Reference Materials Area
Write ONE-VARIABLE, TWO-STEP EQUATIONS AND INEQUALITIES TO REPRESENT CONSTRAINTS OR CONDITIONS WITHIN PROBLEMS Including, but not limited to: Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol Variable – a letter or symbol that represents a number One on one side of the equation or inequality Ex: Coefficient – a number that is multiplied by a variable(s) Whole numbers Integers Decimals (less than or greater than one) Fractions (proper, improper, and mixed numbers) Ex: Constant – a fixed value that does not appear with a variable(s) Whole numbers Integers Decimals (less than or greater than one) Fractions (proper, improper, and mixed numbers) Ex: Solution set – a set of all values of the variable(s) that satisfy the equation or inequality Constraints or conditions Ex: Minimum, maximum, up to, no more than, no less than, etc. Distinguishing between equations and inequalities Characteristics of equations Equates two expressions Equality of the variable One solution Characteristics of inequalities Shows the relationship between two expressions in terms of >,≥, ≤,or ≠ Inequality of the variable One or more solutions Equality and inequality words and symbols Equal to, =
Write a corresponding real-world problem given a one-variable, two-step equation or inequality. Supporting Standard Write a corresponding real-world problem given a one-variable, two-step equation or inequality. Supporting Standard Write A CORRESPONDING REAL-WORLD PROBLEM GIVEN A ONE-VARIABLE, TWO-STEP EQUATION OR INEQUALITY Including, but not limited to: Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol Variable – a letter or symbol that represents a number One variable on one side of the equation or inequality Ex: Coefficient – a number that is multiplied by a variable(s) Whole numbers Integers Decimals (less than or greater than one) Fractions (proper, improper, and mixed numbers) Ex: Constant – a fixed value that does not appear with a variable(s) Whole numbers Integers Decimals (less than or greater than one) Fractions (proper, improper, and mixed numbers) Ex: Solution set – a set of all values of the variable(s) that satisfy the equation or inequality Constraints or conditions Ex: Minimum, maximum, up to, no more than, no less than, etc. Distinguishing between equations and inequalities Characteristics of equations
Use DATA FROM A RANDOM SAMPLE TO MAKE INFERENCES ABOUT A POPULATION Including, but not limited to: Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are whole numbers and b ≠ 0, which includes the subsets of whole numbers and counting (natural) numbers (e.g., 0, 2, etc.). Various forms of positive rational numbers Whole numbers Decimals (less than or greater than one) Fractions (proper, improper, and mixed numbers) Data – information that is collected about people, events, or objects Inference – a conclusion or prediction based on data Population – total collection of persons, objects, or items of interest Sample – a subset of the population selected in order to make inferences about the entire population Ex: Ex: Random sample – a subset of the population selected without bias in order to make inferences about the entire population Random samples are more likely to contain data that can be used to make predictions about a whole population. Data from a random sample given or collected in various forms Verbal Tabular (vertical/horizontal) Graphical Inferences based on random sample Qualitative – a broad subjective description (e.g., the probability of an event occurring is certain, more likely, not likely, equally likely, or impossible.) Quantitative – a narrowed objective description associated with a quantity (e.g., the probability of selecting a consonant from the word EXPERIMENT is 1.5 times as likely as selecting a vowel from the same word, etc.) Statistical analysis of data in a random sample to make inferences about a population Ex: Ex: Note(s):
amount, r represents the interest rate in decimal form, and t represents the number of years the amount is deposited or borrowed Ex: Comparing simple and compound interest earnings Ex: Note(s): Grade Level(s): Grade 4 compared the advantages and disadvantages of various savings options. Grade 8 will calculate and compare simple interest and compound interest earnings. Various mathematical process standards will be applied to this student expectation as appropriate. TxRCFP: Financial Literacy TxCCRS: I. Numeric Reasoning VIII. Problem Solving and Reasoning IX. Communication and Representation X. Connections Analyze and compare monetary incentives, including sales, rebates, and coupons. Supporting Standard Analyze and compare monetary incentives, including sales, rebates, and coupons. Supporting Standard Analyze, Compare MONETARY INCENTIVES, INCLUDING SALES, REBATES, AND COUPONS Including, but not limited to: Monetary incentives Sale – a reduced amount or price of an item