* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Unit 1, Concept 1 – Number Sense, Fractions, and Algebraic
John Wallis wikipedia , lookup
Abuse of notation wikipedia , lookup
Infinitesimal wikipedia , lookup
History of mathematics wikipedia , lookup
History of logarithms wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Law of large numbers wikipedia , lookup
History of mathematical notation wikipedia , lookup
Location arithmetic wikipedia , lookup
Real number wikipedia , lookup
Large numbers wikipedia , lookup
System of polynomial equations wikipedia , lookup
Positional notation wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
History of algebra wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Unit 1, Concept 1 – Number Sense, Fractions, and Algebraic Thinking Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 7AF 1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)². 7AF 1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. 7AF 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Concept (and related skills) Understand the composition of whole numbers • Represent algebraic situations using pictures • Use the order of operations • Relate factors, multiples, and divisibility • Identify numbers as prime or composite • Use powers and exponents to write repeated multiplication • Recognize and use the commutative and associative properties of multiplication • Recognize and use the multiplicative identity Textbook Connections Vocabulary Lessons 1.1 Introduction to Picture Algebra 1.2 Factors and Multiples 1.3 Least Common Multiple 1.4 Prime and Composite Numbers 1.5 Prime Factorization 1.6 Powers and Exponents 1.7 Greatest Common Factor Associative Property of Multiplication Base Common factor Common Multiple Commutative Property of Multiplication Composite factor Composite Number Divisible Exponent Expression Factor Factor pair Greatest common Factor Least common Multiple Multiple Multiplicative Identity Order of operations Power Prime factor Prime factorization Prime number Variable Variable expression 86 Unit 1, Concept 2 – Number Sense, Fractions, and Algebraic Thinking Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: Note: This concept does not explicitly address a seventh grade standard, but it is necessary for the development of future concepts. Concept (and related skills) Understand the composition of fractions • Use fractions to represent parts of a whole • Use fractions to divide more than one whole into equally sized parts • Write equivalent fractions • Write fractions in simplest form • Compare and order fractions Textbook Connections Vocabulary Lessons 2.1 2.2 2.3 2.4 2.5 2.6 Dividing a Whole into Fractional Parts Dividing More Than One Whole into Parts Dividing Groups into Fractional Parts Equivalent Fractions Simplifying Fractions Comparing and Ordering Fractions Completely simplified Denominator Equation Equivalent fractions Fraction Greater than Least common denominator Less than Numerator Reasonable solution Simplest form Simplest terms 87 Unit 1, Concept 3– Number Sense, Fractions, and Algebraic Thinking Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 7NS 2.2 Add and subtract fractions by using factoring to find common denominators. Concept (and related skills) Textbook Connections Vocabulary Lessons Understand operations on fractions and mixed numbers • Add and subtract like fractions • Add and subtract unlike fractions • Convert between improper fractions and mixed numbers • Multiply and divide fractions • Add, subtract, multiply, and divide mixed numbers 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Adding and Subtracting Fractions with Like Denominators Adding and Subtracting Fractions with Unlike Denominators Improper Fractions and Mixed Numbers Multiplying Fractions Dividing Fractions Adding and Subtracting Mixed Numbers Multiplying and Dividing Mixed Numbers Working with Customary Units Capacity Cup, pint, quart, Gallon Improper fractions Inch, foot, yard, mile Least common Denominator Length Like fractions Metric system Mixed number Multiplicative Identity Multiplicative Inverse Ounce, pound, ton Reciprocal Remainder Unlike fractions Weight 88 Unit 2, Concept 1– Decimals, Ratios and Proportions, and Percents Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 7NS 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Concept (and related skills) Understand the composition of decimals and operations on decimals • • • • • • • Textbook Connections Vocabulary Lessons 4.1 Decimals as Special Fractions 4.2 Place Value and Expanded Form 4.3 Decimals as Fractions: Comparing and Write decimals as fractions Represent decimals using a placeRounding Decimals value chart, in word form, and with 4.4 Adding and Subtracting Decimals expanded form 4.5 Multiplying Decimals 4.6 Dividing Decimals Compare and order decimals 4.7 Working with Metric Units Round decimals Add, subtract, multiply, and divide decimals Represent decimals using base-ten pieces Use metric units to measure length, mass, capacity Decimal Dividend Divisor Expanded form Gram Liter Meter Metric system Place-value Product Quotient Rounding off Standard form 90 Unit 2, Concept 2 – Decimals, Ratios and Proportions, and Percents Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7AF 4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. 7MG 1.1 Compare weights, capacities, geometric measures, times, and temperatures within an between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters). 7MG 1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer. Concept (and related skills) Understand ratio and proportion Write ratios as fractions Compare ratios Write proportions Solve proportions Find the means and extremes of a proportion • Find unit rates • Solve problems using proportions • • • • • Textbook Connections Vocabulary Lessons 5.1 5.2 5.3 5.4 Ratios and Fractions Writing and Solving Proportions Using Ratios and Rates Using Proportions to Solve Problems Extremes Means Proportion Rate Ratio Unit Rate Variable 91 Unit 2, Concept 3 – Decimals, Ratios and Proportions, and Percents Instructional Resources: Carnegie Learning: Bridge to Algebra Standard: 7NS 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. 7NS 1.6 Calculate the percentage of increases and decreases of a quantity. 7NS 1.7 Solve problems that involve discounts, markups, commissions, and profit, and compute simple and compound interest. Concept (and related skills) Understand percents • Write percents as decimals and fractions • Write decimals and fractions as percents • Use benchmark percents (1% and 10%) to find the percent of a number • Use a proportion to find the percent of a number and to find a number that corresponds to 100% • Use a proportion to find percent, given two numbers • Find simple interest • Find the percent increase or decrease of a quantity Textbook Connections Vocabulary Lessons 6.1 6.2 6.3 6.4 6.5 6.6 Percents Making Sense of Percents Finding the Percent of a Number Finding One Whole, or 100% Finding Percents Given Two Numbers Percent Increase and Percent Decrease Benchmark percent Commission Discount Markup Percent Percent decrease Percent increase Principal Simple interest 92 Unit 3, Concept 1 – Integers, Algebraic Problem Solving and Linear functions (Including Graphing) Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7NS 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation. 7NS 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 7NS 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base. 7NS 2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. 7AF 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Concept (and related skills) Understand integers and operations on integers. • Write integers to represent reallife situations • Graph integers on a number line • Compare integers • Add, subtract, multiply, and divide integers • Write the absolute value of a number • Represent numbers using powers of 10 • Multiply and divide by powers of 10 • Read and write numbers using scientific notation Textbook Connections Vocabulary Lessons 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Negative Numbers in the Real World Adding Integers Subtracting Integers Multiplying and Dividing Integers Absolute Value and Additive Inverse Powers of Ten Scientific Notation Absolute value Additive inverse Difference Expanded form Exponent Integer Integer addition Integer subtraction Negative exponent Negative integer Number line Opposites Positive integer Power Power of ten Product Quotient Scientific notation Sum 94 Unit 3, Concept 2 – Integers, Algebraic Problem Solving and Linear functions (Including Graphing) Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7AF 1.1 Use variables and appropriate operations to write an expression, and equation, an inequality, or a system or equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). 7AF 4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. 7AF 4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. Concept (and related skills) Textbook Connections Understand algebraic problem Lessons solving (linear applications) and plotting sets of points to 8.1 Picture Algebra represent solutions 8.2 Writing, Evaluating, and Simplifying Expressions 8.3 Solving One-Step Equations • Use picture algebra to represent 8.4 Solving Two-Step Equations and solve problems 8.5 Plotting Points in the Coordinate Plane • Evaluate expressions • Solve one-step equations • Solve two-step equations • Identify and graph points in the coordinate plane • Make tables of values • Create a graph of ordered pairs • Use equations, tables, and graphs to solve problems Vocabulary Algebraic expression Cartesian coordinate system Coordinate plane Equation Evaluate Inverse operations One-step equation Ordered pair Origin Solve Two-step equation Variable x-axis and y-axis x-coordinate and y-coordinate 95 Unit 3, Concept 3 – Integers, Algebraic Problem Solving and Linear functions (Including Graphing) Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7AF 1.1 Use variables and appropriate operations to write an expression, and equation, an inequality, or a system or equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). 7AF 1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. 7AF 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. 7AF 3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities. Concept (and related skills) Understand linear functions and graphing lines • • • • • • • Textbook Connections Vocabulary Lessons 8.7 Use tables and graphs to 13.1 represent functions 13.2 Use function notation 13.3 Make input-output tables for 13.4 linear functions and use these to 13.5 graph those functions Find the slope of a line as a ratio and as a rate of change Find the x- and y-intercepts of a line Graph lines using slopes and intercepts Use the slope-intercept form of the equation of a line to graph the line Solving Problems Using Multiple Representations Relations and Functions Linear Functions Slope and Rates of Change Finding Slope and Y-Intercepts Using Slope and Intercepts to Graph Lines “Rise” and “Run” Dependent variable Domain and range Function Function notation Graph Independent variable Input-output table Linear equation Linear function Multiple representation Rate of change Relation Slope Slope-intercept form Table x-intercept and y-intercept 96 Unit 4, Concept 1 – Probability and Statistics, Number Systems, Area, and the Pythagorean Theorem Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7NS 1.5 Know that every rational number is either a terminating or a repeating decimal and be able to convert terminating decimals into reduced fractions. 7NS 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base. 7NS 2.3 Multiply, divide, and simplify rational numbers by using exponent rules. Concept (and related skills) Understand rational and irrational numbers • Use the number line to compare and order rational numbers • Learn about and classify different types of numbers • Find powers of rational numbers • Multiply and divide powers of rational numbers • Identify decimals as terminating or repeating • Write repeating decimals as fractions • Use and apply the distributive property Textbook Connections Vocabulary Lessons 14.1 14.2 14.3 14.4 14.5 Rational Numbers Powers of Rational Numbers Irrational Numbers Real Numbers and their Properties The Distributive Property Distributive property Integer Irrational number Natural number Power Rational number Real number Repeating decimal Terminating decimal Venn diagram Whole number 98 Unit 4, Concept 2 – Probability and Statistics, Number Systems, Area, and the Pythagorean Theorem Instructional Resources: Carnegie Learning: Bridge to Algebra Standards: 7 SDAP 1.1 Know various forms of display for data sets, including a stem-and-leaf or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data. 7 SDAP 1.3 Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the maximum of a data set. Concept (and related skills) Understand elements of probability and statistics • Find the probability of an event • Distinguish between independent and dependent events • Find the probability of a compound event • Find the mean, median, mode, and range of a set of data • Create and interpret frequency tables and histograms • Create and interpret stem-andleaf plots • Create and interpret box-and whisker plots • Create and interpret circle graphs Textbook Connections Vocabulary Lessons 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Finding Simple Probabilities Finding Probabilities of Compound Events Mean, Median, Mode, and Ranges Histograms Stem-and-Leaf Plots Box-and Whisker Plots Circle Graphs Box-and-whisker plot Circle graph Complementary events Compound event Event Frequency table Histogram Independent and dependent events Mean Median Mode Outcome and favorable outcome Probability of an event Random Range Sample space Stem-and-leaf plot Theoretical and experimental probability Upper and lower quartiles 99 Unit 4, Concept 3 – Probability and Statistics, Number Systems, Area, and the Pythagorean Theorem Instructional Resources: Carnegie Learning: Bridge to Algebra Standard: 7MG 3.3 Know and understand the Pythagorean Theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths and the lengths of other line segments and, in some situations, empirically verify the Pythagorean Theorem by direct measurement. Concept (and related skills) Understand areas and the Pythagorean Theorem with applications • Find squares of numbers and find and estimate the square roots of numbers • Prove the Pythagorean Theorem and use it to solve problems • Use the converse of the Pythagorean Theorem • Find Pythagorean triples Textbook Connections Vocabulary Lessons 10.4 10.5 10.6 Squares and Square Roots The Pythagorean Theorem Using the Pythagorean Theorem Converse Hypotenuse Leg Perfect square Pythagorean theorem Pythagorean triple Radical sign Radicand Square Square root 100