# Purposeful Design Publications. Intermediate Mathematics Series Scope and Sequence

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2 PO Box Colorado Springs, CO

3 I. NUMBER THEORY A. Classification of Numbers Counting numbers Whole numbers Integers Rational numbers Irrational numbers Real numbers B. Whole Numbers Place value Primes and composites Divisibility rules Factorization Greatest common factor Multiples Least common multiple Fundamental Theorem of Arithmetic C. Exponents, Powers, and Roots Positive exponents Square roots Scientific notation Negative exponents Zero exponents Product of powers Quotient of powers Power of a power Square Root of a Product Property Square Root of a Quotient Property D. Operations Order of operations Inverse operations Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Distributive Property Identity Property of Addition Identity Property of Multiplication Purposeful Design Intermediate Mathematics Series Scope and Sequence 1

4 Inverse Property of Addition Inverse Property of Multiplication Equality Properties II. FRACTIONS AND DECIMALS A. Concepts Naming equivalent fractions Determining least common denominator Naming equivalent decimals Terminating and repeating decimals Modeling fractions and decimals Converting fractions to decimals Converting decimals to fractions Comparing and ordering Using inequality and equality signs Using reciprocals Rounding and estimating B. Operations with Rational Numbers Add and subtract fractions and mixed numbers Add and subtract decimals Multiply and divide fractions and mixed numbers Multiply and divide decimals III. RATIO, PROPORTION, AND PERCENT A. Ratio and Rate Writing ratios in simplest form Comparing ratios Equivalent ratios Ratio tables Determining rate, unit rate, and unit price Using conversion factors Sine, cosine, and tangent ratios B. Proportion Defining and recognizing proportions Solving proportions using cross products Property of Cross Products Related to maps and scale drawings Related to similar figures Using the constant of proportionality 2 Purposeful Design Intermediate Mathematics Series Scope and Sequence

5 C. Percent Relating fractions and decimals to percent Relating ratio to percent Converting fractions to percents and percents to fractions Converting decimals to percents and percents to decimals Estimating percents Finding a percent of a number Finding what percent one number is of another Finding a number when a percent is known Writing and solving equations involving percent Percent increase and decrease Taxes and discounts Interest IV. ALGEBRA A. Integers Properties of integers Opposites Absolute value Comparing and ordering integers Graphing on a number line Adding and subtracting integers Multiplying and dividing integers Simplifying and evaluating expressions Solving equations B. Properties Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Distributive Property Identity Property of Addition Identity Property of Multiplication Inverse Property of Addition Inverse Property of Multiplication Equalities Properties Property of Cross Products Square Root of a Product Property Square Root of a Quotient Property Purposeful Design Intermediate Mathematics Series Scope and Sequence 3

6 Properties of Exponents C. Expressions, Equations, Inequalities 4 Purposeful Design Intermediate Mathematics Series Scope and Sequence Constants and variables Identifying terms Translating numerical and algebraic expressions Simplifying and evaluating expressions Writing and solving addition and subtraction equations Writing and solving multiplication and division equations Writing and solving two-step equations Solving absolute value equalities Writing and solving equations using square roots Solving equations with variables on both sides Writing and solving inequalities Graphing inequalities Writing and solving compound inequalities Solving absolute value inequalities Graphing absolute value inequalities D. Functions and Relationships Identifying numerical patterns Function tables Writing functions Domain, range, and ordered pairs Graphing functions Rate of change and slope Slope intercept form of a linear equation Writing and graphing linear equations Identifying slopes of parallel lines Writing and graphing linear inequalities Solving a system of linear equations by graphing Solving a system of linear equations by substitution E. Polynomials Identifying and classifying polynomials Modeling polynomials Simplifying polynomials Adding and subtracting polynomials Multiplying and dividing by a monomial Multiplying binomials Factoring polynomials

7 Expressing geometric measurements V. MEASUREMENT A. Linear Measurement Customary units of length Metric units of length Comparing customary and metric units B. Capacity Customary units of capacity Metric units of capacity Comparing customary and metric units C. Weight and Mass Customary units of weight Metric units of mass Comparing customary and metric units D. Temperature Customary units of temperature Metric units of temperature Using formulas to convert temperature E. Conversion Factors Converting rates Converting units VI. GEOMETRY A. Plane Geometry Identifying points, lines, and angles Identifying polygons Circles and parts of circles Finding perimeter of polygons Finding perimeter of irregular figures Finding area of polygons Finding area of irregular figures Finding circumference of a circle Finding area of a circle Sum of the angles in a polygon Sum of the exterior angles of a polygon Constructing a congruent line segment Constructing a perpendicular bisector Constructing a congruent angle Constructing triangles Purposeful Design Intermediate Mathematics Series Scope and Sequence 5

8 Constructing an angle bisector Constructing a perpendicular line through a point Constructing parallel lines Constructing trefoil designs B. Solid Geometry Identifying polyhedrons Identifying cylinders, cones, and spheres Finding surface area of polyhedrons Finding volume of polyhedrons Finding surface area of cylinders Finding volume of cylinders and cones Finding surface area of cones and spheres Finding volume of spheres Conic sections Spatial visualization C. Formulas Area and perimeter of a rectangle Area and perimeter of a square Area and perimeter of a parallelogram Area and perimeter of a triangle Area of a trapezoid Area and circumference of a circle Euler s formula Volume and surface area of a rectangular prism Volume and surface area of a cube Volume of a triangular prism Surface area of a triangular prism Volume of a pyramid Surface area of a pyramid Volume and surface area of a cylinder Volume of a cone Surface area of a cone Volume and surface area of a sphere Pythagorean Theorem Distance Formula Midpoint Formula Sine, cosine, and tangent ratios 6 Purposeful Design Intermediate Mathematics Series Scope and Sequence

9 D. Relationships Parallel, perpendicular, and intersecting lines Angle relationships Congruent triangles Similar figures Tessellations Pythagorean Theorem Irrational spiral Sine, cosine, and tangent ratios VII. COORDINATE SYSTEM A. Coordinate Grid Using a four-quadrant grid Graphing ordered pairs Graphing linear equations Graphing horizontal and vertical lines Identifying slope of a line Determining rate of change Writing the equation of a line Graphing linear inequalities Graphing systems of linear equations Distance Formula Midpoint Formula B. Transformations Translations Reflections Dilations Rotations Reflections over parallel lines Reflections over intersecting lines VIII. PROBLEM SOLVING A. Problem-solving Guide Read and analyze word problem Select a strategy Apply the selected strategy using the appropriate operations Check for reasonableness B. Problem-solving Strategies Choose an operation Draw a diagram/picture Purposeful Design Intermediate Mathematics Series Scope and Sequence 7

10 8 Purposeful Design Intermediate Mathematics Series Scope and Sequence Find a pattern Identify sub-goals Make a table or graph Make an organized list Solve a simpler problem Try and check Use a formula Use a graph Use a coordinate grid Use estimation or mental math Use logical reasoning Work backward Write an expression, equation, or inequality C. Applications Greatest common factor and least common multiple Integers Positive exponents and powers Negative exponents and powers Square roots Fractions and mixed numbers Decimals Ratios, rates, unit rates, and unit prices Proportions Percents Taxes and discounts Interest Measurements Measurement conversions Map skills Perimeter and area Surface area and volume Geometric relationships Sine, cosine, and tangent ratios Permutations and combinations Probability of events Measures of central tendency Tables, charts, and graphs Transformations

11 Slope and rate of change Linear equations Polynomials IX. PROBABILITY A. Counting Methods Fundamental Counting Principle Sample spaces and tree diagrams Permutations Combinations B. Determining Probabilities Relating favorable outcomes to the number of possible outcomes Making predictions Theoretical and experimental probabilities Expressing probability as a fraction Expressing probability as a percent or decimal Mutually exclusive events Independent events and dependent events Simulations Geometric probability Determining odds Fair and unfair games X. STATISTICS A. Graphs Bar graph Histogram Circle graph Line graph Line plot Stem plot (stem-and-leaf plot) Box plot (box-and-whisker plot) Scatterplot Double-line graph Double-bar graph Pictograph Pictorial model Interpreting graphs Selecting an appropriate graph Identifying visually misleading graphs Purposeful Design Intermediate Mathematics Series Scope and Sequence 9

12 B. Collecting and Analyzing Data Sampling Surveys Tally charts Frequency tables Spreadsheets Trend lines Correlation Interpreting statistics Making predictions C. Statistical Measurements Frequency Mean Median Mode Range Quartiles 10 Purposeful Design Intermediate Mathematics Series Scope and Sequence

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