Middle School Math
Philosophy
The Middle School math curriculum builds on the precepts presented in the general math philosophy of the school. It consists of three high school honors level courses: Pre-Algebra, Algebra 1, and Algebra 2, with delivery appropriate for children of middle school age.
The curriculum is intended to provide students with a solid foundation to address more advanced courses in secondary schools. The rigorous and fast-paced program encourages our gifted students to reach their potential and pushes the limit of their abilities.
Students, upon transition to the Middle School, are initially placed in either Pre-Algebra or Algebra 1 for about a year and a half. These two courses are followed by a year and a half of Algebra 1 or Algebra 2 respectively. However, additional topics can be added as needed to create an expanded curriculum and to address students’ needs and interests.
The curriculum is demanding and is enhanced through reading, writing, preparation to math contests, and usage of the graphing calculator. Connections to other disciplines (e.g. humanities, science, art, etc.) are critical in involving students who are hesitant in their application of mathematics and/or who have low self-esteem in their application of mathematics. Mathematics is fascinating and our aim is to appreciate and foster varied facets of mathematical ability.
Content
The curriculum follows the NTCM (National Council of Teachers of Mathematics) Standards for School Mathematics. The Standards portion of the NTCM Principles and Standards for School Mathematics center upon ten areas of mathematics curriculum development. The number assigned to each standard is for easy reference and is not part of each standard’s official.
- Numbers and operations
- Algebra
- Geometry
- Measurement
- Data Analysis and Probability
- Problem Solving
- Reasoning and proof
- Communication
- Connections
- Representation
These ten major areas are intertwined in the instructional programs. Middle school students explore these areas in increasing degree of complexity as they move from Pre-Algebra to Algebra 1 and Algebra 2. The following description appears in the Glencoe books cited in the "Materials" section. This should enable all students to:
1. Number and operations
- Understand numbers, ways of representing numbers, relationships among numbers, and numbers systems
- Understand the meaning of operations and how they relate to each other
- Compute fluently and make reasonable estimates
2. Patterns, relations, and Algebra
- Understand patterns, relations, and functions
- Represent and analyze mathematical situations and structures using algebraic symbols
- Use mathematical models to represent and understand quantitative relationships
- Analyze change in various contexts
3. Geometry
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
- Specify locations and describe spatial relationships using coordinate geometry and other representational systems
- Apply transformations and use symmetry to analyze mathematical situations
- Use visualization, spatial reasoning, and geometric modeling to solve problems
4. Measurement
- Understand measurable attributes of objects and the units, systems, and processes of measurement
- Apply appropriate techniques, tools, and formulas to determine measurements
5. Data analysis, statistics, and probability
- Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
- Select and use appropriate statistical method to analyze data
- Develop and evaluate inferences and predictions that are based on data
- Understand and apply basic concepts of probability
6. Problem Solving
- Build new mathematical knowledge through problem solving
- Solve problems that arise in mathematics and in other contexts
- Apply and adapt a variety of appropriate strategies to solve problems
- Monitor and reflect on the process of mathematical problem solving
7. Reasoning and Proof
- Recognize reasoning and proof as fundamental aspects of mathematics
- Make and investigate mathematical conjectures
- Develop and evaluate mathematical arguments and proofs
- Select and use various types of reasoning and methods of proof
8. Communication
- Organize and consolidate their mathematical thinking through communication
- Communicate their mathematical coherently and clearly to peers, teachers, and others
- Analyze and evaluate the mathematical thinking and strategies of others
- Use the language of mathematics to express mathematical ideas precisely
9. Connections
- Recognize and use connections among mathematical ideas
- Understand how mathematical ideas build on one another to produce a coherent whole
- Recognize and apply mathematics in contexts outside of mathematics
10. Representation
- Create and use representations to organize, record, and communicate mathematical ideas
- Select, apply, and translate among mathematical representations to solve problems
- Use representations to model and interpret physical, social, and mathematical phenomena
Pre-Algebra
The intent of the Pre-Algebra program is to build the groundwork for success in Algebra 1 by providing a rigorous curriculum that moves beyond arithmetic and prepares students for the transition to Algebra. The ten content areas described above are spread throughout the five units that form the Pre-Algebra course:
- Algebra and integers
- Algebra and rational numbers
- Linear equations, inequalities, and functions
- Applying algebra to geometry
- Extending algebra to statistics and polynomials
Algebra and Integers
Students learn about the basic tools of algebra including variables, integers, and equations. They use a four-step plan to solve problems, translate verbal phrases and expressions into algebraic expressions, and use mathematical properties and the order of operations. They compare and order integers and perform operations with integers. Students evaluate and simplify algebraic expressions and write and solve two-step equations. Students use a coordinate plane to locate and graph ordered pairs and represent algebraic relationships.
Algebra and rational numbers
Students explore rational numbers. They learn to use prime factorization and Greatest Common Factor to solve real-world problems and simplify algebraic equations. Students examine exponents, multiply and divide monomials, and use scientific notation. Students also compute and solve equations with rational numbers. They use measure of central tendency. Students apply ratios and proportions to problems involving fractions, decimals, and percents. Students use percent equations, calculate percent of change, and make predictions based on probability.
Linear equations, inequalities, and functions
Students learn about linear equations, inequalities, functions, and graphing. They analyze data to discover whether a linear relationship exists. They learn that inequalities can help describe relationships between mathematical expressions. Students analyze how equations and graphs are used to describe mathematical relationships. They expand their knowledge in the use of x- and y- intercepts, slope, and rate of change.
Applying algebra to geometry
Students examine square roots, and the real number system. They classify angles and triangles, and explore the Pythagorean Theorem, the Distance Formula, Midpoint Formula, and the properties of similar figures. Students also use trigonometric ratios to solve problems. In addition, students classify, find the angle measures, and find the area of various polygons. Students examine the circumference and area of circles. Additionally, they find the volume and surface area of three-dimensional objects and use precision and significant digits to describe measurements.
Extending Algebra to statistics and polynomials
In this last unit, students display and interpret data in various plots and graphs and use measure of variations to compare data. Students also count outcomes and use permutations and combinations. Probability is extended to using simulations, finding experimental probability, and finding the probability of multiple events. Students also identify and classify polynomials and find the degree of a polynomial. They learn to use operations involving monomials and polynomials. Students finish the unit with an examination of linear and nonlinear functions extending to the graphing of quadratic and cubic functions.
Algebra 1
The intent of the Algebra 1 program is to build the groundwork for success in Algebra 2 by providing a rigorous curriculum. The five content strands described above are spread throughout the five units that form the Algebra 1 course:
- Expressions and equations
- Linear Functions
- Polynomials and non-linear functions
- Radical and rational functions
- Data analysis
Expressions and equations
Students explore using variable to represent data. They learn to write, evaluate, and simplify variable expressions. They build on this to write and solve linear equations. Students perform operations with real numbers. They use algebraic expressions to model and analyze real world situations.
Linear Functions
Students explore data to determine whether a linear relationship exists. They learn to represent a linear relationship as points on a coordinate plane and as an equation representing a line. Students analyze how the equation and the graph of a line are related. They extend their knowledge of linear graphing to inequalities and systems of linear equations.
Polynomials and Nonlinear Functions
Students are introduced to nonlinear functions. They first learn about polynomials and operations involving monomials and polynomials. Students learn various methods of factoring and are finally introduced to quadratic and exponential functions.
Radical and Rational Functions
Students are introduced to additional nonlinear functions such as radical and rational equations. They learn how to simplify radical and rational expressions, and how to solve equations involving these expressions. Students also explore triangles through the Pythagorean Theorem and trigonometric ratios.
Data Analysis
Students learn how to analyze data using statistical analysis. This includes understanding sampling techniques, histograms, and box-and-whisker plots. Students then learn how to use probability to predict outcomes.
Algebra 2
The intent of the Algebra 2 program is to build the groundwork for success in Pre-Calculus and Geometry by providing a rigorous curriculum. The five content strands described above are spread throughout the five units that form the Algebra 2 course:
- First-degree equations and inequalities
- Polynomial and radical equations and inequalities
- Advanced functions and relations
- Discrete mathematics
- Trigonometry
First-degree equations and inequalities
In this unit, students begin by applying the properties of real numbers to expressions, equalities, and inequalities, including absolute value inequalities and compound inequalities. Throughout the unit, students explore the relationship between linear equations and their graphs. These explorations include modeling data with scatter plots and lines of regressions, as well as linear programming and solving systems of equations. The unit concludes with instruction about operations on matrices and using matrices to solve systems of equations.
Polynomial and radical equations and inequalities
Students extend their knowledge of first-degree equations and their graphs to radical equations and inequalities, Then they graph quadratic functions and solve quadratic equations and inequalities by various methods, including completing the square and using the quadratic formula. The unit concludes with methods for evaluating polynomial functions including the Remainder and Factor Theorems. Students graph polynomial functions and investigate their roots and zeros. They finally study the composition of two functions and then find the inverse of a function.
Advanced functions and relations
At the beginning of this unit, students are reacquainted with the Midpoint and Distance Formulas before exploring conic sections. They then learn to combine rational expressions, which leads to graphing rational functions where they examine asymptotes and holes. This knowledge of functions is applied to direct, joint, and inverse variations. The unit concludes with an investigation of exponential and logarithmic functions. Finally, logarithms with base e and natural logarithms are investigated an applied to real-world situations involving investigating growth and decay.
Discrete Mathematics
In this unit, students explore various topics of discrete mathematics, including arithmetic and geometric sequences and series, as well as recursion and fractals. They also apply the Binomial Theorem, and prove statements using mathematical induction. The unit concludes with an investigation of probability and statistics, including permutations, combinations, and the normal distribution. Finally, students apply their mathematical skills in a simulation as well as to sampling and to test hypotheses.
Trigonometry
In this unit, students investigate the six trigonometric functions, both as ratios in right triangles and as circular functions, which they graph. The Law of Sines and the Law of Cosines are used to solve problems as are the inverse trigonometric functions. The unit concludes with lessons in which students verify and use trigonometric identities, and solve trigonometric equations.
Skills
The following are the minimum math standards used to guide instruction in Middle School mathematics at The Sage School. Although students at The Sage School are not grouped by age, they do follow a traditional "grade sequence" and The Sage School instructional groups have corresponding "grade levels."
This scope and sequence of skills was developed by our own faculty and is based on the study and review of the Mathematically Correct and Virginia Standards and the Massachusetts Frameworks for instruction. Our scope and sequence provides a foundation on which the total math experience is based. It also assures that students are provided with a comprehensive program by avoiding duplication of skills or missing skills.
The skill sets are divided into broad topics which are re-visited each year. However, each year the complexity and demand for performance and understanding increases. The skill sets presented are only a portion of the total math program. Our overall goal is to prepare students not just to be learners of math, but mathematicians. This part of the program is based on the development of higher level thinking skills. It includes being able to approach novel problems in creative ways, being able to apply knowledge/ math skills, and understanding why and how the answer is correct. Although there is a demand to show mastery of each new skill, this is not the end of real learning in that area. Being able to demonstrate the skill once, for the test is not enough. We are looking to achieve a level of mastery which imparts the skills permanently and allows students to recall them quickly and comfortable in many different situations.
Middle School Algebra
Pre-Algebra Algebra 1 Algebra 2
The student will evaluate expressions in one, two or three variables when given specific numerical values for each variable using any or all of the processes covered above.
The student will simplify expressions involving one, two or three variables such as m3p + mpm2 -m3+ 3pmp2 or a2bc/ab3
The student will solve linear equations and inequalities in one variable such as 2x/5 = -10 or 5y -2.7 = -12.7y or 3r - 6/5 = 9/3
The student will recognize linear equations with two variables in the form y=mx+b
The student will determine the slope of a line given two points on the line.
The student will evaluate a system of equations in the form y=mx+b by using substitution.
The student will determine parallel and perpendicular lines by the slope of the lines.
The student will, given various "word" or "real world" problems, write and solve linear equations of a single variable and convert these solutions into answers to the problems.
| The student will translate between problem situations, verbal expressions, and mathematical expressions with variables, and extend this process to include problems involving exponents and simple radicals, polynomials, absolute values and inequalities. For expressions that are not written as equations, the student will evaluate these expressions for given replacement values of the variables. For equations or sets of equations, the student will determine the values of the variables that constitute the solution set of the equation or set of equations.
The student will solve linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable and apply these skills to solve practical problems.
The student will translate freely among various representations of linear equations, including the slope of the line and a point on it; two points on a line; a point on the line and the condition that the line is parallel or perpendicular to another given line; a graph of the line or of points on it; a problem situation or word problem representing a line; and an equation for the line in slope-intercept form, standard form, or arbitrary form.
The student will translate an equation for a line in any form to an equation for either variable in terms of the other variable, and use the equation to find values for one variable given replacement values for the other variable. This includes using a linear equation in a problem situation to solve the problem, and recognizing when a linear equation cannot be used to solve a problem.
The student will determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line. The student will also describe the slope as a rate of change and identify slopes as positive, negative, zero, or undefined.
The student will solve systems of two linear equations in two variables by graphical estimation and by algebraic techniques including substitution and the addition and subtraction of equations (with and without a multiplication step). These techniques will be applied to solve practical problems. Students will extend this work to graphing the solution set of two linear inequalities. Students will also solve systems of addition and subtraction equations in three variables with three unknowns by substitution.
The student will use matrices to organize and manipulate data, including matrix addition, subtraction, and scalar multiplication. Data will arise from business, industrial and consumer situations.
The student will factor completely binomials and trinomials, including quadratics and expressions having coefficients for the highest order term greater than one, in one or two variables when they are factorable over the rational numbers.
The student will factor special forms of (factorable) polynomials, including those requiring regrouping or repeated factorization and those having coefficients for the highest order term greater than one. This will include the extraction of monomial and binomial factors from expressions in the third or fourth degree.
The student will simplify expressions involving radicals, including the square roots of expressions involving constants and variables, and expressions involving the sum, difference, and products of radicals of both real numbers and monomial and binomial expressions, and use radical expressions to solve problems. | Students solve equations and inequalities involving absolute value.
Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
Students are adept at operations on polynomials, including long division.
Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.
Students prove simple laws of logarithms.
Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.
Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.
Students use the definition of logarithms to translate between logarithms in any base.
Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.
Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.
Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.
Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.
Students use fundamental counting principles to compute combinations and permutations.
Students use combinations and permutations to compute probabilities.
Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.
Students apply the method of mathematical induction to prove general statements about the positive integers.
Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.
Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.
Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.
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Middle School Geometry
Pre-Algebra Algebra 1 Algebra 2 Geometry
The student will calculate the area and perimeter of rectangles, parallelograms, trapezoids, triangles and circles. The student will also calculate the area of more complicated shapes by breaking them into pieces such as rectangles, triangles and fractions of circles.
The student will calculate the volume and surface area of right angle boxes, cylinders, and spheres. The student will also calculate the volume and surface area of rectangular prisms with known heights and irregular bases.
The student will use the Pythagorean Theorem to calculate the length of the missing side of a right triangle and to determine if a triangle is a right triangle.
The student will recognize and name polygons with up to 10 sides as well as dodecagons.
The student will recognize congruent and similar polygons and will use proportions to determine the lengths of unlisted sides.
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The student will use the Pythagorean Theorem and its converse to find distance measures in the special case of right triangles, and use the representation of this theorem in the coordinate plane, the distance formula, to find the distance between any two points or the length of a specified line segment between two points. The student will also explain each step when given a proof of the Pythagorean Theorem.
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| The student will recognize the three undefined terms "point", "line", and "plane", and their symbols, and be able to use these to define other terms such as "space", "ray", "angle" and so on.
The student will know the five major "existence" postulates about points lines and planes: 1) A line contains at least two points, a plane contains at least three points not all on one line; a space contains at least four points not all on one plane; 2) There is exactly one line through two points; 3) There is exactly one plane through three points not on one line; 4) If two points lie in a plane, then the line joining them lies in that plane; 5) If two planes intersect, their intersection is a line. The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; diagramming arguments involving quantifiers (all, no, none, some), using Venn diagrams; using valid forms of deductive reasoning, including the law of syllogism; and recognizing logical errors in faulty arguments.
The student will use the relationships between angles formed by two lines cut by a transversal to determine if two lines are parallel and verify, using algebraic and coordinate methods as well as deductive proofs.
The student will Identify congruence and similarity relationships between triangles; and Prove two triangles are congruent or similar given information in the form of a figure or statement, using algebraic and coordinate as well as deductive proofs.
The student will be able to state and be able to use the Triangle Inequality Theorem. Given information concerning the lengths of sides and/or measures of angles, the student will apply the triangle inequality properties to determine whether a triangle exists and to order sides and angles. These concepts will be considered in the context of practical situations.
The student will solve practical problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. Because special right triangles and their properties recur, the student will commit to memory the side ratios of special right triangles and use them to solve triangle problems.
The student will Identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals; Prove these properties of quadrilaterals using algebraic and coordinate as well as deductive proofs; Use properties of quadrilaterals to solve practical problems.
The student will understand the meaning of the term "regular polygon" and be able to determine the measures of the interior and exterior angles of regular polygons. In addition, the student will use measures of interior and exterior angles of polygons to solve problems.
The student will use the properties of angles, arcs, chords, tangents, and secants to solve problems involving circles. Problems will include finding the area of a sector and doing constructions. This will include constructing inscribed or circumscribed circles given a triangle; locating the center of a circle; and constructing the tangent to a circle from a point on the circle and from a point not on the circle.
The student will construct, using a compass and straightedge, a line segment congruent to a given line segment, the bisector of a line segment, a perpendicular to a given line from a point not on the line, a perpendicular to a given line at a point on the line, the bisector of a given angle, an angle congruent to a given angle, and a line parallel to a given line from a point not on the line.
The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems.
The student, given similar geometric objects, will use proportional reasoning to solve practical problems; investigate relationships between linear, square, and cubic measures; and describe how changes in one of the measures of the object affect the others. |
Materials
As with most high school math courses, our Middle School math curriculum is based on high quality texts, which allows continued reference by the students to further their ability. In addition, other curricular materials (internet sources, literature, and enrichment media) are used as appropriate. A full list of the currently used texts is given below.
Malloy, C., Price, J., Willard, D., Sloan, L. L. 2005. Pre-Algebra . Glencoe McGraw-Hill
Holliday, B., Cuevas, G. J., Moore-Harris, B., Carter, J. A., Marks, D., Casey, R. M., Day, R., Hayek, L. M. 2005. Algebra 1. Glencoe McGraw-Hill
Holliday, B., Cuevas, G. J., Moore-Harris, B., Carter, J. A., Marks, D., Casey, R. M., Day, R., Hayek, L. M. 2005. Algebra 2. Glencoe McGraw-Hill
