Download MATHEMATICS

Middle School
MATHEMATICS
2012
Curriculum Summary
MIDDLE SCHOOL MATHEMATICS
Philosophy
In order to ensure that every student will be successful and challenged, yet, not over-paced, we have developed
three separate courses of mathematics study in the Upper School curriculum. Students differ in the pace at which
they learn mathematical concepts. Therefore, we have developed these three programs to match students’
individual learning styles. All programs are either on-grade level or advanced.
Program
3
2
1
6th grade
Pre-Algebra
Math 6
Math 6
7th grade
Algebra I
Pre-Algebra
Math 7
8th grade
Geometry
Algebra I
Pre-Algebra
Mathematics Program 1 – This program prepares students to take all of the math courses generally required for
admission to college by the end of high school. Students who complete this sequence will study Algebra I,
Geometry, Algebra II, and Trigonometry/Pre-calculus in high school.
Mathematics Program 2 -- The goal of this accelerated program is to enable students to take Advanced Placement
(AP) Calculus in high school. Students seeking admission into this program must display superior math ability, as
measured at the end of sixth grade by report card grades, standardized test scores, and teacher observations.
Students who complete the sequence will take Geometry, Algebra II, Trigonometry/Pre-calculus, and AP Calculus
AB in high school.
Mathematics Program 3 -- This accelerated sequence is designed for students with mathematical ability in the 97th
percentile or above. Students seeking admission into this program must not only demonstrate superior
mathematical ability, but they must also possess the necessary maturity, study habits, and work ethic to keep pace
with the demands of each course. At the end of 5th grade, all students take a placement test to determine
admission into this track. In addition to the results of the placement test, we consider the student’s standardized
test scores, as well as their overall performance in math during the year. Students who complete this sequence
will take Algebra II, Trigonometry/Pre-calculus, AP Calculus AB, AP Calculus BC, and possibly multi-variable
Calculus in high school.
We are now approaching our tenth year of offering these sequences of mathematics courses. We have found that
the program has been very successful for students at every level. They are effectively challenged and able to
transition smoothly into the next level.
6th Grade Mathematics
Course Description
The sixth grade math course prepares sixth graders by teaching and reinforcing their arithmetic skills so that they
will be successful in Math 7 or Pre-Algebra (which is a one-year advance placement), when they enter seventh
grade. The course provides them with a solid foundation in mathematical reasoning by constructing their new
learning from a basis of prior knowledge and experience.
Text and Materials
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Middle School Math, Scott Foresman – Addison Wesley
Student Manipulative Kits
Teacher Resource Kits and Planner, including CD ROMs and Websites
Upper School Technology lab and various math web sites
Course Objectives
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Develop problem-solving skills for everyday situations;
Develop automaticity with basic number facts, arithmetic skills, and algebra;
Master skills through practice with math games;
Share ideas about mathematics through discussion;
Learn cooperatively through group and partner activities;
Develop number sense (recognize a reasonable answer, explain where numbers come from and what they
mean, and detect errors in results);
7. Develop operation sense (understand and explain what the four operations mean and how they are
interrelated);
8. Develop measure sense (a knowledge of what measurement means, what kinds of measures are
appropriate in different situations, and what range of results is reasonable in a given situation);
9. Develop spatial sense (mentally manipulate 1-, 2-, and 3-dimensional objects and perceive relationships
between and among them).
Student Objectives
Students will be able to:
Chapter One - Graphing
1. Read and interpret bar graphs, pictographs, and line graphs. They will also identify trends suggested by
scatterplots.
2. Organize data and determine its shape. Students construct bar graphs and stem-and-leaf diagrams to
display information.
3. Determine the mean, median, and mode to describe a set of data. They also determine if outliers affect the
analysis of the data set.
Chapter Two – Connecting Arithmetic to Algebra
1. Write numbers in word from, number-word form, and with exponents. They also learn the rules for
rounding, comparing, and ordering large numbers.
2. Use mental math and estimation.
3. Use the rules for order of operations as the tools necessary to solve many problems.
4. Work with variables and variable expressions. They are also introduced to equations.
Chapter Three – Decimals
1. Name and write numbers using decimal notation. They will also learn the rules for rounding, comparing,
and ordering decimals, and how to express numbers in scientific notation.
2. Estimate solutions to problems involving decimals. They will also add and subtract decimals and learn to
solve addition and subtraction equations involving decimals.
3. Multiply and divide with decimals and where to place the decimal point in the solution.
4. Use mental math to solve multiplication and division equations involving decimals.
Chapter Four – Measurement
1. Find perimeter, convert units within the metric system, and use the customary system of measurement for
length, weight and capacity.
2. Find the area of squares, rectangles, parallelograms, and triangles.
3. Use pi to calculate the circumference and area of circles. Then they learn to find the area of irregular
shapes.
Chapter Five – Patterns and Number Theory
1. Use divisibility rules to find the prime factorization of a number. They will also learn to find the least
common multiple of two numbers.
2. Write different fractions have equivalent values.
3. Express numbers in decimal notation and in fraction notation, and to convert between the two forms.
4. Compare and order fractions.
Chapter Six – Adding and subtracting Fractions
1. Add and subtract fractions. They will also learn to solve fraction equations by adding and subtracting
fractions.
2. Add and subtract mixed numbers.
3. Estimate sums and differences of fractions and mixed numbers.
Chapter Seven – Multiplying and Dividing Fractions
1. Estimate products and quotients of fractions and to multiply by fractions.
2. Divide by fractions and to solve equations by multiplying or dividing by fractions.
Chapter Eight – The Geometry of Polygons
1. Describe different kinds of lines and angles. They will also learn to measure the size of angles using a
protractor.
2. Classify triangles by the measure of their angles and by the lengths of their sides. They will also classify
polygons and quadrilaterals.
3. Investigate polygons whose shapes consist of repeated patterns. They will also examine symmetry and
tessellations: figures that are flipped over a line, rotated around a point, and slid to a new position.
Chapter Nine – Integers and the Coordinate Plane
1. Use the rules for addition, subtraction, multiplication, and division of integers so that they can use integers
to solve problems.
2. Locate points on the coordinate plane. They will then learn to graph translations, reflections, and
equations.
Chapter Ten – Ratio, Proportion, and Percent
1. Use ratios to compare quantities.
2. Express equal ratios as a proportion and to solve proportions.
3. Express ratios as percents. They will also learn to relate fractions, decimals, and percents.
Procedures
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Teacher-led instruction
Class discussions
Group and individual journal activities
Group and individual projects
Evaluation
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Tests and quizzes
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Homework and preparedness for class
Written assignments such as Study Links or Math Boxes
Class participation
7th Grade Mathematics
Course Description
The seventh grade math course is designed for students of average math ability. It prepares them so that they will
be successful in Pre-Algebra when they enter eighth grade. The course provides them with a solid foundation in
mathematical reasoning by constructing their new learning from a basis of prior knowledge and experience.
Text & Materials
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Mathematics (Adisson Wesley, 1993)
Workbooks
Multiplication Tables
Manipulatives
Course Objectives
1. To help students understand the link between math and real life;
2. To help students become better problem solvers and view math as a language which can be used to
explain and understand the world around them;
3. To help students acquire the base of knowledge in math they need to be successful in Pre-algebra and the
SSAT;
4. To help students develop proficiency with technology, including calculators and computer software.
Student Objectives
Students will be able to:
Numbers & Operations
1. Describe and demonstrate ways to visualize quantity and math operations;
2. Memorize the order of operations in multi-operation math problems and solve multi-operation math
problems;
3. Describe the characteristics of a base 10 number system and identify the place value of all the digits in any
positive whole number;
4. Describe and demonstrate how rounding off and estimation;
5. Define and demonstrate Commutative, Associative, Distributive, and Identity Properties;
6. Describe and demonstrate how to perform operations on numbers represented by different units (i.e. feet
and inches).
Decimals & Decimal Operations
1. Describe how decimals fit into the base 10 number system, identify the place value of all digits in any
positive integer, and compare and order decimals;
2. Memorize the rules associated with adding, subtracting, multiplying and dividing decimals and solve such
math problems;
3. Describe and demonstrate how to round off decimals and estimate answers to math problems involving
decimals.
Fractions & Fraction Operations
1. Define numerator and denominator, describe the purpose of fractions, and compare and order fractions;
2. Define factors, identify the factors of a number, reduce fractions by finding common factors until the
fraction is completely reduced, find the Greatest Common Factor of two or more numbers, and use the
GCF to completely reduce fractions in a single step;
3. Define multiples, identify the multiples of a number, and find the Least Common Multiple of two or more
numbers;
4. Memorize the rules associated with adding, subtracting, multiplying and dividing fractions and solve such
math problems;
5. Convert fractions to decimals and decimals to fractions and convert mixed numbers to improper fractions
and improper fractions to mixed numbers;
6. Describe and demonstrate how to round off fractions and estimate answers to math problems involving
fractions.
Introduction to Algebra
1. Define key English words (is, of, and) in terms of their math symbols (=, x, +) and translate problems from
English words to math symbols;
2. Identify the inverse operation of each math operation and the reverse of the order of operations, explain
how they are used to isolate a variable, and solve 1 and 2 step, single variable algebra problems;
3. Describe what it means to rewrite a math equation by cross-multiplying, identify the circumstances in
which you must cross-multiply and solve algebra problems that require cross-multiplication.
Ratio, Rate & Scale
1. Identify types of math problems that must be solved by creating an equation with two ratios;
2. Describe and demonstrate what it means to use consistent units in the numerators and denominators of
ratios;
3. Solve math problems requiring the determination and comparison of rates, unit prices, and scale.
Percent & Percent Operations
1. Define percent in terms of how the word is translated into a math operation, describe the purpose of
percent, and compare and order percents;
2. Convert fractions, decimals, and percents from one to the other;
3. Solve math problems involving percent and the calculation of simple interest, discount or sale prices, taxes
or tips, and percent change.
Charts & Statistics
1. Define and identify circle graphs, bar graphs, line graphs and scattergrams;
2. Respond to questions about information displayed in circle graphs, bar graphs, line graphs and
scattergrams, including questions that require the ability to make inferences and predictions;
3. Define and identify mean, median, and mode and calculate the mean, median and mode from a series of
numbers.
Exponents & Exponent Operations
1. Define exponent and root, find the square and cube root of perfect squares and cubes, and estimate the
square root of numbers that are not perfect squares;
2. Describe where exponents fit into the order of operations, multiply and divide numbers with exponents,
and describe and demonstrate how to isolate a variable with an exponent;
3. Define scientific notation and convert large numbers to and from scientific notation.
Negative Numbers & Negative Number Operations
1. Display negative numbers on a number line;
2. Memorize the rules associated with adding, subtracting, multiplying and dividing negative numbers and
solve such math problems.
Pre-Algebra
1. Solve 3 -step, single variable equations;
2. Graph inequalities on a number line and solve 1, 2 and 3-step, single variable inequalities;
3. Graph two variable equations on a coordinate plane using a table of ordered pairs;
4. Define and describe function, domain and range, describe the purpose of functions, and solve math
problems involving functions.
Geometry
1. Define parallel and perpendicular lines, bisectors and complimentary and supplementary angles, memorize
the angle sum relationships of lines, triangles and quadrilaterals, and identify missing angles;
2. Define 2-dimensional shapes in terms of their number of sides, length of sides, and degree of interior
angles and draw 2-dimensional shapes;
3. Define perimeter and circumference and memorize the formula used to calculate the perimeter of
rectangles and parallelograms and the circumference of circles;
4. Define area, memorize the formulas used to calculate the area of triangles, rectangles, parallelograms,
trapezoids, and circles, and describe and demonstrate how to calculate the area of irregular shapes;
5. Memorize the Pythagorean theorem, identify when it is used and solve math problems using it;
6. Define 3-dimensional objects in terms of the shape of their base and whether they have roof or a point on
top and construct 3-Dimensional objects;
7. Define volume and memorize the formulas used to calculate the volume of prisms, cylinders, pyramids and
cones.
Probability
1. Define probability and express it as a ratio, percent and decimal;
2. Analyze problem situations such as games of chance, estimate the probability of various events, and use
experiments to test the estimate.
Procedures
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Reading, discussing and working problems in the mathematics text and worksheets
Classroom activities and experiments using manipulatives
Lessons using the TI 82 calculator
Independent projects addressing consumer math and the history of math
Evaluation
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Tests
Homework
Class work and participation
Oral or written reports on projects
Pre-Algebra (6th, 7th, 8th Grades)
Course Description
The students in Pre-Algebra 6 at LCDS have been chosen to participate in our two-year accelerated mathematics
program. The students in Pre-Algebra 7 have been chose to participate in our one-year accelerated mathematics
program. The students in Pre-Algebra 8 are in the standard math track. All students are taught the same course,
from the same textbook, in single-grade classrooms. In other words, students from different grades are not mixed.
The course covers all concepts necessary to provide a thorough preparation for Algebra I.
Text and Materials
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Middle School Math Course 3, Scott-Foresman-Addison Wesley (8th grade only)
Mathematics: Explorations and Applications, Prentice Hall, 1995.
Teacher resource materials provided by the publishers
Scientific Calculators
Middle Grades Mathematics Project, Dale Seymour Publications
Manipulatives
Course Objectives
1. To build student confidence through problem solving, mathematical reasoning, communicating
mathematically , and connecting to other knowledge and subject areas.
2. To prepare students for Algebra 1 by helping them gain a firm understanding of and proficiency with
variables and algebraic equations.
3. To recognize and promote effort, willingness to take risks, perseverance, and independent thinking.
4. To encourage students to assess the reasonableness of their answers.
5. To increase students’ ability to apply mathematics to real-life situations.
6. To increase students’ ability to understand and solve problems.
7. To increase students’ ability to verbalize and define concepts .
8. To encourage flexibility in problem solving and the willingness to listen to the ideas of others.
9. To encourage cooperative group work.
10. To promote the integration of mathematics, computers, and calculators.
Student Objectives
Students will be able to:
Data Analysis
1. Understand data with line plots and stem-and-leaf diagrams.
2. Analyze and represent data using mean, median, and mode.
3. Analyze and represent the spread and distribution of data using box-and-whisker plots.
4. Understand data using bar graphs and line graphs.
5. Read and interpret circle graphs.
6. Make scatterplots and determine trends shown in the scatterplot.
Integers
1. Compare and order integers.
2. Find the absolute value of integers.
3. Add, subtract, multiply, and divide integers.
4. Evaluate expressions involving integers and use the distributive property.
5. Plot pairs of integers on a coordinate grid.
6. Convert large numbers between standard notation and scientific notation.
Variables, Expressions, and Equations
1. Use familiar formulas as an introduction to algebra.
2. Apply algebraic expressions in a wide variety of situations.
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Translate verbal expressions into algebraic expressions.
Write an equation for a word problem or model.
Solve equations by using addition or subtraction properties of equality.
Solve equations by using multiplication or division properties of equality.
Solve simple two-step equations.
Solve equations by combining like terms and using the distributive property on one side.
Solve equations with an unknown on both sides.
Linear Equations and Inequalities
1. Describe patterns produced by relationships between two variables.
2. Determine whether a pair of values is a solution of a two-variable equation.
3. Graph a two-variable relationship.
4. Find the slope of a line.
5. Graph an equation and then find the slope and the intercepts.
6. Find a single solution of a pair of linear equations.
7. Write and graph simple inequalities with one variable.
8. Solve one-step inequalities using properties for inequalities.
9. Solve simple two-step inequalities.
10. Express two-variable inequalities graphically.
Ratio and Proportion
1. Estimate ratios and rates from pictures and data.
2. Use ratios and rates to compare quantities.
3. Create and identify equal ratios and rates.
4. Test for proportionality.
5. Use mental math and cross multiplication to solve proportions.
6. Find unit rates and use rate formulas to solve proportion problems.
7. Solve problems with rates and proportions.
8. Understand the concept of similar triangles, and find the length of an unknown side by solving a
proportion.
9. Use scale and create scale drawings.
Percent
1. Convert among fractions, decimals, and percents.
2. Use proportions and equations to solve percent problems.
3. Estimate percents of numbers and what percent one number is of another.
4. Use the formula for percent of change to solve problems involving percent increase and percent decrease.
5. Calculate simple and compound interest.
Number Sense, Rational Numbers, and Irrational Numbers
1. Use rules for determining whether a number is divisible by another number.
2. Find the greatest common factor of two or more numbers.
3. Find the least common multiple of two or more numbers.
4. Compare and order rational numbers.
5. Add, subtract, multiply, and divide rational numbers.
6. Simplify expressions with negative exponents.
7. Solve equations involving addition and subtraction of rational numbers.
8. Solve equations involving multiplication and division of rational numbers.
9. Compute square roots and identify perfect squares.
10. Identify square roots that are irrational numbers.
11. Use the Pythagorean Theorem with right triangles.
Geometry and Measurement
1. Choose an appropriate unit of measurement.
2. Identify more precise measurements.
3. Locate places using map coordinates and latitude and longitude.
4. Draw, measure, and identify angles.
5. Recognize and construct parallel and perpendicular lines.
6. Classify polygons.
7. Represent three-dimensional shapes in drawing.
Area and Volume
1. Find the area and perimeter of polygons.
2. Dilate rectangles and predict the resulting perimeters and areas.
3. Find the area and circumference of circles.
4. Find surface area of prisms and cylinders.
5. Find the surface area of pyramids and cones.
6. Determine the volume of rectangular prisms.
7. Dilate rectangular prisms and predict their volume.
8. Find the volume of prisms and cylinders.
9. Find the volume of pyramids and cones.
Functions and Relationships
1. Recognize a function and find the input and output values of a function.
2. Represent functions using tables, graphs, and equations.
3. Represent quadratic functions as graphs, tables, and equations.
4. Graph and evaluate other types of functions.
5. Evaluate polynomials.
6. Add, subtract, multiply, and divide polynomials.
Similarity, Congruence, and Transformations
1. Identify similar figures.
2. Identify congruent figures.
3. Identify congruent triangles.
4. Use ratios in order to find missing side lengths of a right triangle.
5. Recognize several types of symmetry.
6. Create endless patterns using transformations.
Counting and Probability
1. Use tree diagrams and develop counting methods.
2. Develop ways of counting in situations for which the order of items is important.
3. Recognize unordered selections as combinations and develop ways of counting in situations for which the
order of items is not important.
4. Compute probability.
5. Use experiments to find probability.
6. Understand what affects the probability of an event.
7. Recognize an independent event.
Procedures
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Reading, discussing and working problems in the mathematics text
Classroom activities and experiments using manipulatives
Web site activities and lessons
Middle Grades Mathematics Project activities and lessons
Calculator activities
Independent projects
Evaluation
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Homework, quizzes, and tests
Projects
Classroom preparation and participation
Algebra I (7th, 8th Grade)
Course Description
The 7th grade students in Algebra I have been chosen to participate in our two-year accelerated mathematics
program. The 8th grade students in Algebra I have been chose to participate in our one-year accelerated
mathematics program. All students are taught the same course, from the same textbook, in single-grade
classrooms. In other words, students from different grades are not mixed. Ours is a traditional Algebra 1 course.
Students will be competent and efficient in manipulating variable expressions, solving equations and inequalities,
and graphing linear and quadratic functions. They will also be able to work with exponential expressions, radical
expressions, and polynomials. Our goal is to use this time to prepare our students with all the algebra they will
need for geometry, as well as a good foundation for the more advanced algebra that is necessary for the
successful study of calculus.
Text and Materials
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Algebra (Prentice Hall, 1998)
Worksheets from practice workbook
MATHCOUNTS handbook
Graphing calculators
Course Objectives
1. To provide a traditional high school level first year Algebra course.
2. To equip students with mathematical skills that can help them solve problems in the natural sciences, the
social sciences, and computer science.
3. To foster students’ general inductive and deductive abilities; to demonstrate the applicability of logical
thought to broad areas of their lives.
4. To demonstrate the fundamental role algebra plays the sciences.
5. To promote enthusiasm and respect for mathematics, scientific achievements, and logical thought in
general.
6. To foster students’ appreciation of the relationship between mathematics and technology.
Student Objectives
Students should be able to:
Tools of Algebra
1. Recognize and name basic forms of graphs; judge which kind of graph to use for a given set of data in a
systematic way.
2. Define, and identify or calculate: mean, median, mode.
3. Define: variable, variable expression, equation, term, coefficient. Name the number of terms in an
expression, and name the coefficients of those terms.
4. Represent simple ‘real-world’ situations with algebraic equations.
5. State the conventional order of operations; evaluate multistep arithmetic expressions, and multistep
variable
6. expressions for different values of the variables.
7. Perform arithmetic with integers and other rational numbers. Distinguish between rational and irrational
numbers in decimal form.
8. State the formula for finding the probability of a particular outcome of a simple event. Distinguish between
theoretical probability and experimental probability. State the formula for finding experimental probability.
Show understanding of the facts that probability is expressed as a fraction or decimal between 0 and 1,
and that a probability of 1 indicates a certain event. Solve related problems.
9. Define matrix. Demonstrate understanding of the relationship between matrices and tabular organization of
data in computer programs.
Introduction to Functions
1. Define scatter plot. Create scatter plots for sets of bivariate data. Show recognition of correlations
indicated in scatter plots, and distinguish between positive and negative correlations. Solve related
problems.
2. Sketch graphs of the movement of objects relating distance traveled, distance from a point, and velocity,
to time. Sketch graphs of velocity over time based on those of distance, and do the reverse. Show
awareness of the fact that the relationship between distance traveled by an object and its speed is a
principal concern of differential and integral calculus.
3. Classify data as continuous or discrete.
4. Demonstrate understanding of the relationship between graphs and tables of the same data. Identify the
independent variable and the dependent variable of bivariate date when appropriate.
5. Define relation and function. Evaluate functions for given values of an independent variable.
6. Demonstrate understanding of conventional function notation. Write function rules for tables of data.
7. Graph functions, given function rules.
8. Distinguish among the graphs of linear, quadratic, and absolute-value functions.
Simple Algebraic Equations
1. Show awareness of the fact that the algebraic properties of equality allow us to compose formal solutions
to algebraic equations. Distinguish among statements of these properties.
2. Solve one-step equations entailing the four basic arithmetic operations.
3. Solve two-step equations entailing the same.
4. Model related ‘real-world’ problems with algebraic equations of these types, and solve them.
5. Simplify equations before inverse operations occur, by using the distributive property and combining like
terms. Solve related problems.
6. Solve equations containing fractions and mixed numbers. Solve related problems.
7. Define: compound event, independent event, dependent event. Find the probability of a compound event.
Find the probability of obtaining certain results while playing the game Risk, ideally using computer
programming to automate calculations.
8. Define percent. Model situations entailing percentages with equations, and solve.
9. State the formula for finding percentage of change in a quantity. Solve related problems.
Equations and Inequalities
1. Define: ratio, proportion, cross products, similar figures. Solve equations using cross-multiplication. Solve
related problems.
2. Solve first-degree equations containing variables on both sides. Model and solve related problems.
3. Solve equations involving expressions of absolute value. Solve related problems.
4. Define literal equation. Transform formulas. Solve related problems.
5. Define inequality. Model appropriate situations with inequalities and graph them accurately.
6. State the multiplication and division properties for inequalities. Solve simple inequalities. Solve related
problems.
7. Solve multistep inequalities. Solve related problems.
8. Solve compound inequalities, including those expressing using absolute value. Solve related problems.
Linear Functions
1. Define slope. Find slope between two points on the coordinate plane. Show understanding of the
relationship between lines of the following slopes, and their graphs: positive, negative, zero, undefined.
Solve related problems, particularly those of constant rate of change.
2. Define linear equation, assuming such is a bivariate function. Show understanding of the slope-intercept
form of a linear equation, and transform linear equations of other forms into this form. Solve related
equations.
3. Define constant of variation for problems of direct variation. State the particulat form of a linear equation
expressing direct variation. Show understanding of the relationship between this and proportionality. Solve
related problems.
4. State the point-slope formula for finding the equation of a line. Write the equation of a line given point and
slope or given two points.
5. Show understanding of the standard form of a linear equation. Model appropriate problems as such, and
graph the equations.
6. Find the equation of a line parallel to, or perpendicular to, another line, given a point through which the
first line passes.
7. Show understanding of the fact that if 0 is substituted for y in a linear function, the solution of the new
equation is the same as the x-intercept of the original function. Solve related problems.
Systems of Linear Equations
1. Solve systems of linear equations by graphing. Solve simple problems modeled as such by guessing and
checking; then confirm the solution by graphing the systems modeling the problems.
2. Solve systems of linear equations by substitution and elimination (linear combination). Model and solve
related problems.
3. Graph linear inequalities accurately. Solve related problems.
4. Graph systems of two or three linear inequalities. Solve related problems.
Operations with Polynomials
1. Name the degree of a term, and the degree of an algebraic expression. Show awareness of the graphs of
the functions based of polynomials of various terms.
2. Classify a polynomial according to its degree and its number of terms.
3. Add and subtract polynomials.
4. Multiply polynomials with multiple terms each. Solve related problems.
5. Factor out the greatest common factor of a polynomial.
6. Factor trinomials of the form x2 + bx + c
7. Factor trinomials of the form ax2 + bx + c
8. Demonstrate thorough understanding of the difference-of-two-squares pattern. Use it backward and
forward.
9. Demonstrate thorough understanding to the patterns related to ‘perfect-square trinomials’ (trinomial
squares). Square binomials mentally, including some of those of the second degree and higher. Factor
perfect-square trinomials using specific patterns. Solve related problems.
Quadratic Equations
1. Graph quadratic functions of the form y = ax2. Show understanding of situations modeled by this form.
2. Show understanding of the roles in a quadratic function y = ax2 + bx + c played by coefficients a, b, and c.
Find the coordinates of the vertex of a quadratic function. Graph quadratic functions with various vertices.
Solve related problems.
3. Demonstrate understanding of the relationship between a quadratic function and the corresponding
equation where y, or f(x), = 0. Show awareness that for the second equation the solution(s) will be the xintercept(s) of the function.
4. Solve quadratic equations by factoring. Solve related problems.
5. Solve quadratic equations by completing the square.
6. Solve ax2 + bx + c by completing the square, deriving the quadratic formula, with guidance as necessary.
7. Solve quadratic equations by the quadratic formula. Solve related problems.
8. Solve mixed sets of quadratic functions, choosing the most efficient method.
Exponents and Exponential Functions
1. Define growth factor. State the general form of an exponential function. Show basic understanding of
situations modeled as such. Solve related problems.
2. Solve problems of compound interest, focusing on credit-card debts and investments.
3. Derive the rules of exponents, with guidance as necessary. Use the rules of exponents to simplify algebraic
expressions.
4. Convert numbers in standard notation into scientific notation, and do the reverse. Solve related problems.
Perform arithmetic with numbers in scientific notation. Solve related problems.
Right Triangles
1. Name the sides of a right triangle. State the Pythagorean theorem. Solve related problems. Identify
Pythagorean triples.
2. Derive the distance formula from the Pythagorean theorem, with guidance as necessary.
3. Derive the midpoint formula. Solve related problems.
4. Simplify radicals.
5. Show understanding of the standard deviation. Show understanding that of the fact that sets of scores
with the same means and medians can nevertheless be very different, as indicated by different standard
deviations. Solve related problems.
Rational Expressions
1. Simplify rational expressions by factoring, noting restrictions on the values of the variables.
2. Add and subtract rational expressions.
3. Solve rational equations by multiplying by the least common denominator. Solve related problems.
Procedures
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Full-class discussions and problem-solving
Small-group discussions and problem-solving
Practice worksheets
Written homework
Evaluation
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Tests
Quizzes
Projects
Completion of homework
Performance in class
Geometry (8th Grade)
Course Description
The students in Geometry at LCDS are a full two years ahead of a “grade level” mathematics sequence. As a result
they are given a rigorous and challenging introduction to the basic definitions, postulates, and theorems of plane
and solid geometry. We expect these students to understand and be able to produce formal proofs. In addition,
we move at a very fast pace to ensure that we meet all the objectives of the course. Students who complete our
geometry course are very well prepared for an advanced level Algebra 2 course in high school.
Text and Materials
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Geometry (McDougal Littell, 2000)
Practice Workbook
Teacher created practice
MATHCOUNTS handbook
Course Objectives
1. To provide a traditional high school level course in geometry.
2. To equip students with mathematical skills that can help them solve problems in the natural sciences, the
social sciences, and computer science.
3. To foster students’ general inductive and deductive abilities; to demonstrate the applicability of logical
thought to broad areas of their lives.
4. To demonstrate the key role geometry plays in physics and other sciences.
5. To promote enthusiasm and respect for mathematics, scientific achievements, and logical thought in
general.
6. To foster students’ appreciation of the relationship between mathematics and technology.
Student Objectives
Students shall be able to:
Points, Lines, Planes, and Angles
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Define and use the term equidistant
Use the undefined terms point, line, and plane.
Draw representations of points, lines, and planes.
Define and use the terms collinear, coplanar, and intersection.
Use symbols for lines, segments, rays, and distances.
Find distances.
Name angles and find their measures.
State and use the Segment Addition and Angle Addition Postulates.
Recognize what can be concluded from a diagram.
Apply postulates and theorems relating to points, lines, and planes.
Deductive Reasoning
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Recognize the hypothesis and conclusion of a conditional statement.
State the converse of a conditional statement.
Use a counterexample to disprove a conditional statement.
Understand statements containing the phrase if and only if.
Use properties from algebra and properties of congruence in proofs.
State and apply the Midpoint and Angle Bisector theorems.
Know the kinds of reasons that can be used in proofs.
State and apply the definitions of complementary and supplementary angles.
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State and apply
State and apply
State and apply
Plan proofs and
the Vertical Angle Theorem.
definition and theorems about perpendicular lines.
the theorems about angles supplementary to, or complementary to, congruent angles.
write them in two-column form.
Parallel Lines and Planes
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Distinguish between intersecting lines, parallel lines, and skew lines.
State and apply the theorem for two parallel planes intersected by a third plane.
Indentify and name the angles formed when two lines are cut by a transversal.
State and apply postulates and theorems about parallel lines.
State and apply the theorems about lines parallel or perpendicular to a given line through a point not on
the line.
Classify triangles by sides and by angles.
State and apply the Angle Sum Theorem and its corollaries.
State and apply the Triangle Exterior Angle Theorem.
Recognize and name convex polygons and regular polygons.
Find the measures of interior and exterior angles of convex polygons.
Apply inductive reasoning.
Congruent Triangles
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Identify corresponding parts of congruent figures.
Prove two triangles congruent by using the SSS, SAS, or ASA Postulates.
Deduce information about segments and angles after proving that two triangles are congruent.
State and apply the Isosceles Triangle Theorem, its converse, and its corollaries.
Use the AAS and the HL theorems to prove two right triangles are congruent.
Prove that two overlapping triangles are congruent.
Prove two triangles congruent by first proving two other triangles congruent.
State and apply the definitions of the median and altitude of a triangle and the perpendicular bisector of a
segment.
9. State and apply the theorem about a point on the perpendicular bisector, and its converse.
10. State and apply the theorem about a point on the bisector of an angle, and its converse.
Quadrilaterals
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State and apply the definition of a parallelogram.
State and apply the properties of a parallelogram.
Prove that certain quadrilaterals are parallelograms.
State and apply theorems about parallel lines the segment that joins the midpoints of two sides of a
triangle.
State and apply the definitions of a rectangle, a rhombus, and a square.
State and apply the properties of a rectangle, a rhombus, and a square.
Determine whether a quadrilateral is a rectangle, a rhombus, or a square.
State and apply the definition and properties of a trapezoid and an isosceles trapezoid.
Inequalities in Geometry
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Apply properties of inequality to positive numbers, lengths of segments, and measures of angles.
State the contrapositive and inverse of a conditional statement.
Understand the relationship between logically equivalent statements.
Draw correct conclusions from given statements.
Write indirect proofs in paragraph form.
State and apply the inequality theorems and corollaries for one triangle.
State and apply the inequality theorems for two triangles.
Similar Polygons
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Express a ratio in simplest form.
Solve for an unknown term in a given proportion.
Express a given proportion in an equivalent form.
State and apply the properties of similar polygons.
Use the AA Similarity Postulate, the SAS Similarity Theorem, and the SSS Similarity Theorem to prove
triangles similar.
6. Use similar triangles to deduce information about segments or angles.
7. State and apply the Triangle Proportionality Theorem and its corollary.
8. State and apply the Triangle Angle-Bisector Theorem.
Right Triangles
1. Determine the geometric mean between two numbers.
2. State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right
triangle.
3. State and apply the Pythagorean Theorem, its converse, and related theorems about obtuse and acute
triangles.
4. Determine the lengths of two sides of a 45-45-90 or a 30-60-90 triangle when the length of the third
side is known.
5. Define the tangent, sine, and cosine ratios for an acute angle.
6. Apply trigonometric ratios to determine side lengths and angle measurements in right triangles.
7. Apply trigonometric ratios to solve problems involving angle of elevation and angle of depression.
Circles
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Define a circle, a sphere, and terms related to them.
Recognize circumscribed and inscribed polygons and circles.
State and apply theorems that relate tangents and radii.
Define and apply properties of arcs and central angles.
State and apply theorems about the chords of a circle.
Solve problems and prove statements involving inscribed angles and angles formed by chords, secants,
and tangents.
7. Solve problems involving lengths of chords, secant segments, and tangent segments.
Constructions and Loci
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Perform fourteen basic constructions.
Use the basic constructions in original construction exercises.
State and apply theorems involving concurrent lines.
Describe the locus that satisfies one or more given conditions.
Apply the concept of locus in the solution of construction exercises.
Area of Plane Figures
1. Understand the meaning of area of a polygon.
2. State and apply postulates and theorems about area of polygons.
3. State and apply the formulas for area of rectangles, parallelograms, triangles, rhombuses, trapezoids, and
regular polygons.
4. State and apply the formulas for circumference and area of circles, arc length, and areas of sectors of a
circle.
5. Find the ratio of the areas of two triangles.
6. Apply relationships between scale factors, perimeters, and areas of similar figures.
7. Use lengths and areas to solve problems involving geometric probabiltity.
Areas and Volumes of Solids
1. Identify the parts of prisms, pyramids, cylinders, and cones.
2. Find the lateral areas, surface areas, and volumes of right prisms, regular pyramids, right cylinders and
right cones.
3. Find the surface area and volume of a sphere.
4. State and apply the properties of similar solids.
Coordinate Geometry
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State and apply the distance formula.
state and apply the general equation of a circle.
State and apply the slope formula.
Determine whether two lines on the coordinate plane are parallel, perpendicular, or neither.
Understand the basic properties of vectors.
State and apply the midpoint formula.
Identify the slope and y-intercept of the line specified by a given equation.
Draw the graph of the line specified by a given equation.
Write an equation of a line given either one point and the slope of the line, or two points on the line.
Determine the intersection of two lines.
Choose a convenient placement of coordinate axes and assign appropriate coordinates to a given polygon.
Prove statements by using coordinate geometry methods.
Transformations
1. Define and apply the terms used with transformations: image, preimage, mapping, one-to-one mapping,
transformation, isometry, and congruence mapping.
2. Locate images of figures by reflection, translation, glide reflection, rotation, and dilation.
3. Describe the symmetry of figures and solids
4. Create a project using translations and rotations.
Procedures
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Full-class discussions and problem-solving
Small-group discussions and problem-solving
Practice worksheets
Reading assignments in texts and on the internet
Written homework and proofs
Evaluation
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Tests
Quizzes
Projects
Completion of homework
Performance in class