Math Outlines: Pre-Algebra, Introduction to Algebra,

Algebra 1 and 2, Geometry

Pre-Algebra
(Recommended for 6th grade students)

1. Natural and whole numbers and number ray. Scale. Comparing numbers in algebraic form.
2. Counting numbers within a range in numeric and algebraic form.
3. Introduction to arithmetic sequence.  
4. Decimal representation, expanded form. Place value. Rounding and approximation.
5. Rules of addition, subtraction, multiplication and division.
6. Powers of 2. Powers of 10. Introduction to exponents (integer positive powers only).
7. Scientific notation.
8. Order of operations. Drills involving all operations.
9. Prime and composite numbers. Factorization and prime factorization.
10. Divisibility by 2, 4, 8, 3, 9, 5, 10, 25. Problems involving proofs of divisibility of two-, three- and four-digit numbers.
11. Common factor. Relatively prime numbers. Greatest Common Factor (GCF) and Least Common Multiple (LCM).
12. Fractions and Decimals (Including negative fractions and decimals):
    • Proper and improper fractions. Mixed fractions. Converting between mixed and improper fractions. Reducing to lowest terms.  Ordering and comparing fractions.
    • Addition, subtraction, multiplication and division of fractions.
    • Decimals. Comparing decimals. Converting between decimals and fractions. Operations with decimals. Mixed Operations with decimals and fractions.  
13. Percent. Expressing a decimal/common fraction as a % and expressing a % as a decimal or common fraction.
14. Ratios and proportions.
15. Representation of positive and negative rational numbers on a number line. Opposite numbers.
16. Rational numbers and their properties.
17. Linear equations.
18. Absolute value of a number. Simple equations and operations with absolute value.
19. Solving linear inequalities (recognizing solutions, graphing solutions on the number line, the addition and subtraction principles, the multiplication and division principles for solving a linear inequality).
20. Simple inequalities involving absolute value.
21. Circumference and area of a circle.
22. Area of a sector of a circle.
23. Areas of compound plane figures.
24. Volumes and surface areas of rectangular prisms, pyramids and cylinders.
25. Perpendicular bisector of a line segment. Bisector of an angle.
26. Square roots  (basics). Pythagorean theorem (as a fact, no proof). The converse of the Pythagorean theorem (as a fact).

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Introduction to Algebra

(Recommended for 6-7th grade students)

1. Review of rational numbers and the decimal positional system.
2. Examples of irrational numbers. Real numbers.
3. Review of linear equations and inequalities and their properties.
4. Solving equations and inequalities with absolute value.
5. Solving word problems involving distance, work, and mixtures.
6. Coordinate plane. Distance between two points.
7. Slope of a line. Equation of a line on a coordinate plane (general form, slope-intercept form, point-slope form).
8. Graphing linear functions.
9. Graphing solutions of linear inequalities.
10. Slopes of parallel and perpendicular lines.
11. Solving systems of linear equations with two variables. Graphic representation on the coordinate plane.
12. Square and cubic roots (basics).
13. Powers and Exponents 
    • Powers with natural exponents.
    • Zero and negative exponents. 
    • The notion of a power with an integer exponent.
    • Roots as powers. Fractional exponents.
    • Multiplying and dividing powers with the same base.
    • Power of a product and a fraction. Power of a power.
14. Monomials.
15. Arithmetic and geometric sequences.  

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Introduction to Geometry
(Recommended for 6-7th grade students just starting geometry)

1.      Points, Lines, Angles, Segments, Rays. Postulates relating points and lines on the plane.

2.      Measuring angles. Classifying angles (acute, obtuse, right, straight) - review.

3.      Intersecting lines. Complementary, Supplementary, Adjacent, and Vertical angles.

4.      Parallel lines and transversals. Alternate interior and alternate exterior angles. Properties of parallel lines. Proving that given lines are parallel.

5.      Perpendicular lines. Perpendicular bisector of a segment.

6.      Angles of a triangle. Angles of a polygon.

7.      Congruent triangles (properties and theorem for congruent triangles).

8.      Using congruent triangles.

9.      Classifying triangles (isosceles, equilateral, right, obtuse).

10.  Theorems about isosceles triangles.

11.  Medians, altitudes and perpendicular bisectors of a triangle. Properties.

12.  Quadrilaterals and their properties (parallelograms, rhombuses, rectangles, squares, trapezoids).

13.  Construction problems.

14.  Similar polygons.

15.  Theorems for similar triangles.

16.  Similar right triangles and their properties.

17.  Areas of similar plane figures.

18.  Pythagorean theorem. The converse of the Pythagorean theorem.

19.  Special right triangles.

20.  Applications of right triangles.

21.   Perimeters and areas of a polygons and circles. Applications.

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Algebra Level 1

(Recommended for 7-8th grade students)

1. Monomials and Polynomials
   • Review of powers and monomials.
   • Adding, subtracting, multiplying, dividing monomials.
   • Factoring monomials.
   • The notion of a polynomial. Degree of a polynomial.
   • Simplifying polynomials.
   • Standard (canonical) form of a polynomial.
   • The sum and the difference of polynomials.
   • The product of polynomials.
   • Dividing a polynomial by a monomial.
   • Dividing polynomials. Long division of polynomials.
   • Factoring polynomials by grouping and simple monomial factoring.
   • Factoring polynomials using special products.
2. Rational Expressions
   • Basic definitions and reductions.
   • Multiplying and Dividing Rational Expressions.
   • Adding and Subtracting Rational Expressions.
3. Equations Involving Rational Expressions.
4. Word problems involving polynomials and rational expressions. Work and Motion problems.
5. Simple quadratic equations. Standard form.
6. Completing the square. Solving quadratic equations by completing the square.
7. Notion of a function and its graph. Domain. Range. Graphing a general linear function.

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Geometry Level 1
(Recommended for 7-8th grade students graduated from intro to geometry class)

1.      Angles, parallel and perpendicular lines. Sum of the angles of a polygon – review.

2.      Congruent triangles, similar triangles, right triangles – review. 

3.      Quadrilaterals and their properties – review.

4.      Perimeters, areas, volumes – review.

5.      Areas and perimeters of similar plane figures.

6.      Surface areas and volumes of solids (prisms, pyramids, cones, cylinders, spheres).

7.      Areas and volumes of similar solids.

8.      Pythagorean theorem. The converse of Pythagorean theorem.

9.      Special right triangles. Applications.

10.  Elements of trigonometry. Sine, cosine and tangent ratios in a right triangle.

11.  Problems involving right triangles.

12.  Inequalities in geometry. Inequalities in a triangle.

13.  Circles – definition, chord, radius, diameter.

14.  Arcs, chords and central angles of a circle.

15.  Inscribed angles of a circle.

16.  Elements of analytical geometry.

17.  The distance formula. The midpoint formula.

18.  Equation of a circle.

19.  Parallel and perpendicular lines.

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Algebra Level 2

(Recommended for 8-10th grade students)

1. Linear equations and matrices
   • Review of linear function and its graph. Slope and x/y-intercepts. Slopes of parallel and perpendicular lines.
   • Review of systems of linear equations and different methods of solving them.
   • Matrix, its dimension. Zero matrix, identity matrix.
   • Adding and Subtracting matrices.
   • Matrix multiplication.
   • Determinants. Cramer’s rule.
   • Inverse matrix.
   • Solving systems of linear equations using matrices.
   • Problems involving matrices
2. Elements of logic
   • Sentence, Phrase, Proposition (statement).
   • The negation of a proposition. Truth table.
   • Conjunction and disjunction of propositions.
   • Truth tables for statements combining different logic operations. Tautology and equivalent statements. De Morgan Law as an example of equivalent statements.
   • Implication. Equivalence. Expressing implication and definition of equivalent statements using equivalence.
3. Square roots and their properties.
   • The definition of a square root. Principal square root and absolute value of a number.
   • Product/quotient/power property of square root(s)
   • Rationalizing the denominator of a fraction.
   • Simplifying expressions with square roots.
   • Transforming expressions containing square roots. Combining different operations with square roots.
4. Quadratic equations
   • Solving quadratic equation by factoring and by completing a square
   • Discriminant. Quadratic formula. Solving quadratic equations by quadratic formula.
   • Solving quadratic equations using different methods.
   • Viet’s theorem.
   • Factoring quadratic trinomials in general case.
   • Solving word problems involving quadratic equations.
5. Graphing quadratic functions.
6. Quadratic inequalities. 
7. Word problems involving quadratic equations and inequalities.
8. Functions.
   • Function, its Domain and Range.
   • Representing a Function by formula.
   • Review of linear and quadratic functions, their domain and range.
   • Square root function. Its basic properties and its graph.
   • Some easy transformations of the graph and examples of solving irrational equations using graphs.
   • Inverse proportionality graph and its properties.
9. Arithmetic and Geometric Series
10. Introduction to Combinatorics.
    • Permutations, Combinations, the Binomial Theorem
11. Introduction to probability and statistics
    • Experimental probability
    • Probability involving permutations and combinations
    • Probability of compound events
    • Conditional probability
    • Independent and dependent events
    • Random variables
    • Binomial, uniform and normal distributions
    • Expected value

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Geometry Level 2
(Recommended for 8-10th grade students graduated from geometry 1 class)

1.      Review of basic plane and solid geometry.

2.      Triangles.

3.      Quadrilaterals.

4.      Similarity.

5.      Circles and their property.

6.      Tangents to a circle.

7.      Vectors (addition, subtraction, dot product).

8.      Transformations (rotations, reflections, dilations). Symmetry.

9.      Angles on a unit circle in the coordinate plane. Definition of sine, cosine.

10.  Definition of tangent, cotangent, secant, cosecant. Connection of cotangent, secant, cosecant with tangent, cosine, and sine.

11.  Basic trigonometric identities.

12.  Reduction formulas.

13.  Sines and altitudes. Formula for the area of a triangle.

14.  The Law of sines (including ambiguous case).

15.  The Law of cosines.

16.  Inverse trigonometric functions.

17.  Solving triangles using trigonometry. Word problems.

18.  Radian measure.

19.  Formulas for sin (a+ß), cos (a+ß), and tan (a+ß).

20.  Formulas for double angles.

21.  Formulas for half-angles.

22.  Trigonometric equations.

23.  Graphs of trigonometric functions.

24.  Odd and even functions.

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