M9 - Table of Contents

Transcription

1 M9 Tt Page 1 M9 - Table of Contents Duotang/Notes/Homework M9 - Table of Contents M9 - Methods M9 - Remember M Symmetry M Real Numbers M Eponents M Algebraic Epressions M Circles M Parallel Lines M Eponent Laws M Negative Eponents M Combine Like Terms M Multiplying Polynomials M Dividing Polynomials M Patterns M Toothpicks M Inequalities M Angles M Inscribed/Central Angles M Tangents/Semi Circles M Opposite Angles M Interior Angles

2 M9 Tt Page 2 M9 - Methods Numbers Real Rational Integers Whole Natural Irrational Circles Shade the Cord/Arc Etend the lines Draw a radius Rotate the page Eponents Laws Change of base: 9 = 3 3 = 3 9 = 3 Linear Relations TOV Equations Slope = = Polynomials Adding and Subtracting Like Terms a + a = 2a 5a 2a = 3a Combine like terms. Add/Subtract Coefficients Circle, square or cloud like terms. Multiply: Multiply coefficients, add eponents. a a = a 2a 3a = 6a 3 5 = 15 Divide: Divide coefficients, subtract eponents = 4 30a 6a = 5a Symmetry: Horizontal Line Symmetry Vertical Flip Vertical Line Symmetry Horizontal flip Oblique-Oblique Inequalities Signs Laws Number Line Equations Similar triangles SAS, ASA, AAS, SSS, AAS, AAA,

3 M9 Tt Page 3 M9 - Remember Algebra a(b) = a b = ab a( b) = a ( b) = ab 2( 3) = 2 ( 3) = 6 a( b) = a ( b) = +ab 2( 3) = 2 ( 3) = +6 a(b) = a (b) = ab 2(3) = 2 (3) = 6 Eponents Laws Need same base! 2 = = Mistakes Never multiply the base by the eponent = 8 Negative numbers with brackets to odd eponents stay negative. ( 2) = 8 ( 2) = 8 2 = 16 Negative numbers with brackets to even eponents become positive ( 2) = 16 ( 2) = 16 ( 2 ) = 8 ( 2) = 8 Negative numbers without brackets stay negative 2 = 8 2 = 16 Unnecessary brackets (2) = 8 (2) = 16 ( 2 ) = 16 ( 2) = 16 Polynomials Remember to circle the term's sign! = = Can only add or subtract Like Terms Or Cannot add or subtract unlike terms Inequality: -The alligator eats the bigger number. -When graphing inequalities/solving treat the inequality sign like an equal sign. -Divide /Multiply by a negative: Change the direction of the sign. Greater than: > Greater than or equal to: Less than: < Less than or equal to: Does not equal:, [ ] Included (closed, square, solid) <, > ( ) (, ) Not Included (open, round, dotted) Circles ΔABC A A B 60 ABC = B = 60 C AC = b C a = CB c = AB B

4 M9 Tt Page 4 M Real/Irrational/Rational/Integers/Whole/Natural Review Irrational Numbers Real Numbers ( 3 1, 5 2, 1 1, 1 7, 0 1, 1 1,, , 1 1, 2 1, π, 2 ) Rational Numbers (, e, π, , 2, ) ( 5 2, 2 1, 3 2, 1 1, 0 1, 1 2, 2 3, 1 1, 3 1, ) Decimal: - Does not end - Does not repeat Cannot be epressed as a fraction OR Decimal: - ends OR - repeats Can be epressed as a fraction Integers ( 3, 2, 1, 0, 1, 2, 3, ) Whole Numbers (0, 1, 2, 3, ) Natural Numbers (1, 2, 3, ) All numbers are real. Irrational numbers cannot be epressed as a fraction. Rational numbers can be epressed as a fraction. Number Line A number can be shown on a number line with a dot and the number above or below.

5 M9 Tt Page 5 1) M Eponents Review Rule: 2 2 = 2 = 2 = 32 Eample: Base = (2 2 2) (2 2) = 2 Eponent Remember: -Never multiply the base by the eponent -Must have same base to use laws. Add eponents 2) = 3 = 3 = 27 3 = = Subtract eponents 3) (2 ) = 2 = 2 = 64 (2 ) = (2 2) (2 2) (2 2) = 2 Multiply eponents (2) = 2 = 8 (2) = (2) (2) (2) = 8 Distribute eponents 3 5 = 3 = = = = Distribute eponents 4) 2 = = Bring to the bottom, make eponent positive 1 3 = 3 = 9 1 Bring to the top, make eponent positive 5) 2 = 2 2 = 4 2 = ( 2 ) ( 2) = ( 2) ( 2) = 4 ( 4) = ( 4) ( 4) ( 4) = 64 Negative numbers WITHOUT brackets stay NEGATIVE Negative numbers with brackets to EVEN eponents become POSITIVE Negative numbers with brackets to ODD eponents stay NEGATIVE 6) 5 = 1 Anything (nonzero) to the 0 is 1. 8 = 8 Anything to the 1 is itself. 1 = 1 1 to the anything is 1. 0 to the (positive) anything is 0. 7) Theory 3 = 27 3 = 9 3 = 3 3 = 1 3 = 1 3 = = = = = = = = = = = = 1 = 1 = 1 = = = = 1 = 1

6 M9 Tt Page 6 M Eponents Table ^

7 M9 Tt Page 7 M Algebraic Epressions Review Coefficient = 4 4 Eponent = 2 Letter/variable = Coefficient: a number in front of a variable Variable: a letter Degree of term: The variable eponent or sum of variable eponents. Eponent: 3 = 3 3 = 9 5 = = 125 Like term: same letter(s), same eponent(s). Term Like terms Term Degree: 2 (y ) 5 8 = Numbers have a degree of "zero" 2 1, 2, 3, 4, 5, 6,. a a, 2a, 3a, 4a,., 2, 3, 4, a, 2a, 3a, 4a, a Leading Term: The term with the highest degree. Degree of polynomial: Degree of highest degree term. Polynomial Leading Term: Degree of Polynomial Adding and Subtracting like terms. a + a = 2a 3a + 2a = 5a 6a 3a = 3a Add or subtract coefficients a + a = 2a 8 y y 3 a + 1 = a + 1 You can only add and subtract like terms. Polynomial: Terms with variables with whole number eponents. Multiplying and Dividing a a = a 2a 3a = 6a 3 5 = 15 You cannot add or subtract unlike terms. Multiply coefficients, add eponents Monomial: One term. Binomial: Two terms. Trinomial: Three terms. Eamples: 2,,, 2y, 5z, , 4, y + 5, 3 + y, , a + b + c 20 5 = 4 30a 6a = 5a Divide coefficients, subtract eponents Polynomial: Monomials, Binomials, Trinomials and more than three terms. 2, + 2, , a + b + c + d + e Not Polynomial:,,, 2

8 M9 Tt Page 8 M Inequalities Review 5 is less than 8. 8 is greater than 5. 5 < 8 8 > 5 The alligator eats the bigger section. 7 is less than or equal to is greater than or equal to < 8, [ ] Included (closed, square, solid) Sketching Inequalities <, > ( ) (, ) Not Included (open, round, dotted) 5 Place a closed circle at 5 on the number line (when a can equal 5) Draw a line with an arrow to the right (greater than) < 3 Place an open circle at 3 (when cannot equal 3) Draw a line with an arrow to the left (less than) < 3 Inbetween

9 M9 Tt Page 9 M Circles Review Parallel Line Rules Angles on a line sum to 180. (supplementary) Opposite angles are equal. Alternate interior angles equal. Corresponding angles equal. Co-Interiors angles add to 180 o Angles on a point add to 360 o Methods Rotate the Page Etend Parallel lines Etend the transversal lines Draw a radius/connect Points Draw a radius to eterior point Identify an unknown as "" Equal/Parallel Lines Tick/Double Tick The Equal Lines Arrow the Parallel Lines If multiple act accordingly Identifying Angles in Circles 1. Make a slice of pie with your left and right hand. 2. Central/inscribed angle is between your inde fingers. 3. Arc/chord is crust of piece of pie. 4. Shade Arc Finding Shared Arcs/Chords Do you see an angle measure? What type of angle is it? Where is its arc/chord? Shade in its arc. Are there any other angles from that arc/chord. Circle Rules Triangles Central angles from same/equal chords are equal. Inscribed angles from same/equal chords are equal. Inscribed angles are half central angles from same/equal chords. Central angles are twice inscribed angles from same/equal chords Draw Triangles 180 in a triangle Isosceles Equilateral Right - Pythagoras Opposite angles in a cyclic quadrilateral sum to 180. Tangent lines are perpendicular to radius. Perpendicular bisector of a chord passes through center of circle. Tangents to eterior points are equal. Inscribed angles in a semi-circle equal 90 o. The angle between the tangent and the chord is equal to the inscribed angle on the opposite side of the chord. Polygons Sum of Interior Angles = (n 2) 180 Interior Angle = Sum = n (n 2) 180 n Interior + Eterior = 180 Sum of all Eterior Angles sum to 360 n = # of sides

10 M9 Tt Page 10 M Parallel Lines Rules Review = 180 o 150 o 30 o Angles on a line sum to 180. (supplementary) Opposite angles are equal. 80 o 1 = 2 50 o Alternate interior angles equal. 1 = 2 80 o OPP 's = 50 o Alt int. 's = 50 o Corresponding angles equal. 120 Corresponding angles equal. 50 o 1 = 2 CORR 's = = 2 CORR 's = 130 Co-Interiors add to 180 o 50 o : parallel lines. Angles on a point add to 360 o = 180 o Angles on a line sum to = 2 OPP 's = = 5 1 = 4 Alt int. 's = CORR 's = = 180 Parallel lines: lines that never touch Transversal: a line passing through parallel lines. Alternate: angles across a transversal. Interior: angles inside parallel lines. Co-interior: angles inside parallel lines on the same side of a transversal : parallel lines. Co Interior add to 180 Etend the Lines