Welcome to the Math S.A.T. Enjoyment Hours
Hosted by the B B & S Brothers Bianco, Bianco & Skeels
Quick Drillsky
#1 43 + 47
#2 180 ÷ 3
#3 145 - 96
#4 (12)2
#5 (2)5
#6 (10)8
#7 √ 169
#8 √ (475)2
#9 (9)9 (3)18
#10 43 + 90 + 47
LET’S √ EM!
#1 43 + 47
#1 43 + 47 90
#2 180 ÷ 3
#2 180 ÷ 3 60
#3 145 - 96
#3 145 - 96 49
#4 (12)2
#4 (12)2 144
#5 (2)5
#5 (2)5 32
#6 (10)8
#6 (10)8 100,000,000
#7 √ 169
#7 √ 169 13
#8 √ (475)2
#8 √ (475)2 475
#9 (9)9 (3)18
#9 (9)9 (3)18 1
#10 43 + 90 +47
#10 43 + 90 +47 180
You can have PSAT/SAT Fun everyday! Go to www.collegeboard.com
Strategy - ! If the sum of 4 consecutive integers is ‘f’, then, in terms of ‘f’, what is the least of these integers? A) f/4 B) (f - 2)/4 C) (f - 3)/4 D) (f - 4)/4 E) (f - 6)/4
Strategy - Substitute! If the sum of 4 consecutive integers is ‘f’, then, in terms of ‘f’, what is the least of these integers? A) f/4 B) (f - 2)/4 C) (f - 3)/4 D) (f - 4)/4 E) (f - 6)/4
Strategy - sdrawkcaB kroW Work backwards!!!! Fill in the answer choices for complex algebra problems. Example: If (a/2)3 = a2, a≠0, then a = A) 2 B) 4 C) 6 D) 8 E) 10 *From last lesson - ran out of time!
Helpful Hint: Remember the answer choices are arranged from least to greatest so it may help start in the middle and proceed in the right direction.
Objectives: To review Geometry concepts on SAT. To introduce Student Produced Response problems.(SPR) To introduce 1 more strategy.
GEOMETRY & MATH WE ALL KNOW FIGURES INVOLVED IN GEOMETRY
GEOMETRY & MATH WE ALL KNOW FIGURES INVOLVED IN GEOMETRY
GEOMETRY & MATH WE ALL KNOW FIGURES INVOLVED IN GEOMETRY
GEOMETRY & MATH BUT WITH A FEW DEFINITIONS WE CAN TACKLE MANY PROBLEMS WHICH OTHERWISE WOULD BE IMPOSSIBLE
ESSENTIALS OF GEOMETRY A RIGHT ANGLE:
ESSENTIALS OF GEOMETRY A RIGHT ANGLE: An angles with a measure of 90°
ESSENTIALS OF GEOMETRY AN ACUTE ANGLE:
ESSENTIALS OF GEOMETRY AN ACUTE ANGLE: An angle which measurement is less than 90°
ESSENTIALS OF GEOMETRY AN OBTUSE ANGLE:
ESSENTIALS OF GEOMETRY AN OBTUSE ANGLE: An angle which measurement is more than 90°
ESSENTIALS OF GEOMETRY PERPENDICULAR LINES:
ESSENTIALS OF GEOMETRY PERPENDICULAR LINES: Two lines that intersect at right angles ( note written as )
ESSENTIALS OF GEOMETRY VERTICAL ANGLES: 2 1
ESSENTIALS OF GEOMETRY VERTICAL ANGLES: Two intersecting lines form 2 pair of vertical angles.
ESSENTIALS OF GEOMETRY VERTICAL ANGLES: 1 2 1 and 2 are vertical
ESSENTIALS OF GEOMETRY VERTICAL ANGLES: ALWAYS HAVE THE SAME MEASURE!
ESSENTIALS OF GEOMETRY SUPPLEMENTARY ANGLES : 2 1
ESSENTIALS OF GEOMETRY SUPPLEMENTARY ANGLES : Two angles whose measures have a sum of 180°
ESSENTIALS OF GEOMETRY COMPLEMENTARY ANGLES : 1 2
ESSENTIALS OF GEOMETRY COMPLEMENTARY ANGLES : Two angles whose measures have a sum of 90°
ESSENTIALS OF GEOMETRY SUM OF THE ANGLES IN A TRIANGLE: 1 2 3
ESSENTIALS OF GEOMETRY SUM OF THE ANGLES IN A TRIANGLE: The sum of the three angles in a triangle is 180°
ESSENTIALS OF GEOMETRY SUM OF THE ANGLES IN A TRIANGLE: 1 m+ m + m = 180 2 3
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a2 + b2 = c2 a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: c a b
ESSENTIALS OF GEOMETRY PYTHAGOREAN THEOREM: NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c A B
GEOMETRY PRACTICE FIND THE VALUE OF X X 3 4
GEOMETRY PRACTICE FIND THE VALUE OF X a2 + b2 = c2 X 3 4
GEOMETRY PRACTICE FIND THE VALUE OF X a2 + b2 = c2 X 3 32 + 42 = X2 4
X 3 4 GEOMETRY PRACTICE a2 + b2 = c2 32 + 42 = X2 9 + 16 = X2 FIND THE VALUE OF X a2 + b2 = c2 X 32 + 42 = X2 3 9 + 16 = X2 4
X 3 4 GEOMETRY PRACTICE a2 + b2 = c2 32 + 42 = X2 9 + 16 = X2 25= X2 FIND THE VALUE OF X a2 + b2 = c2 X 32 + 42 = X2 3 9 + 16 = X2 4 25= X2
±5 = X X 3 4 GEOMETRY PRACTICE FIND THE VALUE OF X A2 + B2 = C2
X = 5 X 3 4 GEOMETRY PRACTICE FIND THE VALUE OF X A2 + B2 = C2
Geometry Tips Often figures are not drawn to scale. Redraw the diagrams more accurately.
Geometry Tips Sometime it is helpful to add extra segments, lines, etc. to a drawing.
1,000,000 words (inflation) Geometry Tips If there is no drawing, make your own. A picture is worth what? 1,000,000 words (inflation)
133° x° GEOMETRY PRACTICE Find the value of x: A 37 B 47 C 57 D 90 E 133 133° x°
133° x° 133° GEOMETRY PRACTICE First you must realize that angle 133° and the angle x° are supplementary angles
GEOMETRY PRACTICE 133° x° 133° Then let: x° + 133° = 180°
133° x° 133° GEOMETRY PRACTICE Then let: x° + 133° = 180° Subtract: -133° -133°
133° x° 133° GEOMETRY PRACTICE Then let: x° + 133° = 180° Subtract: -133° -133° Finally : x° = 47°
133° x° GEOMETRY PRACTICE Find the value of x: A 37 B 47 C 57 D 90 E 133 133° x°
GEOMETRYPRACTICE Find the value of x: A 23 B 33 C 43 D57 E 90 x° 57°
90° GEOMETRY PRACTICE Find the value of x: x° 57° A 23 B 33 C 43 D57 E 90 x° 90° 57°
GEOMETRY PRACTICE x° 57° x° + 57°+ 90° = 180°
GEOMETRY PRACTICE x° 57° x° + 147° = 180°
GEOMETRY PRACTICE x° 57° x° + 147° = 180° -147° -147°
GEOMETRY PRACTICE x° 57° x° + 147° = 180° -147° -147° x° = 33°
GEOMETRYPRACTICE Find the value of x: A 23 B 33 C 43 D57 E 90 x° 57°
GEOMETRY PRACTICE FIND THE VALUE OF X 17 x 8 12
GEOMETRY PRACTICE FIND THE VALUE OF X 17 y x 8 12
GEOMETRY PRACTICE FIND THE VALUE OF X 17 y x 8 12 82 + y2 = 172
GEOMETRY PRACTICE FIND THE VALUE OF X 17 y x 8 12 y = 15
GEOMETRY PRACTICE FIND THE VALUE OF X 17 x 15 8 12
GEOMETRY PRACTICE FIND THE VALUE OF X 17 x 15 8 12 x2 + 122 = 152
GEOMETRY PRACTICE FIND THE VALUE OF X 17 x 15 8 12 x2 + 122 = 152
GEOMETRY PRACTICE FIND THE VALUE OF X 17 x 15 8 12 x = 9
GEOMETRY PRACTICE The complement of an angle is 44more than the angle. What is the sum of the angle’s complement and its supplement?
x + 44 x
x + x + 44 = 90 x + 44 x
x + x + 44 = 90 2x + 44 = 90 x + 44 x
x + x + 44 = 90 2x + 44 = 90 2x = 46 x + 44 x
x + x + 44 = 90 2x + 44 = 90 2x = 46 x = 23 x + 44 x
GEOMETRY PRACTICE The complement of an angle is 44more than the angle. What is the sum of the angle’s complement and its supplement?
Solution Angle is 23 Complement is 90 - 23 = 67 Supplement is 180 - 23 = 157 Sum of comp & supp is 224
GEOMETRY Coordinate Geometry Lines and angles Triangles and Polygons Perimeter Area Volume
Coordinate Geometry Distance formula: d = √(x2 - x1)2 + (y2 - y1)2
Coordinate Geometry Distance formula: d = √(x2 - x1)2 + (y2 - y1)2 Slope: ∆y = (y2 - y1) ∆x (x2 - x1)
Lines and Angles Adjacent angles 3 2 4 1
Lines and Angles Adjacent angles - 2,3 ; 3,4 1,2 ; 1,4 3 2 4 1
Lines and Angles Adjacent angles - 2,3 ; 3,4 1,2 ; 1,4 Vertical angles
Lines and Angles Adjacent angles - 2,3 ; 3,4 1,2 ; 1,4 Vertical angles 1,3 ; 2,4 3 2 4 1
Parallel Lines: m || n 1 5 m 2 6 3 7 n 4 8 t
Triangles Interior angles always have a sum of
Triangles Interior angles always have a sum of 180°. Exterior angles always have a sum of
Triangles Interior angles always have a sum of 180°. Exterior angles always have a sum of 360°. (1 at each vertex) Each exterior angle is equal to the sum of the 2
Triangles Interior angles always have a sum of 180°. Exterior angles always have a sum of 360°. (1 at each vertex) Each exterior angle is equal to the sum of the 2 remote interior angles. Similar triangles have corresponding sides which are proportional. (CSSTP)
Triangles Area of a ∆ = 1/2 base times height ∆ Inequality Thm - The sum of any two lengths must be greater than the third length. Isosceles ∆- 2 or more congruent sides. (Angles opposite those sides are also congruent.) Equilateral ∆ - all sides and angles are congruent.
Right Triangles a b c Pythagoras said “In a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. or a2 + b2 = c2
Rt. ∆s - Perfect Triples a b c 3, 4, 5; 5,12,13; 8, 15, 17 7, 24, 25
Rt. ∆s - Perfect Triples 3, 4, 5; 5,12,13; 8, 15, 17 7, 24, 25 a b c 3, 4, 5; 5,12,13; 8, 15, 17 7, 24, 25 All multiples of these are also perfect triples.
Special Right Triangles 1, √3, 2 1, 1, √2 30-60-90 ∆ 45-45-90 ∆ 2x x√2 60° x x 30° x x√3
Other Polygons Define and give area for each. Parallelogram Rectangle Square The sum of the interior angles for any convex polygon is
CIRCLES Circumference C = 2πr Area A = πr2 Arc lengths and sectors, multiply by portion of circumference or area used.
SOLIDS Surface area and Volume Use formula sheet. Know these before the test.
Strategy:BoD On Geometry problems be careful of figures that are “not drawn to scale”, redraw as acurate a figure as you can. Feel free to extend lines, rays, etc., or draw extra segments as needed. Example: Find the value of x.
Strategy:BoD 32° x° Note: The figure is not drawn to scale.
Strategy:BoD (#2) The trapezoid shown below has a height of 12. Find the length of the base not given. 20 17 Note: The figure is not drawn to scale. 13
Practice Work with your neighbor to complete the 6 practice problems. Try to use some of the strategies presented today to help you. You have 12 minutes starting now.
On your mark, get set.....
START!
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Time’s Up!!!!
Example 1: In the figure, l m, and x is 20° less than y. What is the value of y? A) 35 B) 45 C) 55 D) 80 E) 100 l y° x° m
Example 2: In the figure,if ∆ABC is the same size and shape as ∆ABD, then the degree measure of <BAD is ___? A) 25 B) 35 C) 45 D) 50 E) 75 B D 40° 70° E A C
Example 3: In right triangle ABC, if the measure of <ABD = 15° ands <A = 30°, what is the length of DB? A) 6 B) 6√3 C) 6√2 D) 6√3 - 6 E) 6√2 - 6 A 30° 12 D 15° B C E
Example 4: If the lengths of two sides of a triangle are 14 and 23, then the perimeter : I. must be between 9 and 37 II. must be between 46 and 74 III. must be greater than 50 A) I only B) I & II only C) I, II, & III D) II only E)None of the above
Example 5: What is the area of a circle with a circumeference of π2?
Example 6: Cube A has an edge of 4. If each edge of cube A is increased by 25%, creating a second cube B, then the volume of cube B is how much greater than the volume of cube A? A) 16 B) 45 C) 61 D) 64 E) 80
Be sure to turn this in to your math teacher the next time you go to math class!
Closing Comments
Today we will.........
Vocabulary Terms
GEOMETRY PRACTICE Find the value of x: A 37 B 47 C 57 D 90 E 133
GEOMETRYPRACTICE Find the value of x: A 23 B 33 C 43 D57 E 90
GEOMETRY PRACTICE FIND THE VALUE OF X x
GEOMETRY PRACTICE The complement of an angle is ___ more than the angle. What is the sum of the angle’s complement and its supplement?
Example 1: In the figure, l m, and x is 20° less than y. What is the value of y? A) 35 B) 45 C) 55 D) 80 E) 100 l y° x° m
Example 2: In the figure,if ∆ABC is the same size and shape as ∆ABD, then the degree measure of <BAD is ___? A) 25 B) 35 C) 45 D) 50 E) 75 B D 40° 70° E A C
Example 3: In right triangle ABC, if the measure of <ABD = 15° ands <A = 30°, what is the length of DB? A) 6 B) 6√3 C) 6√2 D) 6√3 - 6 E) 6√2 - 6 A 30° 12 D 15° B C E
Example 4: If the lengths of two sides of a triangle are 14 and 23, then the perimeter : I. must be between 9 and 37 II. must be between 46 and 74 III. must be greater than 50 A) I only B) I & II only C) I, II, & III D) II only E)None of the above
Example 5: What is the area of a circle with a circumeference of π2?
Example 6: Cube A has an edge of 4. If each edge of cube A is increased by 25%, creating a second cube B, then the volume of cube B is how much greater than the volume of cube A? A) 16 B) 45 C) 61 D) 64 E) 80