Welcome to the Math S.A.T. Enjoyment Hours

Welcome to the

Math S.A.T. Enjoyment

Hours

Hosted bythe

B B & S Brothers

Bianco, Bianco & Skeels

Quick Quick DrillskyDrillsky

#1#143 + 4743 + 47

#2#2180 ÷ 3 180 ÷ 3

#3#3145 - 96145 - 96

#4#4(12)(12)22

#5#5(2)(2)55

#6#6(10)(10)88

#7#7√ 169√ 169

#8#8√ (475)√ (475)22

#9#9 (9) (9)99

(3)(3)1818

#10#1043 + 90 + 4743 + 90 + 47

LET’S LET’S √ EM!√ EM!

#1#143 + 4743 + 47

#1#143 + 4743 + 47

90

#2#2180 ÷ 3 180 ÷ 3

#2#2180 ÷ 3 180 ÷ 3

60

#3#3145 - 96145 - 96

#3#3145 - 96145 - 96

49

#4#4(12)(12)22

#4#4(12)(12)22

144

#5#5(2)(2)55

#5#5(2)(2)55

32

#6#6(10)(10)88

#6#6(10)(10)88

100,000,000

#7#7√ 169√ 169

#7#7√ 169√ 169

13

#8#8√ (475)√ (475)22

#8#8√ (475)√ (475)22

475475

#9#9 (9) (9)99

(3)(3)1818

#9#9 (9) (9)99

(3)(3)1818

1

#10#1043 + 90 +4743 + 90 +47

#10#1043 + 90 +4743 + 90 +47

180

You can have You can have PSAT/SAT Fun PSAT/SAT Fun

everyday!everyday!Go to Go to

www.collegeboard.comwww.collegeboard.com

Strategy - !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 - Strategy - SubSubstitute!stitute! 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 kroWStrategy - 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: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:Objectives: To review Geometry concepts on

SAT. To introduce Student Produced

Response problems.(SPR) To introduce 1 more strategy.

GEOMETRY & MATHWE ALL KNOW FIGURES INVOLVED IN GEOMETRY

GEOMETRY & MATHWE ALL KNOW FIGURES INVOLVED IN GEOMETRY

GEOMETRY & MATHWE ALL KNOW FIGURES INVOLVED IN GEOMETRY

GEOMETRY & MATHBUT 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:

1 2

ESSENTIALS OF GEOMETRY

VERTICAL ANGLES: Two intersecting lines form 2 pair of vertical angles.

ESSENTIALS OF GEOMETRY

VERTICAL ANGLES:

1 21 and 2 are vertical

ESSENTIALS OF GEOMETRY

VERTICAL ANGLES: ALWAYS HAVE

THE SAME MEASURE!

ESSENTIALS OF GEOMETRY

SUPPLEMENTARY ANGLES :

1 2

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:

13 2

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

2 3

m+ m + m= 180

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

a2 + b2 = c2

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

a cb

ESSENTIALS OF GEOMETRY

PYTHAGOREAN THEOREM:

A cB

NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE

GEOMETRY PRACTICE

FIND THE VALUE OF X

34

X

GEOMETRY PRACTICE

FIND THE VALUE OF X

34

Xa2 + b2 = c2

GEOMETRY PRACTICE

FIND THE VALUE OF X

34

Xa2 + b2 = c2

32 + 42 = X2

GEOMETRY PRACTICE

FIND THE VALUE OF X

34

Xa2 + b2 = c2

32 + 42 = X2

9 + 16 = X2

GEOMETRY PRACTICE

FIND THE VALUE OF X

34

Xa2 + b2 = c2

32 + 42 = X2

9 + 16 = X2

25= X2

GEOMETRY PRACTICE

FIND THE VALUE OF X

34

XA2 + B2 = C2

32 + 42 = X2

9 + 16 = X2

25 = X2

±5 = X

GEOMETRY PRACTICE

FIND THE VALUE OF X

34

XA2 + B2 = C2

32 + 42 = X2

9 + 16 = X2

25 = X2

X = 5

Geometry TipsOften figures are not drawn to scale. Redraw the diagrams more accurately.

Geometry TipsSometime it is helpful to add extra segments, lines, etc. to a drawing.

Geometry TipsIf there is no drawing, make your own. A picture is worth what?

1,000,000 words (inflation)

GEOMETRY PRACTICEFind the value of x:A 37 B 47 C 57 D 90 E 133

133°x°

GEOMETRY PRACTICE

133°

133°x°

First you must realize that angle 133° and the angle x° are supplementary angles

GEOMETRY PRACTICE

133°

133°x°

Then let: x° + 133° = 180°

GEOMETRY PRACTICE

133°

133°x°

Then let: x° + 133° = 180°Subtract: -133° -133°

GEOMETRY PRACTICE133°

133°x°

Then let: x° + 133° = 180°Subtract: -133° -133°Finally : x° = 47°

Find the value of x:A 37 B 47 C 57 D 90 E 133

GEOMETRY PRACTICE

133°

133°x°

GEOMETRYPRACTICEFind the value of x:

A 23 B 33 C 43 D57 E 90

x°57°

GEOMETRY PRACTICEFind the value of x:

A 23 B 33 C 43 D57 E 90

x°57°90°

GEOMETRY PRACTICE

x° + 57°+ 90° = 180°

x°57°

GEOMETRY PRACTICE

x° + 147° = 180°

x°57°

GEOMETRY PRACTICE

x° + 147° = 180°

x°57°

-147° -147°

GEOMETRY PRACTICE

x° + 147° = 180°

x°57°

-147° -147° x° = 33°

GEOMETRYPRACTICEFind the value of x:

A 23 B 33 C 43 D57 E 90

x°57°

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

8

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

8y

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

8y

82 + y2 = 172

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

8y

y = 15

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

815

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

815

x2 + 122 = 152

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

815

x2 + 122 = 152

GEOMETRY PRACTICEFIND THE VALUE OF X

x12

17

815

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?

xx + 44

x + x + 44 = 90

xx + 44

x + x + 44 = 90

xx + 44

2x + 44 = 90

x + x + 44 = 90

xx + 44

2x + 44 = 902x = 46

x + x + 44 = 90

xx + 44

2x + 44 = 902x = 46x = 23

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

GEOMETRYGEOMETRY 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

1

23

4

Lines and Angles Adjacent

angles - 2,3 ; 3,4 1,2 ; 1,4 1

23

4

Lines and Angles Adjacent

angles - 2,3 ; 3,4 1,2 ; 1,4

Vertical angles

1

23

4

Lines and Angles Adjacent

angles - 2,3 ; 3,4 1,2 ; 1,4

Vertical angles 1,3 ; 2,4

1

23

4

Parallel Lines: m || n1

2

3

4

5

6

7

8

m

n

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

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

ab

c

Rt. ∆s - Perfect Triples

3, 4, 5; 5,12,13; 8, 15, 17 7, 24, 25

ab

c

Rt. ∆s - Perfect Triples

3, 4, 5; 5,12,13; 8, 15, 17 7, 24, 25

ab

c

All multiples of these are also perfect triples.

Special Right Triangles 1, √3, 2 1, 1, √2

30-60-90 ∆ 45-45-90 ∆

x

x√2x x

2x

x√330°

60°

Other Polygons Define and give area for each. Parallelogram Rectangle Square The sum of the interior angles

for any convex polygon is

CIRCLESCIRCLES 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:BoDStrategy: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:BoDStrategy:BoD32°

x°Note: The figure is not drawn to scale.

Strategy:BoD (#2)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!START!

1212minutes minutes

remainingremaining

1010minutes minutes

remainingremaining

55 minutes minutes

remainingremaining

22 minutes minutes

remainingremaining

11minutes minutes

remainingremaining

Time’s Time’s Up!!!!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

mx°y°

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 70°40°

A

B

C

D

E

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

15°

30°

A

BC

D

E

12

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 Closing CommentsComments

Today we will.........

Vocabulary Terms

GEOMETRY PRACTICEFind the value of x:A 37 B 47 C 57 D 90 E 133

GEOMETRYPRACTICEFind the value of x:

A 23 B 33 C 43 D57 E 90

GEOMETRY PRACTICEFIND 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

mx°y°

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 70°40°

A

B

C

D

E

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

15°

30°

A

BC

D

E

12

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