welcome to the math s.a.t. enjoyment hours
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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