Download GEOMETRY CURRICULUM - St. Ignatius College Preparatory

GEOMETRY CURRICULUM – ST. IGNATIUS COLLEGE PREP Part 1: Course Enduring Understandings Part 2: Unit Essential Questions and Learning Outcomes Part 3: Suggested Pacing Part 4: Two Sample Common Assessments Part 1: Geometry Enduring Understandings 1. Geometry is omnipresent in the physical world; it can be used to solve problems in real life. 2. Geometry knowledge is used in many branches of mathematics. 3. Geometry uses standard vocabulary and symbols to communicate facts and relationships about geometric figures. 4. Geometric figures are ruled by known relationships of measures, often expressed as theorems and/or algebraic formulas. 5. Proofs, constructions and visual observations demonstrate why geometric relationships are true. 6. Logic, in combination with facts, theorems and formulas can be used to draw conclusions about geometric figures. 7. A proof is a formal argument supported by postulates, theorems and definitions; it uses logical reasoning to come to its conclusion. Part 2: Geometry Essential Questions & Performance Objectives by Unit Basic Skills
1. What symbols, formulas and vocabulary are conventional for communicating within the context of Geometry? Students will explore the general concepts of patterns, points, lines, planes and angles. They
will review midpoints and distances from Algebra, and will begin constructions. By the end of
this unit, students will be able to:
• Identify inductive reasoning
• Find and describe visual and numeric patterns
• Understand basic undefined terms of geom.
• Identify intersections of lines and planes
• Classify angles
• Know and apply distance formula
• Know and apply Segment Addition Postulate
• Know and apply Angle Addition Postulate
• Understand Geometry vocabulary
• Know and apply Midpoint Formula or equivalent method
• Perform constructions: bisect an angle, bisect a segment
• Use a protractor to draw & measure angles
• Identify vertical & linear pairs of angles, and use their relationships algebraically
• Find areas & perimeters of common plane figures
• Use area & perimeter formulas to solve problems
Reasoning
1. How and why is deductive reasoning used in geometric proof? 2. How can traditional constructions deepen understanding and illustrate geometric relationships? Students will begin an exploration of the deductive system in Geometry by exploring patterns,
the geometric structure of theorems and postulates, conditional statements, and algebraic
properties. Students will also explore segment and angle relationships. By the end of this unit,
students will be able to:
• Analyze & write conditional & biconditional statements
• Understand basic point, line and plane postulates
• Use symbolic notation for conditional statements
• Form conclusions by using laws of logic
• Use properties of length and measure to justify segment/angle relationships
• Recognize algebraic properties of equality/properties of congruence
• Write reasons for steps in a proof
• Use deductive reasoning to prove statements about segments and angles
• Perform constructions: copy a segment, copy an angle
Perpendicular & Parallel Lines
1. What algebraic and geometric conditions are sufficient and necessary to prove lines parallel or perpendicular? 2. What are the angle relationships when parallel lines are cut by a transversal? 3. What are the conventional forms of proof? Students will explore lines and planes, using algebraic connections and logic to lead them into
different proof styles. Properties of parallel and perpendicular lines are introduced. By the end
of this unit, students will be able to:
• Construct lines parallel or perpendicular to a given line
• Identify relationships between lines
• Identify/use relationships of angle pairs formed by lines and a transversal
• Prove & use angle relationships involving parallel lines and a transversal
• Prove lines are parallel, given angle relationships
• Use slopes to identify parallel/perpendicular lines in the coordinate plane
• Write equations of lines, given point and/or slope information
• Graph lines from equations
• Become familiar with different types of proof
• Use properties of lines to prove statements
• Complete Geometer’s Sketchpad tutorial unit
Congruent Triangles
1. What are the different classifications for triangles?
2. How can triangles be proven congruent?
3. How can congruent triangles be used to solve problems?
Triangles are explored extensively, with emphasis on congruent triangles and proof. Students
will use congruent triangles and will extend their work with constructions. By the end of this
unit, students will be able to:
• Classify triangles by sides and angles
• Solve problems based on interior and exterior angle relationships
• Identify congruent figures and corresponding parts
• Understand and apply postulates and theorems proving triangles congruent
• (SSS, SAS, AAS, ASA, HL)
• Prove triangles congruent with given information
• Use congruent triangles to solve problems
• Copy a triangle by construction
• Use congruent triangles to prove segment or angle relationships
• Understand & use properties of isosceles and equilateral triangles
Properties of Triangles
1. What segments have special purposes in understanding triangles and solving problems?
2. What are some traditional constructions involving special segments in triangles?
3. What is indirect proof and how is it different from direct proof?
Students will explore special segments in triangles as well as triangle inequalities. They will
further explore constructions. By the end of this unit, students will be able to:
• Use properties of perpendicular bisectors & angle bisectors to solve problems
• Identify special segments in a triangle
• Construct circle circumscribing a triangle
• Construct centroid of a triangle
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Solve problems using properties of a centroid
Solve problems using properties of medians & altitudes of a triangle
Solve problems using properties using the midsegments of a triangle
Write and solve inequalities using properties of sides and angles of one triangle
Write and solve inequalities comparing sides/angles of two triangles
Write a paragraph-style indirect proof
Quadrilaterals 1. By what characteristics can one classify quadrilaterals? 2. What are necessary and sufficient conditions for proving a quadrilateral is a parallelogram? 3. How can algebra be used to classify quadrilaterals? Students will explore polygons, with a special emphasis on quadrilaterals. By the end of this
unit, students will be able to:
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Identify & describe polygons
Solve problems using the sum of interior angles of a quadrilateral
Know and use properties of parallelograms, rectangles, rhombi & squares
Prove that a quadrilateral is a parallelogram or special parallelogram
Know and use properties of trapezoids & kites
Find areas of quadrilaterals
Use area formulas of quadrilaterals to solve problems
Use coordinate geometry in conjunction with quadrilaterals to solve problems
Transformations
1. What are the types and characteristics of geometric transformations? 2. What are the traditional ways to represent vectors? Students will explore rigid motion in a plane. By the end of this unit, students will be able to:
• Identify the three basic rigid transformations – reflection, translation, rotation
• Use transformations to solve problems
• Identify the two types of symmetry – line and rotational
• Use a vector to describe a translation
• Write a vector in component form
• Use vectors to solve problems
Similarity 1. How are ratio and proportion related to geometric figures? 2. What information is needed to prove triangles similar? 3. How is knowledge of similar figures applicable to real-­‐world problems? Students will explore and use ratio and proportion. Students will explore similar polygons with an
emphasis on similar triangles. By the end of the unit, students will be able to:
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Compute the ratio of two numbers
Use proportions to solve problems
Use properties of proportions
Identify and Define similar polygons and find their scale factor
Use similar polygons to solve problems
Identify similar triangles
Use similar triangles in coordinate geometry
Use the AA, SSS, and SAS similarity theorems to prove two triangles are similar
Use similar triangles to solve real-life problems
Use proportionality theorems to solve problems
Identify a dilation and write the scale factor of a dilation
Use dilations to solve problems
Right Triangles 1. What theorems and other rules apply specifically to right triangles? 2. What information is needed in order to apply these rules and theorems? 3. How are vectors and trigonometry used to solve real-­‐world problems? Students will explore right triangles, special right triangles, trigonometric ratios in right triangles,
and vectors in right triangles. They will explore models involving right triangles. By the end of the
unit, students will be able to:
• Prove right triangles are congruent using HL congruence theorem
• Use properties of right triangles
• Compare similar right triangles using the altitude drawn to the hypotenuse
• Use the Pythagorean Theorem and its converse to solve problems
• Find the lengths of sides of special right triangles
• Use special right triangles to solve problems
• Find the sine, cosine, tangent, secant, cosecant and cotangent of any angle measured
in degrees (using a calculator)
• Use trigonometric ratios to solve problems
• Use coordinate geometry in the exploration of vector problems
• Find the magnitude and direction of a vector
• Add vectors and sketch vectors
• Solve a right triangle
• Use right triangles to solve problems
Circles 1. What vocabulary is used to describe circles as they relate to lines and angles? 2. How can circles give us information about angle measures and segment lengths? 3. How are the equation of a circle and its graph on the Cartesian Plane related? Students will explore circles, their parts and their equations. By the end of the unit, students will be
able to:
• Use all vocabulary associated with circles
• Use the properties of circles in problems
• Use properties of tangents to solve problems in geometry
• Name minor and major arcs of a circle
• Find measures of central angles and arcs of circles
• Use the measures of central angles and their arcs to solve problems
• Use properties of chords and arcs to solve problems
• Find the lengths of segments and chords in a circle
• Use the properties of inscribed angles to solve problems
• Use properties of the inscribed angles of a quadrilateral to solve problems
• Calculate angles formed by tangents, chords, and secants
• Use angle measures to solve real-life problems
• Write the equation of a circle and use it to solve real-life problems
• Sketch a circle on the coordinate plane given its equation
Planar Measurements
1. How are area formulas for plane figures derived? 2. How are area and perimeter used in real-­‐world applications? Students will explore area and perimeter of polygons and area and circumference of circles. By
the end of the unit, students will be able to:
• Find the perimeter of a polygon
• Find the area of a square, a rectangle, a parallelogram, a triangle, and a trapezoid
• Find the area of an un-classified quadrilateral whose diagonals are perpendicular
• Find the area of a regular polygon
• Use areas to solve problems
• Find the measures of and the sum of the interior and exterior angles of a polygon
• Find the area and circumference of a circle
• Find the length of an arc of a circle
• Find the area of a sector of a circle (regions of a circle)
• Compare the areas and perimeters of similar polygons
• Calculate probabilities based on area
Spatial Measurements
1. How are surface area formulas for 3-D figures derived?
2. How are volume formulas for 3-D figures derived?
3. How are surface area and volume formulas used in real-­‐world applications? Students will explore surface area and volume of three-dimensional solids. By the end of the
unit, students will be able to:
• Identify solids that are polyhedrons
• Use polyhedrons to solve real-life problems
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Find the surface area of a prism, cylinder, pyramid, cone and sphere
Find the volume of a prism, cylinder, pyramid, cone and sphere
Use volume and surface areas to solve real-life problems
Part 3: Suggested Geometry Pacing (Regular) Chapter # of class days, Counting assessments & Review days (Based on Larson Geometry text, 2004) Emphasis 1 10 Notation, segments, angles 2 12 Bisectors, two-­‐column proof, reasoning 3 9 Parallel Lines, two-­‐column proof (Sometimes we can do that much in the first quarter, and sometimes we need to finish Ch 3 in the second quarter. A common cumulative quiz may be given at midterm time or in the second quarter as agreed-­‐upon. Traditionally, we have had the students learn Geometer’s Sketchpad during two class days in the first or second quarter.) 4 10 Congruent triangles with proof 5 10 Special segments in triangles; some construction (In past years, we have gotten through some of Chapter 6 before December final exams. In ’09-­‐
’10, we got through Ch. 5, and some classes did some of Ch. 6.) 6 12 parallelograms, area 7 5 Rotations, reflections, translations, vectors (7.1-­‐7.4) 8 11 Similar triangles, proportion 9 15 Pythagorean Theorem, Special Rt. Triangles, Basic Trig Trig Supp. 6 Radians, Unit Circle, Exact trig values (Some years, some teachers have included 9.6 & 9.7 with the trig supplement and assessed them together, so Ch. 9 was 13 days, and the trig supp was 8 days. Also, we have typically given a common cumulative quiz on the midterm day that ends the third quarter.) 10 7 De-­‐emphasize segments/geom. mean in circle 11 7 Time crunch? Keep problems fairly simple 12 7 Give all formulas for test but emphasize understanding (In recent years, some have combined Ch. 11 and Ch. 12 for assessments.) Part 4: Geometry Fall Final Exam 2011
70 points
A.M.D.G.
Multiple Choice Portion – 1 point each
1. True or False: Skew lines can be
perpendicular.
[A] true
[B] false
3. Find the slope of the line given the
points 6, −7 and 9, −9 .
(
)
(
[A] 0
2
[C] −
3
)
[B] undefined
3
[D] −
2
5. Which of the statements is FALSE,
given ΔABC ≅ ΔMNO ?
[A] AC ≅ MO
[B] CB ≅ ON
[C] ∠A ≅ ∠M
[D] CA ≅ NM
7. What additional information do you
need to prove ΔABC ≅ ΔADC by the SAS
Postulate?
Name________________________
December 2011
2. True or False: If two lines are perpendicular
to a third line, they are perpendicular to each
other.
[A] true
[B] false
4. Classify ΔQVB by its side lengths.
V
[A]
[B]
[C]
[D]
8
Q
equilateral
scalene
isosceles
obtuse
B
8
6. Identify the property: If m∠ABC=m∠DOG
and m∠DOG= m∠MAP, then m∠ABC=
m∠MAP.
[A] reflexive
[B] symmetric
[C] transitive
[D] distributive
8. Given: CD is the perpendicular bisector
of HJ.
Which statement is false?
[A] CH ≅ CJ
C
[B] DH ≅ IJ
[A] AB ≅ AD
[B] ∠ABC ≅ ∠ADC
H
I
D
[C] BC ≅ DC
[C] ∠ACB ≅ ∠ACD
9. Which side lengths allow you to
J
[C] m∠CIJ = 90°
[D] ΔCIH ≅ ΔCIJ
10. Which statement is a true biconditional?
construct a triangle?
[A] 2, 3 and 7
[B] 5, 4 and 9
[C] 8, 2 and 4
[D] 6, 4 and 7
[A] Angles are congruent if and only if they are
vertical angles.
[B] Lines intersect if and only if they are not
parallel.
[C] Lines are perpendicular if and only if they
intersect to form right angles.
[D] Geometry is enjoyable if and only if you
are a math teacher.
11. In the diagram, use the given information to find the value of x.
CU = QT; UQ = QT
U
CU = 10x – 6
C
UQ = 7x + 2
[A] 8/3
[B] 4/3
[C] 10.8
T
Q
[D] none of these
12. Refer to figure below.
Given:
AF ≅ FC , ∠ABE ≅ ∠EBC. Which segment is the median?
B
G
A
D E
F
C
[A]
BD
[B]
BE
[C] BF
[D] GF
13. Which statement is FALSE based on the diagram?
A [A]
[B]
[C]
[D]
Lines h and i are parallel.
Points D and G are collinear.
A, D, and K are coplanar.
Points

FK intersects plane J at point K.
\$ Gdno orj joc`m r\tn oj g\]`g Kg\i` J)
14. Which
equation is accurate based on the given information and diagram. A, B, and C are
]$ Gdno ji` joc`m r\t oj g\]`g gdi` h)
collinear.
^$ <m` K \i_ F ^jggdi`\m: <m` oc`t ^jkg\i\m:
_$ <m` E, B \i_ F ^jkg\i\m:
`$ Gdno ajpm kjdion oc\o \m` iji(^jggdi`\m)
Njgpodji5
[A] AB = 7x + 1
[B] AC = 10x − 3
[C] BD = 10x − 2
[D] Both B & C
\$ Kg\i` BDG' Kg\i` KAG' \hjib n`q`m\g joc`mn) <it ^jh]di\odji ja ocm`` ^jkg\i\m kjdion oc\o \m` ijo
^jggdi`\m rjpg_ ]` ^jmm`^o)
←
→
]$ AB jm \it ^jh]di\odji ja orj ja oc` g`oo`mn A, C jm B di \it jm_`m)
^$ T`n' oc`t gd` ji oc` n\h` gdi`)T`n' tjp i``_ ocm`` kjdion oj ^m`\o` \ kg\i`' nj \it orj jm ocm`` kjdion
\m` ^jkg\i\m)
_$ T`n' \it ocm`` kjdion \m` ^jkg\i\m)
15. Which of the following is an angle bisector in the diagram?

[A] EF

[B] AH

[C] EB

[D] EH
FREE RESPONSE PORTION. POINTS AS NOTED.
SHOW WORK.
2 points each
1. Describe the pattern below and find the next two terms.
−7, −1, 5,11,...
2. Find the area and perimeter of ΔABC .
3. This rectangle has a perimeter of 52 feet. Find CD and the area of the rectangle.
4. Write the third sentence using the Law of Syllogism:
If I get detention, then I won’t go to practice. If I don’t go to practice, then I won’t play in the game. 5. Given that m ⊥ n, find the values of x and y.
m
x° y°
25°
n
6. One side of an equilateral triangle measure 2 y + 3 units. If the perimeter of the triangle is
33 units, what is the value of y?
7. C is the centroid of ΔGHJ and CM = 8. Find HM and CH.
6 points each
8. Solve for x, y, and z in the following diagram. Explain your reasoning using
definitions/postulates/theorems.
x = _______ because of ________________
y = _______ because of ________________
z = _______ because of ________________
9. Write each form of the following conditional statement. For each form, determine if it is
true or false. If the answer is false, provide a counterexample.
Original: If I’m sick, then I don’t go to school.
Converse:______________________________________________
TRUE
TRUE / FALSE
Counerexample?
_______________________________________________________
Inverse:______________________________________________
TRUE / FALSE
Counerexample?
_______________________________________________________
TRUE / FALSE
Contrapositive:___________________________________________ Counerexample?
_______________________________________________________
2 points each
10. Given that m || n, find the value of x.
11. What value of p ensures that m || n?
(8x–2)°
m
m
(p+20)°
(2p–98)°
n
n
(5x+13)°
5 points each
12. G, E, and F are the midpoints of the sides of ΔABC .
If EF = 6 and FG = 5, find AB.
If EG =
5
x − 3 and AC = 3x + 8 find x.
2
If m∠GCF = 36° , find m∠BGE . EXPLAIN.
13. Complete the proof using congruent triangles.
Given: P is the midpoint of RS
TR  SQ
Prove: TR ≅ SQ
14. Write an indirect proof.
Given: m∠1 = 64° and m∠115°
j
Prove: l is not  to m
1
l
2
m
3 points each
15. Find m∠K .
16. Find m∠Q .
R
(2x+15)°
x°
P
Q
17. Mark the figure with the given information and find the following.
Given: FGHJ is a parallelogram, m∠JHG = 68°, JH = 34, FK = 19.
G
F
K
H
J
A. Find m∠FJH.
B. Find FH.
C. Find FG.
1 point
18. The perpendicular bisectors of ΔXYZ intersect at W .
If WX = 10 and AX = 8 , find WZ.
Y
B
A
W
Z
X
Extra Credit (2 points): Use your compass and straightedge to bisect this angle.
Spring 2010 Final Exam
Geometry
Total Points: 76
Part I: Multiple Choice. 16 pts, 1 point each. Answer on the Scantron form.
Name:
2°, 7°
Date:
Good luck everyone!
1. Consecutive angles in a parallelogram are always ________.
[A] complementary angles
[C] congruent angles
[B] vertical angles
[D] supplementary angles
2. Which statement is true?
[A] All quadrilaterals are squares.
[B] All rectangles are squares.
[C] All rectangles are quadrilaterals. [D] All quadrilaterals are rectangles.
3. Which type of quadrilateral has no parallel sides?
[A] rhombus
[B] trapezoid
[C] kite
[D] rectangle
4. Mr. Jones has taken a survey of college students and found that 70 out of 76 students are liberal arts
majors. If a college has 7613 students, what is the best estimate of the number of students who are
liberal arts majors?
[A] 70,120
[B] 8266
[C] 41
[D] 7012
5. One way to show that two triangles are similar is to show that ______.
[A] two angles of one are congruent to two angles of the other
[B] two sides of one are proportional to two sides of the other
[C] a side of one is congruent to a side of the other
[D] an angle of one is congruent to an angle of the other
6. If the side lengths of a triangle are 7, 6, and 9, the triangle _____.
[A] is obtuse
[B] is right
[C] is acute
[D] cannot be formed
7. Find the equation of a circle with center (1, 3) and passes through (1, 6). [A] (x − 1) + (y − 3) = 3 [B] (x + 1) + (y + 3) = 9
[C] (x + 1) + (y + 3) = 3
2
2
2
2
2
2
[D] (x − 1) + (y − 3) = 9
2
2
8. Find the area of an equilateral triangle that has side length 8 in. [A] 8 [B] 8 3 [C] 16 [D] 16 3 9. The surface area of the right cone shown is _____. 2
2
[A] 44π in. [B] 36π in. [C] 16 33π in.2
2
[D] 112π in.
 . 10. If m∠ACB = 48° , find mAB
A112
[A] 24º [B] 48º 123°
C
[C] 96º [D]12º 80°
76°
B
x°
11. Find the value of x. [A] 31 [B] 32 [C] 33 [D] 34 12. An aquarium in a restaurant is a rectangular prism and measures 3.5 feet by 4 feet by 4 feet. What
is the volume of the aquarium?
[A] 56 cubic feet
[B] 19.5 cubic feet
[C] 11.5 cubic feet
[D] 48 cubic feet
13. Find the surface area of a sphere with a diameter of 12 cm. Express your answer in terms of π .
[A] 288π cm2
cm
[B] 144π
cm2
cm2
[C] 576π
[D] 36π
2
14. Which diagram shows a rotation of approximately –200° in standard position?
[A]
[B]
[C]
[D]
15. Which of the following shows a triangle and its reflection image in the x-axis?
[A]
[B]
[C]
[D]
y
y
y
x
16. ΔABC is a right triangle. AB = _____.
x
y
x
x
[A] 3 5
[B] 3 13
[C] 117
[D]
3 6
Part II: Free Response – 60 points
3 points each unless otherwise stated.
Centers of circle are shown with a point.
Use radical form for special triangles;
otherwise round to hundredths.
1. Refer to the figure below.
Given: UVWX is a parallelogram,
A. Find m
WVU.
B. Find m
XUV.
UY = 15, UX = 24, XW = 21.
C. Find UW.
2. Write a two column proof.
Given:
and
Prove:
is a parallelogram
Statement
Reason
3. Trapezoid ABCD has midsegment EF .
A
B
If AB = 6 and EF = 8 , find the length of DC .
E
F
If m∠EFB = 82° then m∠DCF =
D
C
4. Find m∠T in the diagram, if m∠R = 120° and m∠S = 80°
S
R
T
U
5. Find the value of x.
6. Graph
PQR with
described by the vector
7. Solve for x and y.
and
.
Graph
after the translation
A
12
D
35
y
E
x
5
B
C
40
8. Solve for the unknown side lengths and angles in the figure below.
AC = ______, CB = _______, m∠B = ________
9. Find the following values.
a) sinA = R
b) tanA = c) Find the P
. Q
10. Ruby wants to find the height of the tallest building in her city. She stands 480 feet away from the
building. There is a tree 32 feet in front of her, which she knows is 17 feet tall.
A How tall is the
building?
33°
12
C
B
11. The length of the diagonal of a square is 22 cm. What is the length of each side? Draw and label a
sketch first.
12. PR and PQ are tangent to the circle and PR = 2x + 15 and PQ = 5x − 6. Find the value of x. 13. If EF = 6 and DF = 8 , find the length of the diameter.  . Express your answer as a simplified fraction in terms of π. 14. Find the length of JK
J
120°
5
K
15. Find the volume of the right prism below. F
E
D
 = 93° and mML
 = 47° , find the value of x. J
16. If mJK
93°
K
x°
M
47° L
17. Find the volume of the right cone. Leave π 18. Find the surface area of the right cylinder.
Leave π in your answer.
in your answer. 19.
5π R
a) Convert
to degrees.
2
b) Convert 60º to radians. c) Find cot A from the diagram. 20. Make a sketch of a 135° angle in standard position for trigonometry and then find the
exact value of sec135°.