MATH GEOMETRY 9-12 - Elk Grove Unified School District · MATH GEOMETRY 9-12 Performance Objective Task Analysis Benchmarks/Assessment ... Which angles must be congruent in order

MATH GEOMETRY 9-12 Performance Objective Task Analysis Benchmarks/Assessment

*1. Students demonstrate understanding

by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

Students: 1. Demonstrate understanding and give

examples of the following: • Undefined terms • Postulates • Theorems

• Give an example of an undefined term, a postulate, and a theorem.

Write the definition of intersecting lines. Underline any undefined terms used in your definition. Using what you know about parallel lines cut by a transversal, show that the sum of the angles in a triangle is the same as the angle in a straight line, 180 degrees. ( FW)

2. Identify and use inductive and

deductive reasoning.

• Describe the difference between inductive and deductive reasoning, and give an example of each.

*2. Students write geometric proofs,

including proofs by contradiction.

Students: 1. Verify geometric assertions using • Two column proofs

• Fill in the missing statements and reasons for the following proof.

Prove: The opposite angles of a parallelogram are congruent.

Statements Reasons 1. Draw AC. 1. Two points determine a line. 2. 2. Given 3. AB || DC and 3. AD || BC 4. ∠ADC ≅∠CAB 4. and ∠DAC ≅∠BAC 5. 5. Reflexive Property 6. ∆ABD ≅∆CDA 6. 7. 7. CPCTC

*Power Standard FW = California Mathematics Framework

H:\DATA\WORD\MATH\S&B\GEOMETRY.DOC2/3/04

1


• Proof by contradiction

• Use an indirect proof to prove the

following: Prove: If two lines intersect, then they intersect in only one point.

Prove: If a figure is a triangle, then it cannot have more than one right angle. If C is the center of the circle in the figure below, prove that angle b has twice the measure of angle a. ( FW)

*3. Students construct and judge the

validity of a logical argument. This includes giving counter examples to disprove a statement.

Students: 1. Use conditional statements and/or

counterexamples to construct and judge the validity of a logical argument.

• Determine whether each conditional

statement is true or false. Explain how you know.

If you smoke cigarettes, then you increase your risk of heart disease or lung cancer. If a figure is a rectangle, then it is a square.

c b

a



2


If a point B is located such that AB = BC then point B is the midpoint of line segment AC.

• Determine whether each bi-conditional statement is true or false. If the statement is false, give a counter example.

x⋅y = 0 if and only if x = 0 or y = 0.

The contra positive of a statement is true if and only if the statement is true.

Three points are coplanar if and only if they are collinear.

Prove or disprove: Any two right triangles with the same hypotenuse have the same area. ( FW)

True or false: A quadrilateral is a rectangle only if it is a square. ( FW)

Suppose that all triangles that satisfy property A are right triangles. True or false: A triangle that doesn’t satisfy the Pythagorean theorem does not satisfy property A. ( FW)



3


*4. Students prove basic theorems

involving congruence and similarity.

Students: 1. Use deductive reasoning to prove basic

theorems.

• Use deductive reasoning to prove the

following theorems:

Supplements of congruent angles are congruent. If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides that include the angles are proportional, then the triangles are similar.

Suppose that triangle PRS is isosceles, with RP = PS. Show that if the segment PQ bisects the angle ∠ RPS, then RQ = QS. ( FW)

R Q

P

S



4


AB is a diameter of a circle centered at 0. CD AB. If the length of AB is 5, find the length of side CD. (CERT Assessment) ( FW)

Suppose that R and S are points on a circle. Prove that the perpendicular bisector of the line segment RS passes through the center of the circle. ( FW)

5. Students prove triangles are congruent

or similar and are able to use the concept of corresponding parts of congruent triangles.

Students: 1. Use basic theorems to prove the

congruence or similarity of triangles.

• Use the given information to complete

the proof. Given: PQ bisects ∠SPT SP ≅ PT Prove: ∆SPQ ≅ ∆TPQ

4 3

O D

C

B A

T

P

Q

S



5


• Use the given information to complete

the proof. Given: F is the midpoint of DH and EG. Prove: DE ≅ GH • Use the information to complete the

proof. Prove that ∆AUL~ ∆MST

E

H G

F

D

U

L

A 65°

35

37

M S

T 28

30 65°



6


In the figure below, the area of the shaded right triangle is 6. Find the distance between the parallel lines, L1 and L2. Explain your reasoning. (CERT Assessment) ( FW)

6. Students know and are able to use the

Triangle Inequality Theorem.

Students: 1. Know and use the Triangle Inequality

Theorem.

• You are enclosing a triangular

playground with a fence. You have measured two sides of the playground to be 80 feet and 150 feet. What is the maximum total length of fence that you need?

State the shortest and longest sides of the triangle.

6

8

L1

L2

65°

35°

B

A

C



7


Solve the inequality AB + AC > BC.

Using a geometric diagram, show that for any positive numbers a and b, a2 + b2 < a + b. ( FW)

*7. Students prove and use theorems

involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.

Students: 1. Prove and use properties of parallel

lines.

•

Which angles must be congruent in order for lines m and n to be parallel?

Which angles must be supplementary in order for lines o and p to be parallel?

If m∠7 = 38°, find the measure of the following: a) m∠5 b) m∠11 c) m∠12 d) m∠13 e) m∠2

13 14 110

8 7

5 6

42

3

1

p

n

m

o

15 16 12 9

B

A

x+2 x+3

3x+2 C



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Give a reason why p is parallel to q.

2. Prove and use properties of quadrilaterals.

• ABCD is a parallelogram. a) If m∠DAB = 65°, find m∠ABC b) If m∠CAB = 28°, find m∠DCA

Draw a parallelogram that satisfies each set of conditions or write not possible. a). The diagonals are perpendicular but not congruent. b). Two opposite angles are right angles, but the parallelogram is not a rectangle.

C

D

B

A

q p



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3. Prove and use properties of circles.

• Prove: If two chords of a circle (that

are not diameters) are congruent, then they are equidistant from its center.

Given: WX ≅ ST, ZY ⊥ WX, ZV ⊥ ST Prove: ZY ≅ ZV

On the diagram below, with distances as shown, prove that if x=y, then the lines L and M are parallel: ( FW)

Prove that if a diagonal of a parallelogram bisects an angle of a parallelogram, then the parallelogram is a rhombus. ( FW)

X

Y

S

V

T

Z



10

M

L y

x

2

2


Prove that if the base angles of a trapezoid are congruent, then the trapezoid is isosceles. ( FW) Prove that the figure formed by joining, in order, the midpoints of the sides of a quadrilateral is a parallelogram. ( FW)

*8. Students know, derive, and solve

problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.

Students: 1. Derive and solve problems involving

common, geometric ( two and three dimensional) figures.

• Perimeter • Area • Volume

• Find the surface area, lateral area, and

volume of the figures.

Find the circumference and area.

9

27 7.3

5.1

8.7

3



11

MATH Performance Objective Task Analysis Benchmarks/Assessment

Find the radius of : a) a circle that has a circumference

of 16π. b) a circle that has an area of 100 π. A string is wound symmetrically around a circular rod. The string goes exactly 1 time around the rod. The circumference of the rod is 4 cm and its length is 12 cm. Find the length of the string. What is the length of the string if it goes exactly 4 times around the rod? (TIMSS, adapted) ( FW)

9. Students compute the volumes and

surface areas of prisms, pyramids, cylinders, cones, and spheres.

Students: 1. Compute the volumes and surface

areas of prisms, pyramids, cylinders, cones, and spheres.

• Compute the volume and surface area

of the following figures.

8 ft 8 ft

3 ft 3 ft

2 cm

6 cm



12


A sphere of radius 1 is inscribed in a cylinder. Find the volume of the cylinder. ( FW) A right prism with a 4 inch height has a regular hexagonal base. The prism has volume 144 in3. Find the surface area of the prism. ( FW)

12 in

5 in

16 in

3 cm

10 cm

8cm



13


*10. Students compute areas of polygons

including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.

Students: 1. Compute areas of polygons including

rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.

• Compute the area of each figure.

3 in

10

7 m

4 m

13

12 ft

5 ft 3 ft

6 mm 20

5 in

8 in

4 cm



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The diagram below shows the overall floor plan for a house. It has right angles at three corners. What is the area of the house? What is the perimeter of the house? (CERT HS Standards) ( FW)

A trapezoid with bases of length 12 and 16 is inscribed in a circle of radius 10. The center of the circle lies inside of the trapezoid. Find the area of the trapezoid. ( FW)

50 ft.

40 ft. 31 ft.

38 ft.



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11. Students determine how changes in

dimensions affect the perimeter, area, and volume of common geometric figures and solids.

Students: 1. Determine how changes in dimensions

affect the perimeter, area, and volume of common geometric figures and solids.

• A 24 in. by 36 in. poster is to be

reduced by a scale factor of 2/3. What are the dimensions of the reduced poster?

The ratio of the volume of the large

cone below is 8 times greater than the volume of the smaller cone. What is the ratio of their surface areas? of their heights?

Brighto soap powder is packed in cube-shaped cartons. A carton measures 10 cm on each side. The company decides to increase the length of each edge of the carton by 10 per cent. How much does the volume increase? (TIMSS) ( FW)



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A developer makes a scale model of the tract house that she is going to build. Each inch of distance on the scale model corresponds to ten inches on the actual houses. If the scale model uses 2 square feet of roofing material, how much roofing material will be needed for one of the actual houses? ( FW)

*12. Students find and use measures of

sides, interior and exterior angles of triangles and polygons to classify figures and solve problems.

Students: 1. Find and use measures of sides,

interior and exterior angles of triangles and polygons to classify figures and solve problems.

• Solve for y.

Solve for x and y.

y° 52°

67°

y 8ft

6.93 ft 30°

120°

x



17


What values of x and y guarantee that each quadrilateral is a parallelogram? Justify your answer.

A regular polygon has exterior angles each measuring 10 degrees. How many sides does the polygon have? ( FW)

13. Students prove relationships between

angles in polygons using properties of complementary, supplementary, vertical, and exterior angles.

Students: 1. Prove relationships between angles in

polygons.

• Prove that m ∠ ABC = m ∠ BCD + m∠CDB.

120° (y + x)°

(4y - x)° a)

33° 75°

5y° (6x + 15)° b)

B D

C

A



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In the figure below, AB=BC=CD. Find an expression for the measure of angle b in terms of the measure of angle a, and prove that your expression is correct. ( FW)

Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. ( FW)

*14. Students prove the Pythagorean

Theorem.

Students: 1. Prove the Pythagorean Theorem.

• Use the diagram below to prove that a2

+ b2 = c2.

B

b

a

C

D A

c

c

c

c

b b

b

b

a

a

a

a



19


15. Students use the Pythagorean

Theorem to determine distance and find missing lengths of sides of right triangles.

Students: 1. Use the Pythagorean Theorem.

• A picket fence is to have a gate 42

inches wide. The gate is 54 inches tall. Find the length of a diagonal brace for the gate, to the nearest inch.

The measure of the sides of a right triangle are x + 9, x + 2 and x + 10. Find the value of x.

The lengths of two sides of right

triangle ∆STU are given. Find the length of the third side. a) ST = 3, TU = 4 b) ST = 7, SU = 10

U T

S



20


Find the length of the side labeled C in the figure below: ( FW) The bottom of a rectangular box is a rectangle with a diagonal whose length is 4 3 inches. The height of the box is 4 inches. Find the length of a diagonal of the box.

*16. Students perform basic

constructions with straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.

Students: 1. Perform basic constructions with

straightedge and compass using Euclidean restrictions.

• Use a straight edge and compass to:

Construct a congruent angle to ∠LMN.

C

7

5

2

L

M N



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Construct a line that is perpendicular to line P and passes through point Q.

Construct the bisector of ∠ABC.

Given a circle, using unmarked straightedge and compass, find the center of the circle. ( FW)

*17. Students prove theorems using

coordinate geometry, including the midpoint of a line segment, distance formula, and various forms of equations of lines and circles.

Students: 1. Prove theorems using coordinate

geometry.

• Prove the Distance Formula by finding

the length AB with the Pythagorean Theorem.

C(x2y1)

B(x2y2)

M

A(x1y1)

A

B C

Q



22


Prove that the relation r2 = x2 + y2 is true for all points P on circle O with center located at (0,0).

The vertices of a triangle PQR are the points P(1,2), Q(4,6), and R (-4,12). Which one of the following statements about triangle PQR is true? A. PQR is a right triangle with right

angle ∠ P. B. PQR is a right triangle with right

angle ∠ Q. C. PQR is a right triangle with right

angle ∠ R. D. PQR is not a right triangle. (TIMSS) ( FW)

r

O(0,0)

P(x,y)



23


*18. Students know the definitions of the

basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them, (e.g., tan(x) = sin(x)/cos(x), (sin (x))2 + (cos (x)) 2 =1).

Students: 1. Use basic trigonometric functions to

solve problems including ratios.

• Express each trigonometric ratio as a

fraction. a) sin A b) cos B c) tan A

2. Use elementary relationships between basic trigonometric functions.

• Use ∆CAT to show that each statement is true.

a) sinA = tanA cosA

b) sin2A + cos2A = 1

12

C

A

5 13

B

A

8 17

15C

T



24

MATH GEOMETRY 9-12

a 1

D

C

Performance Objective Task Analysis Benchmarks/Assessment

Below is a semicircle of radius 1, and center C. Use similar triangles to express the unknown length D in terms of the angle a: ( FW) If α is an acute angle and cosα = 1/3, find tanα. ( FW) Without using a calculator, determine which is larger, tan(60°) or tan(70°) and explain why. ( FW)

*19. Students use trigonometric functions

to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.

Students: 1. Use trigonometric functions to solve

for an unknown side of a right triangle.

• Solve for each variable (round to

nearest hundredth).

x

36°

14



25


Find the length of side C below, if ∠ a measures 70 degrees: ( FW)

20. Students know and are able to use

angle and side relationships in problems with special right triangles, such as 30-60-90 triangles and 45-45-90 triangles.

Students: 1. Use angle and side relationships in

special right triangle problems.

• Complete each table using exact

values only.

5

a C

8

cba

4 2

6 2

c b

a

45°

45°

8

5

z y x

6 3

x z

60° y

30°



26


Each side of the regular hexagon ABCDEF is 10 cm long. What is the length of the diagonal AC? (TIMSS) ( FW) Express the perimeter of the trapezoid ABCD in simplest exact form. Angle DAB measures 30 degrees and angle ABC measures 60 degrees.

*21. Students prove and solve problems

regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.

Students: 1. Prove and solve problems with circles

including chords, secants, tangents, and polygons.

• Use the figure to answer the

following questions. Given: VK is tangent to circle O at K, mUK = 82°, and mUT = 70°. a) Name an inscribed angle b) Name a secant line c) Name a minor arc d) Find m ∠USK e) Find m∠UOK f) Find m TSK g) Find m ∠OKV h) m∠V

4”

A

C D

6”

B



27


Complete the proof. Given : Figure as shown. Prove: m∠CAD = 1/2(mCD - mBE)

T

S

O

K

V

U 70°

82°

AE

B

D

C

Statements Reasons 1. m∠CBD = m∠CAD+ 1. ∠BDA 2. m∠CBD - m∠BDA = 2. m∠CAD 3. m∠CBD = 1/2mCD, 3. The measure of an inscribed angle m∠BDA = 1/2mBE half the measure of its intercepted arc. 4. 1/2mCD - 1/2mBE = 4. m∠CAD 5. m∠CAD = 5. Algebra; Symmetric Property of 1/2(mCD - mBE) Equality, Distributive Property.



28

MATH GEOMETRY 9-12

P

B

Q

10 cm 7cm A

Performance Objective Task Analysis Benchmarks/Assessment

Two circles with centers A and B as shown below, have radii of 7 cm and 10 cm respectively. If the length of the common chord is 8 cm, what is the length of AB? Show all your work. (TIMSS) ( FW) Use the perimeter of a rectangular hexagon inscribed in a circle to explain why π>3. (ICAS) ( FW)



29


*22. Students know the effect of rigid

motions on figures in the coordinate plane and space, including rotations, translations, and reflections.

Students: 1. Know the effect of transformations on

figures in a plane and in space.

• For the figure ABCDEFG, do each of

the following. a) Draw its reflection across the x-

axis and list the coordinates for each image point.

b) Draw its translation of (-5,1), and list the coordinates for each image point.

c) Draw the image of its 180° rotation.

d) Draw its reflection across the line y = x.

A translation maps A (2,-3) onto A1 (-3,-5). Under the same translation, find the coordinate of B1. the image of B (1,4). (TIMSS) ( FW)

G A

B

C

D E

F

-2 -3 -4 -5 -6 1 2 3 4 5 6 -1

-1

-2 -3 -4 -5 -6

1 2 3 4 5 6



30


The rectangle labeled Q cannot be obtained from the rectangle P by means of a: A. reflection (about an axis in the

plane of the page) B. rotation (in the plane of the page) C. translation D. translation followed by a

reflection (TIMSS) ( FW)

Q

P



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MATH GEOMETRY 9-12 - Elk Grove Unified School District · MATH GEOMETRY 9-12 Performance Objective Task Analysis Benchmarks/Assessment ... Which angles must be congruent in order