Geometry
Benchmark 4
2012-2013
Learner Objectives
Linked to GLEs and/or Standards
1. Determine the surface area and volume of geometric
figures, including cones, spheres and cylinders and judge
the reasonableness of numerical computations and their
results. (CLE: M2CG, MA2, 1.10, DOK 2, N3DG, MA1,
MO3.2, DOK3, ACT: MEA 601, MEA 702)
(Ch. 12 sec 2- 6)
(CCSS G-MG1 Unit 6)
(CCSS G-GMD 1 Unit 9)
(CCSS G-GMD 3 Unit 9)
Evidence Needed to Prove Mastery of the Objective
What are the expectations?
I expect students to be able to develop an informal
arguments for the formulas for volume of a cylinder,
pyramid, cone, and Cavalieri’s principle. I expect
students to be able to calculate the surface area and/or
volume for prisms, pyramids, cones, cylinders and
spheres. I also expect students to be able to solve
application type problems including those of composite
figures for these shapes, for example modeling a tree
trunk or human torso as a cylinder, applying concepts of
density based on area and volume as persons per square
mile. I also expect students to apply geometric methods
to solve design problems such as building a bridge
building or minimizing cost.
(CCSS – G-MG2 Unit 9)
(CCSS – G-MG3 Unit 9)
2. Draw representations of 3-dimensional objects from
different perspectives. (CLE: G4AG, MA2, 4.1, DOK 3)
Sec 12-1
Sec 8-5 Extension
(G-GMD 4 Unit 9 )
3. Use real numbers to solve problems involving area and
perimeter. Apply operations to real numbers, using
mental computation or paper and pencil calculations for
simple cases and technology for more complicated cases
in solving area and perimeter problems. Apply properties
of exponents to simplify area expressions and solve area
problems. (CLE: N1BG, MA5, 3.3, DOK 3, N2DG, MA1, 1.10,
DOK 2, ACT: MEA 301, MEA 302, MEA 401, MEA 402, MEA
I expect students to be able to draw a net of a 3dimensional object and vice versa. I also expect students
to draw isometric and/or orthographic drawings of 3dimensional shapes. Students should be able to
determine the number of faces, edges and vertices of a
polyhedron.
I expect the students to be able to identify the shapes of
two-dimensional cross-sections of three-dimensional
cross objects, and identify three-dimensional objects
generated by rotations of two-dimensional objects.
I expect students to be able to solve area, perimeter and
circumference problems using real numbers by mental
calculations as well as a calculator for more complicated
problems. This would include, square, circles, rectangles,
triangles, parallelograms, kites, trapezoids, and/or
rhombuses with real numbers as well as algebraic
expressions for parts of the figures which may include foil
type problems, rules of exponents (in relation to
multiplying like bases) and/or adding like terms.
501, MEA 502, MEA 503, MEA 702, PPF 703)
Sec 1-7 Ch. 11 sec 1,2,4,5
4. Make conjectures and solve area and perimeter
problems involving two dimensional objects represented
with Cartesian Coordinates. (CLE: G2AG, MA2, 3.3, DOK 3,
ACT: GRE 704, MEA 501)
Sec 1-7, Ch 11 sec 1,2,4, & 5
I expect students to be able to find the area of 2dimensional shapes represented in the Cartesian
coordinate system. Given vertices of a 2-dimensional
shape, I expect students to be able to plot the vertices of
the shape and find its area and/or perimeter.
(CCSS G-GPE 7 Unit 1)
Additional CCSS that are currently in our curriculum but not tested on this
placement that need to be incorporated
I also expect students to use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure defined by four given points in
the coordinate plane is a rectangle; prove or disprove that the point (1,3) lies of the circle
centered at the origin and containing the point (0,2).
Unit 8 -Circles
optional
G-C 2
Understand and apply theorems
about circles
Identify and describe relationships among
inscribed angles, radii, and chords. Include
relationships between central. Inscribed, and
circumscribed angles; inscribed angles on a
diameter are right angles; the radius of a circle
is perpendicular to the tangent where the
radius intersects the circle.
G-C 1
Prove that all circles are similar
G-C 4
(+)Construct a tangent line from a point outside
a given circle to the circle.
G-GPE 1
Derive the equation of a circle of given and
radius using the Pythagorean Theorem;
complete the square to find the center and
radius of a circle given by an equation.
Translate between the geometric
description and the equation for a
conic section
optional
G-C 5
Derive using similarity the fact that the length of
the are intercepted by an angle is proportional
to the radius, and define the radian measure of
Find arc lengths and areas of sectors the angle as the constant of proportionality;
of circles
derive the formula for the area of a sector
G-C 3
(+)Derive the equations of ellipses and
hyperbolas given the foci, using the fact that the
sum or differences of distances from the foci is
constant.
Translate between the geometric
description and the equation for a
conic section
Additional CCSS that are currently not in our curriculum and not tested on this
placement that need to be incorporated in future:
Unit 10 -Statistics
S-CP 1
Understand independence and
conditional probability and use
them to interpret data.*(model)
Describe events as subsets of a sample space
(the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions,
intersections, or complements of other events
("or", "and", "not").
S-CP 2
Understand that two events A and B are
independent if the probability of A and B
occuring together is the product of their
probabilities, and use this characterization to
determine if they are independent.
S-CP 3
Understand the conditional probability of A
given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the
conditional probability of A given B is the same
as the probability of A, and the conditional
probability of B given A is the sames as the
probability of B.
S-CP 4
Construct and interpret two-way frequency
tables of data when two categories are
associated with each object being classified.
Use the two-way table as a sample space to
decide if events are independent and to
approximate conditional probabilities. For
example, collect data from a random sample of
students in your school on their favorite subject
among math, science, and English. Estimate the
probability that a randomly selected student
from your school will favor science given that
the student is in tenth grade. Do the same for
other subjects and compare the results.
S-CP 5
Recognize and explain the concepts of
conditional probability and independence in
everyday language and everyday situations. For
example, compare the chance of having lung
cancer if you are a smoker with the chance of
being a smoker if you have lung cancer.
S-CP 6
S-CP 7
Use the rules of probability to
compute probabilities of compound
events in a uniform probability
model.*(model)
Find the conditional probability of A given B as
the fraction of B's outcomes that also belong to
A, and interpret the answer in terms of the
model.
Apply the Addition Rule, P(A or B)=P(A)+P(B)P(A and B), and interpret the answer in terms of
the model.