Topic: Geometry |
Common Core Mathematics 9-12 |
| Cluster Experiment with transformations in the plane |
| | Grade 9-12 |
| | | G.CO.1. | Know precise definitions of angle, circle, perpendicular line, parallel
line, and line segment, based on the undefined notions of point, line,
distance along a line, and distance around a circular arc. |
| | | G.CO.2. | Represent transformations in the plane using, e.g., transparencies
and geometry software; describe transformations as functions that
take points in the plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to those
that do not (e.g., translation versus horizontal stretch). |
| | | G.CO.3. | Given a rectangle, parallelogram, trapezoid, or regular polygon,
describe the rotations and reflections that carry it onto itself. |
| | | G.CO.4. | Develop definitions of rotations, reflections, and translations in terms
of angles, circles, perpendicular lines, parallel lines, and line segments. |
| | | G.CO.5. | Given a geometric figure and a rotation, reflection, or translation,
draw the transformed figure using, e.g., graph paper, tracing paper, or
geometry software. Specify a sequence of transformations that will
carry a given figure onto another. |
| Cluster Understand congruence in terms of rigid motions |
| | Grade 9-12 |
| | | G.CO.6. | Use geometric descriptions of rigid motions to transform figures and
to predict the effect of a given rigid motion on a given figure; given
two figures, use the definition of congruence in terms of rigid motions
to decide if they are congruent. |
| | | G.CO.7. | Use the definition of congruence in terms of rigid motions to show
that two triangles are congruent if and only if corresponding pairs of
sides and corresponding pairs of angles are congruent. |
| Cluster Understand similarity in terms of similarity transformations |
| | Grade 9-12 |
| | | G.SRT.1. | Verify experimentally the properties of dilations given by a center and
a scale factor:
a. A dilation takes a line not passing through the center of the
dilation to a parallel line, and leaves a line passing through the
center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio
given by the scale factor. |
| | | G.SRT.2. | Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality
of all corresponding pairs of angles and the proportionality of all
corresponding pairs of sides. |
| | | G.SRT.3. | Use the properties of similarity transformations to establish the AA
criterion for two triangles to be similar. |
| Cluster Prove theorems involving similarity |
| | Grade 9-12 |
| | | G.SRT.4. | Prove theorems about triangles. Theorems include: a line parallel to one
side of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity. |
| | | G.SRT.5. | Use congruence and similarity criteria for triangles to solve problems
and to prove relationships in geometric figures. |
| Cluster Define trigonometric ratios and solve problems involving right
triangles |
| | Grade 9-12 |
| | | G.SRT.6. | Understand that by similarity, side ratios in right triangles are
properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles. |
| | | G.SRT.7. | Explain and use the relationship between the sine and cosine of
complementary angles. |
| | | G.SRT.8. | Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems. |
| Cluster Apply trigonometry to general triangles |
| | Grade 9-12 |
| | | G.SRT.9. | (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by
drawing an auxiliary line from a vertex perpendicular to the opposite
side. |
| | | G.SRT.10. | (+) Prove the Laws of Sines and Cosines and use them to solve
problems. |
| | | G.SRT.11. | (+) Understand and apply the Law of Sines and the Law of Cosines
to find unknown measurements in right and non-right triangles (e.g.,
surveying problems, resultant forces). |
| Cluster Understand and apply theorems about circles |
| | Grade 9-12 |
| | | G.C.1. | Prove that all circles are similar. |
| | | G.C.2. | Identify and describe relationships among inscribed angles, radii,
and chords. Include the relationship 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.3. | Construct the inscribed and circumscribed circles of a triangle, and
prove properties of angles for a quadrilateral inscribed in a circle. |
| | | G.C.4. | (+) Construct a tangent line from a point outside a given circle to the
circle. |
| Cluster Find arc lengths and areas of sectors of circles |
| | Grade 9-12 |
| | | G.C.5. | Derive using similarity the fact that the length of the arc intercepted
by an angle is proportional to the radius, and define the radian
measure of the angle as the constant of proportionality; derive the
formula for the area of a sector. |
| Cluster Translate between the geometric description and the equation for a conic section |
| | Grade 9-12 |
| | | G.GPE.1. | Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and
radius of a circle given by an equation. |
| | | G.GPE.2. | Derive the equation of a parabola given a focus and directrix. |
| Cluster Use coordinates to prove simple geometric theorems algebraically |
| | Grade 9-12 |
| | | G.GPE.4. | 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
on the circle centered at the origin and containing the point (0, 2). |
| | | G.GPE.5. | Prove the slope criteria for parallel and perpendicular lines and use
them to solve geometric problems (e.g., find the equation of a line
parallel or perpendicular to a given line that passes through a given
point). |
| | | G.GPE.6. | Find the point on a directed line segment between two given points
that partitions the segment in a given ratio. |
| | | G.GPE.7. | Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula. |
| Cluster Explain volume formulas and use them to solve problems |
| | Grade 9-12 |
| | | G.GMD.1. | Give an informal argument for the formulas for the circumference of
a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use
dissection arguments, Cavalieri’s principle, and informal limit arguments. |
| | | G.GMD.3. | Use volume formulas for cylinders, pyramids, cones, and spheres to
solve problems. |
| Cluster Apply geometric concepts in modeling situations |
| | Grade 9-12 |
| | | G,MG.1. | Use geometric shapes, their measures, and their properties to describe
objects (e.g., modeling a tree trunk or a human torso as a cylinder). |
| | | G.MG.2. | Apply concepts of density based on area and volume in modeling
situations (e.g., persons per square mile, BTUs per cubic foot). |
| | | G.MG.3. | Apply geometric methods to solve design problems (e.g., designing
an object or structure to satisfy physical constraints or minimize cost;
working with typographic grid systems based on ratios). |
| Cluster Understand independence and conditional probability and use them
to interpret data |
| | Grade 9-12 |
| | | S.CP.1. | 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 occurring 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 same 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. |