Download Instructional Guide: Semester 2 13-14

Geometry Semester 2
The focus of Geometry for second semester is on circles with and without coordinates, right triangle trigonometry and proof, and
applications of probability. Circles, with their quadratic algebraic representations connect algebra to geometry. The study of similarity
leads to an understanding of right triangle trigonometry and connects to quadratics through Pythagorean relationships. The link between
probability and data is explored through conditional probability and counting methods, including their use in making and evaluating
decisions. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that
students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem
situations.
The attached Instructional Guide is a recommendation and has organized the standards according to the units outlined in Appendix A of
the CCSS. The ordering of the units does not necessarily imply a particular sequence of instruction, but rather allows the “clustering” of related concepts.
Critical Areas of Study
Circles
In this unit students prove basic theorems about circles, such as a
tangent line is perpendicular to a radius, inscribed angle theorem, and
theorems about chords, secants, and tangents dealing with segment
lengths and angle measures. In the Cartesian coordinate system,
students use the distance formula to write the equation of a circle
when given the radius and the coordinates of its center, and the
equation of a parabola with vertical axis when given an equation of its
directrix and the coordinates of its focus. Given an equation of a
circle, they draw the graph in the coordinate plane, and apply
techniques for solving quadratic equations to determine intersections
between lines and circles or a parabola and between two circles.
Students develop informal arguments justifying common formulas for
circumference, area, and volume of geometric objects, especially
those related to circles.
Student Outcomes
Understand and apply theorems about circles.
Find arc lengths and areas of sectors of circles.
Translate between the geometric description and
the equation for a conic section.
Use coordinates to prove simple geometric
theorem algebraically.
Explain volume formulas and use them to solve
problems.
Right Triangle Trigonometry
Students apply similarity in right triangles to understand right triangle
trigonometry, with particular attention to special right triangles and
the Pythagorean Theorem.
Define trigonometric ratios and solve problems
involving right triangles.
Prove and apply trigonometric identities.
Applications of Probability
Building on probability concepts that began in the middle grades,
students use the languages of set theory to expand their ability to
compute and interpret theoretical and experimental probabilities for
compound events, attending to mutually exclusive events,
independent events, and conditional probability.
Students should make use of geometric probability models wherever
possible. They use probability to make informed decisions.
Understand independence and conditional
probability and use them to interpret data.
Use the rules of probability to compute
probabilities of compound events in a uniform
probability model.
Use probability to evaluate outcomes of decisions.
PLCs should be given the flexibility to organize 2 nd semester in the manner that each PLC determines to be most beneficial to developing
their students’ understanding/mastery of concepts that are required to be successful in the next mathematics course. For PLC planning
purposes, there will only be one Performance Task for second semester with a flexible window for implementing the task.
Formal Geometry 2 Instructional Guide Draft
Semester 2
Essential Question: How can I derive the equation of a circle or parabola using the Pythagorean theorem?
How can I use coordinates to prove simple geometric theorems algebraically?
How can I use the distance formula and Pythagorean theorem to compute distances and solve problems?
CCSS citation
Possible Resources
G.GPE.1 Derive the equation of a circle of given center and radius using Math Vision Project- Connecting Algebra to Geometry
the Pythagorean Theorem; complete the square to find the center and
Math Vision Project-Circles and other conics
radius of a circle given by an equation.
National Science Digital Library- Lessons
G.GPE.2 Derive the equation of a parabola given a focus and directrix.
Sketchpad Circle – GPE.1
G.GPE.4 Use coordinates to prove simple geometric theorems
Sketchpad Parabola – GPE.2
algebraically. For example, prove or disprove that a figure defined by
Equations of Circles 1 (MARS)– GPE.1
four given points in the coordinate plane is a rectangle; prove or
Human Conics (NCTM- Parabolas only) – GPE.1, 2
disprove that the point (1,√3) lies on the circle centered at the origin and Illustrative Math-Midpoint Miracle --GPE.4,5
Tilted Square – GPE.4
containing the point (0, 2).
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and Parallel Lines – GPE.5
Perpendicular Lines – GPE.5
uses them to solve geometric problems (e.g., find the equation of a line
Shmoop Explanation and Sample Problems – GPE.6
parallel or perpendicular to a given line that passes through a given
A Shade Crossed – GPE.7
point).
Finding Equations of Parallel and Perpendicular Lines (MARS)– GPE.5
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.★
Notes:
This unit connects geometry to algebra through the coordinate plane. Students should possess a solid understanding of the connection
between the Pythagorean theorem and the distance formula. The midpoint formula may be used to derive the section formula as illustrated
in the Shmoop example above. Students should have a variety of experiences applying these formulas in novel ways to build a robust
understanding of coordinate geometry and geometric proof.
Assessment resources for the entire semester can be found on Illustrative Mathematics, Nrich and Smarter Balanced.
Essential Questions: What properties of circles can I prove using similarity, angle relationships, and inscribed quadrilaterals?
How can I use the fact that all circles are similar to derive properties of arc lengths in terms of radii?
CCSS citation
G.C.1 Prove that all circles are similar. (CA Framework Example)
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.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.
Possible Resources
Math Vision Project-Circles from a geometric perspective
National Science Digital Library- Lessons
Illustrative Math- Right triangles inscribed in circles I-C.2
Illustrative Math– Mutually Tangent Circles-C.5
Cyclic quadrilaterals – C.3
Partly Circles – C.2
Circle Theorems – C.2
Folding Circles: Exploring Circle Theorems Through paper Folding
(NCTM) – C.1, 2
Circles in Squares – C.2,3
Circles in Triangles – C.2,3
Notes:
Geometer’s Sketchpad (or other mathematics technology) may be useful to illustrate some of the situations presented in C.2. Allow students
to make conjectures and test them against examples before divulging postulates and theorems. Emphasize similarities and differences
between the angle and segment theorems, it helps in committing them to memory (e.g. All angles are either congruent (central), ½ the
measure, sum, or difference, OR supplementary in the case of the opposite angles of an inscribed quadrilateral).
Accelerated
G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle.
Geometry 2 Instructional Guide (draft 12/5/2013)
Semester 2
Essential Questions: Where do the formulas for circumference and area of a circle come from?
How can I determine and utilize the formulas for the volume of a cylinder, pyramid, and cone?
What shape will result from projecting an object from two-dimensions to three-dimensions, and vice-versa?
What is the effect of a scale factor on the length, area, and volume of an object?
How can I estimate the relative size of angles and sides of a triangle and use these estimates to solve problems?
CCSS citation
Possible Resources
G.GMD.1 Give an informal argument for the formulas for the
Math Vision Project-Circles from a geometric perspective
circumference of a circle, area of a circle, volume of a cylinder, pyramid, National Science Digital Library- Lessons
and cone. Use dissection arguments, Cavalieri’s principle, and informal Illustrative Math- Area of a circle-GMD.1
limit arguments. (CA math framework example)
Illustrative Math- Circumference of a circle-GMD.1
G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and
Illustrative Math-Doctor's Appointment-GMD.3
LearnZillion -- Informal proof of circle area – GMD.1
★
spheres to solve problems.
LearnZillion – Volume of prisms/cylinders related to pyramids /cones –
G.GMD.5 Know that the effect of a scale factor k greater than zero on
GMD.1
length, area, and volume is to multiply each by 𝑘, 𝑘 , and 𝑘 ,
Growing Rectangles (Nrich)– GMD.5
respectively; determine length, area and volume measures using scale
Evaluating Statements About Enlargements (MARS) – GMD.3,5
factors. CA
G.GMD.6 Verify experimentally that in a triangle, angles opposite longer Volumes of Compound Objects (MARS) – GMD.3, 5
sides are larger, sides opposite larger angles are longer, and the sum of
any two side lengths is greater than the remaining side length; apply these
relationships to solve real-world and mathematical problems. CA
A.SSE.1 (b) Interpret complicated expressions by viewing one or more of
their parts as a single entity. For example, interpret P(1+r)^n as the
product of P and a factor not depending on P. (literal equations and
formulas).
Notes:
Students are expected to justify the formulas mentioned in GMD.1. GMD.3 is a modeling standard so some problems/tasks should be realworld in nature. You can incorporate SSE.1 into your lessons throughout by having students solve for specific values, like r in 𝐴 = 𝜋𝑟 , within
area and volume formulas.
Will be taught in Integrated III:
G.GMD.4 Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by
rotations of two-dimensional objects.
Geometry 2 Instructional Guide (draft 12/5/2013)
Semester 2
Essential Questions: How do I define trigonometric ratios in terms of side ratios and angle measures of similar right triangles?
How are the sine and cosine ratios of complementary angles related?
How can I use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems?
CCSS citation
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.★ (CA Framework Example)
G.SRT.8.1 Derive and use the trigonometric ratios for special right
triangles (30°,60°,90°and 45°,45°,90°). CA
Possible Resources
Math Vision Project- Similarity and Right triangle trigonometry
National Science Digital Library- Lessons
Illustrative Math-Shortest line segment from a point P to a line L-SRT.8
Illustrative Math-Ask the Pilot-SRT.8.1
Trigonometry Gets to Work (Nrich)– SRT.8
Trigonometry for Solving Problems (NCTM)– SRT.8
Hopewell Geometry – SRT.6,7,8
Notes:
This unit presents introductory trigonometry with a focus on the relationship between acute angles of right triangles and their side ratios.
Build on students understanding of slope as a ratio and similarity as a relationship that preserves proportional sides. Mathematics
technology is useful for demonstrating the connection between angle measurements and side ratios in right triangles.
Will be taught in integrated III:
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).
Geometry 2 Instructional Guide (draft 12/5/2013)
Semester 2
Essential Questions: How is conditional probability related to independent probability?
How can I use data analysis and probability to make predictions?
How can I compute the probabilities of compound events?
CCSS citation
Possible Resources
S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using
Math Vision Project- Probability
characteristics (or categories) of the outcomes, or as unions, intersections, or complements National Science Digital Library- Lessons
of other events (“or,” “and,” “not”).
Illustrative Math- The Titanic 1-CP.1,4,6
S.CP.2 Understand that two events A and B are independent if the probability of A and B
Illustrative Math-The Titanic 2-CP.2,3,4,5,6
occurring together is the product of their probabilities, and use this characterization to
Illustrative Math- Rain and Lightning-CP.2,3,5,7
determine if they are independent. (CA math framework example) (CA math framework
Illustrative Math- Cards and Independence-CP.2,3
example-dependent)
Illustrative Math-Lucky Envelopes-CP.3
S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and
Illustrative Math-How Do You Get to School?-CP.4,6
interpret independence of A and B as saying that the conditional probability of A given B is Illustrative Math-Coffee at Mom's Diner-CP.7
the same as the probability of A, and the conditional probability of B given A is the same as Combining probabilities – CP.2-7
the probability of B. (CA math framework example)
Conditional Probability – CP. 2-7
S.CP.4 Construct and interpret two-way frequency tables of data when two categories are
Modeling Conditional Probabilities 1 (MARS)– CP 1-5
associated with each object being classified. Use the two-way table as a sample space to
Modeling Conditional Probabilities 2 (MARS)– CP 1-5
decide if events are independent and to approximate conditional probabilities. For
Medical Testing (MARS) – CP.1-7
example, collect
Stick or Switch (NCTM) – CP.1-5
data from a random sample of students in your school on their favorite subject among
Explorations with Chance (NCTM) – MD.6
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 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.
S.CP.7 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.
Notes: Teach the following standards if you have time.
S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in
terms of the model.
S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a
game).
Geometry 2 Instructional Guide (draft 12/5/2013)
Semester 2
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Geometry 2 Instructional Guide (draft 12/5/2013)
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Geometry 2 Instructional Guide (draft 12/5/2013)
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(ACCESSED FROM PAGE 30 OF THE CA MATH FRAMEWORK: CHAPTER- INTEGRATED 2)
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Geometry 2 Instructional Guide (draft 12/5/2013)
Semester 2