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CCSS Geometry PPS Curriculum Guide 2014-2015
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
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Introduction As a result of the ongoing implementation of the Common Core State Standards (CCSS) in Portland Public Schools, the high school Geometry course was aligned to the CCSS and had initial implementation during the 2013-2014 school year. At the high school level, the CCSS for Mathematics are organized by conceptual category (number and quantity, algebra, functions, geometry and statistics & probability). The high school Geometry course consists of standards from the categories Geometry and Statistics & Probability. In addition, the eight Standards for Mathematical Practices (see Common Core State Standards for Mathematics, p. 6-8) along with the focus on Modeling (see Common Core State Standards for Mathematics, p. 72-73) and strategic use of technology, are expected throughout a students’ classroom experience. This document is the result of work by a group of Teacher Leaders in PPS that worked over the course of two school years to continue the work of aligning high school courses to the CCSS. Their work was greatly informed by the work from Achieve that is represented in CCSS for Mathematics Appendix A: Designing High School Mathematics Courses. In addition to alignment of sections from the CPM textbook and CCSS Geometry Supplement to CCSS, this document also includes instructional notes, guidance for Learning Targets and specific examples & explanations. The Learning Targets are in a separate document posted on the 6-12 Math website, but this document provides a reference to the appropriate Measurement Topic(s) in the latter document. Listed in the instructional notes are suggested Standards for Mathematical Practice to focus on during the identified lesson(s). It is NOT intended for a lesson(s) to include all suggested Standards for Mathematical Practice listed. Instead, use this list to help determine one or possibly two Standards for Mathematical Practice to include with the lesson(s) or the entire unit. During planning, determine instructional practices that will provide opportunities for students to engage with the selected Standard for Mathematical Practice. Resources Used: • •
Common Core State Standards for Mathematics (including Appendix A) http://www.ksde.org/Default.aspx?tabid=4754 Correlation of CCSS High School Geometry to CPM Geometry Connections with CCSS Geometry Supplement http://www.cpm.org/teachers/CCSS_Supplements.htm
• •
Arizona Department of Education – Academic Content Standards http://www.azed.gov/standards practices/mathematics-standards/ Ohio Department of Education- Mathematics Model Curriculum http://education.ohio.gov/GD/Templates/Pages/ODE/ODEDetail.aspx?page=3&TopicRelationID=1704&ContentID=83475&Content=126 91
• • •
North Carolina Department of Education – Common Core Instructional Support Tools http://www.dpi.state.nc.us/acre/standards/common-core-tools/#unmath Illustrative Mathematics http://illustrativemathematics.org/ Smarter Balanced Assessment Consortium – Mathematics Content Specifications http://www.smarterbalanced.org/smarter-balanced-assessments/#item
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
3
G1: Transformations Chapter 1 Lesson Number and Title
Instructional Notes and Mathematical Practice Focus
Standard with Examples and Explanations
GC 1.2.1-1.2.4
Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g. translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle.
G.CO.1 Examples: • Have students write their own understanding of a given term. • Give students formal and informal definitions of each term and compare them. • Develop precise definitions through use of examples and non-examples. • Discuss the importance of having precise definitions.
Students may use geometry software and/or manipulatives to model transformations.
Distinguish between transformations that are rigid (preserve distance and angle measure-reflections, rotations, translations, or combinations of these) and those that are not (dilations or rigid motions followed by dilations).
Standards for Mathematical Practice MP#3 MP#5 MP#6 MP#7
G.CO.2 Use various technologies such as transparencies, geometry software, interactive whiteboards, and digital visual presenters to represent and compare rigid and size transformations of figures in a coordinate plane. Comparing transformations that preserve distance and angle to those that do not.
G.CO.3 Describe and illustrate how a rectangle, parallelogram, and isosceles trapezoid are mapped onto themselves using transformations. Calculate the number of lines of reflection symmetry and the degree of rotational symmetry of any regular polygon. Students may use geometry software and/or manipulatives to model transformations. Example: For each of the following shapes, describe the rotations and reflections that carry it onto itself:
G.CO.4
Examples: • Perform a rotation, reflection, and translation with a given polygon and give a written explanation of how each step meets the definitions of each transformation using correct mathematical terms. • Construct the reflection definition by connecting any point on the preimage to its corresponding point on the reflected image and describe the line segment’s relationship to the line of reflection (e.g., the line of reflection is the perpendicular bisector of the segment). Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
4 G.CO.5 Create sequences of transformations that map a geometric figure on to itself and another geometric figure. Draw a specific transformation when given a geometric figure and a rotation, reflection or translation. Predict and verify the sequence of transformations (a composition) that will map a figure onto another. Example: The triangle in the upper left of the figure below has been reflected across a line into the triangle in the lower right of the figure. Use a straightedge and compass to construct the line across which the triangle was reflected.
Supplemental Lesson 1.2.5
Supplement 1.2.5 located on PPS 6-12 Math website, https://sites.google.com/sit e/ppshighschoolmath/geo metry
Standards for Mathematical Practice MP#3 MP#5 MP#6 MP#7
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
5
G2: Lines & Angles Chapter 2 Lesson Number and Title GC 2.1.1 - 2.1.3
Instructional Notes and Mathematical Practice Focus Make sure to include problems 2-2, 2-11, 214 and 2-15 to address G.CO.6. In addition, problem 2-6 is a good precursor to 2.1.4 and gives students some work with G.CO.9. Some problems needed as back fill to cover CCSS 7.G.5 and 8.G.5
GC 2.1.4
Standards for Mathematical Practice MP#3 MP#5 MP#7 Standards for Mathematical Practice MP#2 MP#3 MP#5
Standard with Examples and Explanations G.CO.6 A rigid motion is a transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures. Example: Δ𝐴𝐵𝐶 has vertices 𝐴(−1, 0), 𝐵(4, 0), 𝐶(2, 6) a. Draw Δ𝐴𝐵𝐶 on a coordinate grid. b. Translate Δ𝐴𝐵𝐶 using the rule (𝑥, 𝑦) → (𝑥 − 6, 𝑦 − 5) to create Δ𝐴′𝐵′𝐶′. Record the new coordinate grid (using a different color if possible). 𝐴′ __ 𝐵′ __ 𝐶′ __ c. Rotate Δ𝐴′𝐵′𝐶′ 90°CCW using the rule (𝑥, 𝑦) → _______ to create Δ𝐴"𝐵"𝐶". Record the new coordinates below and add the triangle to your coordinate grid (using a different color if possible). 𝐴" __ 𝐵" __ 𝐶" __ d. Write ONE rule below that would change Δ𝐴𝐵𝐶 to Δ𝐴"𝐵"𝐶" in one step.
G.CO.10 Order statements based on the Law of Syllogism when constructing a proof. Interpret geometric diagrams by identifying what can and cannot be assumed. Students may use geometric simulations (computer software or graphing calculator) to explore theorems about triangles. Example: • Given that ∆ABC is isosceles, prove that ∆ABC ≅ ∆ACB.
GC 2.3.3
Portland Public Schools
This lesson has been moved to in between Chapters 4 & 5 to assess along with Trigonometry
CCSS Geometry Curriculum Guide
Revised June 2014
6
G3: Similarity Chapter 3 Lesson Number and Title GC 3.1.1 - 3.1.3
Instructional Notes and Mathematical Practice Focus Students may use geometric simulation software to model transformations. Students may observe patterns and verify experimentally the properties of dilations. Include an explicit discussion about dilation and scale factor connection.
GC 3.1.4
Standards for Mathematical Practice MP#3 MP#5 MP#7 Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence/similarity.
Standard with Examples and Explanations G.SRT.1 A dilation is a transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from the center by a common scale factor. Example: Apply a dilation by a factor of 2, centered at the point P to the figure below.
P ●
l a. In the picture, locate the images A’, B’, and C’ of the points A, B, C under this dilation. b. What do you think happens to the line l when we perform the dilation? c. What appears to be the relationship between distance A’B’ and distance AB? d. Can you prove your observations in part c? G.SRT.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L.
Standards for Mathematical Practice MP#3 MP#4 MP#6 MP#7 Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
GC 3.2.1 – 3.2.2
7 Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures.
G.SRT.2 Use the idea of dilation transformations to develop the definition of similarity. A similarity transformation is a rigid motion followed by a dilation.
Explicitly address similarity transformations.
G.SRT.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L.
G.SRT.3 Examples: • Are all right triangles similar to one another? How do you know? • What is the least amount of information needed to prove two triangles are similar? How do you know? • Using a ruler and a protractor, prove AA similarity.
Standards for Mathematical Practice MP#3 MP#7 GC 3.2.4
Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures.
G.SRT.2 Use the idea of dilation transformations to develop the definition of similarity. A similarity transformation is a rigid motion followed by a dilation. G.SRT.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L.
Explicitly address similarity transformations. Standards for Mathematical Practice MP#3 MP#7
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
8
G4: Trigonometry Chapters 4 & 5 Lesson Number and Title GC 4.1.1 - 4.1.4
Instructional Notes and Mathematical Practice Focus Critical to cover all four sections to make sure students develop an understanding of the connection between similarity and the trigonometric relationships.
Standard with Examples and Explanations G.SRT.6 Students may use applets to explore the range of values of the trigonometric ratios as θ ranges from 0 to 90 degrees. hypotenuse
opposite of
Adjacent to
Standards for Mathematical Practice MP#2 MP#6 MP#7 MP#8
opp hyp adj cos of θ = hyp opp tan of θ = adj sin of θ =
hyp opp hyp sec of θ = adj adj cot of θ = opp csc of θ =
Example: • Explain why the sine of x is the same regardless of which triangle is used to find it in the figure below.
G.SRT.8 Example: • Find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28° and the shadow of the tree is 50 ft.
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
9
GC 4.1.5
Standards for Mathematical Practice MP#1 MP#4 MP#5
G.SRT.8 Example: • Find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28° and the shadow of the tree is 50 ft.
GC 2.3.3
Standards for Mathematical Practice MP#3 MP#5
G.SRT.4 Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures. Examples: • Prove that if two triangles are similar, then the ratio of corresponding attitudes is equal to the ratio of corresponding sides. • How does the Pythagorean Theorem support the case for triangle similarity? View the video at the link and create a visual proving the Pythagorean Theorem using similarity. http://www.youtube.com/watch?v=LrS5_l-‐gk94
GC 5.1.1 – 5.1.2
To address G.SRT.7, include some additional problems like the examples listed in the far column. Create a table for sine and cosine of complementary angles similar to the table created in Chapter 4 for tangent. If students need more work with G.SRT.7, then include CCSS Supplement: Sine and Cosine of Complementary Angles Students may use graphing calculators or programs, tables, spreadsheets, or
Portland Public Schools
G.SRT.6 Students may use applets to explore the range of values of the trigonometric ratios as θ ranges from 0 to 90 degrees. hypotenuse
opposite of
Adjacent to
opp hyp adj cos of θ = hyp opp tan of θ = adj sin of θ =
hyp opp hyp sec of θ = adj adj cot of θ = opp csc of θ =
Example: • Explain why the sine of x is the same regardless of which triangle is used to find it in the figure below. CCSS Geometry Curriculum Guide
Revised June 2014
10 computer algebra systems to solve right triangle problems. Standards for Mathematical Practice MP#2 MP#6 MP#7 MP#8
G.SRT.7 Calculate sine and cosine ratios for acute angles in a right triangle when given two side lengths. Use a diagram of a right triangle to explain that for a pair of complementary angles A and B, the sine of angle A is equal to the cosine of angle B and the cosine of angle A is equal to the sine of angle B. Examples: • What is the relationship between cosine and sine in relation to complementary angles? • Construct a table demonstrating the relationship between sine and cosine of complementary angles. ◦ • Find the second acute angle of a right triangle given that the first acute angle has a measure of 39 . ◦ • Complete the following statement: If sin 30 = 1/2, then the cos______ = 1/2. G.SRT.8 Example: • Find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28° and the shadow of the tree is 50 ft.
GC 5.1.3 – 5.1.4
Students may use graphing calculators or programs, tables, spreadsheets, or computer algebra systems to solve right triangle problems.
G.SRT.8 Example: • Find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28° and the shadow of the tree is 50 ft.
Standards for Mathematical Practice MP#1 MP#4 MP#5
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
CCSS Supplement: Trig Formula for Area of Triangle
11 This lesson covers a (+) standard. Thus it is an optional lesson.
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.
With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles.
G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
OPTIONAL GC 5.3.2-5.3.5 OPTIONAL
These lessons cover (+) standards. Thus these are optional lessons.
Portland Public Schools
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). Example: • Tara wants to fix the location of a mountain by taking measurements from two positions 3 miles o apart. From the first position, the angle between the mountain and the second position is 78 . From o the second position, the angle between the mountain and the first position is 53 . How can Tara determine the distance of the mountain from each position, and what is the distance from each position?
CCSS Geometry Curriculum Guide
Revised June 2014
12
G5: Triangles & Quadrilaterals: Application & Proof Chapters 6 & 7 Lesson Number and Title GC 6.1.1 – 6.1.3
Instructional Notes and Mathematical Practice Focus Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures.
Standard with Examples and Explanations G.CO.7 A rigid motion is a transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures. Example: Are the following triangles congruent? Explain how you know.
Standards for Mathematical Practice MP#3 MP#7
G.SRT.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L.
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
13 GC 7.2.1 – 7.2.6
Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of G.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3
G.CO.8 List the sufficient conditions to prove triangles are congruent. Map a triangle with one of the sufficient conditions (e.g., SSS) onto the original triangle and show that corresponding sides and corresponding angles are congruent. Example: Josh is told that two triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 share two sets of congruent sides and one pair of congruent angles: 𝐴B is congruent to 𝐷𝐸, 𝐵𝐶 is congruent to 𝐸D, and angle 𝐶 is congruent to angle 𝐹. He is asked if these two triangles must be congruent. Josh draws the two triangles below and says, “They are definitely congruent because they share all three side lengths”! • Explain Josh’s reasoning using one of the triangle congruence criteria: ASA, SSS, SAS. • Give an example of two triangles 𝐴𝐵𝐶 and 𝐷EF, fitting the criteria of this problem, which are not congruent.
Make sure to include problem 7-96 to address G.CO.9.
Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures. .
Standards for Mathematical Practice MP#2 MP#3 MP#5
G.CO.10 Order statements based on the Law of Syllogism when constructing a proof. Interpret geometric diagrams by identifying what can and cannot be assumed. Students may use geometric simulations (computer software or graphing calculator) to explore theorems about triangles. Example: • Given that ∆ABC is isosceles, prove that ∆ABC ≅ ∆ACB. G.CO.11 Example: • Suppose that ABCD is a parallelogram, and that M and N are the midpoints of 𝐴𝐵 and 𝐶𝐷, respectively. Prove that MN =AD, and that the line 𝑀𝑁 is parallel to 𝐴𝐷. G.SRT.4 Examples: • Prove that if two triangles are similar, then the ratio of corresponding attitudes is equal to the ratio of corresponding sides.
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
14 • How does the Pythagorean Theorem support the case for triangle similarity? View the video at the link and create a visual proving the Pythagorean Theorem using similarity. http://www.youtube.com/watch?v=LrS5_l-gk94
CCSS Supplement: Pythagorean Theorem Proof
CCSS Supplement: Proportions in Triangles Proof
Portland Public Schools
Students may use geometric simulation software to model transformations to show congruence or similarity of figures. Standards for Mathematical Practice MP#2 MP#3 MP#5 Students may use geometric simulation software to model transformations to show congruence or similarity of figures. Standards for Mathematical Practice MP#2 MP#3 MP#5
G.SRT.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L. G.SRT.4 Examples: • Prove that if two triangles are similar, then the ratio of corresponding attitudes is equal to the ratio of corresponding sides. • How does the Pythagorean Theorem support the case for triangle similarity? View the video at the link and create a visual proving the Pythagorean Theorem using similarity. http://www.youtube.com/watch?v=LrS5_l-gk94 G.SRT.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L. G.SRT.4 Examples: • Prove that if two triangles are similar, then the ratio of corresponding attitudes is equal to the ratio of corresponding sides. • How does the Pythagorean Theorem support the case for triangle similarity? View the video at the link and create a visual proving the Pythagorean Theorem using similarity. http://www.youtube.com/watch?v=LrS5_l-gk94 G.SRT.5 Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L.
CCSS Geometry Curriculum Guide
Revised June 2014
15
G6: Coordinate Geometry Chapter 7 Lesson Number and Title
Instructional Notes and Mathematical Practice Focus
Standard with Examples and Explanations
GC 7.3.1 – 7.3.3
Make sure to include problem 7-105 to address G.CO.9.
G.GPE.4 Examples: • Use slope and distance formula to verify the polygon formed by connecting the points (-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram. • Prove or disprove that triangle ABC with coordinates A (- 1, 2), B (1, 5),
Students may use geometric simulation software to model figures and prove simple geometric theorems.
CCSS Supplement: Slope Proofs
Portland Public Schools
C (- 2, 7) is an isosceles right triangle. • Take a picture or find a picture that includes a polygon. Overlay the picture on a coordinate
plane (manually or electronically). Determine the coordinates of the vertices. Classify the polygon. Use the coordinates to justify the classification.
Standards for Mathematical Practice MP#2 MP#3 MP#5 MP#6
G.GPE.7 This standard provides practice with the distance formula and its connection with the Pythagorean Theorem. Use the coordinates of the vertices of a polygon graphed in the coordinate plane and use the distance formula to compute the perimeter. Use the coordinates of the vertices of triangles and rectangles graphed in the coordinate plane to compute the area. Examples: • Find the perimeter and area of a rectangle with vertices at C (-1, 1), D (3, 4), E (6, 0), F (2, -3). Round your answer to the nearest hundredth when necessary. • Calculate the area of triangle ABC with altitude CD, given A (-4, -2), B (8, 7), C (1, 8) and D (4, 4). • Find the area and perimeter of a real-world shape using a coordinate grid and Google Earth. Select a shape (yard, parking lot, school, etc.). Use the tool menu to overlay a coordinate grid. Use coordinates to find the perimeter and area of the shape selected. Determine the scale factor of the picture as related to the actual real-life view. Then find the actual perimeter and area.
Lesson 7.3.2 in Algebra Connections covers finding the equation of lines parallel or perpendicular to a given line that passes through a given point. Based on students’ mastery of this, may need to provide more
G.GPE.5 Lines can be horizontal, vertical, or neither. Students may use a variety of different methods to construct a parallel or perpendicular line to a given line and calculate the slopes to compare the relationships. Examples: • Find the equation of a line perpendicular to 3x + 5y + 15 through the point (-3. 2). • Find an equation of a line perpendicular to y = 3x - 4 that passes through (3, 4). • Verify that the distance between two parallel lines is constant. Justify your answer.
CCSS Geometry Curriculum Guide
Revised June 2014
16 practice. Standards for Mathematical Practice MP#3 MP#8 CCSS Supplement: Segments Divided Proportionally
Students may use geometric simulation software to model figures or line segments. Standards for Mathematical Practice MP#2 MP#7 MP#8
Portland Public Schools
G.GPE.6 Examples: • Given A (3, 2) and B (6, 11), • Find the point that divides the line segment AB two-thirds of way from A to B. • Find the midpoint of line segment AB. • For the line segment whose endpoints are (0, 0) and (4, 3), find the point that partitions the segment into a ratio of 3 to 2.
CCSS Geometry Curriculum Guide
Revised June 2014
17
G7: Circles Chapters 8 & 10 Lesson Number and Title GC 8.2.1 – 8.2.2
GC 8.3.1 – 8.3.2
Instructional Notes and Mathematical Practice Focus Standards for Mathematical Practice MP#3 MP#4 MP#6 MP#7
Standard with Examples and Explanations
Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course.
G.GMD.1 Explain the formulas for the circumference of a circle and the area of circle by determining the meaning of each term or factor.
The definition for radian will be addressed later in Advanced Algebra (F.TF.1 and G.C.5) For additional practice with finding the area of sectors, see Geometry Portland Public Schools
G.SRT.2 Determine that two figures are similar by verifying that angle measure is preserved and corresponding sides are proportional.
Examples: • Use the idea of a regular polygon with infinite sides to informally describe the formula for finding the area of a circle. • Use the diagram to give an informal argument for the formula for finding the area of a circle. (This concept was introduced in Grade 7).
G.C.5 Examples: • The amusement park has discovered that the brace that provides stability to the Ferris wheel has been damaged and needs work. The arc length of steel reinforcement that must be replaces is between the two seats shown below. If the sector area is 28.25 ft2 and the radius is 12 feet, what is the length of steel that must be replaced? Describe the steps you used to find your answer. CCSS Geometry Curriculum Guide
Revised June 2014
18 Connections Extra Practice: Lesson #15. http://www.cpm.org/tea chers/extraByCourseG C.htm
• Find the area of the sectors below. What general formula can you develop based on this information?
Standards for Mathematical Practice MP#3 MP#4 MP#5 MP#6 MP#7 GC 10.1.1 – 10.1.5
Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures. Standards for Mathematical Practice MP#1 MP#3 MP#4 MP#5 MP#6
G.MG.1 Focus on situations that require relating two- and three- dimensional objects. Apply the properties of geometric figures to comparable real-world objects (e.g., The spokes of a wheel of a bicycle are equal lengths because they represent the radii of a circle). G.C.2 Identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents. Describe the relationship between a central angle and the arc it intercepts. Describe the relationship between an inscribed angle and the arc it intercepts. Describe the relationship between a circumscribed angle and the arcs it intercepts. Recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right angle. Recognize that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Examples: • Given the circle below with radius of 10 and chord length of 12, find the distance from the chord to the center of the circle.
• Find the unknown length in the picture below.
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
CCSS Supplement: Similar Circles
19 Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures.
G.C.1 Using the fact that the ratio of diameter to circumference is the same for circles, prove that all circles are similar. Prove that all circles are similar by showing that for a dilation centered at the center of a circle, the preimage and the image have equal central angle measures. Examples: Show that two given circles are similar by stating the necessary transformations from C to D. • C: center (2, 3) radius 5 and D: center (-1, 4) radius 10
Standards for Mathematical Practice MP#3 MP#5 10.3.1 and CCSS Supplement: AC2 lesson 4.3.2 More Completing the Square
In Grade 8 the Pythagorean theorem was applied to find the distance between two particular points. In high school, the application is generalized to obtain formulas related to conic sections.
G.GPE.1 Examples: • Write an equation for a circle with a radius of 2 units and center at (1, 3). • Write an equation for a circle given that the endpoints of the diameter are (-2, 7) and (4, -8). 2 2 Find the center and radius of the circle 4x + 4y - 4x + 2y – 1 = 0.
Quadratic functions and the method of completing the square are studied in Algebra 1-2. The methods are applied here to transform a quadratic equation representing a conic section into standard form. Standards for Mathematical Practice MP#2 MP#3 MP#7 MP#8
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
20
G8: Geometric Modeling & Constructions Chapter 9 Lesson Number and Title CCSS Supplement: Modeling Density
Instructional Notes and Mathematical Practice Focus Students may use simulation software and modeling software to explore which model best describes a set of data or situation.
Standard with Examples and Explanations G.MG.2 Examples: • Wichita, Kansas has 344,234 people within 165.9 square miles. What is Wichita’s population density? • Consider the two boxes below. Each box has the same volume. If each ball has the same mass, which box would weight more? Why
A King Size waterbed has the following dimensions 72 in. X 84 in. X 9.5in. It takes 240.7 gallons of water to fill it, which would weigh 2071 pounds. What is the weight of a cubic foot of water? G.CO.12 Examples: • Construct a triangle given the lengths of two sides and the measure of the angle between the two sides. • Construct the circumcenter of a given triangle. • Construct the perpendicular bisector of a line segment. • You have been asked to place a warehouse so that it is an equal distance from the three roads indicated on the following map. Find this location and show your work. •
GC 9.2.1 – 9.2.4
Need to include problems 9-57 and 9-90 to address G.CO.13. For problem 9-57, explicitly include language about equilateral triangles. Make sure to include problem 9-54 to address G.CO.9.
In order to meet G.C.3, review 10.1.5 Math Notes and supplement with additional problems that cover inscribed and circumscribed circles, as well as properties of angles involving a Portland Public Schools
• •
Show how to fold your paper to physically construct this point as an intersection of two creases. Explain why the above construction works and, in particular, why you only needed to make two CCSS Geometry Curriculum Guide
Revised June 2014
21 quadrilateral inscribed in a circle. Standards for Mathematical Practice MP#5 MP#6
creases G.CO.13 Example: • Construct a regular hexagon inscribed in a circle. G.C.3 Construct an inscribed and circumscribed circle. Using definitions, properties, and theorems, prove properties of angles for a quadrilateral inscribed in a circle. Examples: • The following diagram shows a circle that just touches the sides of a right triangle whose sides are 5 units, 12 units, and 13 units long. Draw radius lines as in the previous task and find the radius of the circle in this 5, 12, 13 right triangle. Explain your work and show your calculations.
• Given the inscribed quadrilateral below prove that ∠B is supplementary to ∠D.
CCSS Supplement: Construction of a Tangent Line
This lesson covers (+) standards. Thus this is an optional lesson.
G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle.
OPTIONAL
Portland Public Schools
CCSS Geometry Curriculum Guide
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22
G9: Solids & Conics Chapters 9, 11 & 12 Lesson Number and Title GC 9.1.3
Instructional Notes and Mathematical Practice Focus Standards for Mathematical Practice MP#1 MP#2 MP#4 MP#5
Standard with Examples and Explanations G.GMD.3 Example: · Janine is planning on creating a water-based centerpiece for each of the 30 tables at her wedding reception. She has already purchased a cylindrical vase for each table. The radius of the vases is 6 cm, and the height is 28 cm. She intends to fill them half way with water and then add a variety of colored marbles until the waterline is approximately three-quarters of the way up the cylinder. She can buy bags of 100 marbles in 2 different sizes, with radii of 9 mm or 12 mm. A bag of 9 mm marbles costs $3, and a bag of 12 mm marbles costs $4. a. If Janine only bought 9 mm marbles how much would she spend on marbles for the whole reception? What if Janine only bought 12 mm marbles? b. Janine wants to spend at most dollars on marbles. Write a system of equalities and/or inequalities that she can use to determine how many marbles of each type she can buy. c. Based on your answer to part b., how many bags of each size marble should Janine buy if she has $180 and wants to buy as many small marbles as possible?
GC 9.1.4 - 9.1.5
Students may use simulation software and modeling software to explore which model best describes a set of data or situation.
G.MG.3 Examples: • Given one geometric solid, design a different geometric solid that will hold the same amount of substance (e.g., a cone to a prism). • This paper clip is just over 4 cm long. How many paper clips like this may be made from a straight piece of wire 10 m long?
GC 11.1.3 – 11.1.5
In Grade 8, students were required to know and use the formulas for volumes of cylinders, cones, and spheres.
G.GMD.1 Explain the formulas for the volume of a cylinder, pyramid and cone by determining the meaning of each term or factor. Cavalieri’s principle is if two solids have the same height and the same cross-sectional area at every level, then they have the same volume. Example: 2 • Explain why the volume of a cylinder is V = πr h.
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
23 Standards for Mathematical Practice MP#1 MP#2 MP#3 MP#4 MP#5
GC 11.2.2 – 11.2.3
Standards for Mathematical Practice MP#1 MP#3 MP#5
G.GMD.3 Missing measures can include but are not limited to slant height, altitude, height, diagonal of a prism, edge length, and radius. Examples: • Find the volume of a cylindrical oatmeal box. • Given a three-dimensional object, compute the effect on volume of doubling or tripling one or more dimension(s). For example, how is the volume of a cone affected by doubling the height? G.C.2 Examples: • Given the circle below with radius of 10 and chord length of 12, find the distance from the chord to the center of the circle.
• Find the unknown length in the picture below.
GC 12.1.1
Students may use geometric simulation software to model figures and create cross sectional views.
G.GMD.4 Examples: • Identify the shape of the vertical, horizontal, and other cross sections of a cylinder. • Identify the shape of the vertical, horizontal, and other cross sections of a rectangular prism.
Slices of rectangular prisms and pyramids were explored in Grade 7. In high school, the concept is extended to a wider class of solids. Students who eventually take calculus will learn how to compute volumes of solids of revolution by a method involving crosssectional disks. Standards for Mathematical Practice MP#4 MP#5 MP#7 Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
GC 12.1.2 and CCSS Supplement Focus/Diretrix Equation of a Parabola
24 Students may use geometric simulation software to explore parabolas. Standards for Mathematical Practice MP#2 MP#3 MP#7 MP#8
Portland Public Schools
G.GPE.2 Examples: • Write and graph an equation for a parabola with focus (2, 3) and directrix y = 1. • A parabola has focus (-2. 1) and directrix y = -3. Determine whether or not the point (2, 1) is part of the parabola. Justify your answer.
CCSS Geometry Curriculum Guide
Revised June 2014
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G10: Probability Supplemental Materials Lesson Number and Title CCSS Supplement: Probability 9.1.1
CCSS Supplement: Probability 9.1.2 9.1.3
Instructional Notes and Mathematical Practice Focus Use Supplement 9.1.1 with students as preassessment for understanding of the probability models. Use lessons 4.2.1-4.2.3 in Geometry Connections as needed to reteach model(s) students need to revisit. Students could use graphing calculators, simulations, or applets to model probability experiments and interpret the outcomes. Probability is introduced in Grade 7. The concepts of independence and conditional probability are the extended topics in domain S-CP. Standards for Mathematical Practice MP#1 MP#2 MP#4 MP#6 MP#7
Standard with Examples and Explanations No standards match from high school standards. These lessons cover standards that are CCSS Grade 7.
S.CP.1 Intersection: The intersection of two sets A and B is the set of elements that are common to both set A and set B. It is denoted by A ∩ B and is read ‘A intersection B’. • A ∩ B in the diagram is {1, 5} • This means: BOTH/AND
U
A
B 1 2 7 5 4 3
8
Union: The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and is read ‘A union B’. • A ∪ B in the diagram is {1, 2, 3, 4, 5, 7} • This means: EITHER/OR/ANY • Could be both Complement: The complement of the set A ∪B is the set of elements that are members of the universal set U but are not in A ∪B. It is denoted by (A ∪ B)’ • (A ∪ B)’ in the diagram is {8} Examples: • Describe the event that the summing two rolled dice is larger than 7 and even, and contrast it with the event that the sum is larger than 7 or even.
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
26 • Create a Venn diagram to display the information in the table. Describe the set of students who have a curfew but don’t do chores as a subset of the group.
CCSS Supplement: Probability 9.2.19.2.5
Standards for Mathematical Practice MP#1 MP#2 MP#3 MP#4 MP#5 MP#6 MP#7 MP#8
S.CP.7 Example: • In a math class of 32 students, 18 are boys and 14 are girls. On a unit test, 5 boys and 7 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student? S.CP.2 Define and identify independent events. Explain properties of Independence and Conditional Probabilities. Use appropriate probability notation for individual events as well as their intersection (joint probability). Calculate probabilities for events, including joint probabilities, using various methods (e.g., Venn diagrams, frequency table). Compare the product of probabilities for individual events (P (A) ● P (B) ) with their joint probability (P (A ∩ B) ). Understand that independent events satisfy the relationship P (A) ● P (B) = P (A ∩ B). Predict if two events are independent, explain reasoning and check. Examples: When rolling two number cubes: o What is the probability of rolling a sum that is greater than 7? o What is the probability of rolling a sum that is odd? o Are the events independent? Justify your response. S.CP.3 Define dependent events and conditional probability. Explain that conditional probability is the probability of an event occurring given the occurrence of some other event and give examples that illustrate conditional probabilities. Explain that for two events A and B, the probability of event A occurring given the occurrence !(!"#$%) of event B is: P (A\B) = and give examples to show how to use the formula. Determine if two events !(!)
are independent and justify the conclusion. Example: • Given the following Venn diagram, determine whether events A and B are independent.
Portland Public Schools
CCSS Geometry Curriculum Guide
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S.CP.4 Students may use spreadsheets, graphing calculators, and simulations to create frequency tables and conduct analyses to determine if events are independent or determine approximate conditional probabilities. Determine when a two-way frequency table is an appropriate display for a set of data. Collect data from a random sample. Construct a two-way frequency table for the data using the appropriate categories for each variable. Calculate probabilities from the table. Use probabilities form the table to evaluate independence of two variables. Pose a question for which a two-way frequency is appropriate, use statistical techniques to sample the population, and design an appropriate product to summarize the process and report the results. Examples: • Collect data from a random sample of students in your school on their favorite subject among math, science, history, and English. Estimate the probability that a randomly selected student from your school will favor science give that the student is in tenth grade. Do the same for other subjects and compare the results. • A two-way frequency table is shown below displaying the relationship between age and baldness. We took a sample of 100 male subjects and determined who is or is not bald. We also recorded the age of the male subjects by categories.
What is the probability that a man from the sample is bald, given that he is under 45? Are the events independent? Justify your answer. S.CP.5 Illustrate the concept of conditional probability using everyday examples of dependent events. Illustrate the concept of independence using everyday examples of independent events. Interpret conditional probabilities and independence in context. Examples: Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
28 • What is the probability of drawing a heart from a standard deck of cards on a second draw, given that a heart was drawn on the first draw and not replaced? Are these events independent or dependent? • At Johnson Middle School, the probability that a student takes computer science and French is 0.062. The probability that a student takes computer science is 0.43. What is the probability that a student takes French given that the student is taking computer science? • Use the data given in the table. Is owning a smart phone independent from grade level?
S.CP.6 Calculate conditional probabilities using the definition: ‘the conditional probability of A given B as the fraction of B’s outcomes that also belong to A’. Interpret the probability based on the context of the given problem. Students could use graphing calculators, simulations, or applets to model probability experiments and interpret the outcomes. Examples: • A teacher gave her class two quizzes. 30% of the class passed both quizzes and 60% of the class passed the first quiz. What percent of those who passed the first quiz also passed the second quiz? • A local restaurant asked 1000 people, “Did you cook dinner last night?” The results of this survey are shown in the table below.
Determine what the probability is of a person chosen at random from the 1000 surveyed. a. Cooked dinner last night b. Was a male and did not cook dinner c. Was a male d. Was a female and cooked dinner last night
Portland Public Schools
CCSS Geometry Curriculum Guide
Revised June 2014
