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Geometry Standards
Transformations and Definitions
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1. know and define angle, circle, perpendicular line, parallel line, line segment, point, line, distance
along a line, distance around a circular arc, rotation, reflection, and translation (G.CO.1, 4)
 Students should be able to define geometric concepts and objects with precision, and to use
these definitions to reason.
 Students should understand a rotation as a motion along a circular arc through a certain
angle; a reflection as a motion along a path perpendicular to the axis of reflection; and a
translation as a motion along a vector. Students should understand the difference between rigid
and non-rigid motions.
2.
show the results of a sequence of transformations, including dilations, on a geometric figure and
specify a sequence of transformations to produce a given figure from another (G.CO.2–3, 5;
G.5.A–C)
 Students will develop a definition of congruence using rigid motions and similarity using
dilations.
 Students should be able to describe the effect of translations, rotations, reflections, and
dilations of points and polygons in words and algebraically.
 Students should be able to write, read, and apply mapping rules for a transformation; i.e.,
(𝑥, 𝑦) → (𝑥 + 3, 𝑦 − 2).
 Students should be able to produce dilations about arbitrary points and scale factors. For EOC:
dilations will be centered at the origin or a point on the figure to be dilated.
 Students should know that dilations leave the center as a fixed point; that line segments are
longer or shorter in the ratio given by the scale factor; and that lines not passing through the
center of dilation are mapped to parallel lines.
 Students should explore symmetry in the context of reflections and rotations; i.e., the line of
symmetry of a figure is the line about which one half of the figure can be reflected to produce
the whole figure.
3.
describe the intersections of lines in the plane and in space, of lines and planes, and of planes in
space (G.2.D)
 Students should know the possible number of intersections of lines and planes and be able to
determine which occurs in a given situation.
4.
describe the symmetries of two-dimensional figures (G.5.D)
 Students should be comfortable describing symmetries using the language of rigid motions.
Students should be able to use the symmetries of a figure to identify other characteristics of the
figure; e.g., an isosceles triangle has reflexive symmetry, which can explain why its base angles
are congruent.
Geometry Standards based on WA 2008 and CCSS
Last updated 8/20/2013
Congruence and Similarity
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5. define congruence in terms of rigid motions and use this definition to determine if two figures are
congruent (G.CO.6)
 Two figures are defined to be congruent if there is a sequence of rigid motions that carry one
figure exactly onto the other (i.e., superposition).
 In developing this, students may assume without proof that rigid motions preserve distance and
angle.
 Students should be able to describe a series of rigid motions that carry a figure onto another.
6.
define similarity in terms of similarity transformations and use this definition to determine if two
figures are similar (G.SRT.2)
 Two figures are defined to be similar if there is a sequence of transformations, including
dilations, that carry one figure exactly onto the other (i.e., superposition).
 In developing this, students may assume without proof that dilations preserve angle measure
and the proportionality of lengths.
 Students should be able to describe a series of transformations that carry a figure onto another.
7.
predict and verify the effect that changing one, two, or three linear dimensions has on perimeter,
area, volume, or surface area on two- and three-dimensional figures (G.6.D)
 Students should know that the resulting changes are exponential. The emphasis for this
standard is on algebraic solutions and understanding.
 For EOC: Questions will use polygons, cones, pyramids, cylinders, and prisms.
 For example: What happens to the volume of a rectangular prism if four parallel edges are
doubled in length?
Triangles
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8. use congruence and similarity criteria (ASA, SAS, SSS, AAS, AA, HL, CPCTC) for triangles to solve
problems and to prove relationships in geometric figures (G.CO.7–8; G.SRT.3, 5–6; G.3.B)
 Students should be comfortable drawing missing lines in figures to uncover hidden triangles.
9.
prove and apply theorems about triangles (G.CO.10; G.SRT.4; G.3.A; G.4.C)
 Students should be able to apply theorems and properties synthetically and analytically.
 The goal is for students to know a wide variety of theorems about triangles, and to use these
theorems to solve problems, reason and write proofs. The list below is not exhaustive, but
represents a minimal set of theorems students should be exposed to. Whenever possible, students
should develop proofs for each theorem.
 The measure of an exterior angle of a triangle is equal to the sum of the measures of the two
remote angles (WA 2008)
 The sum of the lengths of any two sides of a triangle is greater than the length of the third side
(WA 2008)
 In a triangle, the longest side is opposite the largest angle, and the converse (WA 2008)
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 The measures of the interior angles of a triangle sum to 180º (CCSS)
 A triangle is isosceles if and only if the base angles are congruent (CCSS, WA 2008)
 The segment joining the midpoints of two sides of a triangle is parallel to the third side and half
the length (CCSS)
 A line parallel to one side of a triangle divides the other two sides proportionally (CCSS)
 A triangle is equilateral if and only if it is equiangular (Holt)
10. find, and apply properties of, the centroid of a triangle (G.CO.10; G.SRT.4; G.3.A)
 Properties of the centroid include: medians of a triangle are concurrent at the centroid (WA
2008); the centroid divides the medians of the triangle in the ratio 2:1.
11. find, and apply properties of, the circumcenter of a triangle (G.CO.10; G.SRT.4; G.C.3; G.3.A)
 Properties of the circumcenter include: perpendicular bisectors of a triangle are concurrent at
the circumcenter (WA 2008); the circumcenter is equidistant from the vertices of the triangle (WA
2008), hence is the center of the circumscribed circle of the triangle; the position of the
circumcenter can be used to classify a triangle as acute, right, or obtuse.
 Students should be able to construct and find the equation of the circumscribed circle of a
triangle.
12. find, and apply properties of, the orthocenter of a triangle (G.CO.10; G.SRT.4; G.3.A)
 Properties of the orthocenter include: altitudes of a triangle are concurrent at the orthocenter
(WA 2008); the position of the orthocenter can be used to classify a triangle as acute, right, or
obtuse.
13. find, and apply properties of, the incenter of a triangle (G.CO.10; G.SRT.4; G.C.3; G.3.A)
 Properties of the incenter include: angle bisectors of a triangle are concurrent at the incenter
(WA 2008); the incenter is equidistant from the sides of the triangle (WA 2008), hence is the center
of the inscribed circle of the triangle.
 Students should be able to construct and find the equation of the inscribed circle of a triangle.
14. explain and use the trigonometric ratios (sine, cosine, and tangent) for acute angles to solve
problems (G.SRT.6–8; G.3.E)
 For example: A ladder is leaning against a wall to form a 63º angle with the ground. How many
feet above the ground is the point at which the ladder touches the wall? Or: Find the area (by
first finding the apothem) of a regular hexagon.
15. use the properties of special right triangles to solve problems (G.3.C)
 Students should know that the ratio of side lengths in a 30°–60°–90° triangle is 1: √3: 2 and the
ratio of side lengths in a 45°–45°–90° triangle is 1: 1: √2. Students should recognize that these are
shortcuts and be able to apply them as such when convenient.
16. prove and apply the Pythagorean theorem and its converse (G.SRT.4, 8; G.3.D)
 Students should be able to use the Pythagorean theorem and its converse to solve problems
involving polygons that can be broken down into right triangles. Students should also be able to
use the Pythagorean theorem in the context of a 3-dimensional figure by making a triangle with
the appropriate dimensions.
 One goal is for students to be exposed to a variety of proofs of the Pythagorean theorem.
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17. (+) derive and find the area of a triangle using the formula 𝐴 = 𝑎𝑏 sin 𝑐 (G.SRT.9)
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 It may be more appropriate for this standard to be included in a fourth-year course.
18. (+) prove and use the Laws of Sines and Cosines to find unknown measures in right and non-right
triangles (G.SRT.10–11)
 It may be more appropriate for this standard to be included in a fourth-year course.
Lines, Angles, Parallelograms, and Polygons
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19. prove and apply theorems about angles (G.CO.9; G.2.B)
 Students should be able to apply theorems and properties synthetically and analytically.
 The goal is for students to know a wide variety of theorems about angles, and to use these
theorems to solve problems, reason and write proofs. The list below is not exhaustive, but
represents a minimal set of theorems students should be exposed to. Whenever possible, students
should develop proofs for each theorem.
 Two lines cut by a transversal are parallel if and only if one of these angle pairs is congruent:
corresponding angles, alternate interior angles, alternate exterior angles (WA 2008)
 Two lines cut by a transversal are parallel if and only if consecutive angles are supplementary
(WA 2008)
 Vertical angles are congruent (CCSS)
 If two angles are complements or supplements of the same angle or congruent angles, then
the angles are congruent (WA 2008)
 A point in the interior of an angle lies on the angle bisector if and only if it is equidistant from the
sides of the angle (WA 2008)
20. prove and apply theorems about lines (G.CO.9; G.2.A)
 Students should be able to apply theorems and properties synthetically and analytically.
 The goal is for students to know a wide variety of theorems about lines, and to use these
theorems to solve problems, reason and write proofs. The list below is not exhaustive, but
represents a minimal set of theorems students should be exposed to. Whenever possible, students
should develop proofs for each theorem.
 A point lies on the perpendicular bisector of a line segment if and only if it is equidistant from
the endpoints of the segment (CCSS, WA 2008)
 Given a line and a point not on the line, there exists exactly one line through the point and
parallel to the given line (WA 2008)
 Given a line and a point not on the line, there exists exactly one line through the point and
perpendicular to the given line (WA 2008)
 Two lines are perpendicular if and only if their intersection forms right angles (WA 2008)
 If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular
(Holt)
 If two lines are both parallel to a third line, then they are parallel to each other (WA 2008)
 If two lines are perpendicular to a third line, then they are perpendicular to each other (WA
2008)
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 If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other parallel
line (WA 2008)
21. prove and apply theorems about parallelograms (G.CO.11; G.3.F; G.4.C)
 Students should know and apply definitions of parallelogram, square, rectangle, and rhombus.
 Students should be able to apply theorems and properties synthetically and analytically.
 The goal is for students to know a wide variety of theorems about parallelograms, and to use
these theorems to solve problems, reason and write proofs. The list below is not exhaustive, but
represents a minimal set of theorems students should be exposed to. Whenever possible, students
should develop proofs for each theorem.
 A quadrilateral is a parallelogram if and only if its opposite sides are congruent (CCSS, WA
2008)
 A quadrilateral is a parallelogram if and only if its opposite angles are congruent (CCSS, WA
2008)
 A quadrilateral is a parallelogram if and only if its consecutive interior angles are
supplementary (WA 2008)
 A quadrilateral is a parallelogram if and only if its diagonals bisect each other (CCSS, WA 2008)
 A parallelogram is a rectangle if and only if its diagonals are congruent (CCSS, WA 2008)
 A parallelogram is a rectangle if and only if at least one angle is a right angle (Holt)
 A parallelogram is a rhombus if and only if its diagonals are perpendicular (WA 2008)
 A parallelogram is a rhombus if and only if its diagonals bisect pairs of opposite angles (WA
2008)
 A parallelogram is a rhombus if and only if one pair of consecutive sides is congruent (Holt)
22. prove and apply theorems about polygons (G.3.G; G.4.C)
 Students should know and apply definitions of convex, concave, trapezoid, kite, and polygon.
 Students should be able to apply theorems and properties synthetically and analytically.
 The goal is for students to know a wide variety of theorems about polygons, with an emphasis
on quadrilaterals, and to use these theorems to solve problems, reason and write proofs. The list
below is not exhaustive, but represents a minimal set of theorems students should be exposed to.
Whenever possible, students should develop proofs for each theorem.
 If a figure is a trapezoid, then consecutive angles between a pair of parallel lines are
supplementary (WA 2008)
 If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid (Holt)
 If a trapezoid is isosceles, then each pair of base angles is congruent (Holt)
 If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid (Holt)
 A trapezoid is isosceles if and only if its diagonals are congruent (Holt)
 The midsegment of a trapezoid is parallel to each base and half the sum of the lengths (Holt)
 If a figure is a kite, then the diagonals are perpendicular (WA 2008)
 If a figure is a kite, then exactly one pair of opposite angles are congruent (Holt)
 The sum of one set of exterior angles of a polygon is 360° (WA 2008)
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 The sum of the interior angles of a polygon is 180(𝑛 − 2), where n is the number of sides of the
polygon (WA 2008)
23. prove and apply the slope criteria for parallel and perpendicular lines (G.GPE.5; G.2.A; G.4.A)
 Students should know that the theorems for the slopes of parallel and perpendicular lines are
biconditionals, and be comfortable moving in both directions.
 Students should be able to use algebra to find appropriate values for a variably-defined slope.
For example: Two lines have slopes 2𝑥 − 3 and 𝑥 + 10. For what value of 𝑥 are the lines parallel?
24. find the point on a directed line segment between two points that partitions the segment in a
given ratio (G.GPE.6)
 As written, this standard does not seem to require students to find the second endpoint given
one endpoint and the point of division.
 For example: A line segment with endpoints 𝐴(3, 6) and 𝐵(– 4, 5) is divided by a point 𝐶 such
that 𝐴𝐶: 𝐶𝐵 = 3: 4. Find the coordinates of 𝐶.
25. determine the coordinates of a point that is described geometrically (G.4.B)
 The goal is for students to find the coordinates, or the set of all possible coordinates, that satisfy
a given geometric relationship. This standard should be embedded within tasks throughout the
year.
 By “described geometrically” this standard means using geometrically-defined words and
concepts, such as: midpoint, circumcenter, orthocenter, parallelogram, bisector, etc.
 For example: Find the midpoint of the line segment with endpoints 𝐴(3, 6) and 𝐵(– 4, 5). Given
three vertices of a parallelogram, find the set of all possible coordinates for the fourth vertex. Find
the coordinates of circumcenter of a given triangle.
26. solve problems using the segment addition postulate
27. solve problems using the angle addition postulate
28. solve problems involving complementary and supplementary angles
 For example: Two angles are supplementary. One angle is 12 less than 3 times its
complement. Find the measure of the two angles.
29. solve problems using the midpoint formula (G.GPE.6; G.4.B)
 Students should be comfortable finding the midpoint of a segment and finding the second
endpoint of a segment given the midpoint and one endpoint.
 This could be derived as a special case of Standard 24 (partitioning a segment in a given ratio)
30. solve problems using the distance formula (G.GPE.7; G.3.D)
 Students should experience, but not necessarily be able to reproduce during an assessment,
the fact that: given one endpoint and the length of the segment, the set of all possible second
endpoints is a circle.
 Students should be able to derive the distance formula using the Pythagorean theorem.
Circles
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31. prove and apply theorems about circles (G.C.1–2, 5; G.3.H)
 Students should be able to apply theorems and properties synthetically and analytically.
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 The goal is for students to know a wide variety of theorems about circles and to use these
theorems to solve problems, reason, and write proofs. The list below is not exhaustive, but
represents a minimal set of theorems students should be exposed to. Whenever possible, students
should develop proofs for each theorem.
 Given two congruent circles (or the same circle), two chords are congruent if and only if they
are equidistant from the center of the circle (WA 2008)
 Given two congruent circles (or the same circle), two minor arcs are congruent if and only if
their corresponding chords are congruent (WA 2008)
 If the diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and
the diameter bisects the arc intercepted by the chord (WA 2008)
 If two secants intersect in the interior of a circle, then the sum of the measures of two vertical
angles formed is equal to the sum of the measures of the corresponding intercepted arcs (WA
2008)
 A line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of
tangency (CCSS, WA 2008)
 All circles are similar (CCSS)
 If two segments are tangent to a circle from the same external point, then the segments are
congruent (Holt)
 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary (CCSS, Holt)
 If a tangent and a secant intersect on a circle at the point of tangency, then the measure of
the angle formed is half the measure of its intercepted arc (Holt)
 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then
the measure of the angle formed is half the difference of the measures of its intercepted arcs
(Holt)
 If two chords intersect in the interior of a circle, then the products of the lengths of the
segments of the chords are equal (Holt)
 If two secants intersect in the exterior of a circle, then the product of the lengths of one secant
segment and its external segment equals the product of the lengths of the other secant segment
and its external segment (Holt)
 If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of
the secant segment and its external segment equals the length of the tangent segment squared
(Holt)
32. write equation the equation of a circle described algebraically or geometrically (G.GPE.1; G.4.D)
 Students should derive the equation for a circle using the Pythagorean theorem.
 Students should be comfortable using the equation of a circle to identify radius and center,
including equations which require completing the square.
 For example: Find the equation of the circle with a given line segment as a diameter. Find the
equation of the circle with a radius of 4 units and center (2, 3). Write the equation of the circle
with a given center and tangent line.
33. find the intersection of a circle and a line (G.4.D)
 For example: Find the points of intersection of the circle described by 𝑥 2 + 𝑦 2 = 4 and the line
𝑦 = 𝑥.
34. derive and apply formulas for arc length and area of a sector of a circle (G.C.5; G.6.A)
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 Students should be able to use these formulas and relationships to determine other measures of
the circle (radius, diameter, circumference, area, etc.)
 Students should know that arc length is proportional to the radius, and this fact should be used
to introduce the radian as a unit of measure. However, radians should not be a focus, as they will
be developed more fully in later courses.
35. prove and apply theorems about central, inscribed, and circumscribed angles (G.3.H, G.C.2)
 Students should be able to apply theorems and properties synthetically and analytically.
 The goal is for students to know a wide variety of theorems about circles and to use these
theorems to solve problems, reason, and write proofs. The list below is not exhaustive, but
represents a minimal set of theorems students should be exposed to. Whenever possible, students
should develop proofs for each theorem.
 Given two congruent circles (or the same circle), two arcs are congruent if and only if their
central angles are congruent (WA 2008)
 If two inscribed angles in a circle intercept the same arc, then they have the same measure
(WA 2008)
 An angle inscribed in a circle is a right angle if and only if its corresponding arc is a semicircle
and the longest side of a resulting triangle is a diameter of the circle (CCSS, WA 2008)
 The measure of an inscribed angle in a circle is half the measure of the intercepted arc (WA
2008)
 The measure of a central angle is equal to the measure of the intercepted arc (WA 2008)
 The measure of a circumscribed angle is equal to 180° minus the measure of the central angle
that forms the same arc
Constructions
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36. make geometric constructions (G.CO.12–13; G.C.3–4; G.2.C; G.3.I)
 The goal is for students to make formal geometric constructions and to understand and
produce the proofs for why those constructions give the desired results. Students should be
exposed to a variety of construction techniques, including: compass and straightedge, string,
reflective devices, paper folding, and dynamic geometry software. This standard should be
embedded within all units throughout the year.
 Construct a line parallel to a given line through a point not on the line (WA 2008)
 Construct a line perpendicular to a given line through a point on the given line (WA 2008)
 Construct a line perpendicular to a given line through a point not on the given line (WA 2008)
 Construct a line perpendicular to a given ray through the endpoint of the ray (WA 2008)
 Construct the perpendicular bisector of a line segment (CCSS, WA 2008)
 Construct the circumscribed circle for a given triangle (CCSS, WA 2008)
 Construct the inscribed circle for a given triangle (CCSS, WA 2008)
 Construct a diameter of a given circle (WA 2008)
 Locate the center of a given circle [given two chords] ([CCSS], WA 2008)
 Construct a line tangent to a given circle through a given point on the circle (WA 2008)
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 Construct lines tangent to a given circle through a given point outside the circle (CCSS, WA
2008)
 Copy a segment (CCSS)
 Copy an angle (CCSS)
 Bisect an angle (CCSS)
 Construct an equilateral triangle (CCSS)
 Construct a square (CCSS)
 Construct a regular hexagon inscribed in a circle (CCSS)
Three-dimensional Geometry
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37. give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, volume of a pyramid, volume of a cone, volume of a sphere, and use these
formulas to solve problems (G.GMD.1–3; G.6.C)
 Students should understand that the formulas for each were derived using logic and facts
about the figures. Students should be exposed to arguments using similarity transformations,
dissection, Cavalieri’s principle, and the informal idea of a limit. This standard should prepare
students for further development in later courses.
38. identify and analyze the shapes of two-dimensional cross-sections of three-dimensional objects
and identify three-dimensional objects generated by rotations of two-dimensional objects
(G.GMD.4; G.3.K)
 Students should be able to name the 3-dimensional object that a 2-dimensional cross-section
comes from; name the 2-dimensional figure formed by a cross section of a given 3-dimensional
object; and name the 3-dimensional object formed by rotating a 2-dimensional figure about
some axis.
39. describe prisms, pyramids, parallelepipeds, tetrahedra, and regular polyhedra in terms of their
faces, vertices, edges, and properties (G.3.J)
 Students should know Euler’s formula for convex polyhedra: 𝑉 − 𝐸 + 𝐹 = 2.
 For example: Describe symmetries of three-dimensional polyhedra and their two-dimensional
faces. Describe the lateral faces that are required for a pyramid to be a right pyramid with a
regular base. Describe the lateral faces that are required for a pyramid to be an oblique pyramid
with a regular base.
40. analyze and apply distance and angle measures on a sphere (G.6.B)
 Students should be exposed to non-Euclidean geometries and have the opportunity to see
what adopting a different set of axioms does to the set of true statements about figures within a
geometry. Spherical geometry should be motivated by thinking about the globe.
 This standard is not tested on the EOC.
Conic Sections
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41. derive and use the equation of a parabola given a focus and directrix (G.GPE.2)
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 For this course, the directrix should be parallel to a coordinate axis.
Probability
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42. describe events as subsets of a sample space using characteristics of the outcomes, including
unions, intersections, and complements (S.CP.1)
43. determine if two events are dependent or independent and use this to find probabilities (S.CP.2)
 A goal for this standard is that students interpret probabilities in a context.
 Students should know that two events are independent if 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵). Students
should be able to determine if two events are independent using this characterization.
 Students should know that two events are independent if 𝑃(𝐴|𝐵) = 𝐴 and 𝑃(𝐵|𝐴) = 𝐵.
44. determine the conditional probability of two events occurring (S.CP.3, 5–6)
 A goal for this standard is that students interpret probabilities in a context.
 Students should know and be able to use the formula 𝑃(𝐴|𝐵) = 𝑃(𝐴 ∩ 𝐵)/𝑃(𝐵). Students should
also recognize the Venn diagram model that leads to the formula; i.e., the fraction of 𝐵’s
outcomes that also belong to 𝐴.
 Students should be able to recognize and explain conditional probability in real-world
situations, such as the probability of developing lung cancer if you are a smoker, and the
converse.
45. use the addition rule to find the probability of two events occurring, and interpret the answer
(S.CP.7)
 A goal for this standard is that students interpret probabilities in a context.
 Students should recognize when two events are mutually exclusive and use the appropriate
addition rule to compute 𝑃(𝐴 ∪ 𝐵).
46. construct and interpret two-way frequency tables of data; use these tables to determine if
events are independent; and use these tables to approximate conditional probabilities (S.CP.4)
 A goal for this standard is that students interpret probabilities in a context.
 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.
47.
(+) use the multiplication rule to find the probability of two events occurring, and interpret the
answer (S.CP.8)
 A goal for this standard is that students interpret probabilities in a context.
 Students should recognize when two events are dependent and use the appropriate
multiplication rule to compute 𝑃(𝐴 ∩ 𝐵).
48.
(+) use permutations and combinations to compute probabilities for compound events (S.CP.9)
 A goal for this standard is that students interpret probabilities in a context.
 Students should know the difference between permutations and combinations, and be able to
decide which best represents a given situation.
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49.
(+) use probabilities to make fair decisions, and analyze decisions and strategies using
probability concepts (S.MD.6–7)
 The goal is for students to use probability to make decisions and defend them by
demonstrating thorough and mathematical analysis. This standard should be embedded within
all tasks throughout the unit.
 This standard will be developed further in Algebra II.
Proof
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50. distinguish between inductive and deductive reasoning (G.1.A)
 Students should be able to identify the use of inductive and deductive reasoning within a
context.
51. use inductive reasoning to make conjectures and to find a counterexample (G.1.B)
 The goal is for students to develop conjectures and determine if those conjectures are true (by
writing a proof) or false (by finding a counterexample).
 This standard should be embedded within all tasks throughout the year.
52. use deductive reasoning to prove that a valid geometric statement is true (G.1.C)
 Students should be able to write two-column, flowchart, and paragraph proofs. Students
should be able to use proof by contradiction.
 This standard should be embedded within all tasks throughout the year.
53. write the converse, inverse, and contrapositive of a valid proposition and determine their validity
(G.1.D)
54. identify errors or gaps in a mathematical argument and develop counterexamples to refute
invalid statements (G.1.E)
 Students should have experience determining if conjectures are true or false, and determining
why given proofs are (in)valid.
 This standard should be embedded within all tasks throughout the year.
55. use coordinates to prove simple geometric theorems algebraically (G.GPE.4)
 The goal is for students to engage in analytic geometry. This standard should be embedded
within all tasks throughout the year.
 For example: Prove or disprove that a figure defined by four given points is a rectangle. Prove
or disprove that the point (2, √3) lies on the circle centered at the origin and containing the point
(0, 2).
Measurement
I can...
56. use different, and appropriate, degrees of precision in measurement (G.6.E)
 The goal is for students to understand the limitations of their measurements in solving problems.
This standard should be embedded within all tasks throughout the year.
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 For example, it would not be appropriate to round an answer to the nearest thousandth if the
measurements were obtained by hand with a ruler.
57. apply estimation strategies to obtain reasonable measurements with appropriate precision for a
given purpose (G.6.E)
 The goal is for students to use approximation and estimation to solve problems and verify their
answers or reasoning. Students should be able to identify errors in computation or reasoning by
estimating the answer and comparing it to their actual answer.
 This standard should be embedded within all tasks throughout the year.
58. solve problems involving measurement conversions within and between systems and analyze
solutions in terms of reasonableness of solutions and appropriate units (G.6.F)
 Students should be able to use proportional relationships to convert between measurement
systems; i.e., dimensional analysis. It is not a goal for students to memorize conversion factors.
59. apply concepts of density based on area and volume in modeling situations (G.MG.2)
 For example: What is the population density of a location if 3 million people live in two square
miles?
60. use geometric shapes, their measures, and their properties to describe objects and solve
problems (G.MG.1, 3; G.6.C)
 One goal is for students to analyze complex shapes in terms of simpler figures for which students
know formulas. Another goal is for students to analyze non-mathematical objects using
geometric figures; i.e., modeling a tree trunk as a cylinder.
 Students should be comfortable modeling a problem situation with geometric figures, and
using geometric methods and properties to solve design problems.
 Problem situations may be purely mathematical or have an applied context.
 Students should be comfortable using cones, pyramids, cylinders, prisms, and spheres and their
associated formulas for surface area and volume.
 For example: A rectangle is 5 inches by 10 inches; find the volume of a cylinder generated by
rotating the rectangle about the 10 inch side. Design a structure satisfying certain physical
constraints while minimizing cost.
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