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 Geometry ● Unpacked Content
For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13.
What is the purpose of this document?
To increase student achievement by ensuring educators understand what the standards mean a student must know and be able to do completely and
comprehensively.
What is in the document?
Descriptions of what each standard means a student will know and be able to do. The “unpacking” of the standards done in this document is an effort to
answer a simple question “What does this standard mean that a student must know and be able to do?” and to ensure that description is helpful, specific
and comprehensive.
How do I send feedback?
We intend the explanations and examples in this document to be helpful, specific and comprehensive. That said, we believe that as this document is used,
teachers and educators will find ways in which the unpacking can be improved and made ever more useful. Please send feedback to us at
[email protected] and we will use your input to refine our unpacking of the standards. Thank You!
Just want the standards alone?
You can find the standards alone at www.corestandards.org.
Congruence
G-CO
Common Core Cluster
Experiment with transformations in the plane.
Common Core Standard
G-CO.1 Know precise definitions of
angle, circle, perpendicular line,
parallel line, and line segment, based
on the undefined notions of point,
line, distance along a line, and
distance around a circular arc.
Unpacking
What does this standard mean that a student will know and be able to do?
G-CO.1 Definitions are used to begin building blocks for proof. Infuse these definitions into proofs and other
problems. Pay attention to Mathematical practice 3 “Construct viable arguments and critique the reasoning of
others: Understand and use stated assumptions, definitions and previously established results in constructing
arguments.” Also mathematical practice number six says, “Attend to precision: Communicate precisely to others
and use clear definitions in discussion with others and in their own reasoning.”
Know that a point has position, no thickness or distance. A line is made of infinitely many points, and a line
segment is a subset of the points on a line with endpoints. A ray is defined as having a point on one end and a
continuing line on the other.
An angle is determined by the intersection of two rays.
A circle is the set of infinitely many points that are the same distance from the center forming a circular are,
measuring 360 degrees.
Perpendicular lines are lines in the interest at a point to form right angles.
Parallel lines that lie in the same plane and are lines in which every point is equidistant from the corresponding
point on the other line.
(Level I)
Ex. How would you determine whether two lines are parallel or perpendicular?
G-CO.2 Represent transformations in
the plane using, e.g., transparencies
and geometry software; describe
transformations as functions that take
points in the plane as inputs and give
other points as outputs. Compare
transformations that preserve distance
and angle to those that do not (e.g.,
translation versus horizontal stretch).
G-CO.3 Given a rectangle,
parallelogram, trapezoid, or regular
polygon, describe the rotations and
reflections that carry it onto itself.
G-CO.2 Describe and compare geometric and algebraic transformations on a set of points as inputs to produce
another set of points as outputs, to include translations and horizontal and vertical stretching.
Using Interactive Geometry Software perform the following dilations (x,y)→(4x,4y) (x,y)→(x,4y) (x,y)→(4x, y)
on the triangle defined by the points (1,1) (6,3) (2,13). Compare and contrast the following from each dilation:
angle measure, side length, perimeter.
G-CO.4 Develop definitions of
rotations, reflections, and translations
in terms of angles, circles,
perpendicular lines, parallel lines, and
line segments.
G-CO.4 Refer to 8.G where students have already experimented with the properties of rigid transformations and
dilations. Students should understand and be able to explain that when a figure is reflected about a line, the
segment that joins the preimage point to its corresponding image point is perpendicularly bisected by the line of
reflection. When figures are rotated, the points travel in a circular path over some specified angle of rotation.
When figures are translated, the segments of the preimage are parallel to the corresponding segments of the image.
G-CO.5 Given a geometric figure and
a rotation, reflection, or translation,
draw the transformed figure using,
e.g., graph paper, tracing paper, or
geometry software. Specify a
sequence of transformations that will
carry a given figure onto another.
G-CO.5 Using Interactive Geometry Software or graph paper, perform the following transformations on the
triangle ABC with coordinates A(4,5), B(8,7) and C(7,9). First, reflect the triangle over the line y=x. Then rotate
the figure 180° about the origin. Finally, translate the figure up 4 units and to the left 2 units.
Write the algebraic rule in the form (x,y)→(x’,y’) that represents this composite transformation.
G-CO.3 Describe the rotations and reflections of a rectangle, parallelogram, trapezoid, or regular polygon that
maps each figure onto itself, beginning and ending with the same geometric shape.
Given combinations of rotations and reflections, illustrate each combination with a diagram.
Where a combination is not possible, give examples to illustrate why that will not work (coordinate arguments).
Congruence
G-CO
Common Core Cluster
Understand congruence in terms of rigid motions.
Common Core Standard
Unpacking
What does this standard mean that a student will know and be able to do?
G-CO.6 Use geometric descriptions
of rigid motions to transform figures
and to predict the effect of a given
rigid motion on a given figure; given
two figures, use the definition of
congruence in terms of rigid motions
to decide if they are congruent.
G-CO.7 Use the definition of
congruence in terms of rigid motions
to show that two triangles are
congruent if and only if
corresponding pairs of sides and
corresponding pairs of angles are
congruent.
G-CO.6 Use descriptions of rigid motions to move figures in a coordinate plane, and predict the effects rigid
motion has on figures in the coordinate plane.
G-CO.6 Use this fact knowing rigid transformations preserve size and shape or distance and angle measure, to
connect the idea of congruency and to develop the definition of congruent.
(Level II)
Consider parallelogram ABCD with coordinates A(2,-2), B(4,4), C(12,4) and D(10,-2). Perform the following
transformations. Make predictions about how the lengths, perimeter, area and angle measures will change under
each transformation.
a. A reflection over the x-axis.
b. A rotation of 270° about the origin.
c. A dilation of scale factor 3 about the origin.
d. A translation to the right 5 and down 3.
Verify your predictions. Compare and contrast which transformations preserved the size and/or shape with those
that did not preserve size and/or shape. Generalize, how could you determine if a transformation would maintain
congruency from the preimage to the image?
G-CO.7 Use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only if
their corresponding sides and corresponding angles are congruent.
This standard connects with GSRT.3. Students should connect that two triangles are congruent if and only if they
are similar with a scale factor of one. Building on their definition of similarity, this means that a rigid
transformation will preserve the angle measures and the sides will change (or not change) by a scale factor of one.
(Level II)
Using Interactive Geometry Software or graph paper graph the following triangle M(-1,1), N(-4,2) and P(-3,5) and
perform the following transformations. Verify that the preimage and the image are congruent. Justify your
answer.
a. (x,y)→(x+2, y-6)
b. (x,y)→(-x,y)
c. (x,y)→(-y,x)
G-CO.8 Explain how the criteria for
triangle congruence (ASA, SAS, and
SSS) follow from the definition of
congruence in terms of rigid motions.
G-CO.8 Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence
criteria; ASA, SSS, SAS, AAS, and HL.
Students should connect that these triangle congruence criteria are special cases of the similarity criteria in GSRT.3.
ASA and AAS are modified versions of the AA criteria for similarity. Students should note that the “S” in ASA
and AAS has to be present to include the scale factor of one, which is necessary to show that it is a rigid
transformation. Students should also investigate why SSA and AAA are not useful for determining whether
triangles are congruent.
(Level II)
Andy and Javier are designing triangular gardens for their yards. Andy and Javier want to determine if their
gardens that they build will be congruent by looking at the measures of the boards they will use for the boarders,
and the angles measures of the vertices. Andy and Javier use the following combinations to build their gardens.
Will these combinations create gardens that enclose the same area? If so, how do you know?
a. Each garden has length measurements of 12ft, 32ft and 28ft.
b. Both of the gardens have angle measure of 110°, 25° and 45°.
c. One side of the garden is 20ft another side is 30ft and the angle between those two boards is 40°.
d. One side of the garden is 20ft and the angles on each side of that board are 60° and 80°.
e. Two sides measure 16ft and 18ft and the non-included angle of the garden measures 30°.
Students can make sense of this problem by drawing diagrams of important features and relationships and using
concrete objects or pictures to help conceptualize and solve this problem.
Congruence
G-CO
Common Core Cluster
Prove geometric theorems.
Using logic and deductive reasoning, algebraic and geometric properties, definitions, and proven theorems to draw conclusions. (Encourage multiple ways
of writing proofs, to include paragraph, flow charts, and two-column format).
These proof standards should be woven throughout the course. Students should be making arguments about content throughout their geometry experience.
The focus in the G-CO.9-11 is not the particular content items that they are proving. However, the focus is on the idea that students are proving geometric
properties. Pay close attention to the mathematical practices especially number three, “Construct viable argument and critique the reasoning of others.”
Common Core Standard
G-CO.9 Prove theorems about lines
and angles. Theorems include:
vertical angles are congruent; when a
transversal crosses parallel lines,
alternate interior angles are congruent
and corresponding angles are
congruent; points on a perpendicular
bisector of a line segment are exactly
Unpacking
What does this standard mean that a student will know and be able to do?
G-CO.9
In 8th grade, students have already experimented with these angle/line properties (8.G.5). The focus at this level is
to prove these properties, not just to use and know them.
(Level I)
• Prove that any point equidistant from the endpoints of a segment lies on the perpendicular bisector of
the line.
• A carpenter is framing a wall and wants to make sure his the edges of his wall are parallel. He is using
a cross-brace as show in the diagram below. What are several different ways he could verify that the
those equidistant from the segment’s
endpoints.
G-CO.10 Prove theorems about
triangles. Theorems include: measures
of interior angles of a triangle sum to
180º; base angles of isosceles
triangles are congruent; the segment
joining midpoints of two sides of a
triangle is parallel to the third side
and half the length; the medians of a
triangle meet at a point.
G-CO.11 Prove theorems about
parallelograms. Theorems include:
opposite sides are congruent, opposite
angles are congruent, the diagonals of
a parallelogram bisect each other, and
conversely, rectangles are
parallelograms with congruent
diagonals.
edges are parallel? Can you write a formal argument to show that these sides are parallel? Pair up
with another student who created a different argument than yours, and critique their reasoning. Did
you need to modify the diagram in anyway to help your argument?
G-CO.10
Using any method you choose, construct the medians of a triangle. Each median is divided up by the centroid.
Investigate the relationships of the distances of these segments. Can you create a deductive argument to justify
why these relationships are true? Can you prove why the medians all meet at one point for all triangles? Extension:
using coordinate geometry, how can you calculate the coordinate of the centroid? Can you provide an algebraic
argument for why this works for any triangle?
Using Interactive Geometry Software or tracing paper, investigate the relationships of sides and angles when you
connect the midpoints of the sides of a triangle. Using coordinates can you justify why the segment that connects
the midpoints of two of the sides is parallel to the opposite side. If you have not done so already, can you
generalize your argument and show that it works for all cases? Using coordinates justify that the segment that
connects the midpoints of two of the sides is half the length of the opposite side. If you have not done so already,
can you generalize your argument and show that it works for all cases?
G-CO.11
Jerry is laying out the foundation for a rectangular foundation for an outdoor tool shed. He needs ensure that it is
indeed fulfills the definition of a rectangle. The only tools he brought with him are pegs (for nailing in the ground
to mark the corners), string and a tape measure. Create a plan for Jerry to follow so that he can be sure his
foundation is rectangular. Justify why your plan works. Discuss your method with another student to make sure
your plan is error proof.
Connect this standard with G-CO.8 and use triangle congruency criteria to determine all of the properties of
parallelograms and special parallelograms.
Congruence
G-CO
Common Core Cluster
Make geometric constructions.
Create formal geometric constructions using a compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software.
Common Core Standard
Unpacking
What does this standard mean that a student will know and be able to do?
G-CO.12 Make formal geometric
constructions with a variety of tools
and methods (compass and
straightedge, string, reflective
devices, paper folding, dynamic
geometric software, etc.). Copying a
segment; copying an angle; bisecting
a segment; bisecting an angle;
constructing perpendicular lines,
including the perpendicular bisector
of a line segment; and constructing a
line parallel to a given line through a
point not on the line.
G-CO.12 Copy a congruent segment
G-CO.12 Copy a congruent angle.
G-CO.12 Bisect a segment
G-CO.12 Bisect an angle
G-CO.12 Construct perpendicular lines, including the perpendicular bisector of a line segment.
Using a compass and straightedge, construct a perpendicular bisector of a segment. Prove/justify why this process
provides the perpendicular bisector.
Given a triangle, construct the circumcenter and justify/prove why the process/construction gives the point which is
equidistant from the vertices.
Given a triangle, construct the incenter and justify/prove why the process/construction gives the point which is
equidistant from the sides of the triangle.
***It makes sense to combine this standard with G-CO.9 and 10 and have students make arguments about why
these constructions work.
G-CO.13 Construct an equilateral
triangle, a square, and a regular
hexagon inscribed in a circle.
G-CO.12 Construct a line parallel to a given line through a point not on the line.
G-CO.13 Using a compass and straightedge or Interactive Geometry Software, construct an equilateral triangle so
that each vertex of the equilateral triangle is on the circle. Construct an argment to show that your construction
method will produce an equilateral triangle. If your method and/or argument are different than another students’,
understand his/her argument and decide whether it makes sense and look for any flaws in their reasoning.
Repeat this process for a square inscribed in a circle and a regular hexagon inscribed in a circle.
(Level I)
Donna is building a swing set. She wants to countersink the nuts by drilling a hole that will perfectly circumscribe
the hexagonal nuts that she is using. Each side of the regular hexagonal nut is 8mm. Construct a diagram that
shows what size drill she should use to countersink the nut.
Similarity, Right Triangles, and Trigonometry
Common Core Cluster
G-SRT
Understand similarity in terms of similarity transformations.
Common Core Standard
G-SRT.1 Verify experimentally the
properties of dilations given by a
center and a scale factor.
a. A dilation takes a line not passing
through the center of the dilation
to a parallel line, and leaves a line
passing through the center
unchanged.
b. The dilation of a line segment is
longer or shorter in the ratio given
by the scale factor.
Unpacking
What does this standard mean that a student will know and be able to do?
G-SRT.1a Given a center of a dilation and a scale factor, verify using coordinates that when dilating a figure in a
coordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to it’s
image when the dilation is preformed. However, a segment that passes through the center does not change.
1
y
x
2
4
6
−1
−2
−3
Rule for the transformation is (x,y)→(⅓x,⅓y).
Comment on the slopes and lengths of the segments that make up the triangles. What patterns do you notice? How
could you be confident that this pattern would be true for all cases?
y
x
2
4
6
8
−2
−4
Rule for the transformation is (x,y)→(½x,½y).
Comment on the slopes and lengths of the segments that make up the triangles.
G-SRT.1b Given a center and a scale factor, verify using coordinates, that when performing dilations of the preimage, the segment which becomes the image, is longer or shorter based on the ratio given by the scale factor.
What patterns do you notice? How could you be confident that this pattern would be true for all cases?
G-SRT.2 Given two figures, use the
definition of similarity in terms of
similarity transformations to decide if
they are similar; explain using
similarity transformations the
meaning of similarity for triangles as
the equality of all corresponding pairs
of angles and the proportionality of all
corresponding pairs of sides.
In 8.G-3 and 8.G-4 students have already had experience with dilations and other transformations.
Note proportional reasoning in 6.RP and 7.RP
G-SRT.2 Use the idea of geometric transformations to develop the definition of similarity.
G-SRT.2 Given two figures determine whether they are similar and explain their similarity based on the
congruency of corresponding angles and the proportionality of corresponding sides.
Instructional note: The ideas of congruency and similarity are related. It is important for students to connect that
congruency is a special case of similarity with a scale factor of one. Therefore these similarity rules can be
expanded to work for congruency in triangles. AA similarity is the foundation for ASA and AAS congruency
theorems.
Knowing from the definition of a dilation, angle measures are preserved and sides change by a multiplication of
scale factor k.
Using a document camera and an LCD projector, Vladimir projects the following image the wall in his classroom.
He wants to know if the original image is similar to the one. How could he verify that they are similar. If the
preimage and image aren’t similar, what could be the reason? If the preimage and image are similar, how could he
find the scale factor?
G-SRT.3 Use the properties of
similarity transformations to establish
the AA criterion for two triangles to
be similar.
G-SRT.3 Use the properties of similarity transformations to develop the criteria for proving similar triangles by
AA, SSS, and SAS.
Connect this standard with standard G-SRT.4
(Level III)
Given that VMNP is a dilation of VABC with scale factor k, use properties of dilations to show that the AA criterion
is sufficient to prove similarity.
Similarity, Right Triangles, and Trigonometry
G-SRT
Common Core Cluster
Prove theorems involving similarity.
Common Core Standard
G-SRT.4 Prove theorems about
triangles. Theorems include: a line
parallel to one side of a triangle
divides the other two proportionally,
and conversely; the Pythagorean
Theorem proved using triangle
similarity.
Unpacking
What does this standard mean that a student will know and be able to do?
G-SRT.4
Connect this standard with G-SRT.3
(Level III)
Ex. Line segment AE is the distance across The French Broad River. John wants to find this distance so he paces
of ED and it is 60 paces, DC is 120 paces and EB is 80 paces. How could John use similar triangles to find the
width of the river? Can you calculate the distance of AB ? Create a formal argument to show that
€
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G-SRT.5 Use congruence and
similarity criteria for triangles to
solve problems and to prove
relationships in geometric figures.
€
€
AE AB
=
ED BC
G-SRT.4 Start with a right triangle and construct the altitude to the hypotenuse. This produces three similar right
triangles whose proportional relationships €
can lead to a proof of the Pythagorean theorem.
Reference students’ experience with the Pythagorean Theorem in 8th grade in 8.G.6-8. €
G-SRT.5 Connect with G.CO.8 and G.CO.11. and use triangle congruency criteria to determine all of the
properties of parallelograms and special parallelograms.
Sam claims that there is a SSSS congruency criteria for quadrilaterals? Do you agree or disagree? Justify your answer. If you disagree, can you provide a counterexample? If you agree, can you prove it?
Similarity, Right Triangles, and Trigonometry
G-SRT
Common Core Cluster
Define trigonometric ratios and solve problems involving right triangles.
Common Core Standard
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.
Unpacking
What does this standard mean that a student will know and be able to do?
G-SRT.6 Using corresponding angles of similar right triangles, show that the relationships of the side ratios are the
same, which leads to the definition of trigonometric ratios for acute angles.
(Level II)
Ex. There are three points on a line that goes through the origin, (5,12) (10,24) (15,36). Sketch this graph. How do
y
compare? Why does this make sense? Call the distance from the origin to each point (“r”). Find the
x
x
y
r for each point. Find the ratios of
and .
r
r
the ratios of
Students should use their knowledge of dilations and similarity to justify why these triangles are congruent. It is not
expected at this stage that they will know the Triangle Similarity Theorems (comes in course 3).
G-SRT.7 Show that the sine of an angle is the same as the cosine of the angle’s complement.
(Level II)
Draw a right triangle XYZ with <X as the right angle.
(i) If m<Y = 65°, what is m<Z? How do you know <Z once you know <Y? Is this always true?
(ii) What is the link between the sine of an angle and the “co”sine of the complementary angle?
(iii) Suppose that <M and <N are the acute angles of a right triangle.
(a) Use a diagram to explain/show why sin M = cos N.
(b) Why can the equation in (a) be re-written as sin M = cos (90 – M)? What would the
equivalent equation be for cos N? G-SRT.8 Use trigonometric ratios
and the Pythagorean Theorem to
solve right triangles in applied
problems.*
G-SRT.8 Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right
triangles.
(Level II)
a. An onlooker stands at the top of a cliff 119 meters above the water’s surface. With a clinometer, she spots
two ships due west. The angle of depression to each of the sailboats is 11° and 16°. Calculate the distance
between the two sailboats.
b. What is the distance from the onlooker’s eyes to each of the sailboats? What is the difference of those
distances? Why or why not?
Similarity, Right Triangles, and Trigonometry
G-SRT
Common Core Cluster
Apply trigonometry to general triangles.
Common Core
Standard
G-SRT.9 (+)
Derive the formula
A=½ ab sin (C) for
the area of a
triangle by drawing
an auxiliary line
from a vertex
perpendicular to
the opposite side.
Unpacking
What does this standard mean that a student will know and be able to do?
G-SRT.9 For a triangle that is not a right triangle, draw an auxiliary line from a vertex, perpendicular to the opposite side and derive the
formula, A=½ ab sin (C), for the area of a triangle, using the fact that the height of the triangle is, h=a sin(C).
Instructional note: The focus is on deriving the formula, not using it.
Show that for each of the following cases, A =
€
1
ab • cosC .
2
Before students derive this problem, pose the following problem to help them see the usefulness of this shortcut.
(Level II)
Ex. A wetland conservation specialist wants to calculate the area of a triangular river delta on the Roanoke River. She can measure the
lengths of sides EF and FG and the measure of ∠F with her surveying equipment. She finds that EF = 117m , FG = 162m and the
measure of ∠F = 112  . How can she calculate the area of this river delta?
€
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G-SRT.10 (+)
Prove the Laws of
Sines and Cosines
and use them to
solve problems.
Derive the law of sines and the law of cosines. Explain the connection between right triangle trigonometry and the law of sines and law of
cosines. Explain the connection between the ambiguous case and triangle congruency theorems (ASA, SAS, AAS and SSS).
Find the length of EF . Can you divide up the following triangle to create smaller right triangles to help you solve this problem? Can you
generalize this method to find a method that works for all cases? Once you have done this challenge yourself to find a different method.
Share your methods with another student. Analyze their argument to see if there are any reasoning errors.
€
See G-SRT.11 for application 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).
A surveyor standing at point C is measuring the length of a property boundary between two points located at A and B. Explain what
measurements he is able to collect using his transit. Create a plan for this surveyor to find the length of the boundary between A and B. How
does the surveyor use the law of sines and/or cosines in this problem? Will your process develop a reliable answer? Why or why not?
Circles
G-C
Common Core Cluster
Understand and apply theorems about circles.
Common Core Standard
Unpacking
What does this standard mean that a student will know and be able to do?
G-C.1 Prove that all circles are
similar.
G-C.1 Circles are all similar because each circle will map to another by some size transformation with a
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.
(Level III)
An archeologist dug up an edge piece of a circular plate. He wants to know what the original diameter of the plate
was before it broke. However, the piece of pottery does not display the center of the plate. How could he find the
original dimensions?
scale factor equal to the ratio of their radii.
Apply what you know about chords in circles. You are given two circles of diameters 24cm and 10cm. Locate two
chords of equal length, 8 cm for each circle. Compare their distances from the centers of the circles.
Using Interactive Geometry Software, investigate measurements of central angles and inscribed angles that
intercept the same arc. What is the relationship of these angle measures? Then, explain why inscribed angles on a
diameter are always 90°. Also, make an argument for why the opposite angles in a quadrilateral inscribed in a
circle are supplementary.
In a circular auditorium like the one shown below, there are lights positioned at points A and B. Currently, the light
at point B floods the entire auditorium contained in the inscribed ∠FBE and the light at point A lights the entire
auditorium contained in the inscribed angle ∠DAC . The lighting technician can adjust the angles for the lights at
points B and A (in other words, he can increase or decrease the flood span for the inscribed angles ∠DAC and
∠FBE ). How can he adjust the lights so that points D and F meet and points C and E meet?
€
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A discus athlete is performing in a local competition. Sketch an overhead view that shows the ball’s circular path,
the target, and the position at the moment that she releases the discus. Help her create a strategy for releasing the
discus and hitting the target.
G-C.3 Construct the inscribed and
circumscribed circles of a triangle,
and prove properties of angles for a
quadrilateral inscribed in a circle.
Given a triangle, construct the circumcenter and justify/prove why the process/construction gives the point which is
equidistant from the vertices.
Given a triangle, construct the incenter and justify/prove why the process/construction gives the point which is
equidistant from the sides of the triangle.
***It makes sense to combine this standard with G-CO.9 and 10 and have students make arguments about why
these constructions work.
Resource from Shell Centre (http://map.mathshell.org.uk/materials/download.php?fileid=764)
Listed under ‘Circles in Triangles.’
(Level III)
Jessica works at a daycare center and she is watching three rambunctious toddlers. One of the toddlers is in a crib
at point A, another toddler is in her high chair at point B and the third toddler is in a play-station at point C.
Where can Jessica position herself so that she is equidistant from each of the children? Construct an argument
using concrete referents such as objects, drawing, diagrams and actions.
In a flat field a team is constructing three straight trails for a cyclo-cross race. These three trails form a triangle.
The construction team wants to position the first-aid stand equidistant from each trail. Where should they build the
first-aid stand and why? Construct an argument using concrete referents such as objects, drawing, diagrams and
actions.
G-C.4 (+) Construct a tangent line
from a point outside a given circle to
the circle.
Construct an argument to show that opposite angles in an inscribed quadrilateral are supplementary.
G-C.4 Construct a tangent line from a point outside a given circle to the circle. Justify why this process works.
***It makes sense to combine this standard with G-CO.9 and 10 and have students make arguments about why
these constructions work.
Circles
G-C
Common Core Cluster
Find arc lengths and areas of sectors of circles.
Unpacking
What does this standard mean that a student will know and be able to do?
Common Core Standard
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.
G-C.5 Use similarity to derive the fact that the length of the arc intercepted by an angle is proportional to the
radius, identifying the constant of proportionality as the radian measure of the angle.
G-C.5 Find the arc length of a circle.
G-C.5 Using similarity, derive the formula for the area of a sector.
G-C.5 Find the area of a sector in a circle.
Since all circles are similar, the ratio of the
radius
will be the scale factor for any circle.
circumference
Two students use different reasoning to find the length of an arc with central angle measure of 45° in a circle with
radius=3cm. Compare the effectiveness of these two plausible arguments:
€
Svetlana says:

“I know that 360 = 2π radians
Since
1
3
45 
1
, the equivalent measure of the length of the arc will be (2π • 3) or π cm.”
 =
8
4
360
8
Michael says:
€
“ 360  = 2π radians therefore 1 =
€
€
€
# π &
3
(⋅ 3cm = π cm.”
$ 180 '
4
be 45  %
# π &
π
radians . 45  = 45  %
( radians . Therefore the measure of the arc will
$ 180 '
180
€
€
Can you use Svetlana’s
€ reasoning to find€an arc length associated with a central angle of 270º of a circle with a
radius of 5ft? Can you use Michael’s reasoning to find the arc length of a circle with a radius of 6m and a central
angle of 120º? Which method do you prefer and why?
Expressing Geometric Properties with Equations
G-GPE
Common Core Cluster
Translate between the geometric description and the equation for a conic section.
Unpacking
What does this standard mean that a student will know and be able to do?
Common Core Standard
G-GPE.1 Derive the equation of a
circle of given center and radius using
the Pythagorean Theorem; complete
the square to find the center and
radius of a circle given by an
equation.
G-GPE.1 Use the Pythagorean Theorem to derive the equation of a circle, given the center and the radius.
G-GPE.1 Given an equation of a circle, complete the square to find the center and radius of a circle.
Given a coordinate and a distance from that coordinate develop a rule that shows the locus of points that is that
given distance from the given point (based on the Pythagorean theorem).
If the coordinate of point H in the diagram below is (x,y) and the length of DH is 4 units. Can yo write a rule that
represents the relationship of the x value, the y value and the radius? Why is this relationship true? As point H
rotates around the circle, does this relationship stay true?
€
In the diagram below, circle D translated 4 units to the right to create circle E. Why is the equation of this new
2
€
2
2
2
€
€
2
2
circle ( x − 4 ) + y = 4 . Why is the equation for circle I x + ( y − 4 ) = 4 . Using similar reasoning, could you
right and equation for a circle with the center at (-4, 0) and a radius of 4? Center of (0, -4) and radius of 4? What is
the equation of a circle with center at (-8,11) and a radius of 5 ? Can you generalize this equation for a circle with
a center at (h,k) and a radius of r?
G-GPE.2 Derive the equation of a
parabola given a focus and a directrix.
Add limitations for course 2 and 3.
G-GPE.2 Given a focus and directrix, derive the equation of a parabola.
Parabola is defined as “the set of all points P in a plane equidistant from a fixed line and a fixed point in the plane.”
The fixed line is called the directrix, and the fixed point is called the focus.
(Level II)
Ex. Derive the equation of the parabola that has the focus (1, 4) and the directrix x=-5.
Ex. Derive the equation of the parabola that has the focus (2, 1) and the directrix y=-4.
Ex. Derive the equation of the parabola that has the focus (-3, -2) and the vertex (1, -2).
G-GPE.3 (+) Derive the equations of
ellipses and hyperbolas given the foci,
using the fact that the sum or
difference of distances from the foci
is constant.
G-GPE.3 Given the foci, derive the equation of an ellipse, noting that the sum of the distances from the foci to any
fixed point on the ellipse is constant, identifying the major and minor axis.
G-GPE.3 Given the foci, derive the equation of a hyperbola, noting that the absolute value of the differences of the
distances from the foci to a point on the hyperbola is constant, and identifying the vertices, center, transverse axis,
conjugate axis, and asymptotes.
Expressing Geometric Properties with Equations
G-GPE
Common Core Cluster
Use coordinates to prove simple geometric theorems algebraically.
Common Core Standard
Unpacking
What does this standard mean that a student will know and be able to do?
G-GPE.4 Use coordinates to prove
simple geometric theorems
algebraically. For example, prove or
disprove that a figure defined by four
given points in the coordinate plane is
a rectangle; prove or disprove that the
point (1, √3) lies on the circle
centered at the origin and containing
the point (0, 2).
G-GPE.4 Use the concepts of slope and distance to prove that a figure in the coordinate system is a special
geometric shape.
G-GPE.5 Prove the slope criteria for
parallel and perpendicular lines and
use them to solve geometric problems
(e.g., find the equation of a line
parallel or perpendicular to a given
line that passes through a given
point).
G-GPE.5 Use the formula for the slope of a line to determine whether two lines are parallel or perpendicular. Two
lines are parallel if they have the same slope and two lines are perpendicular if their slopes are opposite reciprocals
of each other. In other words the product of the slopes of lines that are perpendicular is (-1). Find the equations of
lines that are parallel or perpendicular given certain criteria.
(Level II)
Ex. The coordinates are for a quadrilateral, (3, 0), (1, 3), (-2, 1), and (0,-2). Determine the type of quadrilateral
made by connecting these four points? Identify the properties used to determine your classification. You must give
confirming information about the polygon. .
Ex. If Quadrilateral ABCD is a rectangle, where A(1, 2), B(6, 0), C(10,10) and D(?, ?) is unknown.
a. Find the coordinates of the fourth vertex.
b. Verify that ABCD is a rectangle providing evidence related to the sides and angles.
(Level I/II)
Ex. Investigate the slopes of each of the sides of the rectangle ABCD (shown below). What do you notice about
the slopes of the sides that meat at a right angle? What do you notice about the slopes of the opposite sides that are
parallel? Can you generalize what happens when you multiply slopes of perperpendicular lines?
Ex. Suppose a line k in a coordinate plane has slope
c
. d
a. What is the slope of a line parallel to k? Why must this be the case? b. What is the slope of a line perpendicular to k? Why does this seem reasonable?
Ex. Two points A(0, -4) , B(2, -1) determines a line, AB.
a. What is the equation of the line AB?
b. What is the equation of the line perpendicular to AB passing through the point (2,-1)?
Ex. There is a situation in which two lines are perpendicular but the product of their slopes is not (-1). Explain the
situation in which this happens.
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.6 Given two points on a line, find the point that divides the segment into an equal number of parts. If
finding the mid-point, it is always halfway between the two endpoints. The x-coordinate of the mid-point will be
the mean of the x-coordinates of the endpoints and the y-coordinate will be the mean of the y-coordinates of the
endpoints. At this level, focus on finding the midpoint of a segment.
(Level III)
Ex. If general points N at (a,b) and P at (c,d) are given. Why are the coordinates of point Q (a,d)? Can you find the
coordinates of point M?
Ex. If you are given the midpoint of a segment and one endpoint. Find the other endpoint.
a. midpoint: (6, 2) endpoint: (1, 3)
b. midpoint: (-1, -2) endpoint: (3.5, -7)
Ex. If Jennifer and Jane are best friends. They placed a map of their town on a coordinate grid and found the
point at which each of their house lies. If Jennifer’s house lies at (9, 7) and Jane’s house is at (15, 9) and they
wanted to meet in the middle, what are the coordinates of the place they should meet?
G-GPE.7 Use coordinates to compute
perimeters of polygons and areas of
triangles and rectangles, e.g., using
the distance formula..
G-GPE.7 Students should find the perimeter of polygons and the area of triangles and rectangles using coordinates
on the coordinate plane.
(Level II)
Ex. John was visiting three cities that lie on a coordinate grid at (-4, 5), (4, 5), and (-3, -4). If he visited all the
cities and ended up where he started, what is the distance in miles he traveled?
Geometric Measurement and Dimension
G-GMD
Common Core Cluster
Explain volume formulas and use them to solve problems.
Common Core Standard
G-GMD.1 Give an informal
argument for the formulas for the
circumference of a circle, area of a
circle, volume of a cylinder, pyramid,
and cone. Use dissection arguments,
Cavalieri’s principle, and informal
limit arguments.
Unpacking
What does this standard mean that a student will know and be able to do?
G-GMD.1 Explain the formulas for the circumference of a circle and the area of a circle by determining the
meaning of each term or factor.
G-GMD.1 Explain the formulas for the volume of a cylinder, pyramid and cone by determining the meaning of
each term or factor.
G-GMD.1 Understanding the formula for the circumference of a circle, you can either begin with the measure of
the diameter or the measure of the radius. Take those measurements and measure around the outside of a circle.
The diameter will go around a little over 3 times, which indicates 𝐶 = 𝜋𝑑. The radius will go around half of the
circle a little over 3 times, therefore 𝐶 = 2𝜋𝑟. This can either be done using pipe cleaners or string and a
measuring tool.
Understanding the formula for the circumference of a circle can be taught using the diameter of the circle or the
radius of the circle. Measure either the radius or diameter with a string or pipe cleaner. As you measure the
distance around the circle using the measure of the diameter, you will find that it’s a little over three, which is pi.
Therefore the circumference can be written as 𝐶 = 𝜋𝑑. When measuring the circle using the radius, you get a little
over 6, which is 2𝜋. Therefore, the circumference of the circle can also be expressed using 𝐶 = 2𝜋𝑟.
Understanding the formula for the area of a circle can be shown using dissection arguments. First dissect portions
of the circle like pieces of a pie. Arrange the pieces into a curvy parallelogram as indicated below.
Understanding the volume of a cylinder is based on the area of a circle, realizing that the volume is the area of the
circle over and over again until you’ve reached the given height, which is a simplified version of Cavalieri’s
principle. In Cavalieri’s principle, the cross-sections of the cylinder are circles of equal area, which stack to a
specific height. Therefore the formula for the volume of a cylinder is 𝑉 = 𝐵ℎ. Informal limit arguments are
not the intent at this level.
Informal arguments for the volume of a pyramid and cone
Ex.
G-GMD.2 (+) Give an informal
argument using Cavalieri’s principle
for the formulas for the volume of a
sphere and other solid figures.
G-GMD.2 Using Cavalieri’s Principle, provide informal arguments to develop the formulas for the volume of
spheres and other solid figures.
G-GMD.3 Use volume formulas for
cylinders, pyramids, cones, and
spheres to solve problems.*
G-GMD.3 Solve problems using volume formulas for cylinders, pyramids, cones, and spheres.
(Level II)
Ex. A cookie factory is making cookies in a pyramid shape with equilateral triangle as a base. We know that the
lateral edge of the cookie is 2 cm long and the base edge of the cookie is 3 cm long.
a. Prove that the height of the cookie is 1 cm.
b. Find the volume of the cookie.
c. Each cookie is wrapped totally in an aluminum foil. Prove that the minimum surface of foil necessary to
wrap 100 cookies is greater than 960 cm2. (Level II)
Ex. Diana’s Christmas present is placed into a cubic shaped box. The box is wrapped in a golden paper.
a. Are there 3 m2 of golden paper enough for wrapping?
b. If 1 m2 of golden paper cost $3, how much would the wrapping material cost?
Could you pour 1 liter of juice in the box? (Know that 1 l=0.001 m3)
Geometric Measurement and Dimension
G-GMD
Common Core Cluster
Visualize relationships between two-dimensional and three-dimensional objects.
Common Core Standard
G-GMD.4 Identify the shapes of twodimensional cross-sections of threedimensional objects, and identify
three-dimensional objects generated
by rotations of two-dimensional
objects.
Unpacking
What does this standard mean that a student will know and be able to do?
G-GMD.4 Given a three- dimensional object, identify the shape made when the object is cut into cross-sections.
G-GMD.4 When rotating a two- dimensional figure, such as a square, know the three-dimensional figure that is
generated, such as a cylinder. Understand that a cross section of a solid is an intersection of a plane
(two-dimensional) and a solid (three-dimensional).
Modeling with Geometry
G-MG
Common Core Cluster
Apply geometric concepts in modeling situations.
Common Core Standard
G-MG.1 Use geometric shapes, their
measures, and their properties to
describe objects (e.g., modeling a tree
trunk or a human torso as a
cylinder).*
G.MG.2 Apply concepts of density
based on area and volume in
modeling situations (e.g., persons per
square mile, BTUs per cubic foot).*
Unpacking
What does this standard mean that a student will know and be able to do?
G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a soda can
or the paper towel roll as a cylinder).*
(Level I/II)
Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep and
extends for 5 feet. Then for 15 feet (horizontally) there is a constant slope downwards to the 10 foot-deep end.
a. Sketch the pool and indicate all measures on the sketch.
b. How much water is needed to fill the pool to the top? To a level 6 inches below the top?
c. One gallon of pool paint covers approximately 75 sq feet of surface. How many gallons of paint are needed
to paint the inside walls of the pool? If the pool paint comes in 5-gallon cans, how many cans are needed?
d. How much material is needed to make a rectangular pool cover that extends 2 feet beyond the pool on all
sides?
e. How many 6-inch square ceramic tiles are needed to tile the top 18 inches of the inside faces of the pool?
If the lowest line of tiles is to be in a contrasting color, how many of each tile are needed?
G-MG.2 Use the concept of density when referring to situations involving area and volume models, such as
persons per square mile.
(Level I)
Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep and
extends for 5 feet. Then for 15 feet (horizontally) there is a constant slope downwards to the 10 foot-deep end.
a. Sketch the pool and indicate all measures on the sketch.
b. How much water is needed to fill the pool to the top? To a level 6 inches below the top?
c. One gallon of pool paint covers approximately 75 sq feet of surface. How many gallons of paint are needed
to paint the inside walls of the pool? If the pool paint comes in 5-gallon cans, how many cans are needed?
d. How much material is needed to make a rectangular pool cover that extends 2 feet beyond the pool on all
sides?
e. How many 6-inch square ceramic tiles are needed to tile the top 18 inches of the inside faces of the pool?
If the lowest line of tiles is to be in a contrasting color, how many of each tile are needed?
G-MG.3 Apply geometric methods to
solve design problems (e.g. designing
an object or structure to satisfy
physical constraints or minimize cost;
working with typographic grid
systems based on ratios).*
G-MG.3 Solve design problems by designing an object or structure that satisfies certain constraints, such as
minimizing cost or working with a grid system based on ratios (i.e., The enlargement of a picture using a grid and
ratios and proportions)
(Level I/II)
Ex. Consider a rectangular swimming pool 30 feet long and 20 feet wide. The shallow end is 3½ feet deep and
extends for 5 feet. Then for 15 feet (horizontally) there is a constant slope downwards to the 10 foot-deep end.
a. Sketch the pool and indicate all measures on the sketch.
b. How much water is needed to fill the pool to the top? To a level 6 inches below the top?
c. One gallon of pool paint covers approximately 75 sq. feet of surface. How many gallons of paint are
needed to paint the inside walls of the pool? If the pool paint comes in 5-gallon cans, how many cans are
needed?
d. How much material is needed to make a rectangular pool cover that extends 2 feet beyond the pool on all
sides?
e. How many 6-inch square ceramic tiles are needed to tile the top 18 inches of the inside faces of the pool?
If the lowest line of tiles is to be in a contrasting color, how many of each tile are needed?