An Update from the NYSED Offices of
Curriculum & Instruction, and
Assessment
December 2014
Sue Brockley, Mathematics Assistant
EngageNY.org
August 2015/2016
Administrations of
the Regents Exam in
Algebra
2/Trigonometry
http://www.p12.nys
ed.gov/assessment/
hsgen/2015/a2trig
memo-aug15.pdf
JUNE 2015
Transition Memo to Common Core Regents Examinations in English Language Arts and Mathematics October 2014
http://www.p12.nysed.gov/assessment/commoncore/transitionccregents1113rev.pdf
General Education and Diploma Requirements Chart
http://www.p12.nysed.gov/ciai/gradreq/revisedgradreq3column.pdf
What about the Waiver…. ?
WAIVER: The US DOE approved another one year waiver regarding the assessment requirements for
students who are accelerated into Regents Mathematics courses in Grades 7 and 8. Under the waiver
(like last year) districts may exempt students who are in Regents Mathematics courses, and who will take
a Regents Mathematics Assessment in June 2015, from the grade level (7 or 8) Mathematics Assessment
administered in April. The Board of Regents were presented with the amendment to the regulations at
the October meeting. Those regulation changes will be put in the public register and will come before
the Board for final approval in January.
http://www.regents.nysed.gov/meetings/2014/October2014/1014p12d2.pdf
THE FOLLOWING UPDATED DOUBLE TESTING FIELD MEMO DATED
2/26/2014 MAY PROVIDE ANSWERS TO WHAT OCCURRED LAST YEAR
BASED ON THE WAIVER. THIS MEMO IS LOCATED AT
http://www.p12.nysed.gov/accountability/documents/UPDATEDDou
bleTestingFieldMemo22614.pdf.
ANY SPECIFIC QUESTIONS ABOUT THE WAIVER CAN BE ADDRESSED
TO THE OFFICE OF ACCOUNTABILITY AT
ACCOUNTINFO@MAIL.NYSED.GOV
What’s New…
Resources
RFP
Modules
PAEMST
Engage
Common Core
Assessments Page
DDI
Algebra II
Geometry
Acceleration
Computer Science
Education Week
Geo Test Blueprint
Graduation Pathways
Thank You
EngageNY.org
How has constructing a sound answer changed with the
Rigor
Common Core ?
Conceptual
Procedural
Mathematical Practice #3
Construct viable arguments and critique the reasoning of
others.
Explain/Justify
Compute
Solve
Identify
Describe how/why . . .
Make clear and/or offer
reason.
Convey an idea,
qualities or background
information.
• Students will
provide/use solid
mathematical
arguments and
language.
• Written paragraph.
• Measurement using
appropriate tools.
• Written proof.
EngageNY.org
CO
SRT
GMD
GPE
C
MG
BACK
Congruence (G-CO)
A. Experiment with the transformations in the plane. (Supporting)
G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
G.CO.A.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.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations
and reflections that carry it onto itself. Regular Polygons
NYSED: Trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”
G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments. Students may describe translations in terms of
vectors, entities that have both magnitude and direction.
G.CO.A.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. Students will need to be able to perform
and describe transformations in the coordinate plane as well, still need to know the “rules”. Might
link this work to what is done in the clusters from Geometric Properties with Equations (GPE).
Direction/description of rotations will be stated. Shorthand notation will be consistent with what has
appeared in the past with. Students may have to provide a sequence of transformations, but
notation f ° g, no.
Congruence (G-CO)
B. Understand congruence in terms of rigid motions. (Major)
G.CO.B.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. A rigid motion of the plane ( also known as an isometry ) is a motion
which preserves distance and angle measure.
There are four basic rigid motions:
(1) Reflection
(2) Glide Reflection
(3) Rotation
(4) Translation
What do they do….
Map lines to lines, rays to rays, segments to segments, angles to angles
Preserve lengths of segments and the measures of angles
G.CO.B.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.
Illustrative Math
G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of
congruence in terms of rigid motions.
To start reasoning about the congruence of the two triangles, Sue and Peter
have created the following diagram in which they have marked an ASA
relationship between the triangles.
1. Based on the diagram, which angles have Peter and Sue indicated are
congruent? Which sides?
2. To convince themselves that the two triangles are congruent, what else
would Peter and Sue need to know?
S
B
C
T
A
R
“I know what to do,” said Peter. “We can translate point A until it maps with point
R, then rotate line segment AB about point R until it maps with
Line segment RS. Finally, we can reflect ΔABC across line segment RS and then
everything maps so the triangles are congruent.” Sue says…”Hmmmm…”
B S
C
Is this enough language, is
T
the argument complete ?
Adapted from 2012 Mathematics Vision Project mathematicsvisionproject.org
A
R
Now LOOK at G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and
to prove relationships in geometric figures.
NYSED: ASA, SAS, SSS, AAS, and Hypotenuse‐Leg (HL) theorems are valid criteria for triangle
congruence. AA, SAS, and SSS are valid criteria for triangle similarity.
Examples:
1. Common Core Sample Question #14 pg. 53
2. June 2012 #35 pg. 56
3.
In the diagram below, P’ is the image of P over l. The points O and R are on l. .
Prove <POR ≅ < 𝑃′ 𝑂𝑅.
P
l
Language of
Transformations
O
R
P’
Reflections: A point P’ is the reflected image of point P
over line l iff l is the perpendicular bisector of
segment PP’, assuming points P and P’ are not on l
Congruence (G-CO)
C. Prove geometric theorems. (Major)
G.CO.C.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 those equidistant from the segment’s endpoints.
NYSED: Theorems include but are not limited to the listed theorems. Example: theorems that involve
complementary or supplementary angles.
G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°;
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.
NYSED: Theorems include but are not limited to the listed theorems.
Example: an exterior angle of a triangle is equal to the sum of the two non‐adjacent interior angles of the triangle.
G.CO.C.11 Prove theorems about parallelograms (trapezoids). 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.
NYSED: Theorems include but are not limited to the listed theorems.
Example: rhombus is a parallelogram with perpendicular diagonals.
These theorems need not be grand theorems, but rather any non-obvious statement that can be justified on the basis
of previously established statements. Proof by contradiction, valid method of proof.
Algebraic problems using theorems- link to G.SRT.B.5: Use congruence and similarity criteria for triangles to solve
problems and to prove relationships in geometric figures.
Use transformational geometry to prove simple angle
theorems…
The congruence of vertical angles (rotations)
If two parallel lines are cut by a transversal, then the
corresponding angles are congruent. (translations)
The sum of the angles of a triangle is 180.
C’
C
3
1
A
2
4
5
B
B’
Prove that in an isosceles trapezoid with AD≅ BC , the straight line which passes
through the diagonals intersection parallel to the bases bisects the angle
between the diagonals.
Congruence (G-CO)
D. Make geometric constructions. (Supporting)
G.CO.D.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.
NYSED: Constructions include but are not limited to the listed constructions.
Example: constructing the median of a triangle or constructing an isosceles triangle with given lengths.
All constructions from 2005 are fair game….
Link to G.C.A.3 Construct the inscribed and circumscribed circles of a triangle.
G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Square
Equilateral triangle Might want to link with standards in the Circles clusters.
Back to Snowman
Similarity, Right Triangles and Trigonometry (G-SRT)
A. Understand similarity in terms of similarity transformations. (Major)
G.SRT.A.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.
Performing dilations in the coordinate plane is within the scope of this standard. The center does not
always need to be the origin. Assessment items will always be clear as to the center of the dilation.
G.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to
decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as
the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles
to be similar. Students will be proving why the AA similarity criteria works. SSS and SAS similarity criteria as
well.
G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
NYSED: ASA, SAS, SSS, AAS, and Hypotenuse‐Leg (HL) theorems are valid criteria for triangle
congruence. AA, SAS, and SSS are valid criteria for triangle similarity.
Scale Drawings :
Ratio and Parallel
Method
The Progression of the
Similarity …
Triangle Splitter Theorem or Triangle
Proportionality Theorem (A line segment
splits two sides of a triangle proportionally iff it is
parallel to the third side. )
Dilation Theorem (If a dilation with center O and scale factor r
sends point P to P’ and Q to Q’, then P’Q’=r (PQ). Furthermore, if r≠1
and O,P and Q are the vertices of a triangle, then PQ//P’Q’)
A.A. Similarity Criteria: (2 figures are
similar if one is ≅to a dilation of the other,
or if the second can be obtained from the first by
a sequence of rotations, reflections, translations
and dilations)
S.A.S and SSS Similarity
Criteria
Similarity, Right Triangles and Trigonometry (G-SRT)
B. Prove theorems using similarity. (Major)
G.SRT.B.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.
NYSED: Theorems include but are not limited to the listed theorems.
Example: the length of the altitude drawn from the vertex of the right angle of a right triangle to its
hypotenuse is the geometric mean between the lengths of the two segments of the hypotenuse.
G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures. Algebraic problems using theorems.
NYSED: ASA, SAS, SSS, AAS, and Hypotenuse‐Leg (HL) theorems are valid criteria for triangle congruence.
AA, SAS, and SSS are valid criteria for triangle similarity.
C. Define trigonometric ratios and solve problems involving right triangles. (Major)
G.SRT.C.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.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems. Students will also have to find angles using inverse trig ratios.
Modeling Example
Back to
snowman
Circles(G-C)
A. Understand and apply theorems about circles. (Supporting)
G.C.A.1 Prove that all circles are similar.
G.C.A.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.
NYSED: Relationships include but are not limited to the listed relationships.
Example: angles involving tangents and secants. (All 2005 circle theorems)
Find the equation of tangent lines, link to G.GPE.5.
G.C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
B. Find arc lengths and areas of sectors of circles. (Supporting)
G.C.B.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.
Use the diagram to show that measure of arc DE=y+x and the
measure of arc FG=y-x and show your work.
What about circle proofs ? Link to G. SRT.5
Common Core Sample Question #13
In the diagram below, secant ACD and tangent AB are drawn
from external point A
to circle O.
Prove the theorem: If a secant and a tangent
are drawn to a circle from an external
point, the product of the lengths of the secant
segment and its
external segment equals the length of the
tangent segment
squared. (AC x AD = AB^2 )
Back to Snowman
A
B
C
. O
D
Consider the circle with equation (𝒙 − 𝟑)𝟐 + (𝒚 − 𝟓)𝟐 = 20. Find the equations of two
tangent lines to the circle that each have slope -1/2.
y-9= -1/2(x-5)
y-1= -1/2(x-1)
Back
Expressing Geometric Properties with Equations (G-GPE)
A. Translate between the geometric description and the equation of a conic
section.
(Supporting)
G.GPE.A.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.
Equations will not solely be seen in center/radius form as in the past. Students will still need to transfer back
and forth between equation and graph. Example
B. Use coordinates to prove simple geometric theorems algebraically. (Major)
G.GPE.B.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). This involves
students using the midpoint, slope and distance formulas.
G.GPE.B.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). Methods
G.GPE.B.6 Find the point on a directed line segment between two given points that partitions the
segment in a given ratio.
G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula.
Snow
Method One: Illustrative Math: When are two lines perpendicular ?
Method Two: Module 4 Connecting Algebra and Geometry Through Coordinates
Topic B Lessons 5-8
Lesson 5: Using the Pythagorean Theorem
OA= 𝑎12 + 𝑎22
OB= 𝑏12 + 𝑏22
AB= (𝑏1 − 𝑎1)2 +(𝑏2 − 𝑎2)2
𝑂𝐴2 + 𝑂𝐵2 = 𝐴𝐵2
0=(𝑎1 )(𝑏1 ) + (𝑎2 )(𝑏2 )
If OA is
perpendicular to OB
then (and common
endpoint at origin) …
Slopes are opposite reciprocals
Lesson
8
Grade 5
Module 6
Lessons 14-17
Back
Derivation of the midpoint formula
(𝑥2 ,𝑦2 )
(𝑥𝑚 ,𝑦𝑚 )
(𝑥1 ,𝑦1 )
𝑥𝑚 - 𝑥1 = ½ (𝑥2 -𝑥1 )
𝑦𝑚 - 𝑦1 = ½ (𝑦2 -𝑦1 )
Given the points A(-1,2) and B(7, 8), find the
coordinates of point P on directed line segment
AB that partitions AB in the ratio 1/3.
B(7, 8)
8-2
1/4 of 6=1.5
X- -1 = ¼ (7 - -1)
X+1=2
X=1
A(-1,2)
7- -1
1/4 of 8=2
B(7, 8)
Y-2= ¼ (8-2)
Y-2=1.5
Y=3.5
1.5
A(-1,2)
P(1,3.5)
2
BACK
Geometric Measurement and Dimension (G-GMD)
A. Explain volume formulas and use them to solve problems. (Supporting)
G.GMD.A.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. Example
Cavalieri
G.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Modeling Example
B. Visualize relationships between two-dimensional and three-dimensional objects.
(Supporting)
G.GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and
identify three-dimensional objects generated by rotations of two-dimensional objects. Example
Back to snowman
Modeling with Geometry (G-MD)
A. Apply geometric concepts in modeling situations. (Major)
G.MG.A.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.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per
square mile, BTUs per cubic foot). Example
G.MG.A.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.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve
right triangles in applied problems.
G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas
of triangles and rectangles, e.g., using the distance formula.
G.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and
spheres to solve problems.
Fluencies:
• Triangle congruence and similarity criteria.
• Use of coordinates to establish geometric results.
• Constructions
BACK
Exploring
Rotational
Symmetry
Find the angle of rotation that will
carry the 12 sided regular polygon to
itself.
How many sides does a regular polygon have
that has an angle of rotation equal to 20
degrees. How many lines of symmetry will it
have ?
If one of the angles of a regular polygon is 160
degrees, find the angle of rotation that will
carry this polygon onto itself.
Classify quadrilaterals based on
transformational properties
180 degree rotation
180 degree rotation
2 lines of symmetry
along diagonals
Back
180 degree rotation
2 lines of symmetry
through midpoint
of sides
90 and 180 degree rotation
4 lines of symmetry
along diagonals and through
midpoints of sides
Adapted from 2012 Mathematics Vision Project mathematicsvisionproject.org
Find the center of the rotation that
takes AB to A’B’.
B’
A’
B
A
Back
Chante claims that two circles given by (𝑥 + 2)2 + (𝑦 − 4)2 =49 and 𝑥 2 +𝑦 2 -6x+16y+37=0
are externally tangent. Justify why she is correct.
(𝑥 + 2)2 + (𝑦 − 4)2 =49
Center (-2,4) and r=7
𝑥 2 +𝑦 2 -6x+16y+37=0
𝑥 2 -6x + 𝑦 2 - 16y = -37
(𝑥 − 3)2 + (𝑦 + 8)2 =36
Center (3,-8) and r=6
BACK
Is the
distance
between the
two radii
equal to the
sum of the
radii ?
Informal Limit Arguments are used…
Area of Circle can be determined by taking the limit of the area
of either inscribed regular polygons or circumscribed polygons as
the number of sides n approaches infinity.
Approximate the area of a disk of radius 𝟐 using an inscribed regular
hexagon.
Approximate the area of a disk of radius 𝟐 using a circumscribed
regular hexagon.
Based on the areas of the inscribed and circumscribed hexagons, what is an approximate area of the given disk?
What is the area of the disk by the area formula, and how does your approximation compare?
Approximate area average
A=1/2(6 3 + 8 3 )
A=1/2(14 3 )=7 3 ≈ 12.12
Actual Area of circle with
radius 2
A=𝜋 𝑟 2 = 4 𝜋 ≈ 12.57
Lesson 4 Module 3
𝐴𝑟𝑒𝑎(𝑃𝑛)=[𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟(𝑃𝑛)] (1/2)(ℎ𝑛 )
Think of the regular polygon when it is inscribed in a circle. What happens to
ℎ𝑛 and 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟(𝑃𝑛) as 𝑛 approaches infinity (𝑛→∞) in terms of the radius
and circumference of the circle?
As 𝑛 increases and approaches infinity, the
height ℎ𝑛 becomes closer and closer to the
length of the radius (as 𝑛→∞, ℎ𝑛 →𝑟).
As 𝑛 increases and approaches infinity,
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟(𝑃𝑛) becomes closer and closer to the
circumference of the circle
(as 𝑛→∞, 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟(𝑃𝑛)→𝐶)
Since we are defining the area of a circle as the limit of the areas of the
inscribed regular polygon, substitute 𝑟 for ℎ𝑛 and 𝐶 for 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟(𝑃𝑛) in
the formulation for the area of a circle:
𝐴𝑟𝑒𝑎(𝑐𝑖𝑟𝑐𝑙𝑒)=(1/2 )𝑟𝐶
=(1/2) 𝑟(2𝜋𝑟) = 𝜋𝑟 2
We are going to show why the circumference of a circle has the
formula 𝟐𝝅𝒓. Circle 𝑪𝟏 below has a diameter of 𝒅=𝟏, and circle 𝑪𝟐
has a diameter of 𝒅=𝟐𝒓.
All circles are similar. What scale factor of the similarity transformation takes 𝑪𝟏 to 𝑪𝟐?
A scale factor of 𝟐𝒓.
Since the circumference of a circle is a one-dimensional measurement, the value of the ratio of two
circumferences is equal to the value of the ratio of their respective diameters. Rewrite the following equation
by filling in the appropriate values for the diameters of 𝑪𝟏 and 𝑪𝟐:
𝐂𝐢𝐫𝐜𝐮𝐦𝐟𝐞𝐫𝐞𝐧𝐜𝐞(𝑪𝟐)/𝐂𝐢𝐫𝐜𝐮𝐦𝐟𝐞𝐫𝐞𝐧𝐜𝐞(𝑪𝟏)=𝐝𝐢𝐚𝐦𝐞𝐭𝐞𝐫(𝑪𝟐)/𝐝𝐢𝐚𝐦𝐞𝐭𝐞𝐫(𝑪𝟏)
𝑪𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆(𝑪𝟐)/𝑪𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆(𝑪𝟏)=𝟐𝒓/𝟏
Since we have defined 𝝅 to be the circumference of a circle whose diameter is 𝟏, rewrite the above equation
using this definition for 𝑪𝟏.
𝑪𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆(𝑪𝟐)/𝝅=𝟐𝒓/𝟏
Rewrite the equation to show a formula for the circumference of 𝑪𝟐.
𝑪𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆(𝑪𝟐)=𝟐𝝅𝒓
Back
Sketch the figure formed if the rectangular (triangular) region is
rotated around the provided axis:
Describe the shape of the cross-section of each
of the following objects.
Right circular cone:
Cut by a plane through the vertex and perpendicular to the base
Triangular Prism :
Cut by a plane parallel to a base
Cut by a plane parallel to a face
Back
Cavalieri’s Principle: A method for finding the volume of any
solid for which cross-sections by parallel planes have equal
area.
Plane Properties Revisited
Using these plane properties/congruent triangles to
informally prove Cavalieri’s Principle showing that for any
prism, no matter what polygon the base is, the cross-sections
are congruent to the base.
Prove that cross sections are similar to the base using dilations (lengths along
edge of pyramid allows us to find scale factor of dilation) and SSS similarity
criteria. The area of the similar region should be the area of the original figure
times the square of the scale factor.
General Cone Cross-Section Theorem:
If two general cones have the same base area and the same height, then cross-sections
for the general cones the same distance from the vertex have the same area.
Scaling and effect on volume
Informal argument and scaling
used to prove volume of
pyramid V=1/3(B)(h)
Cavalieri’s to
prove volume of
cylinder and cone
Back
VS
It is given that point D is the image of point A after a reflection in line CH.
It is given that line CH is the perpendicular bisector of segment BE at
point C. Since a bisector divides a segment into two congruent segments
at its midpoint, segment BC is congruent to segment EC . Point E is the
image of point B after a reflection over the line CH, since points B and E
are equidistant from point C and it is given that line CH is perpendicular
to BE.
Point C is on line CH therefore, point C maps to itself after the reflection
over line CH.
Since all three vertices of triangle ABC map to all three vertices of
triangle DEC under the same line reflection, then ∆ABC≅ ∆ DEC because
a line reflection is a rigid motion and triangles are congruent when one
can be mapped onto the other using a sequence of rigid motions.
Back
Exercises 1–3
Each exercise below shows a sequence of rigid motions that map a pre-image onto a final image. Identify each rigid
motion in the sequence, writing the composition using function notation. Trace the congruence of each set of
corresponding sides and angles through all steps in the sequence, proving that the pre-image is congruent to the final
image by showing that every side and every angle in the pre-image maps onto its corresponding side and angle in the
image. Finally, make a statement about the congruence of the pre-image and final image.
Given that ∆𝐴𝐵𝐶 ≅ ∆ A’’’B’’’C’’’, state a sequence of
rigid motions that will map ∆𝐴𝐵𝐶 to ∆ A’’’B’’’C’’’
Pg. 161
Students may use
function notation
(vectors) to state a
sequence of rigid
motions, or in words
describe each
transformation of the
sequence in
order…complete
language being important
here, e.g. stating line of
reflection, fully describing
the translation (direction
and length (maps point B’
to B’’) and then stating
center, direction and
degree of rotation. Any
rotations that need to be
performed will not
involve the use of a
protractor.
Back
13 grant awards to 10 school districts
Common Core Institute with Sponsored Common Core Institute Fellowship
The primary purpose of this request for proposals (RFP) is to grant school districts, Board of Cooperative
Education Services (BOCES), and charter schools, from across the state, resources to allow the organization to
serve as a Common Core Institute (CCI) and sponsor selected educators as Common Core Institute Fellows to
support professional development and capacity-building, specifically through the enhancement of the optional
and supplemental curricular modules currently posted on EngageNY.org.
Each eligible application must nominate one full-time educator or two part-time educators (each 50 percent of
an FTE) for one of the grade levels in Grades K-12 Mathematics or Grades 3-12 ELA, or one full-time or two
part-time ELL educators for two grades in an ELA grade band (3-4, 5-6, 7-8, 9-10, 11-12).
Applications must be received by: October 7, 2014
Anticipated Preliminary Award Notification: December 2014
Anticipated Project Period: January 2015 - June 30, 2015
http://usny.nysed.gov/rttt/rfp/sa-18/home.html
http://www.nysed.gov/Press/Common%20Core%20Institute
%20Grants
Grades for Math: K,1,3,4,7 and 8
Any Ideas or Suggestions
Please share…
• more precise and comprehensive
scaffolds and supports for ELLs and
SWDs
• more effective formatting and usability
• modular organization to support local
pacing decisions
• bridging supports for students who
require remedial reinforcement
• Additional performance tasks and DDI
supports
BACK
EngageNY.org
What is New ?
Answer Keys for Grades 1-5, Modules 1-3
More coming….
BACK
EngageNY.org
New York State Certified Teacher Participation
Opportunities with the New York State Education
Department for…
Item development, test form review, range finding, and
standard setting.
http://www.p12.nysed.gov/assessment/teacher/home.html
50 % of 2014
3-8 test items released
PLD’s
http://www.p12.nysed.gov/irs/pressRelease/20140814/home.html
2014 Grades 3-8 ELA and Math Test Results
Information and Reporting Services (IRS)
Release of Data - August 14, 2014
Equating Explained FAQ
Educational Testing Service (ETS)
http://www.ets.org/Media/Research/pdf/RD_Connections16.pdf.
Raw Score/Percent Score/Scale Score/Equating Process/Anchor Items
EngageNY.org
Performance Level Definitions
NYS Level 5
Students performing at this level
exceed Common Core expectations.
NYS Level 4
Students performing at this level
meet Common Core expectations.
NYS Level 3
Students performing at this level
partially meet Common Core
expectations (required for current
Regents Diploma purposes).
NYS Level 2 (Safety Net)
Students performing at this level
partially meet Common Core
expectations (required for Local
Diploma
purposes).
NYS Level 1
Students performing at this level do
not demonstrate the knowledge and
skills required for NYS Level 2.
… used in Assessment
PLDs are essential in setting standards for the New York State Regents Examinations. Standard setting
panelists use PLDs to determine the threshold expectations for students to demonstrate the knowledge
and skills necessary to attain just barely a Level 2, Level 3, Level 4, or Level 5 on the assessment. These
discussions then influence the panelists in establishing the cut scores on the assessment. PLDs are also
used to inform item development, as each test needs questions that distinguish performance all along
the continuum.
EngageNY.org
… used in Instruction
PLDs help communicate to students, families, educators and the public the
specific knowledge and skills expected of students to demonstrate proficiency and
can serve a number of purposes in classroom instruction. They are the foundation
of rich discussion around what students need to do to perform at
higher levels and to explain the progression of learning within a subject area.
EngageNY.org
BACK
18 shared standards with Algebra I
GAISE Report
Guidelines for Assessment and Instruction in
Statistics Education
(American Statistical Association)
Four Components of the Statistical
Problem Solving Process and the role
of Variability
•
Formulate Questions
•
Collect Data
BACK
•
Analyze Data
•
Interpret Results
Assessment Limits for Standards Assessed on More
Than One End-of-Course Test/EOY Evidence Tables
EngageNY.org
Test Blueprint
EngageNY.org
Standards Clarifications
In an effort to ensure that the standards can be interpreted by teachers and used effectively to inform classroom instruction, several standards of
the Geometry curriculum have been identified as needing some clarification. These clarifications are outlined below.
• G-CO.3
Trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”
• G-CO.9, G-CO.10, G-CO.11, G-SRT.4
Theorems include but are not limited to the listed theorems.
• G-CO.12
Constructions include but are not limited to the listed constructions.
• G-SRT.5
ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence.
AA, SAS, and SSS are valid criteria for triangle similarity.
Mathematics Tools
• G-C.2
Relationships include but are not limited to the listed relationships.
Reference Sheet
Same as Algebra I
for the
Regents
Examination in
Geometry (Common
Core)
•
Graphing Calculator
•
Straightedge
•
Compass
In response to field feedback, the Educator Guide to the Regents Examination in Geometry (Common Core) has been
updated.
One important refinement is that there will be two 6-credit constructed-response questions: one 6-credit question
will require students to develop multi-step, extended logical arguments and proofs involving major content, and one
6-credit question will require students to use modeling to solve real-world problems. This refinement balances field
expectations with what the standards require, and will allow students the opportunity to exhibit the knowledge and
skills associated with both types of questions.
The guide also provides additional information about the types of questions that will appear on the test in June
2015.
Updates are shown as highlighted text in the revised guide posted at: https://www.engageny.org/resource/regentsexams-mathematics-geometry-test-guide
Regents Examination in Geometry (Common Core) Design
Test
Component
Number of
Questions
Credits per
Question
Total Credits
in Section
Part I
24
2
48
Part II
8 7
X
2
16
X
14
Part III
4
16
X
Part IV
4
X 3
1 2
X
6
X6
12
12
Total
X
37
36
86
BACK
What the STANDARDS say about
ACCELERATION
• Students who are capable of moving more quickly
deserve thoughtful attention, both to ensure that
they are challenged and that they are mastering the
full range of mathematical content and skills.
• Rather than skipping or rushing through content,
students should have appropriate progressions of
foundational content…the continuity of the learning
progression is not disrupted.
• Skipping material to get students to a particular
point in the curriculum will likely create gaps…which
may create additional problems later.
EngageNY.org
• Placing students into tracks too early should be
avoided at all costs, it is not recommended to
compact the standards before grade 7.
• Districts are encouraged to have a well-crafted
sequence of compacted courses, which require a
faster pace to complete…compacting 3 years of
content into 2 years.
• Decisions to accelerate are almost always a joint
decision between the school and the family,
serious efforts must be made to consider solid
evidence of student learning.
EngageNY.org
I want my
AP Calculus
• Unfortunately, many parents and
community leaders look upon pre-CCSS
grade 8 courses as mostly “skippable.”
• They think of CCSS grade 8 in that old
paradigm and push for “skipping” the
grade again in order to reach Calc AP by
12th grade.
• But there’s a problem with that…
EngageNY.org
20 days
Grade 6
Grade 7
M1:
Ratios and Unit Rates
(35 days)
M1:
Ratios and Proportional
Relationships
(30 days)
20 days
20 days
20 days
M2:
Arithmetic Operations Including
Division of Fractions
(25 days)
M3:
Rational Numbers
(25 days)
M2:
Rational Numbers
(30 days)
M3:
Expressions and Equations
(35 days)
20 days
20 days
20 days
20 days
20 days
M4:
Expressions and Equations
(45 days)
M5:
Area, Surface Area, and Volume
Problems
(25 days)
M6:
Statistics
(25 days)
M4:
Percent and Proportional
Relationships
(25 days)
M5:
Statistics and Probability
(25 days)
M6:
Geometry
(35 days)
Grade 9 -- Algebra I
Grade 8
M1: Integer Exponents and the
M1:
days
Scientific Notation 20 days 20Relationships
Between
(20 days)
Quantities and Reasoning
with Equations and Their
Graphs
M2:
20 days 20 days
(40
days)
The Concept of Congruence
(25 days)
M3:
Similarity
(25 days)
20 days
Grade 11 -- Algebra II
M1:
Congruence, Proof, and
Constructions
(45 days)
M1:
Polynomial, Rational, and
Radical Relationships
(45 days)
M2:days
Descriptive Statistics
20
(25 days)
20 days
Grade 10 -- Geometry
M3:
daysand Exponential
20Linear
Functions
State Examinations
(35 days)
M4:
Linear Equations 20 days 20 days
(40 days)
M2:
Similarity, Proof, and
Trigonometry
(45 days)
State Examinations
M2:
Trigonometric Functions
(20 days)
Grade 12 -- Precalculus
M1:
Complex Numbers and
Transformations
(40 days)
20 days
20 days
M2:
Vectors and Matrices
(40 days)
M3: Functions
(45 days)
State Examinations
20 days
20 days
State Examinations
M3:
Rational and Exponential
Functions
(25 days)
20 days
20 days M4:
M4: Connecting Algebra
Polynomial and Quadratic
and Geometry through
M4: Trigonometry
Expressions, Equations and
Coordinates (25 days)
(20 days)
Functions
M4:
20 days 20 days(30 days)
Inferences and Conclusions
M5:
from Data
M5:
M5:
Circles with and Without
(40 days)
Probability and Statistics
A Synthesis of Modeling
Coordinates
(25 days)
with Equations and
20 days 20 days Approx. test
(25 days)
Functions (20date
days)
for
20 days
20 days
M3: Extending to Three
Dimensions (10 days)
M5: Examples of Functions from
Geometry (15 days)
M6:
Linear Functions
(20 days)
20 days
20 days
M7:
Introduction to Irrational Numbers
Grades 6-8
Using Geometry
Review and Examinations Review and Examinations Review and Examinations Review and Examinations
20 days 20 days
20 days
(35 days)
EngageNY.org
Protecting grade 8
It’s a marketing
problem, but it appears
to have possible
solutions…
Pathways not endorsed by NYSED
BACK
EngageNY.org
Common Core Sample Question #12
Trees that are cut down and stripped of their branches for timber
are approximately cylindrical. A timber company specializes in a
certain type of tree that has a typical diameter of 50 cm and a
typical height of about 10 meters. The density of the wood
is 380 kilograms per cubic meter, and the wood can be sold by
mass at a rate of$4.75 per kilogram. Determine and state the
minimum number of whole trees that must be sold to raise at least
$50,000.
Back
Goal: By 2015, NYS will have an established set of pathways to graduation
that are grounded in CCLS, increase student engagement and achievement.
• Allow for student choice
• Have demonstrated effective outcomes for students
• Similarly rigorous
Stay tuned…
Math Studio Talk Video Series on Engageny: Math coach Nick Timpone takes us from standards in
kindergarten through Grade 5 and demonstrates hands-on ideas, games, activities and models that teachers can take back to their
classrooms or parents can use as a tool as they help their children with their homework. Since the standards move students from a
basic understanding of numbers to more complex math like decimals and fractions, you will also see how these concepts and
strategies build upon each other to help students' math knowledge progress from grade to grade.
CC,OA K-4, NBT K-5, NF 3-5
https://www.engageny.org/content/math-studio-talk-common-coreinstruction-video-series
Scaffolding Instruction for English Language
Learners: A Resource Guide for Mathematics
https://www.engageny.org/resource/scaffolding-instruction-englishlanguage-learners-resource-guides-english-language-arts-and
The resource guides were developed by national experts in ELL
instruction, Diane August and Diane Staehr Fenner, who have
developed these ELL scaffolds for New York State that are aligned to
the Common Core and are research-based instructional strategies for
developing content and language with ELL students. The resource
guides first provide a description of each scaffolding strategy used, and
explain the research basis for such approaches. The guides then
provide examples of lessons from each partner organization that has
worked with NYS educators to develop optional curriculum modules
on EngageNY, embedding research-based scaffolds into the lessons.
The examples include instructions for teachers, actions for students,
and additional resources to facilitate implementing each scaffolding
technique.
Kindergarten, Module 3, Lesson 3: Make Series of Longer Than and
Shorter Than Comparisons
Grade 4, Module 5, Lesson 16: Use Visual Models to Add
and Subtract Two Fractions With the Same Units
Grade 8, Module 3, Lesson 6: Proofs of Laws of
Exponents
Algebra I, Module 3, Lesson 5: The Power of
Exponential Growth
Education Week November 14, 2014
http://www.edweek.org/ew/articles/2014/11/12/12ccell.h34.html
Back
EngageNY.org
2014-2015 Award Cycle
7-12 Grade Level Teachers
For more information about the
PAEMST program visit the New York
State’s Education Department’s
website at
http://www.p12.nysed.gov/ciai/mst/
paemstaward.html,
PAEMST NYSED Coordinators:
Math – Sue Brockley email:
Susan.Brockley@nysed.gov
Science – Ann Crotty email:
Ann.Crotty@nysed.gov
April 1, 2015
May 1, 2015
New York State Teacher of the Year
2016 Information
Back
Purpose
New York State wishes to recognize and celebrate exceptionally skilled and passionate educators. This year,
we will identify five teachers to serve as ambassadors for New York State teachers. Of these five teachers, one
will be selected as the New York State Teacher of the Year and nominated for the National Teacher of the
Year program.
Selection Criteria
A nominee must:
• be rated “effective” or “highly effective” on his or her annual Professional Performance Review (APPR) or,
if the APPR is not applicable to his or her current teaching area, have a record of superior teaching
performance as evidenced by student learning gains, assessments, and recognition of work;
• demonstrate exceptional educational talent as evidenced by effective instructional practices and student
learning results in the classroom and school;
• be an exceptionally skilled and dedicated teacher;
• be appropriately credentialed within his or her current teaching area and work directly with students in a
public school at any grade level or in any subject area from pre-kindergarten through grade 12;
• plan to remain in the field of education during and after his or her year of recognition;
•
have a minimum of five years of current teaching experience;
• demonstrate leadership through active roles in the school and community; and
• be poised and articulate, with the energy and equanimity to manage a busy schedule.
While all of these qualifications are considered, the most important qualification is the superior ability to
inspire learning in students of all backgrounds and abilities.
Teacher of the Year Responsibilities
Being selected as the New York State Teacher of the Year and nominee for the National Teacher of the Year
program is a great honor and responsibility. Selection as New York State Teacher of the Year carries with it an
obligation to appear as a keynote speaker and make public appearances. The New York State Teacher of the
Year is given the opportunity to participate in professional development trainings around the country (with
other state teachers of the year), and traditionally meets the President of the United States at the White
House. Schools and districts may also experience several of the other benefits of the Teacher of the Year
program, such as:
• access to information and resources gathered by the New York State Teachers of the Year as a result of
participation in professional development at the state and/or national levels;
• coaching for other teachers on instructional approaches;
• information and technical support for principals about how to prepare and support excellent teachers;
and
• a visit by the Commissioner of Education to the classroom and school of the New York State Teacher of the
Year.
Nomination and Selection Process
Any parent, community member, student, or educator can nominate an individual who meets the selection
criteria outlined in the section above to be a New York State Teacher of the Year. Nominees must complete
and submit the application by Monday, February 2, 2015. For more information on the timeline process,
please visit the website. http://www.p12.nysed.gov/ciai/toty/home.html
Computer Science Education
Week
December 8-14
The Hour of Code is organized
by Code.org, a public 501c3 non-profit
dedicated to expanding participation in
computer science by making it available in more
schools, and increasing participation by women
and underrepresented students of color.
An unprecedented coalition of partners have
come together to support the Hour of Code, too
— including Microsoft, Apple, Amazon, Boys and
Girls Clubs of America and the College Board.
Drag/drop programming
JavaScript and Python
http://hourofcode.com/us
Using Courses in Computer Science to Meet the Requirements for a
Regents or Local Diploma
New York State’s current graduation requirements call for 22 units of credit at the commencement
level, including three units of credit in both mathematics and science. Although courses in computer
science can be used for elective credit, there are provisions in Section 100.5 of the Commissioner’s
Regulations through which courses in computer science may be used to meet the mathematics or
science diploma credit requirements…specialized courses.
http://www.p12.nysed.gov/ciai/documents/ComputerScienceMemo.pdf
Back
Data Driven Instruction Components
Require the First
Step, First
Essential First Step
Data
Analysis
and Action
High Quality,
Common CoreAligned
Assessments
https://www.engageny.org/resource/ddi-and-assessments-in-mathematics-designing-assessments-that-provide-meaningful-data
Balance of Rigor
Procedural Fluency
Application
Conceptual
Understanding
From the Publishers’ Criteria:
https://www.engageny.org/resource/driven-by-data-increasing-rigor-throughout-the-lesson
Write “know/do” objectives: Students will
know _______ by doing _______.
Go to the nouns/verbs of the standards…
Write an assessment of the skills immediately after
the objective, at the top of the lesson plan
Spiral objectives/ skills/ questions from everything
previously learned to keep student learning sharp.
Use the progressions/previous year’s domains
Fold It In…
Add how/why questions (e.g., Why did you choose this
answer? How do you know your answer is correct?) for
different levels of learners and to push thinking.
Move from “Ping Pong” to “Volleyball:” instead of
teacher responding to every student answer, get other
students to respond to each other: “Do you agree with
him?” “Why is that answer correct/incorrect?” “What
would you add?”
Write questions in plan to specific students who are
struggling with a standard; jot down their
responses in the plans during class.
After getting to the right answer, have student articulate
their original error and how to avoid making the same
error in the future.
Create leveled questions for assessments
Create weekly skills check with a tracking chart:
students track their own progress on each skill.
Follow up data from Exit Ticket with following
day’s Do Now
Create leveled homework (student-specific)
Back
Also ties to…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)
In a two-chamber sand timer like the one shown a the right, sand
passes from one chamber to the other at a rate of 5 𝒊𝒏.𝟑 /min. The
sand forms a pile shaped like a right cone whose diameter is twice
The figure shows a cone inscribed in a cube. The
it’s height. Suppose all of the sand is in one chamber. You turnlength of each edge of the cube is 8 in. Find the
the timer and the sand begins to fall into the empty chamber atvolume of the space between the cone and the
the bottom. What is the height of the pile in the bottom of thecube to the nearest cubic inch.
chamber after three minutes ?
Volume of Cube
V= 𝑠 3 =83 =512 𝑖𝑛3
Volume of Cone
V=1/3 𝜋 𝑟 2 h = 1/3 𝜋(128) 𝑖𝑛3
Volume of Cube – Volume of Cone ≈ 378𝑖𝑛3
Back
Southwestern Geometry: An integrated approach
The angle of elevation of a hot air balloon,
climbing vertically, changes from 25 degrees
at 10:00 am to 60 degrees at 10:02 am.
The point of observation of the angle of
elevation is situated 300 meters away from
the take off point.
ℎ𝑇
What is the upward speed, assumed
constant, of the balloon?
Give the answer in meters per second and
round to two decimal places.
60°
h1= 300 (tan (25)) ℎ 𝑇 = 300 (tan (60))
ℎ2 = ℎ 𝑇 - ℎ1
𝑚
Speed ℎ2 /2 min x (1 min./60 sec.) ≈ 3.15 𝑠𝑒𝑐
Back
Thank You
Office of Curriculum and Instruction
Mary Cahill, Director mary.cahill@nysed.gov
Susan Brockley susan.brockley@nysed.gov
John Svendsen john.svendsen@nysed.gov
Office of State Assessment
emscassessinfo@mail.nysed.gov
EngageNY.org