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Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 1 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
TEXTBOOK - Geometry for Enjoyment and Challenge published by McDougal Littell in 1991
UNIT I - Introduction to Geometry G.G.1 Recognize special
types of polygons (e.g.,
isosceles triangles,
parallelograms, and
rhombuses). Apply
properties of sides,
diagonals, and angles in
special polygons; identify
their parts and special
segments (e.g., altitudes,
mid-segments);
determine interior angles
for regular polygons.
Draw and label sets of
points such as line
segments, rays, and
circles. Detect
symmetries of geometric
figures.
Students will KNOW: �How to recognize points, segments, lines,
rays, angles, and triangles
�How to measure segments and angles (in
degrees, minutes and seconds)
�How to classify angles by size or measure
�How to identify midpoints
�How to identify segment and angles bisectors
and trisectors
Students will be able to DO: �Identify and differentiate between segments,
lines and rays
�Recognize and name angles in three ways
�Classify angles according to their
size/measure
�Add and subtract angle measurements in
degrees, minutes and seconds
�Understand the definition of midpoints and
its implication
�Understand the definitions of angle and
segment bisectors and trisectors
�Apply midpoints, bisector and trisector
definitions to two column proofs
• Define the terms point, segment, line, ray,
angle, midpoint, angle bisector, trisector, and
triangle
• Demonstrate how to measure segments and
angles
• Work through many examples of measuring
segments and angles
• Work through many examples involving
midpoint and angle bisectors and trisectors
• Student can clearly
define what a segment,
line, ray, angle,
midpoint, angle
bisector, trisector, and
triangle is
• Student can accurately
measure segments and
angles
• Student can correctly
apply the definitions in
this unit to a variety of
examples Sept
G.G.2 Write simple
proofs of theorems in
geometric situations,
such as theorems about
congruent and similar
figures, parallel or
perpendicular lines.
Distinguish between
postulates and theorems.
Use inductive and
Students will KNOW: �How to recognize congruent segments and
angles
�How to recognize collinear and non-collinear
points and the concept of betweenness
�How to interpret a diagram
�How to write a simple two-column proof
�How to write a paragraph proofs
�That geometry is based on a deductive
structure
• Define meaning of congruent segments,
collinear points, non-collinear points, inverse,
converse, contrapositive, deductive structure,
and angles
• Explain the proper terminology to use when
constructing a mathematical proof
• Walk through several examples of writing
mathematical proofs
• Explain to the students how one should
interpret a geometric diagram through
• Student can correctly
define and apply the
following terms:
congruent segments,
collinear points, non-
collinear points,
inverse, converse,
contrapositive,
deductive structure, and
angle
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 2 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
deductive reasoning, as
well as proof by
contradiction. Given a
conditional statement,
write its inverse,
converse, and
contrapositive.
�How to identify undefined terms, postulates
and definitions
�How to write an inverse, converse, and
contrapositive when given a conditional
statement
Students will be able to DO: �Understand the use of tic marks in diagrams
�Understand the concept of congruency
through angles and segments with the same
measure
�Able to recognize collinear and non-collinear
points in a diagram
�Understand the concept of betweenness
�Interpret a diagram by assuming: straight
lines and angles, collinearity of points,
betweenness of points, and relative position of
points
� understand that right angles, congruent
segments & angles, and relative size of angles
and segments cannot be assumed from a
diagram
� begin to write a two-column proof matching
a reason for every statement made
�Practice how to reach a conclusion and
convince others of its validity through
paragraph form
�Understand the structure of deductive
reasoning based on definitions, postulates and
theorems
�Understand conditional statements: if a then
b
�Write converses of a � p then p�a
�Write the inverses of "a � p which is not a
� not p"
�Write the contrapositive of “a � p which is
not p � not a”
example
• Walk through several examples of
conditional statements
• Walk through several examples of calculating
angles and segments with algebraic
relationships
• Describe the concept of collinear and non-
collinear points
• Walk through examples of collinear and non-
collinear points
• Students can accurately
calculate angle sums
and conversions
• Students can correctly
use algebraic ratios to
calculate segment ratios
• Students can correctly
apply the terms of
collinear and non-
collinear points to
mathematical proofs
Oct
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 3 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
G.G.15 Draw the results,
and interpret
transformations on
figures in the coordinate
plane, e.g., translations,
reflections, rotations,
scale factors, and the
results of successive
transformations. Apply
transformations to the
solution of problems.
(10.G.9)
Students will KNOW:
�How to draw a reflection over a given line
�How to draw a rotation around a given point
Students will be able to DO:
�Recognize that a reflection is a mirror image
�Know the difference between clockwise and
counter-clockwise
�Know how to move an object 900, 1800 or
270 o
• Define the terms of reflection and rotation
• Walk through several examples of drawing
reflections over a given line and rotations
centered on a given point
• Students understand
reflection and rotation
• Students can correctly
draw reflections around
a given line
• Students can correctly
draw a rotation around a
given point
UNIT II - Basic Concepts and Proofs G.G.2 Write simple
proofs of theorems in
geometric situations,
such as theorems about
congruent and similar
figures, parallel or
perpendicular lines.
Distinguish between
postulates and theorems.
Use inductive and
deductive reasoning, as
well as proof by
contradiction. Given a
conditional statement,
write its inverse,
converse, and
contrapositive.
Students will KNOW: �The concept of perpendicularity
�How to write a simple proof
Students will be able to DO: �Recognize perpendicular lines/segments and
the use of the symbol, ⊥
�Understand that perpendicularity implies
right angles (900)
�Write proofs using the procedures for
drawing conclusions
�Use definitions, theorems or postulates to
justify each statement
• Define the concept of perpendicularity
• Walk through several examples of
perpendicular segments
• Look at the implications of perpendicularity
• Show examples of simple formal proofs
• Walk through several examples of
mathematical proofs
• Students can define the
concept of
perpendicularity
• Students can correctly
apply the concept of
perpendicularity to a
mathematical proof
• Students can correctly
construct simple
mathematical proofs
October
G.G.6 Apply properties
of angles, parallel lines,
arcs, radii, chords,
tangents, and secants to
solve problems.
Students will KNOW:
�Recognize complementary and
supplementary angles and the implication of
their relationships
�How to apply the addition, subtraction,
• Define complementary and
supplementary angles
• Calculate these angle relationships
• Walk through examples of the addition
and multiplication properties for angles
• Students will be able to
differentiate between an
angle’s complement and
its supplement
• Students will calculate
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 4 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
multiplication and division properties of
angles and segments
�How to apply the transitive and substitution
properties of angles and segments
�Recognize vertical angles
Students will be able to DO: �Understand the definitions of complementary
and supplementary
�Apply the complementary and
supplementary relationships in problem
solving
�Write proofs where students justify
statements by the addition, subtraction,
multiplication and division properties
�Differentiate between the transitive and the
substitution properties
�Use the transitive and substitution properties
as part of their proofs.
�Identify vertical angels in their diagrams
�Use the vertical angles congruency theorem
in their proofs
and segments
• Walk through examples of the
subtraction and division properties for
angles and segments
• Differentiate between the transitive and
the substitution properties of angles and
segments
• Have students practice applying each of
the properties in their proofs as they
pertain to angles and segments
• Investigate vertical angles and prove their
congruency as based on each being a
supplement to a given angle
• Walk through examples of using that
vertical angles are congruent in a formal
proof
the complements and
supplements of a given
angle
• Students will find the
size of an angle
algebraically based on
angle relationships
• Students will be able to
apply the addition and
multiplication
properties to their
proofs
• Students will be able to
apply the subtraction
and division properties
to their proofs
• Students can
appropriately apply
both the transitive and
substitution properties
• Students will recognize
vertical angles and the
fact that they are
congruent in their
proofs
G.G.15 Draw the results,
and interpret
transformations on
figures in the coordinate
plane, e.g., translations,
reflections, rotations,
scale factors, and the
results of successive
transformations. Apply
transformations to the
solution of problems.
(10.G.9)
Students will KNOW: �How to draw a reflection over a given line
�How to draw a rotation around a given point
Students will be able to DO: �Recognize that a reflection is a mirror image
�Know the difference between clockwise and
counter-clockwise
�Know how to move an object 900, 1800 or
270 o
• Use a visual model to demonstrate a
reflection over a given line
• Use a visual model to demonstrate a
rotation around a given point
• Students will draw examples of both
reflections and rotations on a coordinate
plane.
• Students will be able to
draw both reflections
(over a given line) and
rotations (around a
given point)
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 5 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
UNIT III - Congruent Triangles G.G.1 Recognize special
types of polygons (e.g.,
isosceles triangles,
parallelograms, and
rhombuses). Apply
properties of sides,
diagonals, and angles in
special polygons; identify
their parts and special
segments (e.g., altitudes,
mid-segments);
determine interior angles
for regular polygons.
Draw and label sets of
points such as line
segments, rays, and
circles. Detect
symmetries of geometric
figures.
Students will KNOW: �How to identify medians and altitudes
Students will be able to DO: �Apply the property of medians(that a median
divides the segment into two congruent
segments)
�Apply the property of altitudes (forms right
angles at the points of intersection)
• Define median and altitudes
• Have students investigate the relevance
of these lines to proving triangles congruent
• Students can accurately
define median and
altitude
• Students can correctly
use definitions of
medians and altitudes
within a geometric
proof
G.G.2 Write simple
proofs of theorems in
geometric situations,
such as theorems about
congruent and similar
figures, parallel or
perpendicular lines.
Distinguish between
postulates and theorems.
Use inductive and
deductive reasoning, as
well as proof by
contradiction.
Students will KNOW:
�How to prove triangles congruent by
applying the SSS, SAS, and ASA postulates
�Apply the principle of CPCTC
�Recognize some basic properties of circles
�Understand why auxiliary lines are used in
proofs
Students will be able to DO: �Analyze the diagrams to determine and use
the appropriate triangle congruency postulates
�Will be able to prove segments and angles
are congruent by recognizing corresponding
parts of congruent triangles
�Use the congruency of radii of a circle in
proofs
• Demonstrate the SAS postulate using a ruler
and protractor
• Have students discover the ASA and the SSS
postulates using a ruler and a protractor
• Define the CPCTC principle
• Have students analyze and write proofs
• Have student draw auxiliary lines as
necessary for their proofs
Students will accurately analyze
and write proofs using SAS, ASA
and the SSS postulates and the
CPCTC principle
November
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 6 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
G.G.4 Draw congruent
and similar figures using
compass, straightedge,
protractor, or computer
software. Make
conjectures about
methods of construction.
Justify the conjectures by
logical arguments.
(10.G.2)
Students will KNOW:
�How to use a ruler and protractor to draw
congruent triangles
Students will be able to DO:
�Draw an angle of a specific measurement
using a protractor
�Draw segments of a specific measurement
using a ruler
�Make conjectures about congruency of
triangles from drawings
• Demonstrate the SAS postulate using a
ruler and protractor
• Have students discover the ASA and the
SSS postulates using a ruler and a
protractor
• Define the CPCTC principle
• Students will accurately
analyze and write
proofs using SAS, ASA
and the SSS postulates
and the CPCTC
principal
November
G.G.5 Apply congruence
and similarity
correspondences (e.g.,
∆ABC ≅ ∆XYZ) and
properties of the figures
to find missing parts of
geometric figures, and
provide logical
justification. (10.G.4)
Students will KNOW: �Understand what congruent figures are
�How to identify the corresponding parts
�Understand the reflexive property
Students will be able to DO:
�Identify congruent geometric figures
�Identify the corresponding parts
�Use the reflexive property in proofs.
• Define congruency and corresponding parts
• Define reflexive property and give examples
of their use
• Have students identify congruent figures and
congruent corresponding parts
Students can accurately work with
congruent figures and their
corresponding parts
November
G.G.6 Apply properties
of angles, parallel lines,
arcs, radii, chords,
tangents, and secants to
solve problems.
Students will KNOW: �Recognize some basic properties of circles
Students will be able to DO: �Use the congruency of radii of a circle in
proofs
• Have students recognize that the radii of a
circle are congruent
• Student properly use
congruent radii in their
proofs
November
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 7 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
G.G.8 Use the properties
of special triangles (e.g.,
isosceles, equilateral,
30º–60º–90º, 45º–45º–
90º) to solve problems.
(10.G.6)
Students will KNOW:
�The various types of triangles and their parts
Students will be able to DO:
�Identify triangles according to congruent
sides
�Identify triangles according to angles
measures
�Apply the definitions of the types f triangles
and angles to solve numerical problems and in
proofs
• Define scalene, isosceles, and equilateral
triangles
• Define acute, right, obtuse and equiangular
triangles
• Have students solve problems using different
types of triangles
• Students will be able to
solve numerical
problems with different
types of triangles
• Students will solve
proofs involving
different types of
triangles
November
UNIT IV - Lines in the Plane G.G.2 Write simple
proofs of theorems in
geometric situations,
such as theorems about
congruent and similar
figures, parallel or
perpendicular lines.
Distinguish between
postulates and theorems.
Use inductive and
deductive reasoning, as
well as proof by
contradiction. Given a
conditional statement,
write its inverse,
converse, and
contrapositive.
Students will KNOW:
�How to incorporate the concepts of:
equidistance and perpendicular bisection in
their proofs.
Students will be able to DO: �use definitions, postulates and theorems
related to equidistance and perpendicularity
and perpendicular bisection in a proof.
• Review the definitions of equidistance,
perpendicularity and perpendicular bisection.
• Review the postulates and theorems related to
equidistance, perpendicularity and
perpendicular bisection.
• Apply these definitions, postulates and
theorems to their two-column proofs.
• Students will
demonstrate an
understanding of the
definitions, postulates
and theorems related to
equidistance and
perpendicularity by
using perpendicular
bisectors in their proofs. December
G.G.5 Apply congruence
and similarity
correspondences (e.g.,
∆ABC ≅ ∆XYZ) and
properties of the figures
to find missing parts of
geometric figures, and
Students will KNOW: �How to use a detour proof to show
congruencies of triangles or missing
angles/sides.
�How to organize information and draw
diagrams for problems presented in words.
• Demonstrate how to prove a pair of triangles
congruent in order to use a pair of
corresponding parts from those triangles to
prove a different pair of triangles congruent
• Have students look at several examples and
decide which of them require a detour proof
• Students will show an
understanding of detour
proofs by recognizing
when they do not have
enough information to
prove given figures
congruent. They will
December
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 8 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
provide logical
justification. (10.G.4)
Students will be able to DO: �To prove more than one pair of triangles
congruent in order to solve the problem.
�Read the problem, identify the givens,
identify what needs to be proven, draw and
label an appropriate diagram and write a
formal or paragraph proof.
• Have students look for the triangles that
would need to be proven first
• Have students work in pairs to complete a
“detour” proof
• Students will practice reading a problem with
no diagram; they will then draw and label a
diagram that depicts the written description
• Once a labeled diagram is drawn, students
will determine a list of the givens and decide
what needs to be proven
• Students will use their diagrams and the
givens to complete their proof
have to prove another
pair of triangles
congruent first in order
to use their congruent
corresponding parts for
the given proof.
• Students will be able to
organize the
information in and draw
appropriate diagrams
for problems presented
in words.
G.G.6 Apply properties
of angles, parallel lines,
arcs, radii, chords,
tangents, and secants to
solve problems.
Students will KNOW: �How to show that two angles are right angles
if they are both supplementary and congruent
Students will be able to DO: �Prove perpendicularity by showing angles
are right angles
• Using an algebraic proof, students will
understand that two angles that are both
supplementary and congruent add up to
ninety degrees and therefore are right angles.
• Students will then apply this relationship
to prove that two lines are perpendicular.
• Students will
demonstrate their ability
to prove that lines are
perpendicular by
finding right angles.
December
G.G.8 Use the properties
of special triangles (e.g.,
isosceles, equilateral,
30º–60º–90º, 45º–45º–
90º) to solve problems.
(10.G.6)
Students will KNOW: �How to apply the equidistance theorems to
prove that triangles are isosceles
Students will be able to DO: �Prove that a triangle is isosceles by using an
equidistance theorem
• Define distance: distance between two points
is the length of the segment that joins them.
• Define equidistant: if two points are the same
distance from a third point, they are said to be
equidistant.
• Define perpendicular bisector: a line that both
bisects and is perpendicular to a segment.
• Students will be
proving that a given
triangle is isosceles by
showing that points are
equidistant from the
endpoints of the base.
December
G.G.12 Using rectangular
coordinates, calculate
midpoints of segments,
slopes of lines and
segments, and distances
between two points, and
apply the results to the
solutions of problems.
Students will KNOW: �The midpoint formula
�Understand the concept of slope
�Determine the slope of a line using a formula
�The relationship between the slopes of
parallel and perpendicular lines
Students will be able to DO: �Calculate the midpoint of segments using the
midpoint formula
• Define midpoint: a point or segment that
divides a segment into two congruent parts.
• Have the students find the midpoint of a line
segment by measuring.
• Have the students use the formula to find the
midpoint of a diagonal line.
x1 + x2
2,y1 + y2
2
• Review interpretation of a graph with slope
• Students will be able to
tell if a line is
horizontal, vertical, or
diagonal by its slope.
• Students will correctly
determine the slope of a
line by applying the
slope formula.
• Students will
demonstrate an
December
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 9 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
�Apply the definition of equidistance to solve
problems
�Recognize that a rising line has a positive
slope, a falling line has a negative slope, a
horizontal line has a zero slope and a vertical
line has an undefined slope
�Use the slope formula to determine the slope
of a line.
�Calculate slope and then determine if the
lines are parallel, perpendicular or neither
that is positive, negative, zero and no slope.
• Have students look at lines drawn on a
coordinate plane. They will find the slope by
counting how far up/down and how far to the
right the line travels
• Define the slope formula as ∆y
∆x
• Students will use the formula:y2 − y1
x2 − x1
= m
• Students will find the slopes of two or more
lines; they will then determine if the lines are
parallel (same slopes) or perpendicular
(slopes are opposite reciprocals)
•
understanding of the
relationship between
slope and lines being
parallel or
perpendicular.
UNIT V - Parallel Lines and Related Figures G.G.1 Recognize special
types of polygons (e.g.,
isosceles triangles,
parallelograms, and
rhombuses). Apply
properties of sides,
diagonals, and angles in
special polygons; identify
their parts and special
segments (e.g., altitudes,
midsegments); determine
interior angles for regular
polygons. Draw and label
sets of points such as line
segments, rays, and
circles. Detect
symmetries of geometric
figures.
Students will KNOW: �How to recognize and name four sided
polygons
�Recognize diagonals
�The names and definitions of special
quadrilaterals
Students will be able to DO:
�Differentiate between figures that are
polygons and those that are not polygons
�Name polygons according to the number of
sides
�Name diagonals in polygons
�Name and identify parallelograms,
rectangles, rhombuses, kites, squares,
trapezoids and isosceles trapezoids
• Teacher will show a number of pictures of
figures for students to differentiate a
polygons or not
• Teacher will define the special quadrilaterals
• Students will use prior knowledge and
definitions to identify special quadrilateral
• Students will draw different quadrilaterals
according to definition and use measurement
of segments and angles to identify properties
• Students will compile a list of the properties
of special quadrilaterals
• Students will be able to
recognize figures as
being a polygon
• Students will be able to
accurately name each of
the special
quadrilaterals
• Students’ abilities to
discover the unique
properties of specific
quadrilaterals
January
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 10 of 21
MA
Standards Priority
Curriculum
Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
G.G.2 Write simple
proofs of theorems in
geometric situations,
such as theorems about
congruent and similar
figures, parallel or
perpendicular lines.
Distinguish between
postulates and theorems.
Use inductive and
deductive reasoning, as
well as proof by
contradiction. Given a
conditional statement,
write its inverse,
converse, and
contrapositive.
Students will KNOW:
�The indirect proof procedure
�The various methods to prove that lines are
parallel
�The parallel line postulate
�The methods to prove that a quadrilateral is a
parallelogram or a rectangle, rhombus, kite,
square, trapezoid or an isosceles trapezoid
Students will be able to DO: �Use contradiction to solve indirect proofs
�Prove that lines are parallel by showing
alternate interior, corresponding, alternate
exterior angles are congruent or that same side
interior, same side exterior angles are
supplementary or that lines are perpendicular
to the same line
�To use the properties of the special
quadrilaterals in their proofs
• The teacher will demonstrate an indirect
proof.
• The students will practice an indirect proof
by assuming the opposite of the “to prove” is
true and then find a contradiction to one of
the givens.
• Students will
demonstrate an
understanding of
indirect proofs by
applying the idea of
contradiction
• Students will correctly
identify all angles
associated with parallel
lines to accurately use a
variety of methods to
prove that lines are
parallel.
January
G.G.5 Apply congruence
and similarity
correspondences (e.g.,
∆ABC ≅ ∆XYZ) and
properties of the figures
to find missing parts of
geometric figures, and
provide logical
justification. (10.G.4)
Students will KNOW: �The properties of each of the special
quadrilaterals
�The methods to prove that a quadrilateral is a
parallelogram or a rectangle, rhombus, kite,
square, trapezoid or an isosceles trapezoid
Students will be able to DO:
�Identify the properties and use these
properties to solve problems
�To use the properties of the special
quadrilaterals in their proofs
• Teacher will demonstrate the use of
numerical and algebraic calculations to find
missing parts of figures
• Students will independently practice finding
missing parts of figures
• Students will work in groups to apply the
properties of special quadrilaterals to their
proofs
• Students will show
mastery of using
numerical and algebraic
calculations to find
missing parts of a figure
• Students will correctly
apply the properties of
the special
quadrilaterals to their
proofs
January
G.G.6 Apply properties
of angles, parallel lines,
arcs, radii, chords,
tangents, and secants to
Students will KNOW: �How to identify pairs of angles formed by a
transversal cutting two parallel lines
�How to prove that angles are congruent
• Define alternate interior angles,
corresponding angles, vertical angles,
alternate exterior angles and same side
interior and same side exterior angles.
• Students will accurately
identify specific pairs of
angles associated with
parallel lines
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 11 of 21
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solve problems. associated with parallel lines.
Students will be able to DO: �Be able to identify alternate interior angles,
corresponding angles, alternate exterior
angles, same side interior and same side
exterior angles
�Prove that corresponding, alternate exterior
angles, and alternate interior angles are
congruent or that same side interior, same side
exterior angles are supplementary or that lines
are perpendicular to the same line
• Students will explore the ways in which to
prove lines parallel by applying the above
definitions.
• Students will apply theorems relating interior
and exterior angles on the same side of the
transversal as being supplementary.
• Students will apply the theorem that states if
two coplanar lines are perpendicular to a
third line they are parallel.
• Students will draw a pair of parallel lines that
are cut by a transversal and they will identify
each of the specific pairs of resulting angles.
January
UNIT VI - Polygons G.G.1 Recognize special
types of polygons (e.g.,
isosceles triangles,
parallelograms, and
rhombuses). Apply
properties of sides,
diagonals, and angles in
special polygons; identify
their parts and special
segments (e.g., altitudes,
midsegments); determine
interior angles for regular
polygons. Draw and label
sets of points such as line
segments, rays, and
circles. Detect
symmetries of geometric
figures.
Students will KNOW: �That the sum of the three angles of a triangle
equals 180°
�How to find the number of diagonals for a
given polygon
�The definition of a regular polygon
�How to find the measure of each exterior
angle of a regular polygon
Students will be able to DO: �Use the formulas to calculate the sum of the
interior and exterior angles of a given polygon
�Use the formula to calculate the number of
diagonals for a given polygon
�Apply the definition of regular polygons
�Find the measure of each exterior angle of a
regular polygon knowing the number of sides
�Find the number of sides of a regular
polygon knowing the measure of the exterior
angle
• Teacher will use the parallel postulate to
demonstrate the sum of the angles of a
triangle are 180 degrees
• Students will work in small groups drawing
polygons and counting the number of
diagonals form one vertex and then total
number of diagonals. After doing this for
several polygons they should be able to write
an equation for the total number of diagonals
for any polygon.
• Given the definition of regular polygon, the
students will to calculate the number of
degrees in each interior angle and then use
supplementary angles to find the number of
degrees in an exterior angle.
• Students will be given a formula to find the
number of degrees in each exterior angle.
• The students will apply the same formula to
find the number of sides of a regular given
the degrees in one of its exterior angles
• Teacher will lead the students to prove the
no-choice theorem and the angle, angle, side
• Students will accurately
calculate a missing
angle of a triangle based
on the total number of
degrees being 180°
• Students will be able to
find the number of
diagonals for any given
polygon by using the
correct formula.
• Students will be able to
calculate the size of an
exterior angle given the
number of sides or the
number of sides when
given the size of the
exterior angle
• Students will be able to
accurately apply the no-
choice theorem and
angle, angle, side
theorem to their proofs.
February
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 12 of 21
MA
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theorem
• Teacher will lead the students to prove
congruency between triangles and to find the
missing angle by applying the no-choice
theorem and the angle, angle, side theorem.
G.G.2 Write simple
proofs of theorems in
geometric situations,
such as theorems about
congruent and similar
figures, parallel or
perpendicular lines.
Distinguish between
postulates and theorems.
Use inductive and
deductive reasoning, as
well as proof by
contradiction. Given a
conditional statement,
write its inverse,
converse, and
contrapositive.
Students will KNOW: �The no-choice theorem
Angle, Angle, Side theorem
Students will be able to DO: �To prove that if two angles of one triangle
are congruent to two angles of another
triangle, the third angles of each are congruent
to each other
�Prove that two triangles are congruent by
showing that two angles and the non-included
side of one triangle are congruent to the
corresponding sides of the other
• Teacher will use the parallel postulate to
demonstrate the sum of the angles of a
triangle are 180 degrees
• Given the definition of regular polygon, the
students will to calculate the number of
degrees in each interior angle and then use
supplementary angles to find the number of
degrees in an exterior angle.
• Students will be given a formula to find the
number of degrees in each exterior angle.
• The students will apply the same formula to
find the number of sides of a regular given
the degrees in one of its exterior angles
• Students will accurately
calculate a missing
angle of a triangle based
on the total number of
degrees being 180°
• Students will be able to
find the number of
diagonals for any given
polygon by using the
correct formula.
• Students will be able to
calculate the size of an
exterior angle given the
number of sides or the
number of sides when
given the size of the
exterior angle.
February
G.G.7 Solve simple
triangle problems using
the triangle angle sum
property, and/or the
Pythagorean theorem.
(10.G.5)
Students will KNOW: �The triangle angle sum property
�The exterior angle of a triangle property
�The midline theorem
Students will be able to DO:
�Use the sum of the measures of the interior
angles of a triangle is 1800 to find the measure
of a missing angle
�Use the measure of the exterior angle of a
triangle is equal to the sum of the two remote
interior angles to solve problems
�Find missing angles and the length of the
third side of a triangle by applying the midline
theorem
• The teacher will lead the students to prove
the midline theorem
• Students will practice applying the midline
theorem to their proofs
• Students will know
when it is appropriate to
apply the midline
theorem to their proofs.
February
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 13 of 21
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UNIT VII - Similar Polygons G.G.2 Write simple
proofs of theorems in
geometric situations,
such as theorems about
congruent and similar
figures, parallel or
perpendicular lines.
Distinguish between
postulates and theorems.
Use inductive and
deductive reasoning, as
well as proof by
contradiction. Given a
conditional statement,
write its inverse,
converse, and
contrapositive.
Students will KNOW: �How to prove triangle similar
�The concept of similarity to establish the
congruence of angles and proportionality of
segments
Students will be able to DO: �Prove that triangles are similar by AA, SAS
similarity, SSS similarity theorems
�Prove that angles are congruent and
segments are proportional by establishing the
similarity of two triangles
• Define similar: figures with the same shape
but not necessarily the same size.
• Have students look around the room to
discover shapes that appear to be similar
• Similar polygons can either be dilations
(enlargements) or reductions.
• Have students look at examples of figures
that are either a dilation or a reduction.
• Define similar polygon: the corresponding
angles are congruent and the corresponding
sides are proportional
• Students will be able to
recognize similar
polygons
• Students will be able to
correctly find the
missing sides of similar
polygons
• Students will be able to
correctly identify figure
that have been dilated
or reduced
March
G.G.5 Apply congruence
and similarity
correspondences (e.g.,
∆ABC ≅ ∆XYZ) and
properties of the figures
to find missing parts of
geometric figures, and
provide logical
justification. (10.G.4)
Students will KNOW: �How to work with ratios and with
proportions
�The definition and characteristics of similar
polygons
�The sum of the measures of the interior
angles of a triangle is 1800
�The side splitter theorem
�The parallel line theorem
�The angle bisector theorem
Students will be able to DO:
�Solve proportion examples
�Find the geometric mean/mean proportional
�Find missing terms of proportions
�Apply the definition of similar polygons to
find a missing length of a side and the
measure of an angle
�Set up a proportions to find the lengths of
• Have students practice finding a missing side
of similar figures by using proportions
• Teacher explanation of the theorem that
states: the ratio of the perimeters of two
similar polygons equals the ratio of any pair
of corresponding sides
• The teacher will define ratio as the quotient
of two numbers
• Show that slope can be shown as a ratio:
rise/run
• Define proportion: an equation between two
ratios, using a colon, or as a product.
• Student will practice solving proportions by
multiplying opposite corners and dividing by
the coefficient of the resulting variable term.
• The teacher will show that “the product of the
means is equal to the product of the
extremes” is another way of saying how the
• Students will be able to
find either a missing
side or perimeter given
the other information
• Students will be able to
calculate arithmetic
ratios and solve
proportions.
• Students will be able to
identify and solve for
the first, second, third
and fourth proportional
• Students will be able to
apply the concept of the
geometric mean/mean
proportional to solve
examples.
• Students will be able to
March
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 14 of 21
MA
Standards Priority
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segments class has been solving proportions.
• Use the four terms:
1st proportional 3rd proportional
2nd proportional 4th proportional
• Define arithmetic mean as an average
• Define geometric mean/mean proportional:
where the means are the same number
• Together the class will prove that if a line is
drawn parallel to one side of the triangle; it
divides the other two sides proportionally
• Students will be given a sheet with three
parallel lines drawn that have been cut by a
transversal; they will measure the resulting
segments.
• They will draw a second transversal and
again measure the resulting segments. They
will form a generalization: the parallel lines
divide the transversals proportionally
• The student will work as a group to prove
that if a ray bisects an angle of a triangle, it
divides the opposite sides into segments that
are proportional.
• Students will review the definition of similar
and apply the AAA postulate to prove that
triangles are similar.
• Students will apply the no choice theorem to
understand that only two angles must be
congruent to have similar triangles.
• Students will read examples that show how to
prove that triangles are similar by using the
SSS similarity and SAS similarity.
• Students will practice proving triangles
similar using a variety of methods.
apply the concept of
similarity to polygons
and then find a missing
side by using
proportions.
• Students will correctly
apply the side-splitter
theorem, the angle
bisector theorem and
the theorem that states
that transversals are
divided proportionally.
• Students will prove
similarity between two
triangles by applying
the methods of AAA,
AA, SSS ~ , and SAS ~
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 15 of 21
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UNIT VIII - Right Triangle Properties G.G.5 Apply congruence
and similarity
correspondences (e.g.,
∆ABC ≅ ∆XYZ) and
properties of the figures
to find missing parts of
geometric figures, and
provide logical
justification. (10.G.4)
Students will KNOW: �how to simplify radicals
�how to solve quadratic equations
�Relationships between the parts of a triangle
when the altitude is drawn to the hypotenuse
Students will be able to DO: �Simplify radical expressions
�Solve quadratic equations either by using the
quadratic formula or by factoring
�Determine similarity of triangles when the
altitude is drawn to the hypotenuse
�Find the measures of segments using a
proportion
• Students will review how to simplify radical
expressions
• Students will review factoring quadratics and
the will solve quadratic equations by
factoring
• Students will draw a right triangle with an
altitude to the hypotenuse. They will make
conjectures based on this drawing.
• The group will prove that the triangles
formed by an altitude to the hypotenuse are
similar.
• The class will prove that one leg is the mean
proportional to the hypotenuse and segment
of the hypotenuse adjacent to it.
• The class will prove that the altitude is the
mean proportional to the hypotenuse and
segment of the hypotenuse adjacent to that
leg.
• Students will practice finding the lengths of
the missing segments by applying these
theorems.
• Students will be able to
accurately simplify
radicals
• Students will be able to
solve quadratic
equations both by
factoring and by using
the quadratic formula
• Students prove triangles
formed by an altitude to
a hypotenuse are similar
• Students will use the
mean proportional
theorems to find the
length of either a leg or
a segment of the
hypotenuse
April
G.G.7 Solve simple
triangle problems using
the triangle angle sum
property, and/or the
Pythagorean theorem.
(10.G.5)
Students will KNOW:
�Pythagorean theorem and its converse
�Recognize groups of whole numbers known
as Pythagorean triples
Students will be able to DO: �Apply the Pythagorean theorem to find the
length of a missing side
�Determine if a triangle is a right triangle by
squaring the measures of the three sides
�Determine the missing side of a right triangle
using a Pythagorean triple
• Students will investigate to find the
relationship between the sum of the square of
the legs of a right triangle and the square of
the hypotenuse.
• Students will look at the physical
demonstration of the Pythagorean
relationship (on-line)
• Students will practice using the Pythagorean
relationship to find the lengths of either leg or
the hypotenuse.
• Students will the converse of this theorem to
prove tat the triangles given are indeed right
triangles.
• Students will recognize the families of whole
• Students will use
radicals and factoring as
hey apply to the
Pythagorean theorem
and to the altitude on
the hypotenuse theorem
• Students will use the
Pythagorean triples as a
shortcut to solve for the
missing sides
April
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 16 of 21
MA
Standards Priority
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numbers known as the Pythagorean triples
• Students will practice solving for the missing
side of a right triangle by using the
Pythagorean triples or their multiples
G.G.8 Use the properties
of special triangles (e.g.,
isosceles, equilateral,
30º–60º–90º, 45º–45º–
90º) to solve problems.
(10.G.6)
Students will KNOW: �The ratio of the side lengths of a 30-60-90
triangle
�The ratio of the side lengths of a 45-45-90
triangle
Students will be able to DO:
�Use the ratios of a 30-60-90 and a 45-45-90
triangle to calculate the missing sides
• Students will draw a 30°, 60°, 90° triangle;
they will measure the shortest side and the
hypotenuse. They should notice that the
hypotenuse is always twice as big as the
shortest side.
• They should apply the Pythagorean theorem
to find the length of the third side (in
simplified form); it will always be x√3.
• Following a similar procedure the students
will investigate the lengths of the sides of a
45°, 45°, 90° triangle; they should discover
that the legs are the same and that the
hypotenuse is always the length of the leg
times √2.
• Students will be able to
find the lengths of
either a leg or a
hypotenuse of a
“special” right triangle
by using the discovered
relationships rather than
having to go through
using the Pythagorean
theorem.
April
G.G.9 Define the sine,
cosine, and tangent of an
acute angle. Apply to the
solution of problems.
Students will KNOW: �The three basic trigonometric relationships
�How to use trigonometric ratios to solve right
triangle problems
Students will be able to DO: �Determine the sine, cosine and tangent of a
given triangle
�Use trigonometric ratios to calculate the
measures of missing sides and angles
• Students will do an exploratory activity to
discover the three basic trigonometric
relationships,
• Students will practice writing the sine, cosine
and tangent for many given triangles as a
fraction.
• Students will learn how to use the table of
trig ratios.
• Students will be given a right triangle with a
side and an angle labeled. They will have to
find the missing part by applying a
trigonometric function.
• Students will expand this to problems where
they must first draw a diagram, label the
given parts and find a missing part by
applying the trigonometric relationships.
• Students will be able to
correctly recognize
which trig function they
need to use for a given
example.
• Students will accurately
find the missing side or
angle by using the trig
function.
• Students will be able to
apply these functions to
answer word problems.
April
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 17 of 21
MA
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G.G.12 Using rectangular
coordinates, calculate
midpoints of segments,
slopes of lines and
segments, and distances
between two points, and
apply the results to the
solutions of problems.
Students will KNOW:
�The distance formula to compute the lengths
of segments in the coordinate plane
Students will be able to DO:
�Calculate the distance between two points
• Students will look at a right triangle drawn on
a coordinate plane. They will label the
points. They will find the vertical and
horizontal lengths by counting spaces and
they will then use the Pythagorean theorem to
find the length of the hypotenuse.
• Based on the above activity students will try
to discover the length of a hypotenuse
without using the Pythagorean theorem.
They will be looking toward the discovery of
the distance formula.
• Once they have found the formula, students
will be practicing finding the lengths of any
side by using the distance formula:
(x2 − x1)2 + (y2 − y1)
2
• Students will apply this formula to a variety
of situations
• Students will accurately
evaluate the distance
between two points and
they will apply this
formula to a variety of
situations
April
G.M.4 Describe the
effects of approximate
error in measurement and
rounding on
measurements and on
computed values from
measurements. (10.M.4)
Students will KNOW: �Rounding before calculations will skew the
results
Students will be able to DO: �Recognize inaccurate results due to rounding
errors or rounding before calculating
• Half the students will be asked to solve an
example without any rounding until they
complete the examples; the other half will
round each time they have decimals within
their work.
• Upon looking at the results of this
investigation, students will have found that if
one were to round their numbers within the
problem rather than waiting to they have an
answer will significantly change the results of
the answers.
• Students will be judged
on their understanding
of the effects of
rounding with the work
they do using the
Pythagorean theorem
and with applying the
distance formula.
April
UNIT IX - Area G.M.1 Calculate
perimeter, circumference,
and area of common
geometric figures such as
parallelograms,
trapezoids, circles, and
triangles. (10.M.1)
Students will KNOW: �Understand the concept of area
�Find the area of rectangles, squares and
related irregular figures
�Determine the area of a variety of polygons
�Determine the area of a circle and portions of
a circle
• Define area
• Students should understand that:
• Every closed region has an area
• If two closed figures are congruent, then
their areas are equal
• If two closed regions share a common
• Students will show that
they understand the
definition of area and its
implications
• Students will be able to
identify the bases and
May
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 18 of 21
MA
Standards Priority
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Students will be able to DO: �Define area
�Estimate areas of irregular figures by
determining the approximate number of square
units
�Find the area of polygons by applying the
given formulas.
�Find the area of circles, sectors and segments
of circles by applying formulas
boundary, then the area of the entire
figure is the sum of the individual areas.
• Students will practice using the formulas
for finding the area of: rectangles,
squares, parallelograms, and triangles.
• Students will need to correctly identify
the base and the heights of the
parallelograms and triangles.
• Students will be able to use either the
radius or the diameter to find the area of
the circle.
• They will find the area of a sector by
multiplying the area of the circle times
the fractional part of the circle (i.e. if an
arc measures 600, then the sector is 600 ÷
3600 or 1/6 of the circle.
• Students will use three different methods
for finding the area of a trapezoid.
• Breaking the trapezoid into three figures
• Using the formula
• Using the product of the average of the
bases and the height.
• Once they have practiced all three they
may determine the one that works best
for them.
• Students will recognize radii vs.
apothems.
• Students will apply three theorems
related to area:
• If two figures are similar, then the ratio of
their areas equals the square of the ratio
of corresponding segments.
• A eq∇ = s
2
43
the heights in order to
correctly apply the
formulas for areas of
rectangles, squares,
parallelograms, and
triangles.
• Students will be able to
use either the radius or
diameter to find the area
of circles and they will
be able to find the area
of a sector of a circle.
• Students will
successfully use the
formulas related to
similar figures and to
regular polygons.
• Students will
demonstrate that they
can use a variety of
methods to find the area
of both trapezoids and
kites.
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 19 of 21
MA
Standards Priority
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• A reg.poly. = 1
2ap where a is the length of
the apothem and p is the perimeter.
• Students will find the area of a kite by
dividing the kite into isosceles triangles
and using the area of a triangle formula
• Students will use an alternate method for
finding the area of a kite: area equals half
the product of the diagonals.
G.M.3 Relate changes in
the measurement of one
attribute of an object to
changes in other
attributes, e.g., how
changing the radius or
height of a cylinder
affects its surface area or
volume. (10.M.3)
Students will KNOW: �How to relate the change of one attribute to
the overall change in the area of a figure
Students will be able to DO: �Increase or decrease the size of the length,
width or height etc. and calculate the overall
effect this will have on the area of reduction or
the dilation of a figure
• Students will continue to practice finding
areas of all figure and they will calculate
the overall effect changing the size of an
attribute has on the area of the new figure
• Students will show that
they understand the
relationship between
changing the size of an
attribute and the overall
change it will have on
the areas of the given
figures.
May
UNIT X - Surface Area and Volume G.G.16 Demonstrate the
ability to visualize solid
objects and recognize
their projections and
cross sections. (10.G.10
Students will KNOW: � Various solid (three dimensional) figures
Students will be able to DO: �Recognize and draw various three
dimensional figures
�To draw representations of various solid
figures
�Visualize the projections and cross sections
of various solid figures
• Students will use visual models of the three
dimensional figures to understand what is
meant by the base of the figure, the lateral
edges, lateral faces.
June
G.M.2 Given the
formula, find the lateral
area, surface area, and
volume of prisms,
pyramids, spheres,
cylinders, and cones, e.g.,
find the volume of a
Students will KNOW: �How to derive and apply formulas for lateral
area, surface area and volume of various
geometric figures
Students will be able to DO: �Students will derive and apply formulas for
• Students will find the lateral surface area of
any prism by finding the sum of the areas of
the lateral faces.
• Students will find the total surface area by first
finding the lateral surface area and then adding
the sum of each of the figures bases.
• Students will be able to
choose the appropriate
formula and to
accurately use the
formulas for lateral
surface area and total
surface areas for prisms,
June
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 20 of 21
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sphere with a specified
surface area. (10.M.2)
lateral area and surface area of prisms,
pyramids, circular solids
�Students will derive and apply formulas for
volume of prisms, cylinders, pyramids, cones
and spheres
• Students will work with cylinders, cones, and
spheres to determine their respective formulas
for lateral and/or total surface areas.
• Students will practice applying the correct
formulas to find the lateral and/or total surface
areas of cylinders, cones, and spheres.
• Volume will be defined as the measure of the
space enclosed by a solid.
• By looking at a physical representation of a
rectangular prism, the students will come to
understand that the volume is simply the area
of the base times the height of the prism.
• Students will translate this understanding to
other prisms and to a cylinder.
• Students will practice using the formulas for
the volume of prisms and cylinders.
• By looking at physical representations of a
pyramid and a prism with the same base and
height, the students will understand that the
volume of this pyramid is one-third that of the
volume of the prism.
• Students will practice using the formulas for
the volumes of pyramids and cones.
• Students will use the formulas for the volume
of the sphere.
cylinders, cones, and
spheres.
• Students will
demonstrate an
understanding of the
concept of volume.
• Students will accurately
apply the formulas for
volume of prisms,
cylinders, pyramids and
cones.
G.M.3 Relate changes in
the measurement of one
attribute of an object to
changes in other
attributes, e.g., how
changing the radius or
height of a cylinder
affects its surface area or
volume. (10.M.3)
Students will KNOW: �How to relate the change of one attribute to
the overall change in the surface area or
volume of a figure
Students will be able to DO: � Increase or decrease size of the length, width
or height etc. and calculate the overall effect
this will have on the surface area or volume of
the reduction or the dilation of a figure.
• Students will continue to practice finding
volumes of all figures and they will calculate
the overall effect changing the size of an
attribute has on the volume of the new figure.
Students will show that they
understand the relationship
between changing the size of an
attribute and the overall change it
will have on the areas of the given
figures. June
Gateway Regional School District
SCOPE & SEQUENCE
Geometry - Standard
Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry
January 2009
Page 21 of 21
MA
Standards Priority
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Benchmarks
Possible Instructional
Strategies
Evidence of Student
Learning (Assessment) Month
G.M.4 Describe the
effects of approximate
error in measurement and
rounding on
measurements and on
computed values from
measurements. (10.M.4)
Students will KNOW:
�Understand the effect of a rounding error on
the overall computed surface areas and
volumes
Students will be able to DO: �See that rounding a measurement too soon or
by making an error in rounding can greatly
affect the overall calculations when finding
surface areas and volumes
• Different students will use the same formulas
with various decimal values (tenth,
hundredth, thousandth). Upon investigating,
the students should recognize rounding to
different places would have an effect on the
answer to surface area and volume examples.
Within their work on surface
areas and volumes, the students
will know when to round and to
what decimal value.
June