Download Bensalem Township School District Geometry Curriculum Based

Bensalem Township School District
Geometry Curriculum
Based upon the Pennsylvania State Standards,
Assessment Anchors and Eligible Content
For the Keystone Exams:
Geometry
Pennsylvania Department of Education
www.education.state.pa.us
2011
Last Updated April 2011
Unit 1 - Basic Terms and Definitions
Standards Link:
2.5.G.B.Use symbols, mathematical terminology, standard notation, mathematical rules,
graphing, and other types of mathematical representations to communicate observations,
predictions, concepts, procedures, generalizations, ideas, and results.
2.8.G.B. Use algebraic representations to solve problems using coordinate geometry
Big Idea:
Points, lines, angles, and planes are the
foundation blocks for the study of
geometry.
Enduring Understanding(s):
The foundation of Euclidean geometry is the
undefined terms point, line, angle and plane.
Parallel lines with transversals create angle
pairs with congruent or supplementary
relationships.
Relationship between parts and wholes of
segments and angels.
Essential Question(s):
What are the foundation blocks for the structure of geometry?
What are the techniques used to measure segments and angles?
How are angle pairs related?
Knowledge:
Skill(s):
There are 3 undefined terms that play
an important role in Geometry.
(2.5.G.B)
Identify points as collinear and/or coplanar
(2.5.G.B)
There are different characteristics of a
line, ray and line segment. (2.5.G.B)
The lengths of a segment can be used to
determine congruency. (2.8.G.B)
The segment addition postulate can be
used to find missing lengths of
segments. (2.8.G.B)
Distinguish the difference between parallel and
skew lines. (2.5.G.B)
Given a picture, identify and name the
following objects: point, line, segment, plane,
ray, angle, parallel, perpendicular, and skew
lines (2.5.G.B)
Find the length of a segment (segment addition
postulate) (2.8.G.B)
Angles can be classified and acute,
right, obtuse or straight, based on its
measure. (2.5.G.B)
The measures of angles can be used to
determine congruency. (2.8.G.B)
The angle addition postulate can be
used to find missing measures of
angles. (2.8.G.B)
Two angles may be classified as
complementary or supplementary if the
sum of their measures is 90 or 180.
(2.5.G.B)
The distance formula can be used to
find the length of a segment. (2.8.G.B)
Find the measure of angle (angle addition
postulate) Problems should require the use of
algebraic equations to solve. 2.8.G.B
Using a protractor, draw various angles of
given measurements. (2.5.G.B)
Find the measure of an angle and/or variable
knowing that two angles are complementary or
supplementary. (2.8.G.B)
Calculate the distance of a segment using the
distance formula. (2.8.G.B)
Determine the midpoint of a segment using the
midpoint formula. (2.8.G.B)
Find the missing endpoint of a segment when
given one endpoint and its midpoint. (2.8.G.B)
The midpoint formula can be used to
find the midpoint and the endpoint of a
segment. (2.8.G.B)
Assessment/Evidence of Learning:
Chapter 1 Common Assessment – found on staff drive
Learning Activities:
Mrs. Fischer had to go to two dinners, one right after the other. Her parents live at
(7, -11) and her brother lives at (4, -5). Her brother’s house is located at the
midpoint between Mrs. Fischer and her parents.
a. Find the coordinates of Mrs. Fischer’s house.
How far she will travel to eat at her parents’ house?
Unit 2 - Reasoning and Proof
Standards Link:
2.4.G.A. Write formal proofs (direct proofs, indirect proofs/proofs by contradiction,
use of counter-examples, truth tables, etc.) to validate conjectures or arguments.
2.4.G.B. Use statements, converses, inverses, and contrapositives to construct valid
arguments or to validate arguments relating to geometric theorems.
Big Idea:
Enduring Understanding(s):
Reasoning and Proof
There are direct and indirect ways of coming
to a conclusion or proving something.
Essential Question(s):
How do proofs support geometric reasoning?
How do you prove theorems and how do you demonstrate conjectures?
Knowledge:
Skill(s):
Conditional statements are the
foundation for theorems and postulates
in Geometry. (2.4.G.B)
Proofs are used to support geometric
reasoning through a logical sequence of
statements and justifications. (2.4.G.A)
Proofs can be written in a 2-column,
paragraph or flow format. (2.4.G.A)
Inductive reasoning is drawing
conclusions based on a pattern.
(2.4.G.A)
Identify the different forms of a conditional
statement. (2.4.G.B)
Write a conditional statement in the different
forms. (2.4.G.B)
Recognize proper definitions using
biconditionals. (2.4.G.B)
Recognize conditional statements and be able
to write the converse, inverse, biconditional
and contrapositive. (2.4.G.B)
Deductive reasoning is drawing
Identify the truth value of each statement.
conclusions based on logical progression (2.4.G.A)
of truths. (2.4.G.A)
Draw conclusions using deductive reasoning,
the law of syllogism and the law of
detachment. (2.4.G.B)
Use equality, transitive, symmetric, reflexive,
substitution and distributive properties to
write formal algebraic proofs (in 2-column
form). (2.4.G.A )
Justify statements about congruent segments
and angles using properties above (including
proof of the vertical angle theorem). (2.4.G.A)
**Honors** Prove all theorems. (2.4.G.A)
Assessment/Evidence of Learning:
Ad Project – Found in Prentice Hall Resources
Chapter 2 Common Assessment – found on staff drive
Unit 3 - Parallel and Perpendicular Lines
Standards Link:
2.3.G.C. Use properties of geometric figures and measurement formulas to solve for a
missing quantity (e.g., the measure of a specific angle created by parallel lines and is a
transversal).
2.4.G.A. Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use
of counter-examples, truth tables, etc.) to validate conjectures or arguments.
2.9.G.A. Identify and use properties and relations of geometric figures; create
justifications for arguments related to geometric relations.
2.9.G.C. Use techniques from coordinate geometry to establish properties of lines,
2-dimensional shapes.
Big Idea:
Enduring Understanding(s):
Properties of angles formed by Parallel
and Perpendicular Lines
Two intersecting lines form angles with
specific relationships.
Parallel lines cut by a transversal form angles
with specific relationships.
Essential Question(s):
How can we use the properties of special pairs of angles to solve algebraic and geometric
problems?
How do we identify and name the angles formed by a transversal crossing two or more
lines?
How is slope used to determine the relationships between pairs of lines?
Knowledge:
Skill(s):
When two or more lines are intersected
by a transversal, there are special angle
relationships that exist. These include
alternate interior angles, alternate
exterior angles, same-side interior
angles, same side exterior angles and
corresponding angles. (2.3.G.C)
When two parallel lines are interested
by a transversal:
a. Corresponding angles are
congruent
b. Alternate interior angles are
congruent
c. Alternate exterior angles are
congruent
d. Same-side interior angles are
supplementary
e. Same-side exterior angles are
supplementary (2.3.G.C)
Lines can be proven parallel when
specific information is known about
angle relationships. (2.9.G.A)
Identify the different angle pair
relationships given 2 or more lines cut by a
transversal: same-side interior, same-side
exterior, alternate exterior, alternate
interior, corresponding, vertical, and linear
pairs. (2.3.G.C)
Identify the relationships between the
angles when the lines are parallel.
(2.9.G.A)
Determine what you need to know to prove
two lines are parallel. (2.9.G.A)
Know what relationships exist when the
two lines are perpendicular to the
transversal. (2.3.G.C)
Write a two-column proof using the
postulates and theorems. Include fill in the
blank. Honors will write a 2-column proof
in its entirety. (2.4.G.A)
Use slope to determine if two lines are
parallel. (2.9.G.A)
Two lines are parallel if and only if they
have the same slope. (2.9.G.A,
Use the slope-intercept form and/or point2.9.G.C.)
slope form of an equation to write equations
for parallel and perpendicular lines. (2.9.G.A,
If two lines are perpendicular, the
2.9.G.C)
product of their slopes is -1. (2.9.G.A,
2.9.G.C)
Given the equation of a line and a point
not on the line, you can write the
equation of a line that is either parallel
or perpendicular to the given line.
(2.9.G.A, 2.9.G.C.)
Assessment/Evidence of Learning:
Chapter 3 Common Assessment – found on staff drive
Parallel and Perpendicular Lines Performance Task – found on staff drive
Unit 4 - Congruent Triangles
Standards Link:
2.1.G.C. Use ratio and proportion to model relationships between quantities.
2.4.G.A. Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use
of counter-examples, truth tables, etc.) to validate conjectures or arguments.
2.5.G.B. Use symbols, mathematical terminology, standard notation, mathematical rules,
graphing, and other types of mathematical representations to communicate observations,
predictions, concepts, procedures, generalizations, ideas, and results.
2.9.G.A. Identify and use properties and relations of geometric figures; create
justifications for arguments related to geometric relations.
Big Idea:
Enduring Understanding(s):
Similar Polygons and Congruent
Triangles
Proof is a justification that is logically valid
and based on definitions, postulates, and
theorems.
Relationships that exist between the angles and
sides of geometric figures can be proved.
Essential Question(s):
What are Sums of Interior and Exterior angles of regular polygons?
How can geometric properties be used to prove relationships between the angles and sides of geometric
figures?
Why is it important to be able to identify congruent triangles?
Knowledge:
Skill(s):
Triangles can be classified as
equilateral, isosceles, scalene, acute,
equiangular, obtuse and right. (2.9.G.A)
Classify triangles as equilateral, isosceles,
scalene, acute, equiangular, obtuse or right.
(2.9.G.A)
There are specific names for the parts
of a triangle. (2.5.G.B)
Identify the vertex, legs, base angles and base
of a triangle. (2.5.G.B)
Congruent figures are figures whose
corresponding parts (sides and angles)
are congruent. (2.1.G.C, 2.4.G.A,
2.5.G.B, 2.9.G.A)
Name corresponding parts of congruent
figures. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A)
If two sides and the included angle of
one triangle are congruent to two sides
and the included angle of another
triangle, then the two triangles are
congruent by side-angle-side (SAS).
(2.4.G.A)
IF two angles and the included side of
one triangle are congruent to two angles
and the included side of another
triangle, then the two triangles are
congruent by angle-side-angle (ASA).
(2.4.G.A)
If two angles and a nonincluded side of
one triangle are congruent to two angles
and the corresponding nonincluded side
of another triangle, then the triangles
are congruent by angle-side-angle
(AAS). (2.4.G.A)
Corresponding parts of congruent
triangles are congruent (CPCTC).
(2.4.G.A)
If the hypotenuse and a leg of one right
triangle are congruent to the hypotenuse
and a leg of another right triangle, then
the triangles are congruent by
hypotenuse-leg (HL). (2.4.G.A)
Determine if two figures are congruent.
(2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A)
Write a formal proof to demonstrate the
process of determining that two figures are
congruent. (2.1.G.C, 2.4.G.A, 2.5.G.B,
2.9.G.A)
Determine if two triangles are congruent.
(2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A)
Identify the theorem (SSS, SAS, ASA, AAS,
HL) that allows us to conclude the two
triangles are congruent. (2.1.G.C, 2.4.G.A,
2.5.G.B, 2.9.G.A)
Determine what information is necessary to
prove the two triangles are congruent by either
SSS, SAS, ASA, AAS or HL. (2.1.G.C,
2.4.G.A, 2.5.G.B, 2.9.G.A)
Write a formal proof to demonstrate the
process of determining that two triangles are
congruent. (2.4.G.A)
Write a formal proof to demonstrate the
process of determining congruence among
corresponding parts of congruent triangles.
(2.4.G.A)
Assessment/Evidence of Learning:
Chapter 4 Common Assessment – found on staff drive
Learning Activities:
Company Logo Project – found on staff drive
Unit 5 - Properties of Triangles
Standards Link:
2.9.G.A Identify and use properties and relations of geometric figures; create
justifications for arguments related to geometric relations.
2.5.G.A Develop a plan to analyze a problem, identify the information needed to solve
the problem, carry out the plan, check whether an answer makes sense, and explain how
the problem was solved in grade appropriate contexts.
2.8.G.B Use algebraic representations to solve problems using coordinate geometry.
2.9.G.C Use techniques from coordinate geometry to establish properties of lines, 2dimensional shapes.
Big Idea:
Enduring Understanding(s):
The classification and properties of
triangles can be determined by their
distinct characteristics.
Knowing properties of triangles helps us solve
various problems.
Essential Question(s):
What are the special properties of triangles?
How can we compare angle measures and side lengths in triangles?
Knowledge:
Skill(s):
The midsegent of a triangle joins the
midpoints of two sides of a
triangle.(2.9.G.A, 2.5.G.A)
The midsegment is parallel to the base
of the triangle(2.9.G.A, 2.5.G.A)
The midsegment is half the length of
the base of the triangle. (2.9.G.A,
2.5.G.A)
The line perpendicular to the base of a
triangle at its midpoint is the
perpendicular bisector(2.9.G.A,
2.5.G.A)
A point equidistant from the endpoints
of a segment lies on the perpendicular
bisector of the segment. (2.9.G.A,
2.5.G.A)
An angle bisector is a ray that cuts an
angle into two equal parts. (2.9.G.A,
2.5.G.A)
A point on the angle bisector is
equidistant from the sides of the angle.
(2.9.G.A, 2.5.G.A)
A interior point equidistant from the
sides of an angle lies on the angle
bisector. (2.9.G.A, 2.5.G.A)
Concurrent lines are three or more lines
that intersect in one point. (2.9.G.A,
2.5.G.A)
The circumcenter is the center of the
circle that circumscribes a triangle.
(2.9.G.A, 2.5.G.A)
The point at which concurrent lines
intersect is the point of concurrency.
(2.9.G.A, 2.5.G.A)
The median is a segment whose
endpoints are a vertex of the triangle
and the midpoint of the opposite side.
Identify and/or construct the midsegment of a
triangle. (2.9.G.A, 2.5.G.A)
Determine the length of the side parallel to the
midsegment of a triangle(2.9.G.A, 2.5.G.A)
Use the properties of midsegments to find
missing lengths(2.9.G.A, 2.5.G.A)
Identify and/or construct a perpendicular
bisector(2.9.G.A, 2.5.G.A)
Determine the length of segments using the
properties of perpendicular bisectors(2.9.G.A,
2.5.G.A)
Identify and/or construct an angle bisector.
(2.9.G.A, 2.5.G.A)
Determine the length of segments and
measures of angles using the properties of
angle bisectors. (2.9.G.A, 2.5.G.A)
Identify and/or construct concurrent lines
(perpendicular bisectors, angle bisectors,
medians, altitudes) within a triangle. (2.9.G.A,
2.5.G.A)
Identify and/or construct points of concurrency
(circumcenter, incenter, centroid, orthocenter)
and match it with the corresponding segments.
(2.9.G.A, 2.5.G.A)
Use coordinate geometry to determine the
coordinates of the points of concurrency.
.(2.8.G.B, 2.9.G.A, 2.5.G.A, 2.9.G.C)
Use coordinate geometry to find the center of a
circle that circumscribes a right triangle.
(2.9.G.A, 2.5.G.A, 2.8.G.B, 2.9.G.C)
Use the properties of the median to find
lengths of segments within a triangle.
(2.9.G.A, 2.5.G.A)
Order the sides/angles of the triangle from
least to greatest. (2.9.G.A, 2.5.G.A)
Determine if the given segment lengths could
(2.9.G.A, 2.5.G.A)
The medians of a triangle are
concurrent at a point that is two thirds
the distance from each vertex to the
midpoint of the opposite side. (2.9.G.A,
2.5.G.A)
The altitude of a triangle is the
perpendicular segment from the vertex
to the line containing the opposite side.
(2.9.G.A, 2.5.G.A)
form a triangle. (2.9.G.A, 2.5.G.A)
The largest side of the triangle lies
opposite the largest angle. (2.9.G.A, 2.5.G.A)
The sum of the lengths of any two sides of a triangle is greater than the length of the third
side. (2.9.G.A, 2.5.G.A)
Assessment/Evidence of Learning:
Chapter 5 Common Assessment – found on staff drive
Learning Activities:
Create a Flag Project – found on staff drive
Unit 6 - Quadrilaterals
Standards Link:
2.9.G.C Use techniques from coordinate geometry to establish properties of lines, 2dimensional shapes.
2.8.G.B Use algebraic representations to solve problems using coordinate geometry.
2.11.G.A Find the measures of the sides of a polygon with a given perimeter that will
maximize the area of the polygon.
Big Idea:
Enduring Understanding(s):
Quadrilaterals
Quadrilaterals can be classified by their
properties.
Polygons and their angles are useful for
solving problems in architecture, construction,
plumbing, engineering, landscaping, etc.
Essential Question(s):
What types of quadrilaterals exist and what properties are unique to them?
Knowledge:
Skill(s):
The slope formula can be used to find
the slope of a line.(2.9.G.C)
Identify, name and describe polygons based on
the number of sides (3-12) through n-gons.
(2.9.G.A)
The slope of a line determines whether
the two lines are parallel, perpendicular
or neither. (2.9.G.C)
Classify polygons as regular or
irregular.(2.11.G.A)
The distance formula can be used to
determine the length of a line. (2.9.G.C)
Classify polygons as convex or concave.
(2.9.G.A)
In a parallelogram, the opposite sides
are congruent and parallel. (2.9.G.A)
A parallelogram is a rhombus if all four
sides are congruent. (2.9.G.A)
Classify quadrilaterals as parallelograms,
trapezoids and kites based on the number of
parallel and congruent sides using coordinate
geometry. (2.9.G.C, 2.8.G.B)
A parallelogram is a rectangle if all four
angles are congruent. (2.9.G.A)
Identify the properties of special
parallelograms. Use those properties to solve
problems. (2.9.G.A)
A parallelogram is a square if all four
sides and four angles are congruent.
(2.9.G.A)
A quadrilateral is a trapezoid if it has
one pair of parallel sides. (2.9.G.A)
Use the properties of a midsegment of a
trapezoid to find lengths of trapezoids.
(Include problems that require multiplying
binomials and factoring in Advanced and
Honors) (2.9.G.A)
A trapezoid is isosceles if the two non
parallel sides are congruent. (2.9.G.A)
Calculate the measure of the interior angles of
a polygon. (2.9.G.A)
A kite is a quadrilateral with 2
congruent adjacent sides and no two
pair of opposite sides congruent.
(2.9.G.A)
Determine the number of sides a polygon has
based on the sum of the angles. (2.9.G.A)
Assessment/Evidence of Learning:
Chapter 6 Common Assessment – found on staff drive
Name that Quadrilateral Open ended question – found on staff drive
Unit 7 - Similarity
Standards Link:
2.9.G.B Use arguments based on transformations to establish congruence or similarity of
2-dimensional shapes.
2.4.G.A Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use of
counter-examples, truth tables, etc.) to validate conjectures or arguments.
2.1.G.C Use ratio and proportion to model relationships between quantities.
Big Idea:
Enduring Understanding(s):
Proportionality and scale factors can be
used to solve practical problems.
We can use congruence and similarity to help
us measure large objects or distances
indirectly.
Objects are similar if they have a common
scale factor for their sides.
We can compare figures that are congruent or
similar to help solve problems.
Essential Question(s):
What is the difference between a ratio and a proportion?
What are the requirements for two shapes to be called similar?
How do you use properties of similar figures to solve practical problems?
How can you use ratios and proportions to connect mathematical ideas?
How do we measure large objects or distances since we cannot measure them directly?
Knowledge:
Skill(s):
A ratio is a comparison of two
quantities.(2.1.G.C)
Define and write ratios.(2.1.G.C)
Identify and solve proportions. (2.1.G.C)
A proportion is a statement that two
ratios are equal. (2.1.G.C)
There exist many properties of
proportions, including the cross-product
property. (2.1.G.C)
The scale compares each length in a
scale drawing to the actual
length.(2.9.G.B)
In order for two figures to be similar,
corresponding sides must be
proportional and corresponding angles
must be congruent.(2.9.G.B)
The similarity ratio is the ratio of
corresponding lengths of two similar
figures.(2.1.G.C)
If two angles of one triangle are
congruent to two angles of another
triangle, then the triangles are similar
by angle-angle (AA).(2.9.G.B)
If an angle of one triangle is congruent
to an angle of a second triangle, and the
sides including the two angles are
proportional, then the triangles are
similar by side-angle-side
(SAS).(2.9.G.B)
If the corresponding sides of two
triangles are proportional, then the
triangles are similar by side-side-side
(SSS).(2.9.G.B)
The altitude to the hypotenuse of a right
triangle divides the triangle into two
triangles that are similar to the original
triangle and to each other.(2.1.G.C,
2.9.G.B)
The side-splitter theorem states that if a
line is parallel to one side of a triangle
Complete the proportion to make the statement
true. (2.1.G.C)
Identify and use properties of similar polygons
to find missing lengths. (2.1.G.C)
Determine the similarity ratio of similar
figures.(2.1.G.C)
Use the similarity ratio to find missing lengths
of similar figures.(2.1.G.C)
Apply similar figures to practical
problems.(2.1.G.C)
Use the definition of similarity to write a
formal proof showing two figures are
similar.(2.9.G.B)
Identify whether the triangles are similar by
AA, SAS, and/or SSS.(2.9.G.B)
Write a formal proof of similar triangles by
AA, SAS, and/or SSS.(2.9.G.B, 2.4.G.A)
Find and use relationships in similar right
triangles to determine missing lengths and
angles.(2.1.G.C)
Use the side splitter theorem to write formal
proofs and find missing sides and angles of a
triangle.(2.4.G.A)
and intersects the other two sides, then
it divides those sides
proportionally.(2.1.G.C, 2.9.G.B)
Assessment/Evidence of Learning:
Chapter 7 Common Assessment – found on staff drive
Learning Activities:
Unit 8 - Right Triangles and Trigonometry
Standards Link:
2.10.G.A Identify, create, and solve practical problems involving right triangles using the
trigonometric ratios and the Pythagorean Theorem.
2.1.G.C Use ratio and proportion to model relationships between quantities.
Big Idea:
Enduring Understanding(s):
Mathematical principals involving right
triangles are the cornerstone for
architectural design.
There are many applications of the
Pythagorean Theorem.
Angles of elevation and depression are needed
to help solve indirect measurement problems.
Essential Question(s):
What are the properties of right triangles and how are they used?
What are the different methods for finding the missing sides/angles of right triangles?
How can we apply these methods to practical situations?
How does the Pythagorean Theorem help us solve authentic applications?
Knowledge:
Skill(s):
The Pythagorean theorem states that the
sum of squares of the lengths of the legs
in a right triangle is equal to the square
of the length of the
hypotenuse.(2.10.G.A)
Use the Pythagorean Theorem to find missing
sides of a right triangle.(2.10.G.A)
The converse of a the Pythagorean
Theorem states that if the square of one
length of a triangle is equal to the sum
of the squares of the lengths of the other
two sides, then the triangle is a right
triangle.(2.10.G.A)
Use the converse of the Pythagorean theorem
to classify a triangle as acute, right, or
obtuse.(2.10.G.A)
Apply the Pythagorean Theorem to practical
situations.(2.10.G.A)
Use the properties of special right triangles to
find missing sides.(2.1.G.C)
Apply properties of special right triangles to
practical situations.(2.1.G.C)
Special right triangles are triangles
whose angles are 30-60-90 and 45-45-
Write the ratios for the tangent, sine, and
cosine or an angle within a right
90.(2.1.G.C)
triangle.(2.10.G.A)
The lengths of the sides of special right
triangles have special
properties.(2.1.G.C)
Use trigonometric ratios to write and solve for
missing sides of a right triangle.(2.10.G.A)
Trigonometric ratios can be used to find
missing sides of right
triangles.(2.10.G.A)
Inverse trig can be used to find the
missing angles of right
triangles.(2.10.G.A)
Use inverse trig to find the missing angles of a
right triangle.(2.10.G.A)
Identify angles of elevation and
depression.(2.10.G.A)
Complete application problems using angles of
elevation and depression.(2.10.G.A)
The angle of elevation is the angle
formed by a horizontal line and the line
of sight above the horizontal
line.(2.10.G.A)
The angle of depression is the angle
formed by a horizontal line and the line
of sight below the horizontal
line.(2.10.G.A)
Angles of elevation and depression can
be used to find heights.(2.10.G.A)
Assessment/Evidence of Learning:
Chapter 8 Common Assessment – found on staff drive
Learning Activities:
Unit 9 - Measurement in Two Dimensions
Standards Link:
2.3.G.E Describe how a change in the value of one variable in area and volume formulas
affect the value of the measurement.
2.10.G.A. Identify, create, and solve practical problems involving right triangles using
the trigonometric ratios and the Pythagorean theorem.
2.11.G.A. Find the measures of the sides of a polygon with a given perimeter that will
maximize the area of the polygon
2.11.G.C. Use sums of areas of standard shapes to estimate the areas of complex shapes.
2.7.G.A. Use geometric figures and the concept of area to calculate probability.
Big Idea:
Enduring Understanding(s):
Measurement in two dimensions
The use of 2-dimensional measurements is
critical for understanding our world.
Essential Question(s):
What do changes in dimensions do to area and volumes of shapes?
What life situations might require us to calculate perimeter/circumference or area?
Knowledge:
Skill(s):
When all of the dimensions of a shape
are changed proportionally, the value of
the perimeter and area are affected by
the changes in all dimensions.
(2.3.G.E)
Compare the area and perimeter when a
change in dimension has occurred.(2.3.G.E)
Formulas for two dimensional figures
can be used to calculate
perimeter/circumference and area.
(2.11.G.A.)
Given perimeter/circumference or area
one missing dimension can be solved
for. (2.11.G.A.)
Relate the ratios of area of two specific
Apply the formulas for perimeter and area of
triangles, quadrilaterals, and circles
(circumference) (2.11.G.A.)
Manipulate the formulas to find missing
lengths. (2.11.G.A.)
Using the correct formula for area for given
shapes, find the probability of getting a
specific outcome. (2.7.G.A.)
shapes to probability. (2.7.G.A.)
Assessment/Evidence of Learning:
Chapter 9 Common Assessment – found on staff drive
Learning Activities:
Unit 10 - Measurements in Three Dimensions
Standards Link:
2.3.G.E Describe how a change in the value of one variable in area and volume formulas
affect the value of the measurement.
2.10.G.A. Identify, create, and solve practical problems involving right triangles using
the trigonometric ratios and the Pythagorean theorem.
2.11.G.A. Find the measures of the sides of a polygon with a given perimeter that will
maximize the area of the polygon
2.11.G.C. Use sums of areas of standard shapes to estimate the areas of complex shapes.
2.5.G.A Develop a plan to analyze a problem, identify the information needed to solve
the problem, carry out the plan, check whether an answer makes sense, and explain how
the problem was solved in grade appropriate contexts.
Big Idea:
Enduring Understanding(s):
Measurements in three dimensions
Changing the parameters of a figure affects the
surface area and volume.
Essential Question(s):
What life situations might require us to calculate surface area or volume?
How can we break a 3-dimensional object into something we can measure with 2dimensional tools?
What effect does a change in the dimensions of an object have on the surface area and
volume?
Knowledge:
Skill(s):
Formulas for three dimensional figures
can be used to calculate surface area
and volume.(2.5.G.A)
Apply the correct formula for surface area and
volume. (2.5.G.A)
When some of the dimensions of a
shape are changed proportionally, the
value of
the surface area and volume are
affected by the changes in the
dimensions. (2.3.G.E)
Compare the surface area and volume when a
change in dimension has occurred.(2.3.G.E)
Assessment/Evidence of Learning:
Chapter 10 common assessment – found on staff drive
Open ended question:
Given two equilateral triangles, find a third one whose area is the sum of the other
two.
What percent of the area of a circle is enclosed by an isosceles triangle one of whose
sides is the diameter of the circle?
Unit 11 - Circles
Standards Link:
2.9.G.A Identify and use properties and relations of geometric figures; create
justifications for arguments related to geometric relations.
Big Idea:
Enduring Understanding(s):
Circles
Basic properties of a circle can be used to
determine measurements in real life situations.
That pi is the relationship between the
circumference and diameter of a circle.
Theorems about circles were developed to help
early astronomers study spatial bodies.
Essential Question(s):
How do you apply the relationships among circles, lines, segments, and the angles they
form?
Knowledge:
Skill(s):
Theorems, postulates and definitions
for arcs, chords, sectors, segments,
secants, tangents, and angles can be
used to find missing information. (
2.9.G.A)
Distinguish between the basic components of a
circle: arc, chord, sector, segment, tangent,
secant, central angle, semi-circle, minor and
major arcs, inscribed angles. (2.9.G.A)
Use properties of tangents, chords, arcs, and
central angles to find missing lengths and
angles (2.9.G.A)
Compare arc length and sectors
Find the arc length of a circle.
Use properties of inscribed angles to find
missing angles
Use properties of secants to find missing
lengths and angles
Assessment/Evidence of Learning:
Unit 12 - Transformations
Standards Link:
2.9.G.B Use arguments based on
transformations to establish congruence
or similarity of two dimensional shapes.
Big Idea:
Enduring Understanding(s):
Transformations of figures in a plane
can preserve distance and/or shape.
Transformations are changes in geometric
figures’ positions, shapes, and/or sizes.
Essential Question(s):
How do you apply the basic properties
of transformations?
What are the results of those
transformations?
Knowledge:
Skill(s):
A transformation is a change in a
geometric figure’s position, shape, or
size. (2.9.G.B)
Identify whether or not the transformation was
an isometry. (2.9.G.B)
The original figure is known as the
preimage. (2.9.G.B)
Name images and corresponding parts.
(2.9.G.B)
The resulting figure is known as the
image. (2.9.G.B)
Given an image and preimage, identify the
transformation that has taken place as a
reflection, rotation, or translation. (2.9.G.B)
An isometry is a transformation in
which the preimage and image are
congruent. (2.9.G.B)
Contruct a geometry preimage and image
using reflections, translations, and rotations.
(2.9.G.B)
A translation maps all points of a figure
the same distance in the same direction
(slide). (2.9.G.B)
Determine whether a figure has point, line,
rotational symmetry or neither. (2.9.G.B)
A reflection is a transformation in
which a figure and its image have
opposite orientations (flip).
A rotation is a transformation that turns
a figure a specified number of degrees
about a point.
Some transformation result in point,
line, or rotational symmetry.
Assessment/Evidence of Learning:
Learning Activities:
Resources: