Course: Geometry
Standards: 4.OA.5, 7.G.5, A.REI.1, 8.G.5, G.CO.9, G.CO.10, G.CO.11
Unit: 3 Proof with Intersecting, Parallel and Perpendicular Lines
21st Century Standard: 20a, 2.b, 2.c, 2.d, 2.e, 2.f, 2.g, 2.h, 2.i, 3.b, 3.d, 4.d, 4.e,
4.g, 5.a, 5.b, 5.c, 5.d, 5.e, 5.f, 5.g
Essential Questions: How can you make a conjecture and prove that it is true? How can a chain of reasoning be organized to communicate proof?
Concepts:
While one counter-example can prove a conjecture false, no number of examples can prove a conjecture true.
Mathematical proof can be communicated through a variety of forms and means.
Content
Skills
Assessment
Instructional Strategies
Observe patterns leading to making
Given a pattern, students will make a
Manipulatives and hands-on pattern
Reasoning and Proof
4.OA.5 – Generate a number or
shape pattern that follows a given
rule. Identify apparent features of
the pattern that were not explicit in
the rule itself.
conjectures through inductive reasoning.
A.REI.1 – Explain each step in solving
a simple equation as following from
the equality of numbers asserted at
the previous step, starting from the
assumption that the original
equation has a solution. Construct a
viable argument to justify a solution
method.
Solve equations, giving reasons for each
step and connecting to simple proof
- Addition Property of Equality
- Subtraction Property of Equality
- Multiplication Property of Equality
- Division Property of Equality
- Reflexive Property of Equality
- Symmetric Property of Equality
- Transitive Property of Equality
- Substitution Property of Equality
- Distributive Property
Write and solve equations using facts
about linear pairs, perpendicular lines,
supplementary, complementary, vertical,
and adjacent angles.
Proof and Angle Relationships
7.G.5 - Use facts about
supplementary, complementary,
vertical, and adjacent
angles in a multi-step problem to
write and solve simple equations for
an unknown angle in a figure.
8.G.5 - Use informal arguments to
Provide counterexamples to prove a
conjecture false.
Write and (Venn) diagram conditional
conjecture, and justify its
reasonableness.
Students will present
counterexamples to disprove
conjectures.
activities
Panel discussion/”debates” to support
or disprove conjectures
Graphic organizers to structure chain of
reasoning
Given an equation in one variable,
students will justify each step taken
to solve for the variable.
With a new set of problems, students
will demonstrate their understanding
of the content and skills by writing
and solving equations involving angle
relationships.
“Can this be assumed?” activities where
students determine whether the
information provided in diagrams can be
assumed to be true, and provide
counter-examples if it cannot be
assumed.
Given a geometric conditional
Graphic organizers and Venn diagrams
establish facts about the angle sum
and exterior angle of triangles, about
the angles created when parallel
lines are cut by a transversal, and the
angle-angle criterion for similarity of
triangles. For example, arrange three
copies of the same triangle so that
the sum of the three angles appears
to form a line, and give an argument
in terms of transversals why this is
so.
G.CO.9 - Prove theorems about lines
and angles. Theorems include:
vertical angles are congruent; when
a transversal crosses parallel lines,
alternate interior angles are
congruent and corresponding angles
are congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
statements and related conditionals
(converse, inverse contrapositive)
regarding angle relationships in if-then
form. Determine whether stated
conclusions are valid.
statement, represent the statement
in the form of a Venn diagram.
Determine whether the conditional
statement is valid, and justify the
determination.
Use the Law of Detachment and the Law
of Syllogism to draw conclusions.
Given a diagram with two intersecting
lines, prove that a pair of vertical
angles is congruent.
Use chain of reasoning or two-column
proof to prove that vertical angles are
congruent.
Given a diagram, with two parallel
lines cut by a transversal and one
angle measure or an algebraic
relationship between two angles,
determine missing angle measures.
Determine those relationships between
angle pairs with a transversal that hold
true iff two lines are parallel. Explain
why these relationships can be used to
prove parallel lines.
G.CO.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.
Use chain of reasoning or two-column
proof to prove theorems about triangles:
Triangle Sum Theorem
Base angles of isosceles
triangles are congruent
Segment joining midpoints of
two sides of a triangle is parallel
to third side and ½ the length
Medians of a triangle meet at a
point (Level 1)
G.CO.11 - Prove theorems about
parallelograms. Theorems include:
opposite sides are congruent,
opposite angles are congruent, the
Discover, justify, and apply
characteristics of slopes (and equations)
of parallel and perpendicular lines.
Given a diagram with angle
relationships provided, determine
whether two lines are parallel, and
justify the response.
Given a diagram with angle and side
measures/relationships provided,
prove facts about specific triangles.
Panel discussion/”debates” to support
or invalidate conditional statements
“Chasing angle” problems and puzzles
“Crook” problems
Use of counter-examples to disprove
information.
Two-column proof protocols.
Proof “puzzles”
Mapping conditional
statements to two-columns
Marking diagrams
Find the error in the proof
Peer collaboration on proofs
Proof gallery walks
Discovery through inductive reasoning
followed by formal proof (deductive
reasoning.)
Use of patty paper and graph paper.
Given a graph, determine
mathematically if lines/line segments
are parallel, perpendicular, or neither.
Justify your answer.
Given two linear equations,
determine if lines are parallel,
perpendicular, or neither. Justify your
answer.
diagonals of a parallelogram bisect
each other, and conversely,
rectangles are parallelograms with
congruent diagonals.
Use chain of reasoning or two-column
proof to prove theorems about
parallelograms:
Opposite sides of a
parallelogram area congruent
Opposite angles of a
parallelogram are congruent
The diagonals of a parallelogram
bisect each other
Rectangles are parallelograms
with congruent diagonals
Given a diagram with angle and side
measures/relationships provided,
prove facts about parallelograms.