Unit
Length and Description:
20
days
In Unit 3, students use exploration and
deductive reasoning to determine relationships that exist between angle sums
and exterior angle sums of triangles, as well as angles created when parallel
lines are cut by a transversal.
Students construct various triangles and find the measures of the
interior and exterior angles. Students make conjectures about the relationship
between the measure of an exterior angle and the other two angles of a
triangle. Students construct parallel
lines and a transversal to examine the relationships between the created
angles. Students recognize vertical angles, adjacent angles and supplementary
angles from 7th grade, and build on these relationships to identify other
pairs of congruent angles. Using these relationships, students use deductive
reasoning to find the measure of missing angles.
In this unit, the students have their first
exposure to the Pythagorean Theorem as well as its converse. Using models, students
explain the Pythagorean Theorem, understanding that the sum of the squares of
the legs is equal to the square of the hypotenuse in a right triangle. Students also understand that given three side
lengths with this relationship forms a right triangle. Allowing
the students to discover for themselves the Pythagorean Theorem by exploring
the proof numerically through perfect squares, geometrically through area,
and then with right triangles will promote a deeper conceptual understanding
and a greater transfer of knowledge to future courses and applications.
Students will use the Pythagorean Theorem to find unknown side lengths in
right triangles in two and three dimensions. This will reinforce and allow the
students an opportunity to apply their understanding of rational and
irrational numbers in a new context. When applying the Pythagorean Theorem to
solve realworld and mathematical problems, the students will use rational
approximations to create a more meaningful solution within the context of the
problem. In addition, students will apply the Pythagorean Theorem to find the distance
between two points on the coordinate plane. The expectation is that students
will be able to explain a proof of the Pythagorean Theorem and be able to
apply it to solve geometric and realworld problems.

Standards:
8.EE.A.2 Use square root and cube root symbols to
represent solutions to equations of the form 𝑥^{2} = 𝑝
and 𝑥^{3} = 𝑝,
where 𝑝 is a positive rational number. Evaluate square roots of small
perfect squares and cube roots of small perfect cubes. Know that √2 is
irrational.
 Standards Clarification: Students will use square roots to
solve for unknown measures in right triangles. It is not the intent of
the standards to teach students how to simplify a radical through
factoring the radicand. The emphasis throughout the standards beginning
in grade 8 and continuing through high school is to be able create a
rational approximation for irrational numbers. Also note that students will work with
radicals throughout grade 8 when writing and solving equations for an
unknown measurement of a threedimensional figure and again when working
with the Pythagorean Theorem.
8.G.A.5 Use informal arguments to 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 angleangle 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.
 Standards Clarification:
The angleangle criterion for similarity of triangles will be
visited in Unit 4.
8.G.B.6 Explain a proof of the Pythagorean Theorem
and its converse.
 Standards Clarification:
This standard does not require the students to prove for themselves
the Pythagorean Theorem or its converse rather explain a proof of it. To
ensure mastery of this standard, multiple proofs of the Pythagorean
Theorem and its converse should be used.
8.G.B.7 Apply the Pythagorean Theorem to determine
unknown side lengths in right triangles in realworld and mathematical
problems in two and three dimensions.
 Standards Clarification:
This unit begins the learning of the Pythagorean Theorem. Students begin the work of finding the
length of a leg or hypotenuse of a right triangle using .
8.G.B.8 Apply the Pythagorean Theorem to find the
distance between two points in a coordinate system.
 Standards Clarification:
This unit begins the connection of distance to the Pythagorean
Theorem.
Focus Standards of Mathematical Practice:
MP.1 Make
sense of problems and persevere in solving them.
MP.2 Reason
abstractly and quantitatively.
MP.3
Construct viable arguments and critique the reasoning of others.
MP.4 Model
with mathematics.
MP.5 Use
appropriate tools strategically.
MP.6 Attend
to precision.
MP.7 Look for
and make use of structure.
MP.8 Look for
and express regularity in repeated reasoning.
Instructional Outcomes:
Full
Development of the Major Clusters, Supporting Clusters, Additional Clusters
and Mathematical Practices for this unit could include the following
instructional outcomes:
8.EE.A.2
 I can use square roots to represent solutions to equations of
the form x^{2} = p, where p is a positive rational number.
8.G.A.5
 I can define and identify transversals.
 I can identify angles created when a parallel line is cut by
transversal (alternate interior, alternate exterior, corresponding,
vertical, adjacent, etc.)
 I can justify that the sum of the interior angles equals 180.
(For example, arrange three copies of the same triangle so that the
three angles appear to form a line.)
 I can justify that the exterior angles of a triangle is equal
to the sum of the two remote interior angles.
8.G.B.6
 I can define key vocabulary: square root, Pythagorean Theorem,
right triangle, legs a and b, hypotenuse, sides, right
angle, converse, base, height, proof.
 I can identify the legs and hypotenuse of a right triangle.
 I can explain a proof of the Pythagorean Theorem.
 I can explain a proof of the converse of the Pythagorean
Theorem.
8.G.B.7
 I can recall the Pythagorean Theorem and its converse in order
to apply it to realworld and mathematical problems using 2 and 3
dimensional figures.
 I can solve basic mathematical Pythagorean Theorem problems and
its converse to find missing length of sides of triangles in two and
three dimensions.
 I can apply the Pythagorean Theorem in solving realworld
problems dealing with two and three dimensional shapes.
8.G.B.8
·
I can recall the
Pythagorean Theorem and its converse and relate it to any two distinct points
on a graph.
·
I can determine how
to create a right triangle from two points on a coordinate graph.
·
I
can use the Pythagorean Theorem to solve for the distance between the two
points.

Enduring
Understandings:
·
When parallel lines are cut by a transversal,
corresponding angles, alternate interior angles, alternate exterior angles,
and vertical angles are congruent.
·
A relationship exists between angle sums and exterior
angle sums of triangles, as well as angles created when parallel lines are
cut by a transversal.
·
The measure of the interior angles of any triangle is
180°.
·
The measure of a right triangle is always 90°.
·
A relationship exists between the measure of an exterior
angle and the other two angles of a triangle.
·
The
number system consists of numbers that are rational and irrational.
·
Irrational
numbers can be represented on a real number line.
·
Every number has a decimal expansion.
The Pythagorean Theorem can be used both
algebraically and geometrically to solve problems involving right triangles
There is a relationship between the
Pythagorean Theorem and the distance formula.
Both the Pythagorean Theorem and distance
formula can be used to find missing side lengths in a coordinate plane and
realworld situation.
Finding the square root of a number is the
inverse operation of squaring that number.
Finding the cube root of a number is the
inverse operation of cubing that number.
Right triangles have a special
relationship among the side lengths which can be represented by a model and a
formula.
Pythagorean Triples can be used to
construct right triangles.
Attributes of geometric figures can be used
to identify figures and find their measures.

Essential
Questions:
·
How can algebraic concepts be applied to
geometry?
·
How is deductive reasoning used in algebra and
geometry?
·
What angle relationships are formed by a
transversal?
·
What is the length of the side of a square of a
certain area?
·
What is the relationship among the lengths of the
sides of a right triangle?
·
Why does the Pythagorean Theorem apply only to
right triangles?
·
How can the Pythagorean Theorem be used to solve
problems?
·
What is the correlation between the Pythagorean
Theorem and the distance formula?
·
How can I use the Pythagorean Theorem to find the
length of the hypotenuse of a right triangle?
·
How do I use the Pythagorean Theorem to find the
length of the legs of a right triangle?
·
How do I know that I have a convincing argument to informally prove the
Pythagorean Theorem?
·
What is the Pythagorean Theorem and when does it apply?
·
How can I determine the length of a diagonal?
·
How can I find the altitude of an equilateral triangle?
·
How could I find the shortest distance from one point to another if
there was an obstacle in the way?
·
Where can I find examples of two and threedimensional objects in the
realworld?
·
How do I simplify and evaluate algebraic equations involving integer
exponents, square roots, and cube roots?
·
How do I know when an estimate, approximation, or exact answer is the
desired solution?
·
How does the knowledge of how to use right
triangles and the Pythagorean Theorem enable the design and construction of
such structures as a properly pitched roof, handicap ramps to meet code,
structurally stable bridges, and roads?
·
How can the Pythagorean Theorem be used for
indirect measurement?
·
How do indirect measurement strategies allow for the measurement of items in the
real world such as playground structures, flagpoles, and buildings?
