Generating and Testing Hypothesis about new Knowledge Grade 8
Strategy
Standard Name
Geometry: Understand and apply the Pythagorean Theorem.
Effectiveness
Synthesis of
Emperical Evidence
Percentile Difference
Benchmark G.8.6
Explain a proof of the Pythagorean Theorem and its converse.
Level of the New Taxonomy of Learning
low
1.14
high
not very
effective
37%
highly
effective
Application
What Do Students Need To Know
Source For Effectiveness
(Knowledge Domain)
Marzano, R.J. (1998). A Theory-Based Meta-Analysis of Research on
Instruction. McREL.org
how to set up an equation to prove a Pythagorean theorem
What Do Students Need To Be Able To Do
(Procedural Domain)
set up an equation to prove a Pythagorean theorem
Vocabulary
right triangle, hypotenuse, proof, converse, pythagorean theorem
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Teaching Strategy: G.8.6
Vocabulary:
right triangle
hypotenuse
proof
converse
pythagorean theorem
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Use outside of this Campus is prohibited and should be reported to [email protected].
Assessment Rubric for G.8.6
Major Parts
Students need to
understand the
meaning of a right
triangle.
Students can
determine the
hypotenuse in a right
triangle.
Students can substitute
numbers in the formula
for Pythagorean
theorem.
Notes
Attributes
c
d
e
f
g
c
d
e
f
g
c
d
e
f
g
Students understand that a
right triangle has one right
angle.
Students understand that
the hypotenuse is the
longest side, and opposite
of the right angle.
Students can solve the
equation of the
Pythagorean theorem.
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Use outside of this Campus is prohibited and should be reported to [email protected].
Instructions for this Strategy
G.8.6
Teacher Directions:
The first thing students need to understand is what is the Pythagorean Theorem. As you teach this standard, be sure that
you explain to your students why this is important to know and be able to do in real life. Give examples of when and how
they would use this knowledge and skill. Additionally, to make certain that students are mastering the concept of proving
theorem, walk them through, not only the knowledge behind the proof, but also show them how to state the proof verbally
and write the proof as if giving directions on how to state the necessary steps to prove the Phyhagorean Theorem and its
converse. If students can verbalize the proof and write it out, they can solve for the mathematical information, as well.
The Pythagorean Theorem states that the square of the hypotenuse (the longest side of a right triangle; opposite of the right
angle) is equal to the sum of the squares of the other two sides.
The converse of this would be if the square of the hypotenuse is equal to the sums of the squares of the other two sides, the
triangle is a right triangle.
Therefore, if we know the length of any two sides in a right triangle, we can find the length of the third side.
Once the students have learned to use the Pythagorean theorem, they can also use the converse. For example if a triangle
has the sides 3,4,7 is it a right triangle?
A²+b² = c²
Therefore
3² + 5²=7²
9 + 25 = 34, but 7x7 =49
Therefore the triangle is not a right triangle.
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Use outside of this Campus is prohibited and should be reported to [email protected].
Major Mistake Territory:
Students will try to do a proof of the Pythagorean theorem on a triangle that is not a right triangle.
Copyright 2017 What Every Teacher Should Know, LLC. This document is licensed for use by Training and Admin, District Office, demo.
Use outside of this Campus is prohibited and should be reported to [email protected].
Interventions for Tier 1
G.8.6
Reteach in a small group setting. Make sure students are correctly setting up and completing the equation.
Have students work with a peer tutor and be a peer tutor. Remember to have the students verbalize their process steps
and/or write the steps out.
In order to master this standard, students will need to master both knowledge and procedural domains. Here are the
domains with the prerequisite skills necessary to master the standard. Find the point where students do not understand and
reteach at that point first. Remember that the domains are written as a hierarchy.
Knowledge
Categories
What They Mean
Vocabulary Terms
Knows and understands the math vocabulary necessary to
master the standard.
Specific Facts
Knows the specific facts necessary to master the standard.
Trends or
Sequences
Knows the sequence of steps necessary to carry out the
math process.
Generalizations
Knows the general tactics, rules, or algorithms necessary to
carry out the mental processes.
Principles
Knows the relationships of the information to mathematics.
Check for
Gaps in
Learning
Reteaching
Record
For strategies that require procedural skills (students do something with the learning), students will demonstrate procedural
knowledge. We have provided a procedural category chart to help you determine if there are gaps in students' procedural
knowledge that would keep them from mastering the skills.
Procedural
Categories
Processes
What They Mean
What are the processes involved in carrying out the task
or producing a product?
Check for
Gaps in
Learning
Can the
student
identify
a right
triangle?
Mental Procedures
Does the
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Use outside of this Campus is prohibited and should be reported to [email protected].
Reteaching
Record
• Tactic
• Algorithms
• Single Rule
The general rules for carrying out a process. The product
will vary.
student
know the
equation for
the proof of
the
Pythagorean
theorem?
Do not vary in application once learned. You will always
get the same result.
Does the
student
construct
an equation
when
solving
for the
Pythagorean
theorem?
A small set of rules without steps.
Can the
student
follow the
teacher’s
directions?
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Use outside of this Campus is prohibited and should be reported to [email protected].
Interventions for Tier 2
G.8.6
Reteach students in a one-on-one setting. Make sure the students are setting up the equation correctly. As they set up the
equation, have them verbalize or explain what they are doing, ste-by-step.
Have students work with a peer tutor and be a peer tutor for additional reinforcement of the learning. Keep in mind that if a
student can teach the standard, he/she has mastered the concept.
Students at Tier Two need the interventions of Tier One. In addition, students at this tier need to fully understand the
assignment. They may need additional examples before they can respond appropriately to this assignment. Here are the
prerequisites in order for mastering this standard. Levels of
the New
Taxonomy
Level 1:
Retrieval
Level 2:
Comprehension
Level 3:
Application
Operation
Level
What It Means
Application to This
Standard
Recognizing
The student can validate correct
information but does not necessarily
understand the structure of the
knowledge.
Does the student know
how to solve the
equation when given any
two sides of a right
triangle?
Recalling
The student can produce features of
the information but not necessarily
understand the structure of the
knowledge.
Can the student recall
how to solve the
equation for the
Pythagorean theorem
regardless of which side
they are solving for?
Executing
The student can perform a procedure
correctly but not necessarily
understand how and why the procedure
works.
Can the student follow
the process of solving for
the equation?
Integrating
The student will be able to identify the
basic structure of the information,
mental procedure or psychomotor
procedure and the critical as opposed
to noncritical characteristics.
Does the student
recognize the
relationship between the
sides of a right triangle?
Symbolizing
The student will be able to construct an
accurate symbolic representation of the
information, mental procedure or
psychomotor procedure differentiating
between critical and noncritical
components.
Can the student create a
visual representation of
the Pythagorean
theorem?
Executing
The student carries out the plan of
action or process skill.
The student can identify
the hypotenuse of a right
triangle.
Copyright 2017 What Every Teacher Should Know, LLC. This document is licensed for use by Training and Admin, District Office, demo.
Use outside of this Campus is prohibited and should be reported to [email protected].
Gaps in
Learning
Table adapted from Anderson and Krathwohl, 2001, pp. 67–68. and from Marzano and Kendall, 2008 Copyright 2017 What Every Teacher Should Know, LLC. This document is licensed for use by Training and Admin, District Office, demo.
Use outside of this Campus is prohibited and should be reported to [email protected].
Interventions for Tier 3
G.8.6
All of the modifications of Tier one and two. Allow the students to use a calculator, if their problem is mathematical accuracy
of math facts and processes. Again, as they go through the process steps, have them verbalize what they are doing and why
they are doing it.
Students at Tier Three may need one to one help with this assignment. By doing that, we can help to fill in the gaps in
learning so that these students can do this type of assignment independently in the future. Use the chart in Tier Two to
determine where the gaps are and then directly teach to fill the gaps. Levels of
the New
Taxonomy
Level 1:
Retrieval
Level 2:
Comprehension
Level 3:
Application
Operation
Level
What It Means
Application to This
Standard
Recognizing
The student can validate correct
information but does not necessarily
understand the structure of the
knowledge.
Does the student know
how to solve the
equation when given any
two sides of a right
triangle?
Recalling
The student can produce features of
the information but not necessarily
understand the structure of the
knowledge.
Can the student recall
how to solve the
equation for the
Pythagorean theorem
regardless of which side
they are solving for?
Executing
The student can perform a procedure
correctly but not necessarily
understand how and why the procedure
works.
Can the student follow
the process of solving for
the equation?
Integrating
The student will be able to identify the
basic structure of the information,
mental procedure or psychomotor
procedure and the critical as opposed
to noncritical characteristics.
Does the student
recognize the
relationship between the
sides of a right triangle?
Symbolizing
The student will be able to construct an
accurate symbolic representation of the
information, mental procedure or
psychomotor procedure differentiating
between critical and noncritical
components.
Can the student create a
visual representation of
the Pythagorean
theorem?
Executing
The student carries out the plan of
action or process skill.
The student can identify
the hypotenuse of a right
triangle.
Copyright 2017 What Every Teacher Should Know, LLC. This document is licensed for use by Training and Admin, District Office, demo.
Use outside of this Campus is prohibited and should be reported to [email protected].
Gaps in
Learning
Table adapted from Anderson and Krathwohl, 2001, pp. 67–68. and from Marzano and Kendall, 2008 Copyright 2017 What Every Teacher Should Know, LLC. This document is licensed for use by Training and Admin, District Office, demo.
Use outside of this Campus is prohibited and should be reported to [email protected].