Applying Standards Based Constructivism:
A Two-Step Guide for Motivating Students
Pythagorean Theorem

Popular Name: Pythagorean Theorem
Grade Level: 11th Grade
Discipline: Mathematics
Standards: Students will understand mathematics and become mathematically confident, by applying mathematics to real-world settings and by solving problems through the study of trigonometry. 
Learning Objectives:
Students will demonstrate an understanding of the Pythagorean Theorem by effectively teaching an alternative proof of the Pythagorean theorem to their classmates. 
EXPLORATORY PHASE:
  • Students will explore a diagram that visually demonstrates the relationships among the sides of a right angle.
  • Students will self-discover the formula that expresses the relationships among the three sides of a right angle.
DISCOVERY PHASE:
Performance Task: Students will be assigned an alternative proof to the Pythagorean theorem, which they are to learn and teach it to their classmates.

Pythagorean Theorem
Popular Name: Pythagorean Theorem
Grade Level: 11th Grade (see Suggestions to the Teacher section for additional thoughts).
Discipline: Mathematics
Standards and Performance Indicators Context
 
MST Standard 3: Mathematics
Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.
  • Students manipulate symbolic representations to explore concepts at an abstract level.
Core Curriculum Outline Connection
  • Pythagorean theorem
Learning Objectives (which will become the dimensions of the assessment’s rubric.)
Students will demonstrate an understanding of the Pythagorean Theorem.
EXPLORATORY PHASE
(Estimated time: 15 to 45 minutes depending on the knowledge base of the students.)
  • Teacher puts students into groups of three.
  • Students are directed to create a right angle triangle with sides of three different lengths using measurements that are to the half-inch. (Protractors and rulers will be needed. Assistance from group members is encouraged when needed.)
  • Teacher directs the students to label the longest side “c” and the other sides “a”
  •       and “b”.
  • The teacher directs the students to indicate on their drawing that "a" side is “n" inches long etc.
  • Students, in groups of three, are directed to brainstorm what they know about right angle triangles. (3 minutes)
  • Students share what they have concluded regarding what they know about right angle triangles. (3 minutes)
  • Teacher records the groups’ conclusions, taking one item per group until all the sharing has been exhausted.
  • The teacher conducts a debriefing/reflection regarding what has been discussed. (Note: the teacher should ask groups if they discussed particular facts/conclusions that other groups have mentioned and tally these. The intent of this reflection piece is to focus attention on the Pythagorean theorem that is expressed in the formula c2 = a2 + b2.)
  • If the formula doesn’t emerge from this process, the teacher introduces it and asks the groups to discuss its meaning.
  • If the meaning emerges from the groups’ deliberations, the teacher has those who understand how to apply it explain it to the others either in small groups or in a whole class setting.
Planned intervention
  • If the meaning of the formula doesn’t emerge from the groups’ discussion, the teacher has the students use the measurements of their triangles to show how the formula gives you the length of the long side from the other two sides.
  • The teacher asks the small groups to discuss a way to demonstrate why the formula works, i.e., demonstrate a proof of the theorem.
  • If a proof emerges through this process, those who have come up with the proof demonstrate it to the others.
Planned intervention
  • If none emerges, the teacher introduces the diagram that appears below and asks the groups to see if this helps them develop a proof of the theorem. 
  • If a proof emerges through this process, those who have come up with the proof demonstrate it to the others. 
  • If none emerges, the teacher directs the students to create similar squares on their original triangle and mark off their squares by the half-inch and connect the marks to create a grid. (This process could be demonstrated on an overhead projector.)
  • The diagram mentioned above and pictured below, is the 3, 4, 5 triangle.  This triangle doesn't require students to deal with fractional parts.  However, when students develop a similar diagram using their original triangles, fractional parts may be involved.  Using their "messy" data to draw conclusions is part of the inquiry process.
  • The application of the c2 = a2 + b2  formula will require a knowledge of how to calculate square root.  This may require an additional intervention either planned or ad hoc/spontaneous.
  • The small groups are then directed to use their diagrams to devise a proof of the theorem.
  • The following reflection centers on the way the diagram demonstrates a proof of the theorem


The unit of measure for this diagram is centimeters.
DISCOVERY PHASE
(Estimated time : four 45 minute class periods at a minimum)

Performance Task (including planned interventions and audience beyond the teacher).
  • Students are to learn an alternative proof to the Pythagorean Theorem and teach this new proof to classmates.
Planned Intervention
  • Students or student groups will conduct a rehearsal/dry-run with the teacher. These dry-run lessons will be a source of student self-evaluation and teacher feedback.  This process may include teacher modeling of teaching techniques and procedures.
Task Specifications for Developing the Student-Generated Product/Process
  • Students or student groups will develop a 20-minute lesson on the alternative proof they had learned.
  • Students will teach this lesson to a group of their classmates.

Assessment of Performance Task
Dimensions of the student-taught lesson on an alternative proof prepared to demonstrate understanding of the Pythagorean theorem
 Criteria for a score of
4
 Criteria for a score of
3
 Criteria for a score of
2
 Criteria for a score of
1
Manipulates
 The proof is taught thru students’ use of manipulatives
The proof is partially taught thru students’ use of manipulatives
The proof is taught thru instructor’s use of manipulatives
Manipulatives are not used.
Directions/ Procedures
 The directions and/or procedures to be followed are clearly explained.
 The directions and/or procedures to be followed are somewhat unclear.
 The directions and/or procedures to be followed are unclear.
 The directions and/or procedures to be followed are inappropriate
Mathematical equations
 The proof is clearly explained thru the use of an appropriate equation.
The proof is partially explained thru the use of an appropriate equation.
The proof is inadequately explained thru the use of an appropriate equation.
The proof is not explained thru the use of an appropriate equation.
Lesson adjustment
 Learner feedback is effectively used to adjust the lesson
Learner feedback is partially used to adjust the lesson
 Learner feedback is not effectively used to adjust the lesson
 Learner feedback is not requested.
Assessment
 Student understanding is to be demonstrated thru use of manipulatives and the use of the formula
 Student understanding is to be demonstrated thru use of manipulatives or the use of the formula
 Students aren’t given adequate opportunity to demonstrate understanding
 Student understanding is not assessed.
Resources to Be Made Available to Students
  • Students will be assigned one of the alternative proofs to the Pythagorean theorem that appear on the cut-the-knot WebPages and whatever other references the teacher decides will be of assistance to the student in preparing his lesson. See http://www.cut-the-knot.org/pythagoras/index.shtml
Suggestions for the Teacher

Grade level
  • This lesson could be adjusted to engage 8th grade students by an increased use of manipulatives and visuals.
Rubric scoring
  • Class discussion regarding the dimensions and criteria found in the rubric will help students gain a clearer picture of what they are being asked to do.
  • If the alternative proofs are taught by student-groups, individual students need to know that the dimensions of the rubric apply to each student not to the group as a whole.
  • The assignment of students to small groups to develop a lesson on the alternative proof will require the teacher to be sure that each student is involved in the development and teaching of the lesson
Assigning an alternative proof
  • Some of the alternative proofs are very abstract, while others can be demonstrated using manipulatives.
    • The teacher may wish to limit the selection of alternative proofs.
    • The teacher may wish to assign particular alternative proofs to specific students.
    • The teacher may wish to allow students who are mathematically proficient to select from some of the more complicated proofs.
References
The teacher may wish to check out the references at the end of the Web site to determine what additional assistance these sources may provide students in the preparation of their lesson.
Timing
The time allotted to each part of this exemplar is a major consideration, since each student or student group will need to have a rehearsal dry-run with the teacher and actually teach their proof. In addition, there will need to be follow–up reflections.