I. Curricular Framework

Concept:

Consistent Rules

Essential Question:

How are the properties and language of Algebra linked to everyday life?

Bridge:

Interface

Content:

Algebraic Properties

Outcomes:

II. Standards Aligned

III. Instruction and Assessment

1. Connect: Connecting to the Concept Experientially

Objective: To create an interest in the "rules" not working and a need to create new truths.

Activity: Each student is given a ping pong ball and a fine-point felt tip marker. They are instructed to draw a small equilateral triangle, and to observe its sides, angles, and the sum of the angles. Next they will draw progressively larger triangles around the first one until frustration is reached.

Assessment: Involvement of each student. Teacher conversations with individual students including comments of "I can't do it," "What do I do with these curved lines?", "Where are the angles?", And observations about the triangle becoming a circle are shared with the rest of the class.

2. Attend: Attending to the Connection

Objective: To analyze the ping pong experience. Students will realize that the term "triangle" is relative to the system and that Euclidean geometry only works in two dimensional space. "Rules" are therefore relative: new situations demand new rules.

Activity: Teacher leads class discussion. What is a triangle? What is the range of the sum of the angles of a spherical triangle? How can "hunches" about this range be tested?

Assessment: Student recognition of the need for developing new rules and testing them.

Assessment, Phase One, Level of Engagement, Fascination:

3. Image: Creating a Mental Picture

Objective: To see how we adapt to the need for creating anew in our own lives. That algebraic language and algebraic properties have transferabilities into everyday language and properties.

Activity: This activity will use metaphor to help students see that algebraic language has transferability to everyday use. Working in small groups, students are asked to brainstorm an example of a time when they needed to describe something for which there was not an adequate word. Have they ever created a new word? What would make a new word or term reach a level of acceptance in society? Teacher gives definition of INTERFACE and its origin in computer terminology. Have a group of students kinesthetically act out "interface." Working in small groups, students brainstorm everyday examples of how "interface" is used. Teacher charts all examples in mindmap format.

Assessment: Ability of students to relate to language experience and "interface" activity.

Assessment, Phase Two, Seeing the Big Picture:

4. Inform: Receiving Facts & Knowledge

Objective: To teach algebraic properties through student involvement, demonstration, and visual images.

Activity: The following properties are introduced and defined by the teacher: Reflexive, Symmetric, Transitive, Identity, Inverse, Commutative, Associative, Distributive, Substitution, and Closure. For each property and definition, a team of students is involved in a kinesthetic or visual representation of that property. For example: The Reflexive property (a=a) is demonstrated by the teacher using a mirror as the "=" sign. The Symmetric property (a=b) (b=a) is demonstrated with hands. If you were to write a=b on the backs of each of your hands, and then fold them together, the a would match with the b and the bîwould match with the a ñ thumb to thumb and pinky to pinky. The Transitive property ñ If (a=b) and (b=c) then (a=c) is likened to crossing a creek. To cross a creek is to transfer from one side to the other by way of steps in between. The Associative property (a+b) + c = a + (b+c) is demonstrated using a boy and two girls from the class. The boy's arms are the parentheses. Associative comes from the root word Associate. This boy "Associates" with two different girls on different days. Have students physically demonstrate this equation. The Distributive property a (b+c+d) = ab + ac + ad is illustrated by a truck distributing Nintendo sets to the various WalMart stores on his route. Students work in small groups to generate their own kinesthetic, metaphorical, and visual examples of each algebraic property as it is introduced.

Assessment: Interest and enjoyment of students in teacher-led activities and ability of students to generate and share their own examples.

Assessment, Phase Three, Success with Acquiring Knowledge:

5. Practice: Developing Skills

Objective: To reinforce student understanding of algebraic properties.

Activity: 1. Divide class into two teams. Each team has a colored set of cards consisting of the following: 2 "a"; 2 "b"; 2 "c"; 3 "="; 2 "+"; 2 "-"; 2 "("; 2 ")"; 1 "if"; 1 "and"; 1 "then"; 2 "ï". As a property is called out by the teacher, teams must assemble to form an algebraic sentence that shows the property. A designated team member raises her hand when the team is ready. Teams face one another and cross correct their sentences. A point system is established for scoring each team's efforts. 2. Students will complete worksheets with arithmetic problems as well as algebraic proofs. They must supply the reasoning for each step in 2-column formal form.

Assessment: Student ability to demonstrate algebraic properties; completion and correctness of assigned work.

Assessment, Phase Four, Success with Acquiring Skills:

6. Extend: Extending Learning to the Outside World

Objective: To create representations of the algebraic properties using various modalities.

Activity: 1. Student teams will find or invent a number system and an operation in which at least one of the properties studied does not hold. They will use a matrix to chart the operations of their system within the set. They will demonstrate whether or not the algebraic properties they have learned hold true in their new system. They may create a new property for their system, name it, and demonstrate it in some way. 2. Working in cooperative groups, students will create a song, poem, rap, sculpture, painting, dance, or role play to demonstrate three of the properties. All properties must be divided up by the class so that each is demonstrated in at least one form.

Assessment: Ability of students to work together on group project. Quality of projects.

7. Refine: Refining the Extension

Objective: To apply the principles of the algebraic properties.

Activity: Students will be given the opportunity to test their number system projects with the teacher to verify workability. Those teams that are not quite on track will be given a second attempt at the correct completion of their project.

Assessment: Individual student contributions to team effort; teacher monitoring for understanding and application.

8. Perform: Creative Manifestation of Material Learned

Objective: To share in final form what has been learned with the rest of the class.

Activity: Student teams present their new system, matrix, and perform their property.

Assessment: Quality of student demonstration of understanding; enjoyment of team presentations.

Assessment, Phase Five,Performance, Creative Use of Material Learned: