Unit Plan - Mathematical Reasoning

Notes:

This unit is designed to consist of mini-lessons interspersed over a full year of math education in the eighth through ninth grades. This unit starts with the application of logic to a "real world" situation, then proceeds to develop basic concepts of logic, particularly mathematical logic.

The unit starts with an exploration well within the grasp of all but the most cognitively challenged students: What would be the results of all the dogs in the world disappearing? From this starting point, students will explore creative thinking to generate possibilities, organizing possibilities into categories to select a concept for development, and viewing the problem set from various points of view to find additional solutions.

The unit continues with the creative use (or non-use) of facts to create false conclusions. The students will be expected to identify missing facts that could lead to different conclusions and the use of questioning to uncover hidden facts.

The unit then helps the student to begin to explore the concepts of completeness and validity of a problem space, the completeness and validity of a solution space, analyzing and combining various solutions to optimize a solution. This leads naturally into mathematical solutions in geometry and number theory.

Stage 1 - Desired Results

Established Goals: What is this?

From National Council of Teachers of Mathematics

Reasoning and Proof:
Instructional programs from pre-kindergarten through grade 12 should enable all students to:
  • Recognize reasoning and proof as fundamental aspects of mathematics
  • Make and investigate mathematical conjectures
  • Develop and evaluate mathematical arguments and proofs
  • Select and use various types of reasoning and methods of proof
Connections:
Instructional programs from pre-kindergarten through grade 12 should enable all students to:
  • Recognize and use connections among mathematical ideas
  • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
  • Recognize and apply mathematics in contexts outside of mathematics
Understandings: What is this?
  • Mathematical reasoning can be used in everyday problems.
  • Mathematical proof has application in everyday life.
 Essential Questions: What is this?
  • "How can I cover a subject completely?"
  • "How can I use mathematical logic to detect (or make) a clever lie?"
  • "How can I tell if a sequence of statements is logical?"
  • "How can I write text that clearly is not a clever lie?"
  • "How can I tell truth from lies?"
Students will know/be able to: What is this?
  • Students will acquire rudimentary skills in developing an idea using math logic.
  • Students will apply mathematical reasoning in areas outside of math.
  • Students will apply mathematical reasoning when reading or listening.
  • Students will acquire a formal knowledge of rudimentary mathematical skills.
  • Students will be able to identify some gross errors in proofs.
Stage 2 - Assessment Evidence
Performance Tasks:
  • Students will write a "what if" paper demonstrating developing an idea through mathematical reasoning.
  • Students will demonstrate the ability to uncover a "clever lie" through mathematical reasoning and questioning.
  • Students will be able to suggest points of view not represented in a news report/advertisement/article.
  • Students will identify assumptions of a news report/advertisement/article.
  • Students will be able to compose an article justifying a pre-selected point of view by selecting facts to represent.
Other Evidence
  • Students will correctly identify valid and invalid proofs by crossing out invalid proofs.
  • Students will be able to describe a process of mathematical logic they can apply in daily life.
  • Students will reflect in their math journal on the use and meaning of mathematical reasoning in their daily lives.
Stage 3 - Learning Plan
Learning Activities:

Activ Type

Lesson Model

Description
Hook Where Evaluate Classroom Discussion Model What if all the dogs in the world disappeared?

This lesson is intended to introduce thinking through all possibilities of a situation. As students explore the possibilities, the teacher will activate knowledge of how students currently identify a problem set. The activities of this unit will form a basis of future lessons. This may be the time to introduce the rubric for this assignment

Note: Inform students that they will be writing a paper on this subject and need to take notes

Equip Revise Evaluate Synectics Version One More Dog-gone Thinking This lesson will introduce students to techniques for exploring possibilities. The Synectics model will be used to help the students to develop a list of realms of further exploration of the disappearing dog question.

Note: Remind students to take notes for a paper they will be writing.

Explore WebQuest Inquiry Model Web is Gone to the Dogs This lesson will allow the students to further develop ideas for their paper. They will be instructed in creative use of keywords to find different information. The students will be encouraged to write down ideas, rather than facts
Formulate Concept Attainment Review of Persuasive Writing This lesson will review the concepts of persuasive writing and how they apply to this assignment. The students will explore the ways they can develop their content to convince their audience that the scenario they are proposing is possible. The rubric for grading the students' paper will be passed out.

Note: The students' assignment for the next lesson will be an outline of their dog paper.

Materials: This lesson will require a worksheet as a tool to help the students develop and record their ideas.

Evaluate Think, Pair, Share Peer Evaluation

The students will pair up to examine each student's outline to verify the following:

  1. The students is sticking to one cohesive possibility.
  2. The student has thoroughly explored the possibility.
  3. The outline is in a logical order

Note: The assignment for this lesson will be a one page paper on 'What if all the dogs in the world disappeared."

Hook Concept Development Model The Clever Lie This lesson will introduce the concept of the clever lie. It will activate the student's prior knowledge by asking how they have created a clever lie. The homework assignment will be to document a clever lie in advertising, politics, or their own life, including what information the audience might want to know that was omitted. This assignment will be used in the next lesson.
Think Formulate Graffiti Model Questioning the Clever Lie - What is Missing? This lesson covers the math concept of identifying all the cases that must be covered. Several clever lies submitted by the students will be examined and the class will be asked, "What question could you ask that will uncover the clever lie?" The teacher should supply a few examples so enough good examples exist to complete the lesson.
Equip Explore Concept Attainment Model Identifying Assumptions This lesson is intended to help the students begin identifying assumptions of a written work. By the end, the students should be able to identify one or more assumptions of a particular statement.
Think Explore Think Pair Share More Assumptions This lesson is intended to reinforce learnings from the previous lesson. The students will be give a set of statements, both mathematical and real-world, of which they are charged to identify the assumptions. The idea that some assumptions are necessary is introduced.
Explore Suchman Inquiry Model Points of View This lesson will explore points of view included/discluded from a written work. The students will learn to identify missing points of view. The homework for this lesson will be to take a set of facts prepared by the teacher and write a paper that supports a point of view pre-selected by the student.
Equip Explore Direct Instruction Intoduction to Mathematical Proofs This lesson will introduce the student to mathematical proofs. The teacher will emphasize identifying assumptions and representing all facts.
Explore Formulate Synectics Excursion My First Proof During this lesson, students will assist each other in developing a simple proof.
Evaluate Concept Attainment Evaluating Proofs During this lesson, students explore how to evaluate a proof.