Unit Plan Assignment

MAED 314A Unit Plan Template

Name: Xi Laura Cang

Title of unit and grade/course:
Math 11 Honours:
- Reasoning and Geometry

1) Rationale and connections:

a) Why do we consider it important for students to learn this topic? Why is it included in the IRPs? (< 150 words)

Logic and reasoning is an integral part of the mathematical process and is vital in that it formally incorporates the skills necessary for higher level, abstract logical reasoning. It is also the precursor to Geometric Proofs and affords the student the practice and skills necessary to succeed in the subsequent topics. Without the background, many students become intimidated by proof exercises that are essential to the development of strong mathematical intuition.

Studying geometry provides a purpose and application to the algebraic exercises that students have been engaged in and adds methodology and mathematical notation to the student's arsenal. Geometric reasoning and terminology provides the basis for many concepts that lead into calculus and as his style of reasoning may be the first introduction to the anatomy of a proof that students receive, it is important to provide a model and a backbone to build upon.


b) What are the historical origins and connections for this topic? (<100 words)

Geometry is a cornerstone of mathematical application as it has been shown that many ancient civilizations utilized their mathematical understanding for building and design: Egyptian pyramids, early studies of astronomy and pretty much all of early mathematics has some relevance to geometric assertions. All of highschool geometry is rooted in Euclid's Elements - a mathematical work of principle importance. In order to understand the basis of mathematics, geometry is an essential area of study.


c) How does this topic connect with life outside mathematics? (<100 words)

Logic and Reasoning skills are vital for all aspects of life and should be a topic studied by all. It helps to enhance numeracy at all levels and improves the problem solving ability in any subject and even provides a general instance of sense for life outside of academics.


2) Balanced teaching, assessment and evaluation plan

a) Describe your balanced assessment and evaluation plan. Consider:
•teacher, peer and self-assessment;
•assessment of student learning, of teaching, and of the unit as a whole
•the weighting of marks to take account of summative and formative assessment, instrumental and relational learning

Logic and Reasoning is a difficult topic for traditional assessment methods as there are many solutions and directions that may be valid for any one problem, thus stock answer keys are undesirable. As such, solutions could require much more scrutiny by the assesser than other topics. However, a topic such as this is ideal for projects and group learning as it is rife with opportunities for discussion.

My assessment plan would be based in a variety of activities for the students:
i) take home assignments
ii) group projects that could require research and problem solving skills that will culminate in a demonstration that highlights multiple solutions and incorporates various strategies

This is a topic that might be reasonably weighted more heavily on the formative side as it emphasizes a strength in relational learning. Students need to feel that the idea is crucial for a satisfactory solution. This section provides a great opportunity for promoting the importance of multiple solutions and I would be inclined to encourage and reward (and maybe even require) students to come up with more than one solution to problems.


b) Project title and 50-word description

Math Project: Problems Showcase!

Similar to a SNAP math fair: students are paired up and are given or may devise their own problems (with approval from the teacher) for which they must find at least 2 solutions. At the end of the unit for the duration of one class, each group may set up a stand for which they have their peers attempt to solve their problem.
Alternative: At the end of each class, one or two groups of students could hand out their problem to the class for their peers to solve. At the start of the next class, they could receive solutions from their classmates and present the solutions that they came up with. In a class of 30 students, this might take up to 7 or 8 classes and may eat 10 - 15 minutes out of the start of each class, but I think this is a skill that is well worth that time.


c) List of 10 lessons with brief topic outline and teaching strategies to be used.

1) Inductive Reasoning

- illustrates how one might draw conclusions based on commonalities and the definitions of conjectures/hypotheses
- provide some examples with the class
- discuss the ramifications of sample size and the importance of defining the parameters of the examples; considerations of the type of data needed in order to draw reasonable conclusions
- problems (may not be strictly mathematical in nature) may be handed out to groups to discuss and practice the thought processes


2) Deductive Reasoning

- difference and definitions between deductive and inductive reasonings
- discussion of definition of 'theorem'
- examples of deductive and inductive reasoning done with the class to infer their understanding
- logic puzzles that may use both deductive and inductive reasoning
- introduction to strong and weak mathematical induction
- partnered take home assignment

3) Conjectures and Counterexamples

- example of conjecturing and a proof for conjecture using induction
- example of conjecturing then a counterexample to disprove conjecture
- discuss causation vs. correlation in terms of statistical facts
- devise a problem requiring research in order to gain the background knowledge for conjecturing; provide different problem per group; present next day in 'jigsaw'


4) Statements Involving "not", "and", "or"

- show how terms are used with examples
- introduce Venn diagrams and the use of them for problem solving
- given problem, have students create a venn diagram to illustrate the concept
- mathematical statements to be understood through number lines - problems involving number lines to highlight differences between 'not' 'and' 'or'
- negation concepts and de Morgan's laws to be discussed

5) Statements involving "If...then"

- introduce 'if...then' statements, contrast with 'if and only if' statements
- develop proper notation for common use
- outline converse, inverse and contrapositive
- describe equivalent value statements; have students decide and discuss (TPS) whether still true or not
- have students create concept maps using the logic connectives "and" "not" "or" "if...then" "if and only if" to define the rules of their favorite sport, game or activity

6) Congruent Triangles

- discuss the conditions for the 'congruence theorems' of triangles: SSS, SAS, ASA
- if students are comfortable have them try to prove why ASS does not count (could use proof by counterexample)
- outline perpendicular bisector theorem and isosceles triangle theorem, have students decide the converse, inverse and contrapositive statements in reference to the theorems to remind them of the process
- take home assignments

7) Indirect Proof & Logic Puzzles

- Example of Proof by contradiction thus proving Parallel Lines Theorem
- inductive or deductive?
- Outline one step progression:
i) state that the result to be proved is either true or false
ii) assume that the result to be proved is false
iii) show that a conclusion can be reached that contradicts the known facts
iv) since a contradiction exists, assumption is incorrect, result is true
- give students the opportunity to try to prove various (simple circle geometry) theorems and present to the class to provide quick insight to theorems to be used in future

8) Mathematical Modelling and Fermi Problems

- Measuring Earth's Circumference: how could we do this?
- have students work in groups to determine their assumptions (spherical or flat? margin of error) and provide a method for which they might be able to discern a solution; have them describe their sources for error.
- have a few groups present solution
- describe Fermi problems, groups to come up with and solve 3 without help of internet, have them hand-in a good copy that outlines assumptions, solution and a 'googled' answer if possible and describe the magnitude and sources of the error

9) Constructions and Proof in Geometry

- introduce geometer's sketchpad (or geogebra if school computer lab not equipped with GS)
- show a few examples of how to use
- lead on some construction problems
- assessment done in class by walking around to see how they are handling the problems, students to show their solutions before progressing to next problem
- emphasize importance of deductive proofs to guarantee result is true in all cases

10) Cumulative Review & Problems Showcase! (SNAP Math fair finale)

- Go over definitions and assign a problem solving review
- Group test could be given for some fairly difficult proofs and puzzles
- have students prepare for the Problems Showcase to be held over an entire class period.


3) In detail:
a) Lesson Plans for three lessons, showing a balanced instructional approach.

Lesson Plan No. 3 - Conjectures and Counterexamples

Lesson Plan No. 5 - Statements involving "If...Then"

Lesson Plan No. 9 - Constructions and Proofs in Geometry

b) Project Plan for the unit project. Include a description, a rationale and a marking scheme:

Problems Showcase Project Description