GT Unit 2: Reasoning & Proof

Using logical statements to generate formal proofs

REASONING & PROOF OVERVIEW

We spent some time in the last unit looking at different simple logic statements.  In this unit, we get further develop our understanding of logical expressions and logic statements as we define new terms and start putting those ideas together into complete proofs.  As we do this, you’ll need to take some good notes about different types of logical expressions, ways they can connect/interact with each other, and  how to put them together into larger ideas.  In this unit, you’ll be expected to do three main things:

  1. Describe situations and ideas using formal logic expressions of different kinds
  2. Prove mathematical ideas about angles and line segments using deductive reasoning
  3. Identify places in the real world where congruent angles and lines can be used to prove ideas

When you’re done, we’ll have the tools for more complex geometry investigations!

PART 1: LOGIC STATEMENTS

(40 pts)  About 2 days

Mathematics and geometry are communicated through formal notation and expressions.  Last unit, we looked at some of the specific notation, and now we get to formalize our communication with various logic statements.  As we do this, you’ll take some notes and practice writing logical expressions of different kinds.

PART 2: REASONING & PROOF

(65 pts)  About 3 days

When we start combining logical expressions, we’re able to develop formal proofs.  A proof is like a legal argument that starts with a generally accepted truth, and then follows through step-by-step as ideas develop more complex and possibly less-obvious conclusions.  The job of geometry is often to prove congruence or similarity of different geometric things.

PART 3: CONGRUENT ANGLES

(30 pts)  About 4 days

To wrap up this unit, we’ll look around us and find various situations of congruent angles and line segments.  In doing so, we can identify ways that the various geometric relationships can be

 Download the part 1 rubric

 Download the part 2 rubric

 Download the part 3 rubric

PART 1: Logic Statements

Writing and interpreting logic statements

In the first part of this unit, your job is to review different kinds of logic statements, as well as to explore some new kinds of logical expressions.  We’ll start by reviewing some information in the presentations shown below and taking some notes about the different kinds of expressions we can write.  You should become familiar with these terms:

  • Conditional, Conclusion, Definition, Theorem, Hypothesis, Truth Value, Counterexample
  • Biconditional, Contrapositive, Converse,

Once these ideas are clear and you’ve completed your homework assignments and quiz, we’ll be ready to move on to writing proofs!

GRADING & PROCESS

 Use your Engineering Notebook to take notes about logic statements of various kinds

 Complete homework assignments 7 and 8 about logic expressions

 Talk to Mr. Benshoof and take Quiz 4: Logic Statements

Have Mr. Benshoof confirm your assignments after they’ve been checked with the key

Logic Introduction

Conditionals

Converse & Biconditionals

Contrapositives & Inverses

What’s Due In Reasoning & Proof Part 1: Logic Statements

  • Logic Statement Notes
  • Homework 7
  • Homework 8
  • Quiz 4: Logic
  • Benshoof’s Check-Off

Here’s what’s due in Part 1: Logic Statements

  1.  Notes on Logic Statements of different kinds
  2. Homework 7: Logic Statements
  3.  Homework 8: Biconditionals
  4.  Quiz 4: Logic Statements
  5.  Have Benshoof check-off your working wiring

PART 2: Design & Draw

Brainstorm ideas for your Bug Bot and draw a careful picture

With the ideas of logical statements under our belt, it’s time to look at what we can do with those logical statements.  In a court room, it’s the lawyer’s job to string together facts in such a way as to prove what happened during a crime.  In a similar way, it’s a good mathematician and engineer’s job to string together mathematical facts (theorems) in a logical way to prove various mathematical relationships.  The ultimate goal is to give you the tools that it takes to look at a geometric structure and identify various shapes/angles/lines that are congruent.

Deductive Reasoning

Algebraic Proofs

Algebraic Proofs 2

Geometry Proofs

What’s Due In Bug Bot Part 2: Design & Draw

  • Brainstorm
  • Research
  • Pick Design & Draw
  • Benshoof’s Check-Off

Here’s what’s due in Part 1: Wiring

  1.  Brainstorm at least 10 ideas for what your Bug Bot might look like or be made of
  2.  Look up “Bug Bot Designs” on Google and see what other designs are out there.  Add 5 more ideas to your brainstorm list.
  3.  Pick what your final design will look like, and make a careful drawing in your engineering notebook
  4.  Have Benshoof check-off your careful drawing

PART 3: Congruent Angles

Applying your knowledge of Congruent Angles

Soldering – how to!  Give some thoughts and notes that summarize the big ideas

Check out this video on how to solder

Check out this video on how to wire your bugbot

Check out this extra video on troubleshooting

What’s Due In Bug Bot Part 3: Build It!

  • Build It
  • (Extra Credit) Decorate it
  • Benshoof’s Check-Off

Here’s what’s due in Part 3: Build It!

  1.  Get into the makerspace and build your Bug Bot!  Make sure it functions properly!
  2.  If you have extra time, give your Bug Bot that special touch by painting, lasering, or making stickers to customize it!
  3.  Have Benshoof check-off your finished, working Bug Bot

Double Check: Unit Expectations

Check what you need to have completed by the unit deadline

The purpose of this unit is to use the Engineering Design Process to wire, design, and build a working Bug Bot.  You should have had the chance to do some soldering, design your own Bug Bot, and use the tools in our makerspace to build your Bug Bot.  If you were really on-task and working hard, you should have also had time to decorate your Bug Bot to make it look super cool.  By the unit deadline, you should have completed and be able to submit the following items:

Engineering Notebook:

 (Part 1) Definition of problem, criteria, constraints, and Bug Bot Overview notes

 (Part 1) Wiring diagram, soldering notes, and wiring reflection

 (Part 2) Bug Bot Design brainstorm, notes, and careful drawing

 (Part 3) Final drawing/picture, half-page written reflection on Bug Bots

Building & Making:

 (Part 1) Wiring should work, soldering should be clean and complete

 (Part 2) Bug Bot design should be carefully drawn out including labels and diagrams

 (Part 3) The Bug Bot should be assembled and should function properly

Benshoof Checkpoints

 (Part 1) Benshoof should have seen your working wiring before you moved on

 (Part 2) Benshoof should have seen your careful drawing before you moved on

 (Part 3) Benshoof should have seen your working Bug Bot