HGT Portfolio Project Math BY HARRISON SCUDAMORE

Inductive Reasoning  Recognizing a pattern to make a conjecture  Reasoning from detailed facts The rule would be 2^x n

Deductive Reasoning  If this is true, and X happens, then Y is true  Based on something else Mammals give live birth Dogs give live birth Dogs are Mammals

Review What type of reasoning is used in the following examples?  1,2,3,4,5…n n=x.  o,t,t,f,f,s,s,e,n,t…n n=the first letter of the number in the sequence.  Birds and bugs fly. A hawk can fly. So a hawk has to be either a bug or bird.

Triangles-angle measurement sum  Every triangle has an angle sum of 180˚ When you divide a triangle into 3 parts, and take the interior angle of each part, then they will line up into a straight line, which is 180˚.

Triangles-congruent angles and sides  In an isosceles triangle, the 2 base angles must be congruent.  In an isosceles triangle, the 2 sides connecting to the base angles are also congruent. A C B D E F If the measure of angle B=the measure of angle C, then the measure of segment D=the measure of segment E.

Triangles-Opposite side  In an Isosceles triangle, the side opposite to the greatest angle measurement has the greatest measurement, the side opposite to the second greatest angle measurement has the second greatest length and the side opposite to the smallest angle measurement has the smallest side measurement.  Same thing goes with side measurements to angles. 10˚30˚ 140˚ 100 in. 75 in 25 in

Triangles-exterior angle  In a triangle, angles 1 and 2 will add up to the supplement of angle 3.  For example, if angles A and C are both 50˚, then using the angle sum conjecture we can determine that angle B would have to be 80˚, since angle D=A+B, and 50˚+80˚=130˚, we can determine that angle D would have to be 130˚. To prove that we can use the supplementary angles conjecture to show that since 50˚+130˚=180˚, we know that this works. A B CD=A+B

Triangles-congruence  There are 4 conjectures that can prove that a triangle is congruent. There is Side Angle Side (SAS), Side Side Side (SSS), Angle Side Angle (ASA), and Angle Angle Side (AAS)  There are 2 conjectures that do not work. Those are Side Side Angle (SSA) and Angle Angle Angle (AAA) These 2 triangles are congruent because of SSS These 2 triangles are congruent because of SAA

Review  Use the learned conjectures about triangles to determine the following answers.  Are these 2 triangles congruent?  In triangle ABC, which segment(s) is the longest? Which segment(s) are the shortest? 40˚ X Y 55˚ A B C 70˚ n

Proofs-flowchart  A flowchart proof involves using boxes and arrows to prove something. In order for your flowchart to be true, you must include what you’re stating in that step and why that’s true.  Here is an example of a scenario and a flowchart proof.  Always put QED at the end of your proof to show that you are done proving what you were trying to prove. Triangle ABC is isosceles Angle A has a measure of 112˚ Show that angles B and C are congruent Given Triangle CongruenceDef. of isosceles Conjecture Given ABC is isosceles Angle A has a measure of 112˚ Segment AB is congruent to segment AC Angles A and B are congruent QED

Proofs-2-column proofs  A 2-column proof involves drawing out two columns, one side for what you’re stating and one for the reasoning behind what you’re stating.  Here is an example of a scenario and a 2-colmn proof.  Always put QED at the end of your proof to show that you are done proving what you were trying to prove. Step Reasoning Angles BCA and angle ACD are 90˚ Segment CA is an angle bisector Prove that Triangles ABC and ADC are congruent and state what congruence conjecture you used A B C D Angles BCA and ACD are congruent Angles BAC and DAC are congruent Segment AC is congruent to segment AC Triangles ABC ADC are congruent Given Def. of angle bisector Same segment ASA QED

Proofs-Paragraph Proof  A paragraph proof is paragraph proof with the same elements as the other proofs.  Start out by stating what you need to prove.  Then state what you have to start with.  Now go through the same steps as a flowchart proof or 2-column proof of stating something that is true and the reasoning behind it.  Make sure that you use good transition vocabulary.  Also make sure that either your last or second to last sentence is what you were trying to prove and how you got there.  Always put QED at the end of your proof to show that you’re done proving what you needed to prove.

Review  Write a proof of the following statement:  Angles TRI and AIR are congruent  Angles TIR and ARI are congruent  Prove that triangles TRI and AIR are congruent and what congruence conjecture can be used to determine that they are congruent. TR I A

Real World Application Reasoning skills  Inductive Reasoning: If you become a boss at work you can look at your employees’ work patterns  Deductive Reasoning: If your boss says, “If you do this extra work, then I’ll give you a raise,” then you can use the reasoning to decide if you want the raise or not.

Real World Application Triangles  Congruent Triangles: If you become an engineer, and you build a bridge, but aren’t sure if the triangles supporting it are of equal dimensions, you can use one of the conjectures to check.  Angles Sum of a Triangle: If you become an architect and are working on seeing what angle measurements of a triangle work best, you need to makes sure that the angles of the triangle you had created add up to 180˚

Real World Application Proofs  Flowchart: If you need to prove something and the person you’re trying to explain that something to is visual, you can use a flowchart to prove whatever it is you need to prove.  2-Column: If you’re in ever in a class or lecture and the teacher or professor is proving something and you need to write it down as they explain it, a 2-column proof is the best way to write the proof.  Paragraph: If you ever make a big discovery and are going to prove it, writing a paragraph proof would be the best way to prove the discovery is true.

Answer key to review-reasoning  1=inductive reasoning  2=inductive reasoning  3=deductive reasoning

Answer Key-triangles  1: x=40˚ y=100˚  2: n=125˚  3: Those 2 triangles are congruent because of SAS  4: In triangle ABC, segment AC is the longest, and segments AB and CB are the shortest.

Answer Key-proofs Angles TRI and AIR are congruent Angles TIR and ARI are congruent Segment IR is congruent to segment IR Given Same Segment Triangles TRI and AIR are congruent ASA conjecture

How each pillar is used  Leadership-Teach the lesson  Creativity-Making diagrams and pictures to demonstrate each type of reasoning and provide visual aid to the visual learners.  Critical Thinking-Real world application  Problem Solving-Solving each review problem  Interdisciplinary-Writing out what each subject we learned about in math class is.