HGT Portfolio ProjectMathBY HARRISON SCUDAMORE

Inductive Reasoning

Recognizing a pattern to make a conjecture

Reasoning from detailed facts

2 4 8 16

The rule would be 2^x

n

Deductive Reasoning

If this is true, and X happens, then Y is true

Based on something elseMammals 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

CB

DE

F

If the measure of angleB=the measure of angle C,then the measure ofsegment 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

C D=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 arecongruent becauseof 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 Congruence Def. 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 ReasoningAngles BCA and angle

ACD are 90˚Segment CA is an angle

bisectorProve that Triangles ABC

and ADC arecongruent and statewhat congruenceconjecture you used

A

BC

D

Angles BCA andACD are congruent

Angles BAC andDAC are congruent

Segment AC iscongruent tosegment AC

Triangles ABCADC are congruent

Given

Def. of anglebisector

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.

T R

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 reasoning2=inductive reasoning3=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

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.