# 3.2: Reasoning and Proof

**At Grade**Created by: CK-12

## Inductive Reasoning

Connections to Music

*This will need to be prepared ahead of time.* Prepare several different examples of repetitive music, such as rap, classical, folk songs, or children’s songs. Make sure that the songs have a refrain (chorus) or a clear consistent pattern.Examples are: Old MacDonald, Puff the Magic Dragon, Let’s Dance (Lady Gaga), excerpts from the Nutcracker (the Russian Dance, Dance of the Reed Flutes, Arabian Dance, Waltz of the Flowers), or Gotta Feeling (Black Eyed Peas). For students to really “hear” the repetition, it might be easiest to start with songs that do not have lyrics.

Play the songs in class and have students develop a rule for each selection. You may have to play the songs 3-5 times before students recognize a pattern. Then, brainstorm a list of possible pattern rules, decide on one and write it on the board. See if students can come up with a counterexample to the rule. After doing one or two examples together as a class, split students into groups and have them listen to remaining selections. Once all the groups are finished, have them share their rules and see if the other groups can find any counterexamples.

As an extension to this activity, you could survey the class to see if any student has an interest in DJ-ing. If one does (and has the appropriate equipment), allow he/she to bring in the necessary equipment and have them give a demonstration on how to mix and dj. Have the class find patterns in their sampling and/or mixing. If no student has the necessary equipment, see if your music department, mass media, or theater classes do.

Connections to Nature

In the computer lab, allow students to search for “nature patterns” in the “images” tab on the search engine. Have students print out 2-3 pictures of plants, fruit, or trees that they feel show a pattern. Give them time to come up with an appropriate rule for each picture. Then, pair up students and have them exchange their pictures with their partner. See if the partner can find any counterexamples.

Challenge

1. Find the next three terms in the sequence and describe the rule.

a) 1, 1, 2, 3, 5, 8, 13, ...

b) 3, 7, 12, 18, 25, ...

2. Plot the points (1, 3), (2, 8), and (3, 13). What do you notice? Can you use algebra to find the rule that maps \begin{align*}x\end{align*} onto \begin{align*}y\end{align*}?

*Answers*

1. a) 21, 34, 55 Each term is made up of the sum of the two numbers before it. This is called a Fibonacci sequence.

b) 33, 42, 52 To find each term, add one more than what was added to the previous term. In other words, we added 4, 5, 6, 7, ... each time.

\begin{align*}^*\end{align*}As an added challenge, have students find the equation for part b. The equation would be \begin{align*}\frac{n(n+5)}{2}\end{align*}. A hint would be to tell students to double every term and see if they can find that pattern, then divide by 2.

2. The three points form a line. The equation of this line is \begin{align*}y=5x-2\end{align*}. Any set of points that are collinear will follow a pattern where the rule is the equation of the line.

## Conditional Statements

Connections to Literature

Give students a copy of the poem “The Road Not Taken” by Robert Frost. Read the poem with the class and discuss the meaning of the poem and the thoughts behind it.Then have the students rewrite the poem in all conditional statements. When finished, ask the students if the meaning of the poem has changed and how conditional statements can impact the different statements. You can have students work in pairs or in groups. Also, decide if you want students to read their new poems aloud to the class.

Connections to Cinema

In the Know What? is a cartoon by Rube Goldberg. You can show students the “Breakfast Machine” scene from Pee Wee’s Big Adventure (1985). This scene is an acted out series of conditional statements. Have students attempt to write the series of if-then statements that complete the breakfast machine.

Challenge

1. Rewrite your Know What? answer as a series of conditional statements.

*Answers*

1. \begin{align*}A \rightarrow B\end{align*}: If the man raises his spoon, then it pulls a string.

\begin{align*}B \rightarrow C\end{align*}: If the string is pulled, then it tugs back a spoon.

\begin{align*}C \rightarrow D\end{align*}: If the spoon is tugged back, then it throws a cracker into the air.

\begin{align*}D \rightarrow E\end{align*}: If the cracker is tossed into the air, the bird will eat it.

\begin{align*}E \rightarrow F\end{align*}: If the bird eats the cracker, then it turns the pedestal.

\begin{align*}F \rightarrow G\end{align*}: If the bird turns the pedestal, then the water tips over.

\begin{align*}G \rightarrow H\end{align*}: If the water tips over, it goes into the bucket.

\begin{align*}H \rightarrow I\end{align*}: If the water goes into the bucket, then it pulls down the string.

\begin{align*}I \rightarrow J\end{align*}: If the bucket pulls down the string, then the string opens the box.

\begin{align*}J \rightarrow K\end{align*}: If the box is opened, then a fire lights the rocket.

\begin{align*}K \rightarrow L\end{align*}: If the rocket is lit, then the hook pulls a string.

\begin{align*}L \rightarrow M\end{align*}: If the hook pulls the string, then the man’s face is wiped with the napkin.

## Deductive Reasoning

Extension

Draw five different parallelograms on the board. Tell students the definition of a parallelogram. Then, have each student make 2-3 conjectures about the angles or sides. Allow students to share their conjectures with the class. As a class, try to find counterexamples or see if the conjectures are true. Then, see if the conjectures hold for other parallelograms.

Extension

After going over the Laws of Detachment, Contrapositive and Syllogism, students might question why there are no Law of Converse or Law of Inverse. These two statements lead to the Converse Error and the Inverse Error. Example 4 would be an example of the Converse Error. You can also extend the use of symbols. See if students can find the conclusion (if there is one) for each of the following:

\begin{align*}& v \rightarrow g && d \rightarrow e && x \rightarrow y && v \rightarrow g\\ & v \rightarrow h && r \rightarrow d && y \rightarrow z && h \rightarrow v\\ & \therefore && \therefore && \therefore && \therefore\\ & None && r \rightarrow d && x \rightarrow z && h \rightarrow g\end{align*}

Logic Challenge

Logic Puzzles are a great way to use the Law of Syllogism and Contrapositive. Below is an example. Have students read the hints and see if they can figure out who left work on what day, what their favorite “vice” is and how they commute. Students should place an “X” where they know something is not a possibility. If something is true, they can either put an “O” in the square or color the square in. These are great activities for students to do after a test or for extra credit. If you need more logic puzzles, go to www.logic-puzzles.org.

*Answers*

Feb \begin{align*}3^{rd} \rightarrow\end{align*} Davis \begin{align*}\rightarrow\end{align*} pizza \begin{align*}\rightarrow\end{align*} 10-speed

Feb \begin{align*}15^{th} \rightarrow\end{align*} Alexa \begin{align*}\rightarrow\end{align*} nachos \begin{align*}\rightarrow\end{align*} scooter

Apr \begin{align*}7^{th} \rightarrow\end{align*} Noe \begin{align*}\rightarrow\end{align*} chocolate \begin{align*}\rightarrow\end{align*} Segway

Sept \begin{align*}1^{st} \rightarrow\end{align*} McKenna \begin{align*}\rightarrow\end{align*} ice cream \begin{align*}\rightarrow\end{align*} skateboard

## Algebraic and Congruence Properties

Connections to Scales

Bring in several different scales for students to work with.Then, prepare an assortment of items for students to work with. For example, apples, bananas, bags of flour, bags of rice, oranges, etc. You can use non-food items too. From here, have students work in groups and come up with collections of items that demonstrate equality. For example, apples and oranges: How many apples “equal” how many oranges (in weight)? Have one member of the group be the recorder and list of the items that equal other items.Then ask students to use the properties from the chapter and write a reflexive statement, a symmetric statement and a transitive statement about two of their equal statements. Allow time for the students to share their work.

Extension

In this lesson, we introduce the two-column proof. If students feel confident with the concept, you can take away the fill-in-the-blank option that is presented throughout this text. See the Differentiated Instruction FlexBook for modifications and alternate proof options.

Challenge

1. Write a two-column proof.

Given: \begin{align*}\overline{AC}\end{align*} bisects \begin{align*}\angle DAB\end{align*}

Prove: \begin{align*}m \angle BAC = 45^\circ\end{align*}

2. Draw a picture and write a two-column proof.

Given: \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} form a linear pair and \begin{align*}m \angle 1 = m \angle 2\end{align*}.

Prove: \begin{align*}\angle 1\end{align*} is a right angle

*Answers*

1.

Statement |
Reason |
---|---|

1. \begin{align*}\angle DAB\end{align*} is a right angle | Given |

2. \begin{align*}m \angle DAB = 90^\circ\end{align*} | Definition of a right angle |

3. \begin{align*}\overline{AC}\end{align*} bisects \begin{align*}\angle DAB\end{align*} | Given |

4. \begin{align*}m \angle DAC = m \angle BAC\end{align*} | Definition of an angle bisector |

5. \begin{align*}m \angle DAB = m \angle DAC + m \angle BAC\end{align*} | Angle Addition Postulate |

6. \begin{align*}m \angle DAB = m \angle BAC + m \angle BAC\end{align*} | Substitution PoE |

7. \begin{align*}m \angle DAB = 2m \angle BAC\end{align*} | Combine like terms |

8. \begin{align*}90^\circ = 2m \angle BAC\end{align*} | Substitution PoE |

9. \begin{align*}45^\circ = m \angle BAC\end{align*} | Division PoE |

2.

Statement |
Reason |
---|---|

1. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} form a linear pair \begin{align*}m \angle 1 = m \angle 2\end{align*} | Given |

2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are supplementary | Linear Pair Postulate |

3. \begin{align*}m \angle 1 + m \angle 2 = 180^\circ\end{align*} | Definition of Supplementary |

4. \begin{align*}m \angle 1 + m\angle 1=180^\circ\end{align*} | Substitution |

5. \begin{align*}2 m \angle 1 = 180^\circ\end{align*} | Simplify |

6. \begin{align*}m \angle 1 = 90^\circ\end{align*} | Division PoE |

7. \begin{align*}\angle 1\end{align*} is a right angle | Definition of a right angle |

## Proofs about Angle Pairs and Segments

Connections to Cooking

In this activity, the students are going to need to prove the following statement: “You must have eggs to make a chocolate cake.”

Assign half of the class the job of proving that this is a true statement. Assign the other half of the class the job of disproving the statement. Allow students to research on the internet for recipes. Recipes that would disprove this statement would be dairy-free or vegan cakes. Students will need at least three sources (websites, cookbooks, or other) to support their ability to prove or disprove. Then, they will need to come up with an argument (this can be done in groups or pairs) to persuade the rest of the class as to why a chocolate cake is better with or without eggs (depending on which they were assigned). Once the groups have shared their proofs with the class, allow time to debate as a whole on which was the best proof.

Challenge

1. Find the measure of the lettered angles in the picture below. *Hint: Recall the sum of the three angles in a triangle is \begin{align*}180^\circ\end{align*}.*

*Answers*

1. \begin{align*}a = 25^\circ, \ b = 75^\circ, \ c = 105^\circ, \ d = 90^\circ, \ e = 50^\circ, \ f = 40^\circ, \ g = 25^\circ, \ h = 130^\circ, \ j = 155^\circ, \ k = 130^\circ\end{align*}