1.2: Reasoning and Proof
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Inductive Reasoning | Conditional Statements |
Continue Conditional Statements Start Deductive Reasoning |
Quiz 1 Finish Deductive Reasoning |
Algebraic and Congruence Properties |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
Quiz 2 Start Proofs about Angle Pairs and Segments |
Finish Proofs about Angle Pairs and Segments |
Quiz 3 Review for Chapter 2 Test |
More Review |
Chapter 2 Test (May need to continue testing on Day 11) |
Inductive Reasoning
Goal
This lesson introduces students to inductive reasoning, which applies to algebraic patterns and integrates algebra with geometry.
Teaching Strategies
After the Review Queue do a Round Robin with difference sequences. Call on one student to say a number, call on a second student to say another number. The third student needs to distinguish the pattern and say the correct number. The fourth, fifth, sixth, etc. students need to say the correct numbers that follow the pattern. Start over whenever you feel is appropriate and repeat. To make the patterns more challenging, you can interject at the third spot (to introduce geometric sequences, Fibonacci, and squared patterns).
Now, take one of the sequences that was created by the class and ask students to try to find the rule. Ask students to recognize the pattern and write the generalization in words.
Additional Example: Find the next three terms of the sequence 14, 10, 15, 11, 16, 12, ...
Solution: Students can look at this sequence in two different ways. One is to subtract 4 and then add 5. Another way is to take the odd terms as one sequence (14, 15, 16, ...) and then the even terms as another sequence (10, 11, 12, ...). Either way, the next three terms will be 17, 13, and 18.
Know What? Suggestion
When going over this Know What? (the locker problem) draw lockers on the front board (as many that will fit). Have students come up to the board and mark x’s to close the appropriate doors, as if they are acting out the problem. This will help students see the pattern.
Real Life Connection
Apply the idea of counterexample to real life situations. Begin by devising a statement, such as, “If the sun is shining, then you can wear shorts.” While this is true for warm weather states such as Florida and California, for those living in the Midwest or Northern states, it is quite common to be sunny and \begin{align*}12^\circ F\end{align*}
Conditional Statements
Goal
This lesson introduces conditional statements. Students will gain an understanding of how converses, inverses, and contrapositives are formed from a conditional.
Teaching Strategies
The first portion of this lesson may be best taught using direct instruction and visual aids. Design phrases you can laminate, such as “you are sixteen” and “you can drive.” Adhere magnets to the back of the phrases (to stick to the white board), or you can use a SMART board. Begin by writing the words “IF” and “THEN,” giving ample space to place your phrases. When discussing each type of conditional, show students how each is constructed by rearranging your phrases, yet leaving the words “IF” and “THEN” intact.
Have students create a chart listing the type of statement, its symbolic form and an example. This allows students anquick reference sheet when trying to decipher between converse, conditional, contrapositive, and inverse. The chart can be added to the Study Guide or place in class notes.
Symbolic Form | Example | True or False? | |
---|---|---|---|
Conditional Statement | \begin{align*}p \rightarrow q\end{align*} | ||
Converse | |||
Inverse | |||
Contrapositive | Logically equivalent to original. |
Spend time reviewing the definition of a counterexample (from the previous section). A counterexample is a quick way to disprove the converse and inverse. Explain to students that the same counterexample should work for both the converse and inverse (if they are false, see Examples 2, 3, and 7).
Use the same setup as the opening activity when discussing biconditionals. Begin with a definition, such as Example 4. Set up your magnetic phrases in if and only if form, then illustrate to students how the biconditional can be separated into its conditional and converse.
Deductive Reasoning
Goal
This lesson introduces deductive reasoning. Different than inductive reasoning, deductive reasoning begins with a generalized statement, and assuming the hypothesis is true, specific examples are deduced.
Teaching Strategies
Start this lesson by writing the Know What? on the board (or copy it onto a transparency). Have the students read each door (either out loud or to themselves) and try to reason which door the peasant should pick. This discussion can lend itself to the definitions of logic and deductive reasoning.
Students may or may not realize that they do deductive reasoning every day. Explain that solving an equation is an example of deductive reasoning. Try to brainstorm, as a class, other examples of deductive reasoning and inductive reasoning.
The best way for students to understand the Laws of Detachment, Contrapositive, and Syllogism is to do lots of practice. Make sure to include problems that do not have a logical conclusion. Like in Examples 7 and 8, it might be helpful for students to put the statements in symbolic form. This will make it easier for them to find the logical conclusion.
Additional Example: Is the following argument logical? Why or why not?
Any student that likes math must have a logical mind.
Lily is logical.
Conclusion: Lily likes math.
Solution: Change this argument into symbols.
\begin{align*}p \rightarrow q\\ q\\ \therefore p\end{align*}
If we were to combine the last statement of the argument and the conclusion, it would be \begin{align*}q \rightarrow p\end{align*} or the inverse. We know that the inverse is not logically equivalent to the original statement, so this is not a logical (or valid) argument.
Algebraic and Congruence Properties
Goal
Students should have some familiarity with these properties. Here we can extend algebraic properties to geometric logic and congruence.
Teaching Strategy
Use personal whiteboards to do a spot check. Write down examples of each property (\begin{align*}4 + a = a + 4\end{align*}, for example) either on the board or overhead. Then have students “race” to see who writes the correct answer on their whiteboard the fastest. If you do not want to make it a competition, just have students show you the answer quickly, 2-3 seconds, and then put their whiteboard down to erase. This could also be done as a competition in groups.
Stress to students that the properties of congruence can only be used with a \begin{align*}\cong\end{align*} symbol and properties of equality with an = sign. Remind students of the difference between congruence and equality that was discussed in Chapter 1.
Have students expand on the properties mentioned in this lesson. Students may come up with the multiplying fractions property, reciprocals, or cross-multiplication.
Prove Move
This lesson introduces proofs. In this text, we will primarily use two-column proofs. Because of the nature of this text, all homework questions and assessment relating to proofs will be fill-in-the-blank. Feel free to explain the concept of a paragraph proof and flow-chart proof, if you feel it would help your students.
Proofs can be very difficult for students to understand. They might ask “why” they have to give a reason for every step. Explain that not everyone reading their proof understands math as well as they or you do. Also, apply proofs (and logical arguments) to the real world. Lawyers use logical arguments all the time. Tell them it might help them they are trying to rationalize something with their parents; a new video game, longer curfew, etc. If they have a logical, fluid “proof” to present to their parents, the parents may be more apt to agree and give them what they are asking for.
Example 4 in the text outlines the basic steps of how to start and complete a proof. Encourage students to draw their own diagrams and mark on them. The bullet list after this example should be gone over several times and addressed as you present Example 5.
Diagrams
The best way to describe what you can and cannot assume is “looks are deceiving.” Reiterate to students that nothing can be assumed. The picture must be marked with notation such as tic marks, angle arcs, arrows, etc. in order for it to be used in a proof. If an angle looks like it is a right angle, it might not be. It needs to be explicitly stated in the Given or the angle must be marked.
Additional Example: Use the diagram to list everything that can be determined from the drawing and those things that cannot. For the latter list, what additional information is needed to clarify the drawing?
Solution: All vertical angles are congruent, \begin{align*}AB=FG\end{align*}, and \begin{align*}\overline{EH} \perp \overleftrightarrow{AD}\end{align*} are given from the markings. From the given statement, we know that \begin{align*}\angle AGF \cong \angle BGA\end{align*}, which can be marked on the drawing.
Things that cannot be assumed are: \begin{align*}DC = HI\end{align*}, \begin{align*}\overline{EH} \perp \overleftrightarrow{FI}\end{align*}, \begin{align*}\overleftrightarrow{AD} \ \| \ \overleftrightarrow{FI}\end{align*}, \begin{align*}\overline{GE}\end{align*} bisects \begin{align*}\angle AGH\end{align*}, \begin{align*}GB = BE\end{align*}. To conclude that these things are true, you must be told them or it needs to be marked. (There may be more, this list will get you started)
In the above example, put the drawing on the overhead or whiteboard. After brainstorming what can and cannot be concluded from the diagram, ask students to correctly mark those things that could not be assumed true. Once they are marked by students, then the statement is validated. This example can lead into a discussion of the different ways you can interpret two perpendicular lines (lines are perpendicular, four right angles, congruent linear pairs, etc). Let students know, in this instance, they only need to be told one of these pieces of information and the others can be concluded from it.
Proofs about Angle Pairs and Segments
Goal
Students will become familiar with two-column proofs and be able to fill out a short proof on their own.
Teaching Strategy
Like with the definitions of complementary and supplementary, the Same Angle Supplements Theorem and the Same Angle Complements Theorem can be confused by students. Remind them of the mnemonic in the Teaching Tips from Chapter 1 (\begin{align*}C\end{align*} is for Complementary and Corner, \begin{align*}S\end{align*} is for Supplementary and a Straight angle).
Prove Move
There are several ways to approach the same proof. The order does not always matter, and sometimes different reasons can be used. For example, the Midpoint Postulate (every line segment has exactly one point that divides it equally in half) and the Definition of a Midpoint (a point that splits a line segment equally in half) can be used interchangeably.
Students may also get stuck on the reasons. Encourage students to not worry about getting the name quite right. If they can’t remember the name of a proof, tell them to write it all out. Depending on your preference, you can also let students use abbreviations for names of theorems as well. For example, the Vertical Angles Theorem can be shortened to the VA Thm. Establish these abbreviations for the entire class so there is no confusion.
In the Proof of the Vertical Angles Theorem, steps 2-4 might seem redundant to students. Explain that they need to completely explain everything they know about linear pairs.
Additional Example: Complete the proof by matching each statement with its corresponding reason.
Solution: The order of the reasons is E, D, G, F, B, A, D