Process of Inductive Reasoning - Students often struggle with the concept of inductive reasoning. It is important to emphasize the three steps of the process: making observations, recognizing a pattern and forming a conjecture. Use real life examples, such as the following:
On Monday the principal decided to play classical music in the cafeteria during lunch. He measured the volume of noise in during lunch and determined it was 80 decibels. On Tuesday he played no music and the volume was 90 decibels. On Wednesday he played pop music and the volume was 85 decibels. The principal repeated this pattern of alternating classical music, no music and pop music a couple more times and noticed similar results. He concludes that classical music reduces the volume of noise in the cafeteria the most during lunch.
This example is very much like a science experiment, which are also typically examples of inductive reasoning. Finding a pattern in a sequence of numbers or figures and continuing the pattern is also inductive reasoning. Having students come up with their own examples of inductive reasoning- either made up examples or examples based on their actually experiences) often helps them to really internalize the concept.
The nth Term - Students enjoy using inductive reasoning to find missing terms in a pattern. They are good at finding the next term, or the tenth term, but have trouble finding a generic term or rule for the number sequence. If the sequence is linear (the difference between terms is constant), they can use methods they learned in Algebra for writing the equation of a line.
Sample: Find a rule for the nth term in the following sequence: 13, 9, 5, 1,...
Sometimes making an input/output table helps students see that the term numbers are the x values and the terms are the y values.
Since this sequence is arithmetic (each term is found by adding or subtracting the same value, or common difference, from the previous term), these points lie on a line. From the table, students can identify two points on the line such as (1, 13) and (2, 9). It may be helpful to do an example where students work in a group and each student in the group chooses a different pair of points. Then they can find the slope between the points and write the equation of the line. Each student should come up with the same equation regardless of which pair of points was chosen. In this case the equation of the line is y=−4x+17 and the rule is −4n+17.
True Means Always True - In mathematics a statement is said to be true if it is always true, no exceptions. Sometimes students will think that a statement only has to hold once, or a few times to be considered true. Explain to them that just one counterexample makes a statement false, even if there are a thousand cases where the statement holds. Truth is a hard criterion to meet, proving a statement false is much less burdensome.
Sequences - A list of numbers is called a sequence. If the students are doing well with the number of vocabulary words in the class, the term sequence can be introduced.
The Advantages and Disadvantages of Non-Math Examples - When first working with conditional statements, using examples outside of mathematics can be very helpful for the students. Statements about the students’ daily lives can be easily broken down into parts and evaluated for veracity. This gives the students a chance to work with the logic, without having to use any mathematical knowledge. The problem is that there is almost always some crazy exception or grey area that students will love to point out. This is a good time to remind students of how much more precise math is compared to our daily language. Ask the students to look for the idea of what you are saying in the non-math examples, and use their powerful minds to critically evaluate the math examples that will follow. One way to help students avoid confusion is to use Euler (pronounced “Oiler”) diagrams to show the relationship between the hypothesis and the conclusion. Here is an example based on the conditional statement, “If today is Saturday, then I will go to the park.”
Figure 1: In this example, the hypothesis is, “today is Saturday.” The hypothesis is written in the inner circle. The conclusion, “I will go to the park,” is written in the outer circle. This diagram is interpreted like a Venn Diagram- if the statement in the inner circle is true, then the statement in the outer circle is also true.
Figure 2: It may help to make an “x” in the inner circle as shown in the figure and say, “You are here, where this statement is true. Does this indicate that you are also inside the circle where it is true that you will go to the park?”
Figure 3: This figure shows an x in the outer circle. Now students should understand that just because you are inside the circle of “going to the park”, that doesn’t not necessary require that you are in the circle of, “it is Saturday.” It could be Saturday, but it doesn’t have to be Saturday.
Figure 4: This final figure illustrates the scenario in which we do not go to the park. If the X is outside of the outer circle, then it is outside the “it is Saturday” circle.
Converse and Contrapositive - The most important variations of a conditional statement are the converse and the contrapositive. Unfortunately, these two sound similar, and students often confuse them. Emphasize the converse and contrapositive in this lesson. Ask the students to compare and contrast them. It is helpful to use the diagrams above to verify the validity of these statements. The Contrapositive can be shown to be true using the last diagram. The statement would be, “If I do not go to the park, then today is not Saturday.” The converse can be shown to be inconclusive using figure 3. The converse statement is, “It I go to the park, then today is Saturday”. Is this true? The correct answer is no, it is not necessarily true.
Converse and Biconditional - The converse of a true statement is not necessarily true! The important concept of implication is prevalent in Geometry and all of mathematics. It takes some time for students to completely understand the direction of the implication. Daily life examples where the converse is obviously not true is a good place to start and making the Euler diagrams should help as well. A good question for students would be, “What would the Euler diagram look like if the converse is true?” They should come to the conclusion that the two circles would completely overlap. In other words there would be one circle with two statements inside as in the example below of the conditional statement, “If an angle is a right angle, then its measure is 90∘.
Both of these statements are equivalent because the definition of a right angle is, “an angle which measures 90∘. In fact, all definitions can be written as true biconditional statements. The students will spend considerable time deciding what theorems have true converses (are bicondtional) in subsequent lessons.
Practice, Practice, Practice - Students are going to need a lot of practice working with conditional statements. It is fun to have the students write and share conditional statements that meet certain conditions. For example, have them write a statement that is true, but that has an inverse that is false. There will be some creative, funny answers that will help all the members of the class remember the material. Encourage students who are struggling to draw the Euler diagram for each conditional statement to help interpret whether or not the other statements are true.
Inductive or Deductive Reasoning - Students frequently struggle with the uses of inductive and deductive reasoning. It is harder for them to see the strengths and weaknesses of each type of thinking, and understand how inductive and deductive reasoning work together to form conclusions. Use situations that the students are familiar with where either inductive or deductive reasoning is being used to familiarize them with the different types of logic. The side by side comparison of the two types of thinking will cement the students’ understanding of the concepts. It would also be beneficial to have the students write their own examples. Some examples follow.
Is inductive or deductive reasoning being used in the following paragraph? Why did you come to this conclusion?
1. The rules of Checkers state that a piece will be crowned when it reaches the last row of the opponent’s side of the board. Susan jumped Tony’s piece and landed in the last row, so Tony put a crown on her piece.
Answer: This is an example of detachment, a form of deductive reasoning. The conclusion follows from an agreed upon rule.
2. For the last three days a boy has walked by Ana’s house at 5 pm with a cute puppy. Today Ana decides to take her little sister outside at 5 pm to show her the dog.
Answer: Ana used inductive reasoning. She is assuming that the pattern she observed will continue.
3. Paul finds the nth term rule for the arithmetic sequence: 5, 9, 13, 17,... to be 4n+1. He then uses this rule to determine that the 100th term is 401.
Answer: This example uses both forms of reasoning. First, Paul uses inductive reasoning to determine the nth term rule. He then uses deductive reasoning when he uses the rule to find the 100th term.
A good rule of thumb for establishing which type of reasoning is being used is to think about what part of the process is occurring. If you are coming up with a conjecture or hypothesis, then it is more likely inductive reasoning. If you are using a known rule, formula or type of argument then it is most likely deductive reasoning. As shown in the previous example, the two work together to form and then prove conjectures or “guesses” about the observed patterns.
Valid Arguments - Students need lots of practice recognizing the valid arguments. The Euler diagrams in the previous section can be used here as well to show that the Law of the Contrapositive and the Law of Detachment are valid. By adding a third circle, the Law of Syllogism can also be illustrated in an Euler diagram.
Converse/Inverse Errors - Students often make the following false conclusions in logical reasoning:
Converse Error: If it rains, then I will bring my umbrella. I bring my umbrella. Therefore, it is raining.
Inverse Error: If it rains, then I will bring my umbrella. It does not rain. Therefore, I do not bring my umbrella.
In these examples, the conclusion is made by assuming that the converse or inverse of the statement is true. We learned in the previous section that they are not necessarily true. This is a good opportunity to review these statements and revisit the Euler diagrams for them again as well.
Algebraic and Congruence Properties
Commutative or Associate - Students sometimes have trouble distinguishing between the commutative and associative properties. It may help to put these properties into words. The associative property is about the order in which multiple operations are done. The commutative is about the first and second operand having different roles in the operation. In subtraction the first operand is the starting amount and the second is the amount of change. Often student will just look for parenthesis; if the statement has parenthesis they will choose associate, and they will usually be correct. Expose them to an exercise like the one below to help break them of this habit.
What property of addition is demonstrated in each of the following statements?
Answer: For example a, it is the commutative property that ensures these two quantities are equal. On the left-hand side of the equation the first operand is the sum of x and y, and on the right-hand side of the equation the sum if x and y is the second operand. In example b, the parentheses are grouping different variables so this is an example of the associative property.
Sometimes it helps students to come up with an expression to remember which is which. One example is: Your group of friends is the people you associate with. This indicates that the associative property refers to a change in grouping. For commutative, think of the word commute-you move from one place to another such as going from home to work. When the variables change position in the expression then it is the commutative property.
Transitive or Substitution - The transitive property is actually a special case of the substitution property. The transitive property has the additional requirement that the first statement ends with the same number or object with which the second statement begins. Acknowledging this to the students helps avoid confusion, and will help them see how the properties fit together. The following statement is true due to the substitution property of equality. How can the statement be changed so that the transitive property of equality would also ensure the statement’s validity?
If ab=cd, and ab=f, then cd=f.
Answer: The equality ab=cd can be changed to cd=ab due to the symmetric property of equality. Then the statement would read:
If cd=ab, and ab=f, then cd=f.
This is justified by the transitive property of equality.
Keeping It All Straight - At this point in the class the students have been introduced to an incredible amount of material that they will need to use in proofs. Laying out a logic argument in proof form is, at first, a hard task. Searching their memories for terms at the same time makes it near impossible for many students. A notebook that serves as a “tool cabinet” full of the definitions, properties, postulates, and later theorems that they will need, will free the students’ minds to concentrate on the logic of the proof. After the students have gained some experience, they will no longer need to refer to their notebook. The act of making the book itself will help the students collect and organize the material in their heads. It is their collection; every time they learn something new, they can add to it.
All Those Symbols - In the back of many math books there is a page that lists all of the symbols and their meanings. The use of symbols is not always consistent between texts and instructors. Students should know this in case they refer to other materials. It is a good idea for students to keep a page in their notebooks where they list symbols, and their agreed upon meanings, as they learn them in class. Some of the symbols they should know at this point in are the ones for equal, congruent, angle, triangle, perpendicular, and parallel.
Don’t Assume Congruence! - When looking at a figure students have a hard time adjusting to the idea that even if two segments or angles look congruent they cannot be assumed to be congruent unless they are marked. A triangle is not isosceles unless at least two of the sides are marked congruent, no matter how much it looks like an isosceles triangle. Maybe one side is a millimeter longer, but the picture is too small to show the difference. Congruent means exactly the same. It is helpful to remind the students that they are learning a new, extremely precise language. In geometry congruence must be communicated with the proper marks if it is known to exist.
Communicate with Figures - A good way to have the students practice communicating by drawing and marking figures is with a small group activity. One person in a group of two or three draws and marks a figure, and then the other members of the group tell the artist what if anything is congruent, perpendicular, parallel, intersecting, and so on. They take turns drawing and interpreting. Have them use as much vocabulary as possible in their descriptions of the figures.
Diagram and Plan - Students frequently want to skip over the diagramming and planning stage of writing a proof. They think it is a waste of time because it is not part of the end result. Diagramming and marking the given information enables the writer of the proof to think and plan. It is analogous to making an outline before writing an essay. It is possible that the student will be able to muddle through without a diagram, but in the end it will probably have taken longer, and the proof will not be written as clearly or beautifully as it could have been if a diagram and some thinking time had been used. Inform students that as proofs get more complicated, mathematicians pride themselves in writing simple, clear, and elegant proofs. They want to make an argument that is undeniably true.
Teacher Encouragement - When talking about proofs and demonstrating the writing of proofs in class, take time to make a well-drawn, well-marked diagram. After the diagram is complete, pause, pretend like you are considering the situation, and ask students for ideas of how they would go about writing this proof.
Assign exercises where students only have to draw and mark a diagram. Use a proof that is beyond their ability at this point in the class and just make the diagram the assignment.
When grading proofs, use a rubric that assigns a certain number of points to the diagram. The diagram should be almost as important as the proof itself.
Refer students to the tips for proof writing that appear between Examples 4 and 5 of this section in the text. They should keep going back to those tips periodically until they become second nature.
Start with “Given”, but don’t end with “Prove” - After a student divides the statement to be proved into a given and prove statements he or she will enjoy writing the givens into the proof. It is like a free start. Sometimes they get a little carried away with this and when they get to the end of the proof write “prove” for the last reason. Remind them that the last step has to have a definition, postulate, property, or theorem to show why it follows from the previous steps. Perhaps reminding them the “prove” is a command statement, not a reason will help them remember that it is just part of the question.
Scaffolding - Proofs are challenging for many students. Many students have a hard time reading proofs. They are just not used to this kind of writing; it is very specialized, like a poem. One strategy for making students accustom to the form of the proof is to give them incomplete proofs and have them fill in the missing statements and reason. There should be a progression where each proof has less already written in, and before they know it, they will be writing proofs by themselves.
Number or Geometric Object - The difference between equality of numbers and congruence of geometric objects was addressed earlier in the class. Before starting this lesson, a short review of this distinction to remind students is worthwhile. If the difference between equality and congruence is not clear in students’ heads, the proofs in this section will seem pointless to them.
Theorems - The concept of a theorem and how it differs from a postulate has been briefly addressed several times in the course, but this is the first time theorems have been the focus of the section. Now would be a good time for students to start a theorem section in their notebook. As they prove, or read a proof of each theorem it can be added to the notebook to be used in other proofs.
Proofs about Angle Pairs and Segments
Mark-Up That Picture - Angles are sometimes hard to see in a complex picture because they are not really written on the page; they are the amount of rotation between two rays that are directly written on the page. It is helpful for students to copy diagram onto their papers and mark all the angles of interest. They can use highlighters and different colored pens and pencils. Each pair of vertical angles or linear pairs can be marked in a different color. Using colors is fun, and gives the students the opportunity to really analyze the angle relationships.
Add New Information to the Diagram - It is common in geometry to have multiple questions about the same diagram. The questions build on each other leading the student though a difficult exercise. As new information is found it should be added to the diagram so that it is readily available to use in answering the next question.
Try a Numerical Example - Sometimes students have trouble understanding a theorem because they get lost in all the symbols and abstraction. When this happens, advise the students to assign a plausible number to the measures of the angles in question and work from there to understand the relationships. Make sure the student understands that this does not prove anything. When numbers are assigned, they are looking at an example, using inductive reasoning to get a better understanding of the situation. The abstract reasoning of deductive reasoning must be used to write a proof.
Inductive vs. Deductive Again - The last six sections have given the students a good amount of practice drawing diagrams, using deductive reasoning, and writing proofs, skills which are closely related. Before moving on to chapter three, take some time to review the first two sections of this chapter. It is quite possible that students have forgotten all about inductive reasoning. Now that they have had practice with deductive reasoning they can compare it to inductive reasoning and gain a deeper understanding of both. They should understand that inductive reasoning often helps a mathematician decide what should be attempted to be proved, and deductive reasoning proves it.
Review - The second section of chapter two contains information about conditional statements that will be used in the more complex proofs in later chapters. Continue to review these variations of the conditional statement in verbal and symbolic form so that students do not forget them.