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.
Key Exercise: Find a rule for the nth term in the following sequence. Answer: The sequence is linear, each term decreases by . The first term is , so the point can be used. The second term is , so the point can be used. Applying what they know from Algebra I, the slope of the line is , and the intercept is , so the rule is .
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.
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.
1. What is the next number in the following number pattern?
Answer: This is the famous Fibonacci sequence. The next term in the sequence is the sum of the previous two terms.
2. What is the missing number in the following number pattern?
Answer: Descending consecutive odd integers are being subtracted from each term, so the missing number is .
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.
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.
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. The students will spend considerable time deciding what theorems have true converses (are bicondtional) in subsequent lessons.
Key Exercise: What is the converse of the following statement? Is the converse of this true statement also true?
If it is raining, there are clouds in the sky.
Answer: The converse is: If there are clouds in the sky, it is raining. This statement is obviously false.
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.
Inductive or Deductive Reasoning – Students frequently struggle with the uses of inductive and deductive reasoning. With a little work and practice they can memorize the definition and see which form of reasoning is being used in a particular example. 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.
Recognizing Reasoning in Action – 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.
Key Exercise: 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 with a cute puppy. Today Ana decides to take her little sister outside at to show her the dog.
Answer: Ana used inductive reasoning. She is assuming that the pattern she observed will continue.
Which is Better? – Students quickly conclude that inductive reasoning is much easier, but often miss that deductive reasoning is more sure and frequently provides some insight into the answer of that important question, “Why?”.
1. What went wrong in this example of inductive reasoning?
Teresa learned in class that John Glenn (the first American to orbit Earth) had to eat out of squeeze tubes, and her mom says the food served in airplanes is not very good. She just had a yummy pizza for lunch. She sees a pattern. Food gets better as one approaches the center of the earth. Therefore the food in a submarine must be delicious!
Answer: She carried the pattern too far.
Commutative or Associate – Students sometimes have trouble distinguishing between the commutative and associate properties. It may help to put these properties into words. The associate 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.
Key Exercise: What property of addition is demonstrated in the following statement?
Answer: 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 and , and on the right-hand side of the equation the sum if and is the second operand.
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.
Key Exercise: 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 , and , then .
Answer: The equality can be changed to due to the symmetric property of equality. Then the statement would read:
If , and , then .
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 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.
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.
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.
Segment and Angle Congruence Theorems
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.
Follow the Pattern – Congruence proofs are a good place for the new proof writer to begin because they are fairly formulaic. Students who are struggling with proofs can get some practice with this style of writing while already knowing the structure of the proof.
State the “if” side in congruence form.
Change the congruence of segments into equality of numbers.
Apply the analogous property of equality.
Change the equality of numbers back to congruence of segments.
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.
1. Prove the following statement.
If , triangle is isosceles.
Definition of congruent segments.
Triangle is isosceles.
Definition of isosceles triangle.
Proofs About Angle Pairs
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 form 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. Since the students did not get to use most of it with these first simple proofs, it would be a good idea to draw their attention to it again and talk briefly about the more complex proof that will be coming.