Algebraic Properties
Commutative or Associate –Students sometimes have trouble distinguishing between the commutative and asso-
ciate 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?
(x+y)+z=z+(x+y)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 ofxandy, and on the right-hand side of the equation the sum ifxandyis 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?
Ifab=cd, andab=f, thencd=f.
Answer: The equalityab=cdcan be changed tocd=abdue to the symmetric property of equality. Then the
statement would read:
Ifcd=ab, andab=f, thencd=f.
This is justified by the transitive property of equality.
Diagrams
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
Chapter 2. Geometry TE - Common Errors