TABLE2.4:(continued)
Statement Reason(^6) HF E∼= (^6) GF B Vertical Angles Theorem
(^6) ABC∼= (^6) GF B Transitive Property of Angle Congruence
←→
ADis parallel to
←→
GE Converse of the Corresponding Angles Postulate.The Converse of the Alternate Exterior Angle Theorem could also be proved using the Converse of the Alternate
Interior Angle Theorem. This would demonstrate to the students that once a theorem has been proved it, can be used
in the proof of other theorems. It demonstrates the building block nature of math.
TABLE2.5:
Statement Reason(^6) ABC∼= (^6) HF E Given
(^6) HF E∼= (^6) GF B
(^6) DBF∼= (^6) GF B
Vertical Angles Theorem
(^6) DBF∼= (^6) GF B Transitive Property of Angle Congruence
←→
ADis parallel to
←→
GE Converse of the Alternate Interior Angles Theorem.Proving the theorem in several ways gives students a chance to practice with the concepts and their proof writing
skills. Similar proofs can be assigned for the other theorems in this section.
Slopes
Order of Subtraction –When calculating the slope of a line using two points it is important to keep straight which
point was made point one and which one was point two. It does not matter how these labels are assigned, but the
order of subtraction has to stay the say in the numerator and the denominator of the slope ratio. If students switch
the order they will get the opposite of the correct answer. If they have a graph of the line, ask them to compare the
sign of the slope to the direction of the line. Is the line increasing or decreasing? Does that match the slope?
Graphing Lines with Integer Slopes –The slope of a line is the ratio of two numbers. When students are asked to
graph a line with an integer slope they often fail to realize what and where the second number is. Frequently they
will make the “run” of the line zero and graph a vertical line. It is helpful to have them write the integer that is the
slope, as a ratio over one, before then do any graphing. Really, they only need to do this a few times on paper before
they are able to graph the lines correctly. They will begin to see the ratio correctly in their heads.
Zero or Undefined –Students need to make these associations:
Zero in numerator – slope is zero – line is horizontal
Zero in denominator – slope is undefended – line is vertical
They frequently switch these around. After the relationships are explained in class, remind them frequently, maybe
have a poster up in the room or write the relationship on a corner of the board that does not get erased.
Use Graph Paper –Making a connection between the numbers that describe a line and the line itself is an important
skill. Requiring that the students use graph paper encourages them to make nice, thoughtful graphs, and helps them
make this connection.
Additional Exercises:
- Find the slope of the line that is perpendicular to the line passing through the points( 5 ,− 7 ), and(− 2 ,− 3 ).
Chapter 2. Geometry TE - Common Errors