2.1 Basics of Geometry
Points, Lines, and Planes
Naming Lines –Students often want to use all the labeled points on a line in its name, especially if there are exactly
three points labeled. Tell them they get to pick two, any two, to use in the name. This means there are often many
possible correct names for a single line.
Key Exercise: How many different names can be written for a line that has four labeled points?
Answer: 12
Student can get to this answer by listing all the combination of two letters. Recommend that they make the list in an
orderly way so they do not leave out any possibilities. This exercise is good practice for counting techniques learned
in probability.
Naming Rays –There is so much freedom in naming lines, that students often struggle with the precise way in which
rays must be named. They often think that the direction the ray is pointing needs to be taken into consideration. The
arrow “hat” always points to the right. The “hat” only indicates that the geometric object is a ray, not the ray’s
orientation in space. The first letter in the name of the ray is the endpoint; it does not matter if that point comes first
or second when reading from left to right on the figure. It is helpful to think of the name of a ray as a starting point
and direction. There is only one possible starting point, but often several points that can indicate direction. Any
point on the ray other than the endpoint can be the second point in the name.
There is only one point B –English is an ambiguous language. Two people can have the same name; one word can
have two separate meanings. Math is also a language, but is different from other languages in that there can be no
ambiguity. In a particular figure there can be only one point labeledB.
Key Exercise: Draw a figure in which
←→
ABintersectsAC.Answer: There are many different ways this can be drawn. There must be a line with the pointsAandB, and a
segment with one endpoint atAand the other endpointCcould be at any location.
Segments and Distance
Number or Object –The measure of a segment is a number that can be added, subtracted and combine arithmetically
with other numbers. The segment itself is an object to which postulates and theorems can be applied. Using the
correct notation may not seem important to the students, but is a good habit that will work to their benefit as they
progress in their study of mathematics. For example, in calculus whether a variable represents a scalar or a vector is
critical. When clear notation is used, the mind is free to think about the mathematics.
Using a Ruler –Many Geometry students need to be taught how to use a ruler. The problems stems from students
not truly understanding fractions and decimals. This is a good practical application and an important life skill.
Measuring in centimeters will be learned quickly. Give a brief explanation of how centimeters and millimeters are
marked on the ruler. Since a millimeter is a tenth of a centimeter, both fractions and decimals of centimeters are
easily written.
Using inches is frequently challenging for students because so many still struggle with fractions. Some may need to
2.1. Basics of Geometry