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?
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 labeled .
Key Exercise: Draw a figure in which intersects .
Answer: There are many different ways this can be drawn. There must be a line with the points and , and a segment with one endpoint at and the other endpoint could 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 be shown how an inch is divided using marks for , and . These fractions often need to be added and reduced to get a measurement in inches.
Review the Coordinate Plane – Some students will have forgotten how to graph an ordered pair on the coordinate plane, or will get the words vertical and horizontal confused. A reminder that the coordinate is first, and measures horizontal distance from the origin, and that the coordinate is second and measures vertical distance from the origin will be helpful. The coordinates are listed in alphabetical order.
1. Points , and are collinear, with located between and .
and . What is ?
(Hint: Draw and label a picture.)
Drawing a picture is extremely helpful when solving Geometry problems. It is good to get the students in this habit early. The process of going form a description to a picture also helps them review their vocabulary.
Rays and Angles
Naming Angles with Three Points – Naming, and identifying angles named with three points is often challenging for students when they first learn it. The middle letter of the angle name, the vertex of the angle, is the most important point. Instruct the students to start by identifying this point and working from there. With practice students will become adept at seeing and naming different angles is a complex picture. Review of this concept is also important. Every few months give the students a problem that requires using this important skill.
Using a Protractor – The two sets of numbers on a protractor are convenient for measuring angles oriented in many different directions, but often lead to errors on the part of the students. There is a simple way for students to check their work when measuring an angle with a protractor. Visual inspection of an angle usually can be used to tell if an angle is acute or obtuse. After the measurement is taken, students should notice if their answer matches with the classification.
1. True or False: A ray can have a measure
Answer: False. A ray extends infinitely on one direction, so it does not have a length.
2. has a measure of . Point is located in the interior of and has a measure of . What is the measure of ?
(Hint: Draw and label a picture.)
3. has a measure of and has a measure of . What is the measure of ?
(Hint: Draw and label a picture.)
Segments and Angles
Congruent or Equal – Frequently students interchange the words congruent and equal. Stress that equal is a word that describes two numbers, and congruent is a word that describes two geometric objects. Equality of measure is often one of the conditions for congruence. If the students have been correctly using the naming conventions for a segment and its measure and an angle and its measure in previous lessons they will be less likely to confuse the words congruent and equal now.
The Number of Tick Marks or Arcs Does Not Give Relative Length – A common misconception is that a pair of segments marked with one tick, are longer than a pair of segments marked with two ticks in the same figure. Clarify that the number of ticks just groups the segments; it does not give any relationship in measure between the groups. An analogous problem occurs for angles.
Midpoint or Bisector – Midpoint is a location, a noun, and bisect is an action, a verb. One geometric object can bisect another by passing through its midpoint. This link to English grammar often helps students differentiate between these similar terms.
Intersects vs. Bisects – Many students replace the word intersects with bisects. Remind the students that if a segment or angle is bisected it is intersected, and it is know that the intersection takes place at the exact middle.
Orientation Does Not Affect Congruence – The only stipulation for segments or angles to be congruent is that they have the same measure. How they are twisted or turned on the page does not matter. This becomes more important when considering congruent polygons later, so it is worth making a point of now.
Labeling a Bisector or Midpoint – Creating a well-labeled picture is an important step in solving many Geometry problems. How to label a midpoint or a bisector is not obvious to many students. It is often best to explicitly explain that in these situations, one marks the congruent segments or angles created by the bisector.
1. Does it make sense for a line to have a midpoint?
Answer: No, a line is infinite in one dimension, so there is not a distinct middle.
Complementary or Supplementary – The quantity of vocabulary in Geometry is frequently challenging for students. It is common for students to interchange the words complementary and supplementary. A good mnemonic device for these words is that they, like many math words, go in alphabetical order; the smaller one, complementary, comes first.
Linear Pair and Supplementary – All linear pairs have supplementary angles, but not all supplementary angles form linear pairs. Understanding how Geometry terms are related helps students remember them.
Angles formed by Two Intersection Lines – Students frequently have to determine the measures of the four angles formed by intersecting lines. They can check their results quickly when they realize that there will always be two sets of congruent angles, and that angles that are not congruent must be supplementary. They can also check that all four angles measures have a sum of .
Write on the Picture – In a complex picture that contains many angle measures which need to be found, students should write angle measures on the figure as they find them. Once they know an angle they can use it to find other angles. This may require them to draw or trace the picture on their paper. It is worth taking the time to do this. The act of drawing the picture will help them gain a deeper understanding of the angle relationships.
Proofs – The word proof strikes fear into the heart of many Geometry students. It is important to define what a mathematical proof is, and let the students know what is expected of them regarding each proof.
Definition: A mathematical proof is a mathematical argument that begins with a truth and proceeds by logical steps to a conclusion which then must be true.
The students’ responsibilities regarding each proof depend on the proof, the ability level of the students, and where in the course the proof occurs. Some options are (1) The student should understand the logical progression of the steps in the proof. (2) The student should be able to reproduce the proof. (3) The student should be able to create proofs using similar arguments.
Vocabulary Overload – Students frequently interchange the words isosceles and scalene. This would be a good time to make flashcards. Each flashcard should have the definition in words and a marked and labeled figure. Just making the flashcards will help the students organize the material in their brains. The flashcards can also be arranged and grouped physically to help students remember the words and how they are related. For example, have the students separate out all the flashcards that describe angles. The cards could also be arranged in a tree diagram to show subsets, for instance equilateral would go under isosceles, and all the triangle words would go under the triangle card.
Angle or Triangle – Both angles and triangles can be named with three letters. The symbol in front of the letters determines which object is being referred to. Remind the students that the language of Geometry is extremely precise and little changes can make a big difference.
Acute Triangles need all Three – A student may see one acute angle in a triangle and immediately classify it as an acute triangle. Remind the students that unlike the classifications of right and obtuse, for a triangle to be acute all three angles must be acute.
Equilateral Subset of Isosceles – In many instances one term is a subset of another term. A Venn diagram is a good way to illustrate this relationship. Having the students practice with this simple instance of subsets will make it easier for the students to understand the more complex situation when classifying quadrilaterals.
1. Draw and mark an isosceles right and an isosceles obtuse triangle.
Answer: The congruent sides of the triangles must be the sides of the right or obtuse angle.
This exercise lays the groundwork for studying the relationship between the sides and angles of a triangle in later chapters. It is important that students take the time to use a straightedge and mark the picture. Using and reading the tick marks correctly helps the students think more clearly about the concepts.
Vocab, Vocab, Vocab – If the students do not know the vocabulary well, they will have no chance at leaning the concepts and doing the exercises. Remind them that the first step is to memorize the vocabulary. This will take considerable effort and time. The student edition gives a good mnemonic device for remembering the word concave. Ask the students to create tricks to memorize other words and have them share their ideas.
Side or Diagonal – A side of a polygon is formed by a segment connecting consecutive vertices, and a diagonal connects nonconsecutive vertices. This distinction is important when student are working out the pattern between the number of sides and the number of vertices of a polygon.
Squaring in the Distance Formula – After subtracting in the distance formula, students will often need to square a negative number. Remind them that the square of a negative number is a positive number. After the squaring step there should be no negatives or subtraction. If they have a negative in the square root, they have made a mistake.
1. Find the length of each side of the triangle with the following vertices with the distance formula. Then classify each triangle by its sides.
a) , and
Answer: The triangle is isosceles with side lengths: , and .
b) , and
Answer: This triangle is scalene with side lengths: , and .
Sometimes students have trouble seeing that they need to take the points two at a time to find the side lengths. By graphing the triangle on graph paper before using the distance formula they can see how to find the side lengths. If they graph the triangle they can also classify it by its angles.
Problem Solving in Geometry
Don’t Panic – Problem solving and applications are particularly challenging for many students. Sometimes they just give up. Let the students know that this is difficult. They are probably going to struggle, have to reread the information several time, and will be confused for a while. It is all part of the process. This section will give them strategies to work through the difficulties.
Highlight Important Information – It is nice when students can actually mark up the text of the exercise, but frequently this is not the case. As they read the paragraph have the students take notes or organize the information into a chart. Otherwise the students can just get lost in all the words. Translating from English to math is often the hardest part.
The Last Sentence – When the students are faced with a sizable paragraph of information the most important sentence, the one that asks the question, is usually at the end. Advise the students to read the last sentence first, then as they read the rest of the paragraph they will see how the information they are being given is important.
Does This Make Sense? – It is so hard to get the students to ask themselves this question at the end of a word problem or application. I think they are so happy to have an answer they do not want to know if it is wrong. Keep reminding them. Sometimes it is possible to not accept work with an obviously wrong answer. The paper can be returned to the student so they can look for their mistake. This is a good argument for the importance of showing clear, organized work.
Naming Quadrilaterals – When naming a quadrilateral the letter representing the vertices will be listed in a clockwise or counterclockwise rotation starting from any vertex. Students are accustomed to reading from left to right and will sometimes continue this pattern when naming a quadrilateral.
The Pythagorean Theorem – Most students have learned to use the Pythagorean Theorem before Geometry class and will want to use it instead of the distance formula. They are closely related; the distance formula is derived from the Pythagorean Theorem as will be explained in another chapter. If they are allowed to use the Pythagorean Theorem remind them that it can only be used for right triangles, and that the length of the longest side of the right triangle, the hypotenuse, must be substituted into the variable if it is know. If the hypotenuse is the side of the triangle being found, the stays a variable, and the other two side are substituted for and .