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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.

Practice Exercise: How many different names can be written for a line that has four labeled points?

Answer: 12, Students can get to this answer by listing all the combinations 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. 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. The “ray” symbol drawn above the two points should be drawn such that the endpoint is over the endpoint of the ray and the arrow is over the second point which indicates the direction.

Example: The ray to the right could be named \overrightarrow{AB} or \overrightarrow{AC}

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 B. A point marks a location and a diagram is like a map. If you have multiple streets with the same name, it is impossible to distinguish between them and find a particular address. Students also need to be reminded that points should always be labeled with capitol letters.

Coplanar Points - Students are often confused by the phrase “three non-collinear points” are necessary to define a plane. They think that none of the points can be on the same line but in reality there is a line through any two points. It must be made clear that the phrase “three non-collinear points” implies that all three of the points are not on the same line but that any two of them may be collinear.

Intersections - Visualizing geometric figures and their intersections can be very difficult for students. Sometimes it helps to use a pencil and paper to illustrate the difference between a line intersecting a plane and a line in a plane. Objects in the room or parts of the building can be used to visualize two planes intersecting in a line (a wall and the ceiling) or three planes intersecting in a point (two walls and the floor). Many high school students still struggle with visualizing these abstract concepts, the more they can be realized in everyday objects, the better students will understand them.

Segments and Distance

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. Students need to practice using a ruler and recognizing centimeters, millimeters and inches. It is very common for students to measure incorrectly, particularly when using inches because they have difficulty interpreting the fractional divisions of inches on a ruler. Some may need to be shown how an inch is divided using the marks for \frac{1}{2}, \frac{1}{4}, \frac{1}{8} and \frac{1}{16}. These fractions often need to be added and reduced to get a measurement in inches.

Another difficulty students encounter is inaccurate measuring because they used the edge of the ruler rather than the mark at zero. Example 1 in the textbook illustrates the correct way to line up the ruler with a segment and the diagram which follows this example shows how to read the partial inches and centimeters on the ruler.

Number or Object - The measure of a segment is a number that can be added, subtracted and combined 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. To be sure students can differentiate between the two concepts, emphasize the notation for the measure of \overline{AB} - Students really struggle with the difference in meaning of \overline{AB}, m\overline{AB} and AB. The first one, \overline{AB}, refers to the object, the segment with endpoints A and B. The latter two expressions refer to the length of the segment and/or the distance from A to B and may be used interchangeably.

Segment Addition - Students should be encouraged to always make a sketch of the segment with particular endpoints and the point between them. Having this concrete diagram will help them avoid setting up an incorrect equation. The process of going from a description to a picture also helps them review their vocabulary. One common mistake with these problems is that once students have been presented with the example in which the point on the segment is the midpoint they sometimes think that the point on the segment in all subsequent examples is also a midpoint. It is important to help them read the questions carefully and note terms such as “in the middle” which indicates a midpoint and “in between” which does not indicate a midpoint.

Review the Coordinate Plane and Horizontal and Vertical Distance - 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 x-coordinate is first, and measures horizontal distance from the origin, and that the y-coordinate is second and measures vertical distance from the origin will be helpful. The coordinates are listed in alphabetical order. When counting these distances on the coordinate plane students occasionally will count the starting point as one and thus end up with a length that is one unit more than the actual distance. Comparing this to when they are playing a board game and counting off squares as they move their token (you don’t count the square you start on as one), or counting laps as they run around the track (you don’t count one until you’ve completed the first lap) may give them a couple of real world example to which they can relate an otherwise abstract concept.

Angles and Measurement

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. Remind students that an angle is made up of two rays and that the three points used to identify the angle come directly from these two rays. With practice students will become adept at seeing and naming different angles in a complex picture. Review of this concept is also important. Every few months give the students a problem that requires using this important skill. It can be difficult for students to learn all of the different notations and labeling practices, especially in the beginning, but practicing these skills will help build a strong foundation for students in geometry. It is crucial that they can read, correctly interpret and correctly communicate in this language in order to be successful in this course.

Using a Protractor - Students need to be shown how to line up the vertex and side of the angle correctly with the protractors available for their use. Not all protractors are the same and students often struggle with this procedure, particularly if the protractor the teacher is using is different from theirs. It is worth taking the time to check with each student to make sure they know how to use their own 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. In fact, encouraging students to always consider whether their answers are reasonable is a good practice to encourage. Students don’t think to do this and often make a small error in calculation that leads to an answer that is clearly wrong. They can help themselves be more accurate in their work if they learn to embrace this habit.

Marking Segments and Angles - Students need to be able to interpret these markings and use them to communicate which angles and segments are congruent and which are not in their own diagrams. It is imperative to practice these skills with students to avoid confusion later in the course.

Midpoints and Bisectors

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.

Midpoint Formula - Students often have a hard time remembering this formula. It helps to make the connection between a midpoint and the “averages” of the x and y coordinates. Students frequently use subtraction rather than addition in this formula and connecting the formula to finding an average helps them to remember that it should be addition. It is also important to remind students that the result is a point, a pair of coordinates and thus it should be written this way.

Angle Pairs

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. There are several ways to help students remember which is which. One is to tell students that it is always right to compliment someone and thus complementary angles add up to 90 degrees or could form a right angle. Another way to remember that complementary angles add up to 90 degrees is to connect the first letter of the word complementary to the first letter of the term corner-typically a corner of a piece of paper is a 90 degree angle. It is also important to present students with examples of each of these pairs of these angles that are separate and adjacent. It is very easy for students to get in the habit of expecting these pairs of angles to occur one way or the other.

Linear Pair and Supplementary - All linear pairs have supplementary angles, but not all supplementary angles form linear pairs. Linear pairs are always adjacent pairs of supplementary angles but not all pairs of supplementary angles are adjacent. Understanding how Geometry terms are related helps students remember them. Linear pairs are a subset of pairs of supplementary angles.

Angles formed by Two Intersecting 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 360 degrees.

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. When students don’t write each angle measure on the diagram they often overlook a relationship between angles that helps them find another measure. It is easy for them to think that they should only find the measures of the angles which are asked for in the problem, when in fact it may be helpful or even necessary to find other, unmarked, angles in the process. 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.

Classifying Polygons

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. It is also important to point out that these subsets are determined by properties exhibited in a figure. In this case, and Equilateral triangle possesses all of the characteristics or properties of an isosceles triangle plus additional properties which make it a subset of the Isosceles triangle category.

Additional Exercises:

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, “caving in” 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. It is also worth noting that all diagonals in a convex polygon are inside the figure and at least one diagonal in a concave polygon lies outside the polygon. This is an additional method to distinguish between concave and convex polygons.

Squaring in the Distance Formula - After subtracting in the distance formula, students will need to square the result. This result is often 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.

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