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1.1: Basics of Geometry

Created by: CK-12

Points, Lines, and Planes

Pacing: This lesson should take approximately three class periods.

Goal: This lesson introduces students to the basic principles of geometry. Students will become familiar with three primary undefined geometric terms and how these terms are used to define other geometric vocabulary. Finally, students are introduced to the concept of dimensions.

Study Skills Tips! Start your students off on the correct foot – vocabulary is a necessity in geometry success! Devote five minutes of each class period to creating flash cards of the major terminology of this text. Use personal whiteboards to perform quick vocabulary checks. Or, better yet, visit Discovery School’s puzzle maker and make your own word searches and crosswords (http://puzzlemaker.discoveryeducation.com/)!

Language Arts Connection! To give an example of why some words are undefined, use the concept of circularity. Students use a dictionary, either electronic or paper (yes, they are still printed!) to complete this activity. Ask students to look up the word point in their reference. Find a key word in that definition. Students should continue this process until the word point is found. Repeat this process for line and plane. The rationale behind this activity is for students to see there is no one way to define these geometric terms, thus allowing them to be undefined but recognizable.

Real World Connection! Have students identify real-life examples of points, lines, planes in the classroom, as well as sets of collinear and coplanar. For example, points could be chairs, lines could be the intersection of the ceiling and wall, and the floor is a great model of a plane. If your chairs are four-legged, this is a fantastic example of why 3 points determine a plane, not four. Four legged chairs tend to wobble, while 3-legged stools remain stable.

To help students understand dimension, use the following table:

Zero-dimensional 1-dimensional (length) 2-dimensional (length and width) 3-dimensional (length, width, and height)

Have students write abstract examples of each dimension (point, line, plane, prism, etc) in the first row. Then have students brainstorm real-life examples of each dimension. Complete the table by gathering the responses of various students.

Segments and Distances

Pacing: This lesson should take one class period

Goal: Students should be familiar with using rulers to measure distances. This lesson incorporates geometric postulates and properties to measurement, such as the Segment Addition Property.

Real World Connection! To review the concept of measurement, use a map of your community. Label several things on your map important to students – high school, grocery store, movie theatre, etc. Have students practice finding the distances between landmarks “as the crow flies.”

Extension! Discuss with your students the rationale of using different units of distance – inch, foot, centimeter, mile, etc. Why are things measured in inches as opposed to fractional feet? This is also a great time to introduce the difference between the metric system and the U.S. measurement system. Have students perform research regarding why the United States continues to use its system while the majority of other countries use the metric system. Provide pros and cons to using each type of system.

Fun tip! Have students devise their own measurement device. Students can use their invention to measure a school hallway, parking lot, or football field. Engage in a whole-class discussion regarding the results.

Refresher! Students may need a refresher regarding multiplying units. Have the students write out the complete unit, as on page 18, and show students how units can be cross-canceled.

Look out! While the Segment Addition Property seems simple, students begin to struggle once proofs come into play. Remind students that the Segment Addition Property allows an individual to combine smaller measurements of a line segment into its whole.

Rays and Angles

Pacing: This lesson should take one class period

Goal: This lesson introduces students to rays and angles and how to use a protractor to measure angles. Several real world models are used to illustrate the concepts of angles.

Real World Connection! Have students Think-Pair-Share their answers to the opening question, “Can you think of other real-life examples of rays?” Choose several groups to share with the class.

Notation Tip! Beginning geometry students may get confused regarding the ray notation. Draw rays in different directions so students become comfortable with the concept that ray notation always points to the right, regardless of the drawn ray’s orientation.

Teaching Strategy! Using a classroom sized protractor will allow students to check to make sure their calculations are the same as yours. Better yet, use an overhead projector or digital imager to demonstrate the proper way to use a protractor.

Teaching Strategy! A good habit for students is to name an angle using of all three letters. This becomes important when labeling vertices of triangles and labeling similar and congruent figures using the similarity statement. Furthermore, stress to students the use of double and triple arcs to denote angles of different measurements. Students can get caught up in the mass amounts of notation and forget this important concept, especially during triangle congruency.

Stress the parallelism between the Segment Addition Property and Angle Addition Property. Students will discover that many geometrical theorems and properties are quite similar, with perhaps one words changed. Yet, the meaning remains the same.

Arts and Crafts Time! Have students take a piece of paper and fold it at any angle of their choosing from the corner of the paper. Open the fold and refold the paper at a different angle, forming two “rays” and three angles. Show how the angle addition property can be used by asking students to measure their created angles and finding the sum – they should equal 90 \;\mathrm{degrees}!

Physical Models! The angle formed at a person’s elbow is a useful physical model of angles. Have the students put their arm straight out, illustrating a straight angle. Then have the student gradually turn their arm up (or down) gradually to demonstrate how the degree changes. Use several students as examples to show that the length of the forearm and bicep do not change the angle measurement.

Segments and Angles

Pacing: This lesson should take one to one and one-half class periods

Goal: The lesson introduces students to the concept of congruency and bisectors. Students will use algebra to write equivalence statements and solve for unknown variables.

Have Fun! Have students do a call-back, similar to what cheerleaders do. You call out “AB” and students would retort, “The distance between!” Continue this for several examples so students begin to see the difference between the distance notation and segment notation.

This is a great lesson for students to create a “dictionary” of all the notation and definitions learned thus far. In addition to the flashcards students are making, the dictionary provides an invaluable reference before assessments.

When teaching the Midpoint Postulate, reiterate to students that this really is the arithmetic average of the endpoints, incorporating algebra and statistics into the lesson.

Visualization! Students have not learned about a perpendicular bisector. Have students complete Example 3 without using their texts as guides. Have students show their bisectors. Hopefully your class will construct multiple bisectors, not simply those that are perpendicular. This helps students visualize that there are an infinite amount of bisectors, but only one that is perpendicular.

Fun Tip! To visualize the angle congruence theorem and provide a means of assessing the ability to use a protractor, give students entering your class an angle measure on a slip of paper (the measurements should repeat). Have the students construct the angle as a warm up. Then have the students find their “matching” partner and check their partner’s angle using a protractor.

Physical Models! Once students have reviewed Example 5, have them copy the angle onto a sheet of notebook paper or patty paper and measure the degree of the bisector. Students will construct a fold at that particular angle measurement to see the angle bisector ray.

Real Life Application! Another method of illustrating angle bisectors is to show a compass rose, as shown below.

http://commons.wikimedia.org/wiki/File:Compass_rose_browns_00.svg

Students can see how directions such as SWS, NNW, etc bisect the traditional four-corner directions.

Angle Pairs

Pacing: This lesson should take one to one and one-half class periods

Goal: Angle pairs are imperative to geometry. This lesson introduces students to common angle pairs.

Inquiry Learning! Students should be encouraged to learn through self-discovery whenever possible. To illustrate the concept of the Linear Pair Postulate, offer several examples of linear pairs. Have students measure each angle and find the sum of the linear pair. Students should discover any linear pair of angles is supplementary.

To further illustrate the idea of vertical angles, extend the adjacent ray of the previous linear pairs to a line. Have students repeat the process of measuring the angles, notating the linear pairs. Students will come to the conclusion that the angles opposite in the “X” are equal.

Students tend to get confused with the term vertical, as in vertical angles. Vertical angles are named because the angles share a vertex, not necessarily because they are in a vertical manner.

Interdisciplinary Connection! NASA has developed many lesson plans that infuse science, technology, and mathematics. The following link will take you to a lesson plan incorporating the seasons and vertical angles. http://sunearthday.nasa.gov/2005/educators/AOTK_lessons.pdf

Classifying Triangles

Pacing: This lesson should take one class period

Goal: Students have previously experienced triangle terminology: scalene, equilateral, isosceles. This lesson incorporates these terms with other defining characteristics.

In Class Activity: Give pairs of students three raw pieces of spaghetti (you can also use non-bendy straws). Instruct one partner to recreate the below table while the second makes two breaks in the spaghetti. It is okay if some breaks away!

The students are to measure the three pieces formed by the two breaks and attempt to construct a triangle using these segments. Students will reach the conclusion that the sum of two segments must always be larger than the third if a triangle is to be formed. The Triangle Inequality Theorem can be found in the lesson entitled Inequalities in Triangles

Segment 1\;\mathrm{Length} (in cm) Segment 2\;\mathrm{Length} (in cm) Segment 3\;\mathrm{Length} (in cm) Can a triangle be formed (Yes/No)

Showing students the difference between line segments and curves, introduce cooked spaghetti. The flexibility of the spaghetti demonstrates to students that segments must be straight in order to provide rigidity and follow the definitions of polygons.

Students can express the concepts presented in this lesson using a Venn diagram or a hierarchy. If students are not familiar with a hierarchy, remind students a hierarchy is an ordering of related objects from the most general to the most specific. An example is shown below.

Classifying Polygons

Pacing: This lesson should take one class period

Goal: This lesson explains the characteristics of a polygon. Students should be able to classify polygons according to its number of sides and whether it’s convex or concave.

Language Arts Connection! Have students find several high school textbooks and internet sites that provide a definition of polygon. Compare each definition for similarities and differences. Devise a workable classroom definition of a polygon, using the ones found as guides.

Study Skills Tip! Flashcards are imperative to geometry success! Students should construct flashcards of the important polygons. One side should be the drawing of the polygon with the reverse naming the polygon, listing the number of sides, and if applicable the sum of the interior angles in a polygon or how to separate the polygon into triangles (useful when determining polygonal area).

We love Pythagoras! When teaching the distance formula, relate this to Pythagorean’s Theorem. The vertical distance (change in y) represents one leg of a right triangle and the horizontal distance (change in x) represents the other leg. Using a^2 + b^2 = c^2, students can derive the distance formula. Many students will use Pythagorean’s Theorem as opposed to the distance formula when determining the length between two points.

What Did You Learn? Use this activity as a culminating activity or perhaps an alternative assessment. Devise a geometry scavenger hunt, listing the major concepts learned thus far. Equip students with a digital camera and their imagination. Ask students to find as many objects as possible, capture them with a photograph and incorporate the photos into a movie or slideshow presentation. Offer bonus points for such things as originality, nature made objects, etc.

Problem Solving in Geometry

Pacing: While this concept should permeate throughout this course, this particular lesson should take one class period.

Goal: Problem solving is necessary is daily life. Drawing diagrams, working backwards, checking multiple options, and answering reasoning questions are imperative for students to learn. This lesson introduces the key questions one should ask when problem solving and some strategies students can use.

Problem solving is essential in daily life. Encouraging students to reflect upon the strategies offered in this lesson and incorporating word problems into your daily routine will help students become successful problem solvers. Allow students to struggle through these problems, facilitating their knowledge rather than providing direct instruction.

Provide a chart for students to use as reference that ask the five essential questions:

  • What is the problem asking for?
  • What do I have that could be used to answer the question?
  • What do I need to know to find the answer?
  • Did I provide the information the problem requested?
  • Does my answer make sense?

By having students begin their problem solving using a chart outlining these questions, students will begin to see when they have answered the problem completely. This helps students later in the textbook by providing a method to answer the age-old question, “When is my proof complete?”

Gather multiple story problems that require various forms of problem solving techniques. Use personal whiteboards to do an immediate check regarding students’ progress.

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