1.1: Basics of Geometry
Author’s Note
This component of the Teacher’s Edition for the Basic Geometry FlexBook is designed to help teacher’s lesson plan. Suggestions for block planning, daily supplemental activities, and study skills tips are also included. It is recommended to hand out the Study Guides (at the end of each chapter and print-ready) at the beginning of each chapter and fill it out as the chapter progresses.
The Review Queue at the beginning of each section in the FlexBook is designed to be a warm-up for the beginning of each lesson and is intended to be done at the beginning of the period. Answers are at the end of each section.
The Know What? at the beginning of each section in the FlexBook is designed as a discussion point for the beginning of a lesson and then answered at the conclusion of the lesson. It can be added to homework or done as an end-of-the-lesson “quiz” to assess how students are progressing.
Throughout the text there are investigations pertaining to theorems or concepts within a lesson. These investigations may be constructions or detailed drawings that are designed to steer students towards discovering a theorem or concept on their own. This is a hands-on approach to learning the material and usually received well by low-level students. It provides them an opportunity to gain ownership of the material without being told to accept something as truth. These investigations may use: a ruler (or straightedge), compass, protractor, pencil/pen, colored pencils, construction paper, patty paper, or scissors. They can be done as a teacher-led activity, as a group, in pairs, or as an individual activity. If you decide to make an investigation teacher-led, have the students follow along, answer the questions in the text, and then draw their own conclusions. In a block period setting, these activities could be done as a group (because activities seem to take longer when students work in groups) with each group member owning a particular role. One or two students can do the investigation, one can record the group’s conclusions, and one can report back to the class.
At the beginning of the Review Questions, there is a bulleted list with the examples that are similar to which review questions. Encourage students to use this list to and then reference the examples to help them with their homework.
In the Pacing sections for each chapter, consider each “Day” a traditional 50-minute period. For block scheduling, group two days together.
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Points, Lines, and Planes |
Continue Points, Lines, and Planes Investigation 1-1 |
Segments and Distance |
Quiz 1 Start Angles and Measurement |
Finish Angles and Measurement Investigation 1-2 Investigation 1-3 |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
Midpoints and Bisectors Investigation 1-4 Investigation 1-5 |
Quiz 2 Start Angle Pairs |
Finish Angle Pairs Investigation 1-6 |
Classifying Polygons |
Quiz 3 Start Review of Chapter 1 |
Day 11 | Day 12 | Day 13 | ||
Continue Review | Chapter 1 Test |
Continue testing (if needed) Start Chapter 2 |
Points, Lines, and Planes
Goal
This lesson introduces students to the basic principles of geometry. Students will become familiar with the terms points, lines, and planes and how these terms are used to define other geometric vocabulary. Students will also be expected to correctly draw and label geometric figures.
Study Skills Tip
Geometry is very vocabulary-intensive, unlike Algebra. Devote 5-10 minutes of each class period to thoroughly defining and describing vocabulary. Use the Study Guides at the end of each chapter to assist you with this. Also make sure that students know how to correctly label diagrams. You can use personal whiteboards to perform quick vocabulary checks. Or, visit Discovery School’s puzzle maker to make word searches and crosswords (http://puzzlemaker.discoveryeducation.com/).
Real World Connection
Have a class discussion to 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 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. During this discussion, have students fill out the following table:
Dimensions | Description | Geometry Representation | Real-Life Example(s) |
---|---|---|---|
Zero | n/a | ||
One | Length | ||
Two | Length and width | ||
Three | Length, width, and height |
Students may have difficulty distinguishing the difference between a postulate and a theorem. Use real-world examples like “my eyes are blue,” for a postulate. That cannot be proven true, but we know that it is by looking (fill in your eye color). A theorem would be something like, “If the refrigerator is not working, then it is unplugged.” We can go through steps to prove (or disprove) that the fridge is unplugged. Students may conclude other reasons that would make the fridge not work (it could be broken, the fan could have gone out, it is old, etc), making this a statement that needs to be proven and cannot be accepted as true.
Segments and Distance
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. There is also an algebraic tie-in, finding the distance of vertical and horizontal lines on the coordinate plane.
Notation Note
Double (and triple) check that students understand the difference between the labeling of a line segment, \begin{align*}\overline{AB}\end{align*}
Relevant Review
Students may need to review how to plot points and count the squares for the horizontal and vertical distances between two points. It might also be helpful add a few algebraic equations to the Review Queue. Problems involving the Segment Addition Postulate can be similar to solving an algebraic equation (Example 9).
Real World Connection
To review the concept of measurement, use an enlarged 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” and using different street routes to determine the shortest distance between the two.
Teaching Strategy
The Segment Addition Postulate can seem simple to students at first. Start with basic examples, like Examples 5 and 6 and then progress to more complicated examples, like 7 and 8. Finally, introduce problems like Example 9. For more examples, see the Differentiated Instruction component. With the Segment Addition Postulate, you can start to introduce the concept of a proof. Use Example 7 and have students write out an explanation of their drawing. Tell students to use language such that the person reading their explanation knows nothing about math.
Angles and Measurement
Goal
This lesson introduces students to angles and how to use a protractor to measure them. Then, we will apply the Angle Addition Postulate in the same way as the Segment Addition Postulate.
Notation Note
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 has the non-arrow end over the endpoint (regardless of the direction the ray points). Reinforce that \begin{align*}\overrightarrow{AB}\end{align*} and \begin{align*}\overleftarrow{BA}\end{align*} represent the same ray.
Real World Connection
Have students Think-Pair-Share their answers to the opening question, “Can you think of real-life examples of rays?” Then, open up the discussion to the whole class.
Teaching Strategies
Using a classroom sized protractor will allow students to check to make sure their drawings are the same as yours. An overhead projector or digital imager is also a great way to demonstrate the proper way to use a protractor.
In this section, we only tell students that they can use three letters (and always three letters) to label and angle. In Chapter 2 we introduce the shortcut. We did not want the confusion that so commonly occurs where students will name any angle by only its vertex.
Stress the similarities between the Segment Addition Property and Angle Addition Property. Students will discover that many geometrical theorems and properties are quite similar.
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. You can also use this opportunity to explain how angles can also be labeled as numbers, \begin{align*}m \angle 1 + m \angle 2 + m \angle 3 = 90^\circ\end{align*}
Student may need additional practice drawing and copying angles. This is the first time they have used a compass (in this course). Encourage students to play with the compass and show them how to use it to draw a circle and arcs. Once they are familiar with the compass (after 5-10 minutes), then go into Investigation 1-3. In addition to copying a \begin{align*}50^\circ\end{align*} angle, it might be helpful to walk students through copying a \begin{align*}90^\circ\end{align*} angle and an obtuse angle.
Midpoints and Bisectors
Goal
The lesson introduces students to the concept of congruency, midpoints, and bisectors. The difference between congruence and equality will also be stressed. Students will use algebra to write equivalence statements and solve for unknown variables.
Teaching Strategies
This is a great lesson for students to create a “dictionary” of all the notations learned thus far. In addition to the Study Guide, 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. Explain the average between two numbers, is the sum divided by 2. The midpoint of two points is the exact same idea.
Ask students to define “bisector” on their own, before discussing a perpendicular bisector (Example 4). Hopefully students will construct multiple bisectors. This will help students visualize that there are an infinite amount of bisectors, and lead them to the fact there is only one perpendicular bisector and the Perpendicular Bisector Postulate.
With Investigations 1-4 and 1-5, students may need to repeat the construction a few times. Copy a handout with several line segments and different angle measures and have them practice the construction on their own or in pairs.
In this lesson and the previous lesson, we have introduced how to make drawings. Encourage students to redraw any pictures that are in the homework so they can mark congruent segments and angles. Also, let students know that it is ok to mark on quizzes and tests (depending on your preference).
Angle Pairs
Goal
This lesson introduces students to common angle pairs, the Linear Angle Postulate and the Vertical Angles Theorem.
Teaching Strategies
Students can get complementary and supplementary confused. A way to help them remember:
- \begin{align*}C\end{align*} in Complementary also stands for Corner (in a right angle)
- \begin{align*}S\end{align*} in Supplementary also stands for Straight (in a straight angle)
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 is supplementary. Also explain that a linear pair must be adjacent. Discuss the difference between adjacent supplementary angles (a linear pair) and non-adjacent supplementary angles (same side interior angles, consecutive angles, or two angles in a drawing that are not next to each other).
To further illustrate the idea of vertical angles, repeatInvestigation 1-6 with two different intersecting lines. Also, encourage students two draw their intersecting lines at different angles than yours. This way, they will see that no matter the angle measures the vertical angles are always equal and the linear pairs are always supplementary. Draw this investigation on a piece of white paper and have students use the whole page. Then, when they are done, have them exchange papers with the students around them to reinforce that the angle measures do not matter.
- \begin{align*}V\end{align*} in Vertical angles also stands for \begin{align*}V\end{align*} in Vertex. Vertical angles do not have to be “vertical” (one on top of the other). Students might get the definitions confused.
In this section there are a lot of Algebra tie-ins (Example 5, Review Questions 17-25). Students might need additional examples showing linear pairs and vertical angles with algebraic expression representations.
Additional Example: Find the value of \begin{align*}y\end{align*}.
Solution: Because these are vertical angles, set the two expressions equal to each other.
\begin{align*}(14y-42)^\circ & = (11y+6)^\circ\\ 3y & = 48^\circ\\ y & =16^\circ\end{align*}
Classifying Polygons
Goal
In this lesson, we will explore the different types of triangles and polygons. Students will learn how to classify triangles by their sides and angles, as well as classify polygons by the number of sides. The definitions of convex and concave polygons will also be explored.
Teaching Strategy
Divide students into pairs. Give each pair three raw pieces of spaghetti. Instruct one partner to break one piece of spaghetti into three pieces 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 is introduced in Chapter 4.
Next, have students create right, obtuse, acute, scalene, isosceles, and equilateral triangles with their pieces of spaghetti. Show them that if the spaghetti pieces’ endpoints are not touching, the polygon is not closed, and therefore not a polygon. You can use pieces of spaghetti on an overhead projector.
To show 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.
After playing with the spaghetti, brainstorm the qualities of polygons and write them on the board (or overhead) and develop a definition. From here, you can compare and contrast convex and concave polygons. Use a Venn diagram to show the properties that overlap and those that are different.
Review
At the end of this chapter there is a Symbol Toolbox with all the labels and ways to mark drawings. Have students make flash cards with the symbols and markings on one side, and what they represent on the other. Students may also want to make flash cards for the definitions for the other words in the (and future) chapters.
In addition to the Study Guide, it might be helpful to go over the constructions from this chapter. You might want to have a Construction Toolbox, where students have one example of each construction they have learned. These construction pages can supplement the Study Guide and should be added to from chapter to chapter. As an added incentive, you might want to grade students’ Study Guides at the end of the chapter. Another option could be to allow students to use their Study Guide on tests and/or allow it to be extra credit. These options can change from test to test or at the teacher’s discretion.