Basic Geometric Concepts

The Lesson Activities will help you meet these educational goals:

  • Content Knowledge—You will know precise definitions of angle, circle, perpendicular line, parallel line, and line segment.
  • Mathematical Practices—You will make sense of problems and solve them, use appropriate tools strategically, look for and make use of structure, and look for and express regularity in repeated reasoning.
  • Inquiry—You will perform an investigationin which you will make observations, analyze results, and communicate your results in written form.
  • 21stCentury Skills—You will employ online tools for research and analysis, apply creativity and innovation, use critical-thinking and problem-solving skills, communicate effectively, and carry out technology-assisted modeling.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-CheckedActivities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

  1. Line Segments and Angles

In this activity, you will use the GeoGebra geometry tool to explore the properties of line segments and angles. Open GeoGebra,and then complete each step below.If you need help, follow these instructions for using GeoGebra.

  1. Plot three points on the coordinate plane and label them A, B, and C. (Be sure that all three points do not lie in a straight line.) Now join the points two at a time using straight paths. How many unique straight paths can you make through the points? Which geometric figure is formed?

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  1. The three paths intersect in pairs at the points A, B, and C. Measure and record the angle formed at each intersection. Also measure and record the lengths of the straight paths, or sides. Capturethe figure showing the three angles and three sides, and paste it in the space below.

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  1. The three straight paths, and meet each other at three points,A, B, and C. How do these points of intersection differ from each other? Explain the differences in terms of the angles that you see. Also look at the length of the side opposite each angle. What pattern do you see regarding the measurements? In what situation would all the points of intersection resemble one another? Modify the triangle in GeoGebra to help you with your answers.

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  1. Use GeoGebra to move points A, B, and C to the locations shown by the ordered pairs in the table. Record the length of each sideand the measure of each angle for the resulting triangle. Note that the length of a line segment is denoted with two letters but no bar on top. For example, AB is read as “the length of ”

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Location / AB / BC / AC / m / m / m
A(3, 4), B(1, 1), C(5, 1)
A(4, 5), B(2, 1), C(7, 3)
A(3, 6), B(3, -2), C(-3, -2)
  1. If you move point C toward point B along how does mchange? How do and change? What happens to these angles if you move point C away from point B along

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  1. Delete and focus on the intersection of and Keeping points B and C fixed, move point A to different locations and observe how changes. What must you do to increase or decrease m? What happens if you move point A to a location directly on ? What is m at this location? Explain.

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How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Circles

In this activity, you will use the GeoGebra geometry tool to explore the properties of circles. Open GeoGebra,and then complete each step below.

  1. Plot the point A(5, 5). Using a line tool, createwitha length of 4 units from point A. Turn on the trace feature at point B, and move point B around point A, keeping the length of fixed. Capture the image,and paste it in the space below.

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  1. Create a circle such that its center is point A and B is a point on the circle. Capture the image, and paste it in the space below. How does the shape you traced in this part of the activity compare with the circle you created in part a?

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  1. In parts a and b, what is the distance from the center of the circle to a point on the circle? Take this idea a step further:what is the relationship between the center of any circle and the points that lie on the circle?

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How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Perpendicular Lines

Use GeoGebra to explore the properties ofperpendicular lines. Then complete each step below.

  1. Move point B to different locations in the coordinate plane. What do you notice about the relationship between and ?Explain in terms of

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  1. Move point B some more. As you move point B, the angle formed betweenand varies. If you want to makeperpendicular to what do you need to do? Explain in terms of

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  1. Describe the relationship betweenandwhen the two lines are perpendicular to Also describe the relationship betweenandwhenis not perpendicular to Zoom in or out on the coordinate plane, if needed.

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How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Parallel Lines

Use GeoGebra to explore the properties of parallel lines. Then complete each step below.

  1. Study the ordered pairs in the table. Move point D to the locations shown, and note m, m, AE, and BF. You can turn on the grid to find the locations more easily. Round angle measurements to the nearest degree.

Type your response here:

Location of D / m / m / BF / AE
(12, 2)
(11, 1)
(4, 2)
(8, 5)
(9, 7)
  1. Based on your observations from part a, what is the relationship between and when and measure something other than 90°? In this situation, what is the relationship betweenand ? Explain.

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  1. What condition is necessary for and to be equal? In this situation, what is the relationship between and ? What can you conclude about parallel lines with regard to the distance between them? Use the word perpendicular in your answer.

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How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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