Sample Lesson

Triangle Transformations (with GeoGebra)

Course/ Grade Level: Geometry/Grade 8

Content Standard(s) Addressed: Understand congruence and similarity using physical models, transparencies, or geometry software. CCSS.MATH.CONTENT.8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: CCSS.MATH.CONTENT.8.G.A.1.A Lines are taken to lines, and line segments to line segments of the same length. CCSS.MATH.CONTENT.8.G.A.1.B Angles are taken to angles of the same measure. CCSS.MATH.CONTENT.8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Focus CCSS-M Practice Standard(s): CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

Content Learning Objective: 

The students will construct triangles and transform the figure by rotating, reflecting, and translating on the GeoGebra software. Through their own conjectures made from exploring their constructions, the students will discover the congruence between the original triangle and its transformation.

Materials/Equipment:

• Computers with GeoGebra installed
• Instructions on constructing triangles with transformations (pg. 11-12)

LAUNCH (5 minutes)

Before exploring with GeoGebra, we start the class with reviewing what "congruence" means as well as the congruence postulates (i.e., SAS, SSS, ASA, etc.). The students will draw on paper, two congruent figures (not necessarily triangles).

EXPLORE (20 minutes)

Now, the students will use GeoGebra. The students are given instructions on 3 different constructions. (Optional: Divide the students in groups of 3 and in will be given one of the constructions each.) They will construct an arbitrary triangle and one of the three transformations for each construction. After constructing the triangle and its rotated transformation, the students will investigate the lengths and angles of each triangle. On paper, the students will write down which sides and angles of one triangle are congruent to the other. Then, they must make a conjecture about the congruence of the triangles and identify any congruence postulates that can be used to prove that the rotation preserves length and angle measure. Similarly, for constructing a reflection, the students will first construct the transformation and then determine length and measurements of sides and angles and make conjectures. Again, the students will do the same for translations.

Questions to consider:

• Will this work for a different triangle?
• What happens when we move this point? this line?
• Are the triangles congruent? Why?
• How can you say they're congruent or not?

DISCUSSION (25 minutes)

After all the exploration and everyone has made a conjecture on their transformed triangles, the teacher will open discussion and select students to share their transformation and conjecture to the class. After careful monitoring, the teacher will select the next student depending on connection towards the previous student’s conjecture.

If the students were placed in expert groups, the students can come back to their group of 3 and teach each other the constructions and the observations and conjectures made while exploring their transformation.

After discussion, the teacher and students will define what the transformations (rotation, reflection, and translation) are and do.

Questions to consider: 

• What can you tell me about these angles/sides?
• How can you say they're congruent or not?
• Do you agree with ____'s conjecture? Why or why not?

CLOSING (10 minutes)

Now that the students have done triangles, the students are open to explore different polygons on GeoGebra. As a "ticket-out-the-door", the students can write up how transformations applies to other polygons, and can the congruence postulates be used to affirm their conjectures.