Lesson 7

Students rotate a copy of a right isosceles triangle four times to build a quadrilateral. It turns out that the quadrilateral is a square. Students are not asked or expected to justify this but it can be addressed in the discussion. The fourth question about rotational symmetry of the quadrilateral will help students conclude that it is a square. 

There are many more opportunities to build figures using rigid transformations in other lessons.

Provide access to geometry toolkits, particularly tracing paper.

Ask students what they notice and wonder about the quadrilateral that they have built. Likely responses include:

• It looks like a square.
• Rotating it 90 degrees clockwise or counterclockwise interchanges the 4 copies of triangle \(ABC\).
• Continuing the pattern of rotations, the next one will put \(ABC\) back in its original position.

Ask the students how they know the four triangles fit together without gaps or overlaps to make a quadrilateral. Here the key point is that the triangle is isosceles, so the rotations match up these sides perfectly. The four right angles make a complete 360 degrees, so the shape really is a quadrilateral. The fact that the quadrilateral is a square can be deduced from the fact that it is mapped to itself by a 90 degree rotation, but this does not need to be stressed or addressed.

The purpose of this activity is to allow students to explore special cases of rotating a line segment \(180^\circ\). In general, rotating a segment \(180^\circ\) produces a parallel segment the same length as the original. This activity also treats two special cases:

• When the center of rotation is the midpoint, the rotated segment is the same segment as the original, except the vertices are switched.
• When the center of rotation is an endpoint, the segment together with its image form a segment twice as long as the original.

As students look to make general statements about what happens when a line segment is rotated \(180^\circ\) they engage in MP8. They are experimenting with a particular line segment but the conclusions that they make, especially in the last problem, are for any line segment.

Watch for how students explain that the \(180^\circ\) rotation of segment \(CD\) in the second part of the question is parallel to \(CD\). Some students may say that they “look parallel” while others might try to reason using the structure of the grid. Tell them that they will investigate this further in the next lesson.

Students may be confused when rotating around the midpoint because they think the image cannot be the same segment as the original. Assure students this can occur and highlight that point in the discussion.

Ask students why it is not necessary to specify the direction of a 180 degree rotation (because a 180 degree clockwise rotation around point \(P\) has the same effect as a 180 degree counterclockwise rotation around \(P\)). Invite groups to share their responses. Ask the class if they agree or disagree with each response. When there is a disagreement, have students discuss possible reasons for the differences.

Three important ideas that emerge in the discussion are:

• Rotating a segment \(180^\circ\) around a point that is not on the original line segment produces a parallel segment the same length as the original.
• When the center of rotation is the midpoint, the rotated segment is the same segment as the original, except the vertices are switched.
• When the center of rotation is an endpoint, the segment together with its image form a segment twice as long.

If any of the ideas above are not brought up by the students during the class discussion, be sure to make them known.

All of these ideas can be emphasized dynamically by carrying out a specified rotation in the applet and then moving the center of rotation or an endpoint of the original line segment. Even if students are not using the digital version of the activity, you may want to display and demonstrate with the applet.

In this activity, students use rotations to build a pattern of triangles. In the previous lesson, students examined a right triangle and a rigid transformation of the triangle. In this activity, several rigid transformations of the triangle form an interesting pattern.

Triangle \(ABC\) can be mapped to each of the three other triangles in the pattern with a single rotation. As students work on the first three questions, watch for any students who see that a single rotation can take triangle \(ABC\) to \(CDE\). The center for the rotation is not drawn in the diagram: it is the intersection of segment \(AE\) and segment \(CG\). For students who finish early, guide them to look for a single transformation taking \(ABC\) to each of the other triangles.

This pattern will play an important role later when students use this shape to understand a proof of the Pythagorean Theorem.

Identify students who notice that they have already solved the first question in an earlier activity. Watch for students who think that \(CAGE\) is a square and tell them that this will be addressed in a future lesson. However, encourage them to think about what they conclude about \(CAGE\) now. Also watch for students who repeat the same steps to show that \(ABC\) can be mapped to each of the other three triangles.

Some students might not recognize how this work is similar to the previous activity. For these students, ask them to step back and consider only triangles \(ABC\) and \(CDE\), perhaps covering the bottom half of the diagram.

Select a student previously identified who noticed how the first question relates to a previous activity to share their observation. Discuss here how previous work can be helpful in new work, since students may not be actively looking for these connections. The next questions are like the first, but the triangles have a different orientation and different transformations are needed. 

Discuss rigid transformations. Focus especially on the question about lengths. A key concept in this section is the idea that lengths and angle measures are preserved under rigid transformations.

Some students may claim \(CAGE\) is a square. If this comes up, leave it as an open question for now. This question will be revisited at the end of this unit, once the angle sum in a triangle is known. The last question establishes that \(CAGE\) is a rhombus.