This month's Growing Problem Solvers department focuses on supporting students' understanding of congruence via the use of rigid motions and transformations.
According to NCTM's Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (2006), initial understandings of congruence and symmetry begin in first grade as students work with different shapes, recognize them from different perspectives, and reflect on how they are similar and different.
The PK–2 grade band task includes an activity asking students to partition a rectangle into two, three, and four equal parts and reflect on equal partitions. Teachers should pay attention to whether students' folds created equal parts. Provide at least 10 cut-out wax-paper rectangles per student, allowing students to use a new piece of wax paper for each different fold so they can keep records of their previous folds. If wax paper is unavailable, lightweight copy paper can be used. This task can be extended by using a variety of sizes of rectangles.
The 3–5 grade band task continues the use of wax paper and folding, this time focusing on the symmetry lines of different shapes. Students are asked to create and recognize a line of symmetry which splits a shape into two congruent parts that are reflections of each other. Teachers should pay attention to whether students' folds divide the figure into two mirror-image halves.
Students are also asked to identify a line that divides each shape into two equal parts that is not a line of symmetry. Because multiple correct solutions exist, the tasks support dialogue among students, especially as students explain their reasoning about how they know they have identified all the lines of symmetry of different polygons.
In the 6–8 grade bandtask, students use GeoGebra activities to apply specific transformations to a shape to map it onto a given image. Teachers can extend the conversation by asking how many different correct sequences of transformations there might be, or how one might decide that one solution is “better” than another. Students will most likely assert that shorter sequences are better (although push-back on that assertion might be interesting: “Why is that?”), but students might also offer other criteria.
In the 9–12 grade band task, students are asked to determine whether the shaded triangle can be mapped onto several other triangles using some combination of the two given transformations. (Teacher note: Triangles I and III are possible, but II and IV are not.) Students are encouraged to consider whether multiple sequences of transformations could accomplish the task. In the case of triangles II and IV, students may come to a rapid consensus that it is impossible (using only the given transformations). The teacher may need to play the part of the skeptic: “How do you know it's not possible? I'm not convinced.” The teacher should push students to try to convince themselves and the rest of the class that it is not possible.
Give students 10 rectangles cut out from wax paper. Students should work in pairs or small groups. Tell students to fold one of their rectangles into two equal parts. Ask them to explain how they know that the rectangle is partitioned into two equal parts.
Have students pick up another rectangle and fold it again to create two equal parts in a different way. Continue to have students fold to create two equal parts in as many different ways as possible.
Have students check the partitions that their group members created and respond to these questions:
- Are the two parts equal?
- Do the parts fit together to form an entire rectangle?
Have students fold another copy of the rectangle into four equal parts. Ask them to explain how they know they created four equal parts. Direct students to pick up another rectangle and fold again to create four equal parts. Instruct students to continue to fold to create four equal parts in as many different ways as possible. Students should again review and discuss group members' folds.
Finally, challenge students to fold a rectangle into three equal parts. Ask students how they know they created three equal parts. Have students pick up another rectangle and fold again to create three equal parts. Instruct students to do this in as many different ways as possible. Students should again review and discuss their group members' folds.
Give students multiple wax-paper rectangles, pentagons, and hexagons like those at right. For more challenge, add wax-paper triangles, squares, parallelograms, and rhombi to the set of wax-paper polygons. Then ask students to fold each polygon into two equal parts. Ask students the following questions to answer either individually or in small groups:
- How do you know you have created two equal parts? Do those parts compose the whole polygon?
- With ink or colored pencil, draw line segments on the wax paper where the creases divide each polygon into two equal parts. How do your line segments compare with your group members' line segments? Does each shape have more than one possible line segment?
- How many lines of symmetry does each polygon have?
- Can you fold each polygon into two equal parts in a way that the crease/fold is not a line of symmetry? If so, how? Discuss with your group the similarities and differences between a line of symmetry and some other line that splits that shape into two equal parts.
For this task, only two transformations can be used:
- Clockwise rotation by 90° about point P
- Reflection over the dotted line
- For each of the four triangles I, II, III, and IV, can you find a sequence of transformations (using only A and B) that maps the shaded triangle onto it?
- If so, what sequence of transformations accomplishes the task? Compare your sequence with that of a classmate. Are they the same? If not, determine whether it is also correct.
- If not, how do you know? How confident are you that no such sequence exists? Why?
S. Asli Özgün-Koca, email@example.com, teaches mathematics and mathematics education courses at Wayne State University in Detroit, Michigan. She is interested in effective and appropriate use of technology in secondary mathematics classrooms along with research in mathematics teacher education.
Matt Enlow, firstname.lastname@example.org, teaches upper-school mathematics at the Dana Hall School in Wellesley, Massachusetts. He is interested both in the art of problem-posing and in mathematical art. He tweets math-y things at https://twitter.com/cmonmattthink.