In this Warm-Up, I want to target students' misconceptions about midpoints and the size of angles. In Problem 1, I want students to explain whether B is a midpoint; students can see that B is a point for which AB=BC, but I want them to explain that B cannot be a midpoint because AC is not a segment. In Problem 2, I want students to clarify what the size of an angle depends on (the distance between the rays, not the length of the rays).
A routine practice in my class is to give students yellow notetakers on which to take notes. While I typically give notes towards the end of a lesson, students need some essential polygons vocabulary before they work on the Special Quadrilaterals Investigation.
Because the topic of transformations is such an important part of geometry, and because our definition of “congruent” has shifted from “all corresponding sides and angles are congruent” to “completely covering another figure after sliding, rotating, and reflecting,” I knew that I had to change all the language I would use when talking to students.
While giving notes on polygons, I decided to use my hands to model some of the ideas I was trying to explain with regard to congruence. For example, I brought my thumbs and index fingers together to form a kite-like shape; I then squished my fingers outwards to form another kite-like shape and asked students to explain whether or not they thought both quadrilaterals were congruent. I then squished my fingers inwards and again, asked whether this quadrilateral was congruent to the first and why. I heard some students say, “they aren’t congruent because even though the sides stayed the same, the angles changed, which means the shapes won’t perfectly cover each other.” Right now, as it is the fourth week of school, this idea of congruence sounds spot on to me.
This investigation builds upon the Defining Angles investigation (See Who's a Widget?). I give students a sheet that contains Special Quadrilateral Examples (trapezoid, kite, parallelogram, rhombus, rectangle, and square). I review the definition of a quadrilateral with the whole class (a polygon with exactly four sides and four angles) and ask groups to explore the sides, angles, diagonals, and symmetries of each special quadrilateral so they can see what differentiates one special quadrilateral from another.
Investigating Special Quadrilaterals Groupwork on the Data Record Sheet--this means one student in each group will be solely responsible for measuring the lengths of the sides and determining how many, if any, are congruent. When all students in the group have finished collecting data for their particular property, they will share their findings with each other to draw conclusions. For example, after all members of a group have shared their findings for a rectangle, they will conclude that a rectangle has two pairs of congruent, parallel sides; four congruent angles (all measuring 90-degrees); two congruent diagonals that bisect each other; two lines of symmetry, and 180 degrees of rotational symmetry.
In the next lesson, students will be mixed into new "expert" groups, coming up with a minimal defining list for one special quadrilateral.
Since the Common Core asks us to completely rethink transformations in the Geometry curriculum, one of my goals this year has been to infuse transformations as a thematic yearlong thread. For this reason, I modified the Data Record Sheet students would use in the investigation. Instead of asking students to use rulers to measure the lengths of sides and diagonals or to use protractors to measure the angles (which is time consuming and sometimes frustrating), I encouraged students to instead use tracing paper.
Before launching the task, I told students that the big idea for the activity was to see how line and rotational symmetry could help them to make sense of the sides, angles, and diagonals of special quadrilaterals.
When debriefing this task, I made sure to use tracing paper to show how one side of a kite, for example, mirrors the other. Since the diagonal connecting the kite's vertex angles acts as a line of symmetry, this helps us understand how one half of the kite maps onto the other half, which is why one pair of angles is congruent and why there are two pairs of congruent sides. Similarly, I used tracing paper to show how a parallelogram's rotational symmetry helps us to understand why opposite sides and opposite angles are congruent. The intersection of a parallelogram's diagonals is the center of rotation, and a rotation of 180 degrees maps the opposite sides and angles onto each other.
During the debrief of this lesson, I have six different groups share out their findings from the Investigation. Each group reports out about their findings for all of the properties for one of the six special quadrilaterals.
I tell students that the goal of the next lesson is to come up with a minimal defining list of information needed to define each of the special quadrilaterals.