Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
I can statements:
Level 1: I can solve problems using congruent triangles.
Level 2: I can explain criteria for triangle congruence.
Level 3: I can determine the congruence of two figures using rigid motions.
Level 4: I can prove theorems about angles in triangles.
Level 1: I can recognize properties of special quadrilaterals.
Level 2: I can use properties of special quadrilaterals to solve problems.
Level 3: I can prove theorems about special quadrilaterals.
Level 4: I can determine sufficient conditions for naming special quadrilaterals.
Congruent Triangle Proofs Using Rigid Motions
Interior and Exterior Angles in Polygons
Parallelograms – Proving Properties
Performance Assessment – PARCC Angle Bisector Proof Floor Pattern
Which Standard for Mathematical Practice did you engage most often during the unit?
Most students chose MP1, make sense of problems and persevere in solving them, and MP 7, look for and make use of structure.
I asked students whether any of the content seemed like repeats of previously learned material.
- I don’t remember working with this information before now, other than knowing that a square has four right angles and four congruent sides
- Of course I already knew what all of the different shapes were, but I’m not going to count that because I’d never actually used the shapes to solve complex and important mathematical problems. We actually used squares and parallelograms instead of simply knowing what they were.
- In each lesson we learned I learned something new that I had not already known
- I felt that I already knew some things about congruent triangles, but the lesson on them led me to a deeper understanding of them. I also knew things about the interior and exterior angles in polygons, but that lesson helped me to reason abstractly about them. I also knew the different types of quadrilaterals
Which lesson helped you the most in this unit?
- I don’t remember exactly which lessons and activities were which. However, I know the extra practice in the packets and the quick polls helped for me to see what I did or did not know.
- In 5A/5B about Interior and Exterior Angles in Polygons, i had never truly understood the ongoing continuation of degree sized of adding a side to a polygon. That sentence didn’t make much sense, but what i mean is like a triangle is 180 degrees, a rectangle is 360, a hexagon..etc.
- 5G helped because it gave me problems to solve with the entire unit, which gave me a perspective on what I understood and what I didn’t.
- As with every and all lessons, the usage of interactive diagrams and pictures provides a visual representation of word descriptions and reasons for accurately defining special polygons, determining the congruency of triangles, and finding the values of interior or exterior angles of any reqular polygon.
- The floor plan activity was very helpful, it really helped me understand what we were actually learning.
What did you learn during this unit?
- I have learned that I must attend to precision when talking about quadrilaterals because of their special characteristics.
- The one part about this unit that stood out was that any polygon with one pair of parallel sides is a trapezoid. I always thought trapezoids had to be isosceles trapezoids until this unit, so that’s very interesting to me.
- That everything you do you have to have proof for it and that you have to have a reason how you got that.
I have recently read Transformative Assessment by James Popham. http://www.amazon.com/Transformative-Assessment-W-James-Popham/dp/141660667X
Popham discusses levels of formative assessment that shift through the teacher using formative assessment to adjust instruction (level 1), to the student using formative assessment to adjust learning strategies (level 2), to the classroom of students using formative assessment to ensure that all students in the class are meeting the standards and making adjustments to help each other when that is not happening (level 3), to implementation of formative assessment throughout the school (level 4). I have talked about this model with my students, and so I am pleased to hear them using language like “helped for me to see what I did or did not know” and “gave me a perspective on what I understood and what I didn’t”. They are beginning to pay closer attention to what they have learned and what they have not learned. They are beginning to make adjustments when they haven’t yet met the learning targets.
And so the journey continues, as we learn how to learn …