A Day With No Foldables, Instead Notice and Wonder

Lately I've been getting a little burned out with how jazzy my Geometry INB is getting. It's a lot of paper. A lot of cutting. A lot of pasting. I'm sure that these are good things to do, but there are other effective ways to teach too.

So for my lesson on exterior angles of a triangles, I tried a take on the Number Strings that I heard about at TMC NYC '16. I used 8 straightforward problems where students could use their prior knowledge on linear pairs and the sum of the interior angles of a triangle to find a particular missing angle (either the exterior angle or one of the remote interior angles). Then I asked them to take a step back and tell me what they noticed and wondered.

Here's what I noticed:

• Of my 3 geometry classes, only one did an awesome job noticing/wondering and was able to tell me the relationship between the exterior angle and the remote interior angles. 
• My sophomore group was cooperative in writing down what they noticed and wondered, but they wrote trivial things, probably just to finish the task quickly in hopes of satisfying me. 
• My other upperclassmen didn't notice or wonder anything. They mostly didn't want to write anything down. Maybe they were just anticipating the final bell. Maybe they didn't feel like thinking. I don't know. 
• The questions generated by the sophomores made me realize that they didn't notice that there was a purpose to which angle I was asking them to find in each question. They didn't notice that I never once asked them to find the adjacent interior angle (though they knew they had to find it as an intermediate step to finding the answer). 

Overall, the lesson went very well and I'm pleased with it. But I'm still bothered by some students' general lack of perseverance and unwillingness to write their thoughts down on paper. 

Here's what I wonder:

• How do you get students to persevere, especially when faced with a low-floor task?
• How do you get students to write in math? 
• How do you convince students that no idea or question is silly? That all ideas are valued and worth using? 

5-30-10, Student-Centered Teaching, Geometry, and Number Strings

Does that title sound like a lot? It should. There are a lot of ideas I've had swirling around my head for the last month or so.

In August I was fortunate enough to be able to attend TMC NYC 16, hosted by New Visions. While I'd come across the phrase "number strings" before through browsing on MTBoS, it wasn't until I saw Michael Pershan demonstrate it that I really understood what number strings really are. It's like a really good problem set designed to get students to discover a concept on their own. And it reminded me of Bowen Kerin's style of problem sets (Important Stuff, Neat Stuff, Tough Stuff) which I got to experience first-hand at MfA's Chancellor's Day Workshop in June. And for a while now I've had this nagging voice in my head telling me to talk less in class. Find a way for the students to take over the talking. And I really want to do this. But how?

Number strings make perfect sense to me on an elementary or middle school level. Important Stuff, Neat Stuff, Tough Stuff makes perfect sense to me for an upper-level mathematics topic. But how do I make this work for Common Core Regents courses, with their mile-wide, inch-deep curricula. (Don't even try to tell me that the new curriculum has fewer topics than the old one so there should be time for more in-depth study. Don't even.)

These days I'm specifically thinking about CC Geometry, because I'm teaching it (for real) and developing this new curriculum for myself for the first time. The stars have aligned, and I have the opportunity to write it the way I want it on the first shot! I want to get it right (or as close to right as one can hope for) this year, so I don't need to make major revisions next year.

The challenge I face in Geometry is that students don't come to the class with much background knowledge. They think they know what a square is, for example. But they don't. How can I get students to define these shapes without me telling them myself? Is this even possible? Is this even the right goal?

So far this year, we've established the most basic vocabulary and we've reviewed geometric transformations. We're standing at the doorstep of the good stuff: angles and triangles. What should be my objective?

Why I'm Loving Open-Notebook Quizzes

Two (almost) full weeks into this school year, and I am loving my decision to give weekly open-notebook quizzes.

• Students are actually using their notebooks. I see them flipping back and forth to find helpful information. I see them doing this duirng quizzes. During class work. I have to believe that they're doing it for the homework, too.*
• Students are bringing their notebooks to class regularly. I only have one or two students who "forgot" to bring their notebook to class so far. I think they're learning pretty quickly that having the notebook is necessary for success. 
• Quiz grades are good to great so far! That makes everyone feel good and willing to keep trying. Nothing breeds success like success. 
• I feel comfortable giving a different kind of help during quizzes. I can remind students to look in their notebook for help. I can point them to a certain part of their notes for help. I can clarify directions by pointing to something they they have written down in their notebooks!

And the big one....

• Quizzes have become learning opportunities just as much as they are assessment opportunities. 

I call this success! It's great to feel like I'm doing the right thing. The time we're investing in creating these Interactive Notebooks is worth it. 

*Unless they're just copying someone else's homework. I'm still working on a way to combat that, while also struggling to decide if it's even worth it since so many studies say there's no point to homework anyway. But more on this another time.

Geometry Course Outline

This is my plan for Common Core Geometry so far. If you've taught this course in New York State, and have any insight, please comment!

CC Geometry Course Outline

1. First Day Business: Class Policies/Norms, Edmodo, Class Jobs, Getting to Know You
2. Set up INB: Pocket, TOC, Class Policies, Math Toolbox & Reference Sheet, Grade Record
3. What is Geometry? How is it different from Algebra and other math classes?
4. Undefined Terms: Point, Line, Plane & Definitions: Segment, Angle, Side, Collinear, Right/Acute/Obtuse Angles
5. Notation and Labeling Figures

Unit 1: Transformations

1. Transformations as Functions (+vocab: image, pre-image)
2. Symmetries
3. Rigid Transformations (Isometries) & Preserved Properties
4. Compositions of Transformations
5. Practice with Translations (with and without coordinates)
6. Practice with Reflections: reflect & find line of reflection
7. Reflections with Coordinates
8. Practice with Rotations: rotate & find center/angle of rotation
9. Rotations with Coordinates

Unit 2: Triangles

1. Segments in Triangles: medians, altitudes, angle bisectors, perpendicular bisectors
2. Linear Pairs, Vertical & Adjacent Angles
3. Complementary & Supplementary Angles
4. Sum of Interior Angles of a Triangle
5. Classifying Triangles by Side (Equilateral, Isosceles, Scalene) and by Angle (Acute, Right, Obtuse)
6. Exterior Angles of a Triangle
7. Isosceles Triangles
8. Triangle Inequalities
9. Pre-Proof Reasoning

Unit 3: Congruence

1. Congruent Figures
2. Rigid Motion and Congruence
3. Methods of Proving Triangles Congruent (SSS, SAS, ASA, AAS, HL)
4. Algebra Proofs (not in curric) as intro to Geometry Proofs
5. Proving Triangles Congruent (Mini Triangle Congruence Proofs)  ← Use Proof Blocks
6. Proving Triangles Congruent (Midpoints and Bisectors)
7. Proving Triangles Congruent (Linear Pair Supplementary Vertical Angles Medians Altitudes)
8. Proving Triangles Congruent (Substitution Transitive Addition Subtraction)
9. Angles and Parallel Lines: corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles
10. Algebraic Problems involving Angles and Parallel Lines + Proving Lines Parallel
11. Perpendicular Lines
12. Perpendicular Bisectors
13. Proving Triangles Congruent (Overlapping Triangles)
14. Corresponding Parts of Congruent Triangles (CPCTC)

Unit 4: Quadrilaterals

1. The Quadrilateral Family - properties of all quadrilaterals (4 sides, sum int angles 360)
2. Properties of Parallelograms
3. Parallelogram Proofs
4. Properties of Rectangles, Rhombuses, and Squares
5. Rectangle, Rhombus, and Square Proofs
6. Properties of Trapezoids and Kites
7. Midsegment of a Trapezoid
8. Mixed Practice Problems

Unit 5: Constructions & Concurrencies

1. Using a Compass and Straightedge: Copy a line segment, copy an angle
2. Bisect a Segment / Perpendicular Bisector, Bisect an Angle
3. Parallel through a Point
4. Perpendicular through a Point not/on a line
5. Construction: Regular Hexagon
6. Construction: Equilateral Triangle
7. Construction: Square
8. Construction: Reflections
9. Construction: Angles 90, 60, 45, 30
10. Construction: Altitude, Median, Angle Bisector in a Triangle
11. Construction: Congruent Triangles - SSS, SAS, ASA
12. Construction: Partition a Line Segment
13. Points of Concurrency
14. Centroid: Concurrent Medians
15. Orthocenter: Concurrent Altitudes
16. Circumcenter: Concurrent Perpendicular Bisectors
17. Incenter: Concurrent Angle Bisectors

Unit 6: Dilations & Similarity

1. Dilations
2. *Horizontal and Vertical Stretches
3. Dilations and Lines
4. Similarity
5. Midsegment of a Triangle
6. Side-splitter Theorem
7. Measurements in Similar Figures (perimeters, areas, volume)
8. Proving Similar Triangles
9. Radicals Review
10. Mean Proportional (Geometric Mean)
11. Mean Proportional in Right Triangles (HLLS & SAAS)
12. Pythagorean Theorem & Triples (Pythag Spiral for fun)
13. Special Right Triangle: 45-45-90
14. Special Right Triangle: 30-60-90
15. Trigonometric Ratios
16. Sine and Cosine of Complementary Angles
17. Trigonometry: Solving for a Side
18. Trigonometry: Solve for an Angle
19. Trigonometry: Word Problems (angle of elevation/depression)

Unit 7: Circles

1. Circle Basics
2. Circumference and Area (or save for unit 9??)
3. Arcs and Chords
4. Inscribed Angles
5. Radius and Tangents
6. Secants and Tangents
7. Angles, Chords, and Secants
8. Review Angle Rules
9. BIG Circles
10. Circle Proofs
11. Segment Rules: Secants and Tangents
12. Segment Rules: Intersecting Chords
13. Arc Length & Radian Measure
14. Area of Sectors
15. Inscribed and Circumscribed Polygons - angles, segments, constructions

Unit 8: Coordinate Geometry & Equations

1. Midpoint
2. Length/Distance
3. Slope
4. Partition a Directed Line Segment
5. Area and Perimeter on the Coordinate Plane
6. Coordinate Geometry Proofs: Triangles
7. Equations of Parallel Lines
8. Equations of Perpendicular Lines
9. Equations of Parabolas (?) as Conics
10. Review Completing the Square
11. Equations of Circles

Unit 9: Extending to 3 Dimensions

1. Types of Polygons (n-sides, regular)
2. Interior Angles of Polygons
3. Exterior Angles of Polygons
4. Area and Perimeter of Polygons
5. Area and Circumference of Circles
6. Right Prisms: Volume
7. Cavalieri’s Principle
8. Pyramids
9. Cylinders
10. Cones
11. Spheres and Hemispheres
12. Solids of Revolution
13. Modeling
14. Density