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

August Planning

There are only a few weeks until school starts again, so it's that bittersweet part of the summer where I'm savoring the last bits of vacation and also getting serious about my back to school planning.

I think (you never know) that I'll be teaching Algebra 1 and Geometry this year. My biggest change over the last year, which I plan to really flesh out this year, is the use of Interactive Notebooks. (I call them INBs, though I know that a lot of teachers call them ISNs.)

I teach in a school where getting students to consistently do homework is very difficult. It's just not part of the norm. I've gone from assigning homework every day, to most days, to twice a week, and now this year I have a new plan. I'm going to make small problem sets with spiraled work which are only given once a week. (Homework is always graded on effort/completion, not accuracy. I firmly believe that homework should be a no-risk opportunity to try and sometimes fail. Tests and quizzes are a higher risk time to try and possibly fail. We have enough of that.) It might sound crazy to only give homework once a week, but I'm hoping to create a mindset shift. I want students to view this problem set as something more like a project - something that they have more time to work on and is therefore worthy of its point value.

Because I want students to learn the value of their INBs, I'm going to make my weekly graded assessment open-notebook. That should motivate every student to do a good job with the notebook. This past year I didn't give tests (to me that means larger, longer, more structured in-class assessments). That did not work out so well. By the time we got to Regents review, students did not seem to be as comfortable handling the practice tests. I think it was because they were overwhelmed by the structure. To combat that this year, I'm going to give a "Marking Period Exam" (oooh, doesn't that sound big and important?). That means there will be a graded test roughly every 6 weeks, and I'll give a practice test about a week in advance so the students can freak out about the practice test instead of the real test. Hopefully this will also minimize the lost instructional time for tests.

I'm putting the finishing touches on my curriculum plans. I'm happy with my Algebra 1 outline, but only time will tell if my Geometry plan is ideal.

I'm still thinking through how I want to organize the classroom, group students, keep track of student work. Maybe seat students in pairs that can quickly turn to become groups of 4? With different classroom jobs for each person in the group? Folders for each student organized by group?

Aside from considering the teaching for the year, I'm anxious about my LIFE this school year. How am I going to maintain my work-life balance? I can't be the workaholic perfectionist I've been in the past. How am I going to figure out how to create/re-work lessons more efficiently and still have them meet my personal standards?

What new ideas are you trying out this year? How do you keep the work-life balance from sliding too far to one side?

Grouping Symbols: Jail for Numbers

I've changed over from teaching PEMDAS to using GEMDAS in Algebra 1. This week I discovered a new reason why this was definitely the right move. My Algebra 2 students had their test on radicals, which included solving radical equations.

I saw a lot of made up math. One student just dropped the radical symbol from the every equation and solved as if it had never been there at all. Others started undoing addition or subtraction that was under the radical. (It always disturbs me to see students do things like this; things I've never modeled in class. Where do these ideas come from? Is it desperation?)

In going over the test with the class, it occurred to me to compare grouping symbols, like radicals, as "jail for numbers." If 2x + 5 is under the radical, "the 5 can't just walk out of jail." For that matter, 2x + 5 can't just walk out of jail! What's the Get-Out-of-Jail-Free card? For square roots, it's squaring.

I think they got the message.

Converting Units Puzzle + Corousel

Much thanks to Sarah Hagan over at Math Equals Love for inspiring this activity. After doing a few examples of converting units (both converting within the same system and converting rates), I gave each group a set of index cards. One card said 'Start,' another said 'End,' and there were 6 smaller cards representing the numerators and denominators of the 3 fractions they would need to multiply by to converting the starting units to the ending units. I had already used tape to make a "template" on one of the desks of each table, so the students would easily be able to see where to place the cards.

Here's what one group's work looked like:

This activity worked out really well. Kids who previously did not understand why we were putting certain numbers and units in certain positions had a lightbulb moment. Other group members were able to explain to them and show by touching and moving cards why a correct solution would work. Students were also able to verify with their calculators that the numbers worked out from the 'Start'  card to the 'End' card.

They didn't all get the cards in the right places on their first try. Instead of checking and correcting each group's work, I had one or two students stay with their work, and the rest of the group rotated over to the next table to check over what they had done. This way the students ended up correcting each other's work, if necessary, and everyone got more reinforcement and confidence.

INBs with CC Algebra 1

Even though I've taught 9th grade algebra many times, this year is my first year teaching Common Core Algebra 1. I've decided I'm ready for some major changes, starting with using Interactive Notebooks (INBs) and seating my classes in groups as their regular seating arrangement.

So far I am loving both of these choices. The kids seem to be loving the INBs.

Pros: 

• Everyone can be successful in my algebra class. (Who can't master cutting and pasting?)
• Students are more organized than ever before. We're integrating adding to the Table of Contents (TOC) into our daily routine. (Will it continue? Will students be able to keep this up all year?)
• The physical demands of the INBs force me to slow down my teaching. Many of my students benefit from a slower pace in class. It also allows me extra time to do extra circulating and helping. 
• The seating arrangement silently implies that collaboration is encouraged. (And it is!)

Cons: 

• Glue stick abuse! Some students feel the need to cover every square inch with glue. I'm afraid I'll run out of glue by the end of October. 
• Perfectionist trap. Some students feel the need to perfectly trim every handout or meticulously decorate and color. At the expense of doing math. 
• The seating arrangement seems to be a distraction for some of my Algebra 2 students. Or maybe they would be talking all the time no matter how the desks were arranged. 

More updates to come soon....

Algebra 2 Mansplaining

Today in Algebra 2 and Trig, I was surprised to witness some mansplaining. A group consisting of 3 girls and 1 boy called me over to settle which student factored a quadratic expression correctly. Of course I wasn't going to say, 'You're right, and you're wrong.' Instead, I asked each person (1 of the girls, and the boy) to explain the work in the hopes that speaking aloud would allow the group to decide who was right and who was wrong.

What followed was a long explanation from the boy in a tone that made it sound like he was sure he was right. In fact, the longer he spoke, the more sure of himself he sounded. When he stopped speaking, the girl said nothing. She made no attempt to present or explain her own work. The group simply seemed ready to accept the boy's solution, though no one looked particularly excited by it or made any comment truly agreeing with it.

Did I mention that the boy was completely wrong, and the girl who shrank back and said nothing was the one who had it right?

I was really hoping that by letting the students speak, I wouldn't have to jump in and save them. Instead, I pointed out that the girls just let the boy speak and went along with his answer without questioning it at all. I reminded the girls that mathematical arguing is an important skill in this class!

It's only the beginning of the school year. But I can see already that I'm going to have to really encourage these girls to not let the boys steamroll them. And I need to encourage ALL of my students to have confidence in their own work!

School Choice? What Happened to NCLB?

I don't want to hear another person endorse the concept of "school choice" for parents without acknowledging the hypocrisy of also being in favor of No Child Left Behind. 

The idea of school choice implies that parents will choose the "better" schools, leaving the "bad" schools with little to no enrollment so that they will eventually disappear. Not only would it not work out that way, but that's the same as giving up on the students who still attend those supposedly terrible schools. 

Teachers don't get to choose their students. We don't say, 'I don't want that one; he's never going to be good enough.' Schools are not allowed to give up on their students. How is it acceptable for society to give up on our schools?

Furthermore, some people need a reality check: this is not Lake Wobegon, "where all the women are strong, all the men are good-looking, and all the children are above average." It is impossible for all children to be above average. It is impossible for all teachers to be above average. It is impossible for all schools to be above average. What we should strive for is giving every child the tools and opportunities for success. That's not only attainable, but meaningful.