Snowflakes (Geometry Review)

My contained special ed class has been following the 8th grade IM curriculum. I decided last week that I would give them a midterm exam on units 1-3 in January which meant we'd need to start reviewing some of the ideas from the geometry units. Folding snowflakes the class before winter break seemed like a good place to start!

I begin each class by projecting an image and asking "How many?" To preview the lesson that day we counted with snowflakes:

(I found this on google images but the source is great - it includes instructions for these and more!)
Students counted snowflakes, holes, points, stars, hearts and snowmen (can you see them in the center snowflake? They're wearing top hats!). They used specifics like first, second and third snowflake as well as small, medium and large holes. They reasoned about totals and made comparisons and wrote equations. Have I mentioned that I love this routine for the rich mathematics and language it invites? Because I do love this routine.

After counting everything we could, it was time to start making our own! I googled something like geometry snowflakes and came across a pdf with an FBI copyright warning which I ignored for my class but I won't post it here. Instead you can have the version with a million ads, sorry. The instructions are the same and that link includes the helpful hint to make sure the smallest triangle is on the outside at the end. I'm glad I tried to make one before doing it with the class or we would have had some asymmetrical snowflakes round 1!

Here's where the geometry review came in. As I walked around the room checking everyone had folded correctly, we discussed a vocab word for each step. Step 1: line segment, Step 2: midpoint, Step 3: vertex, Step 4: name that triangle!, Step 5: congruent. I was impressed with how close to congruent most students angles were. Often when I've done origami in class before students struggle to line things up precisely but this set of instructions was quite accessible.

Once everyone had their folds done it was time to cut. I was surprised to find most of them had never done this before and were hesitant to make a cut. I’m glad I had made a few samples the day before and could fold them back up to show how I cut. Once they saw an example they hesitantly made a cut. It was cool to see that several students unfolded after their first cut to see what happened, then refolded and cut some more. The reactions of “did that cut do what I expected?” were great. Students were definitely invested and expressed enthusiasm or dismay about their creations.

After they finished cutting I asked them to notice and wonder about their own result. Then we gathered all the samples together to discuss notices and wonderings. They were able to name the base shape as a pentagon, describe methods to create stars on the inside and outside, and how to make a tree (one not pictured in the close ups had a tree in the snowflake, the small decorated trees were a happy accident when the para cut too much and her snowflake turned into 5 parts!). Then we discussed aesthetics - we seemed to like the ones with more paper cut out. Kids were eager to cut more of their first one or try again. Everyone who didn’t know where to start cutting the first time was eager to try something new out after our discussion!

Geometry Resources

It appears to be around the time when people find out about next year's schedule. Just as I move away from teaching geometry (I will most likely have Algebra 1 and PreCalc) Sam is taking on a geometry course. He put out a call for resources. While I've posted about many of my geometry activities, I don't know that reading through blog posts gives a sense of organization or how my course really works. I have all of my resources well organized in dropbox, but that doesn't help someone else. Even with my co-workers, I share my entire dropbox folder with them but they tend not to search through it for resources unless we have a conversation which I can conveniently end with "that file is in the dropbox" rather than "I'll try to remember to send it to you later."

So I thought maybe I could add a page to this blog where I linked to prior posts in the order that I teach them along with other projects I've taken from other people. A chronological Virtual Filing Cabinet of sorts. But Michael is always bold in his thoughts and he said:

Which reminded me of a few summers ago when Matt Enlow started a conversation about rethinking geometry. We started a wiki. And I had this crazy idea that I would make an iPad friendly website that would have all of my favorite activities built in. It would be the interactive textbook of the next generation. Can you tell I had just finished PCMI and was brimming with ideas and a whole month with nothing planned? Needless to say that project never came to fruition. But I fully believe in the potential for awesome if I had infinite time for such projects:

However, I do not have time for such a thing. So, what would be a helpful way for me to share my geometry resources as I move on from this course?

Circle Graphs

Our state test is approaching and I'm trying to find ways to mix review in with new material. I only mentioned circles in my geometry class on pi day plus it never hurts to practice data analysis, hence circle graphs! As students walked in I had four questions up on the white boards that I had them walk around and answer  (number of siblings, favorite color, type of pets, number of times moved). They all found the mean, median, mode, range for the number of siblings then completed the following assignment.

Circle Graphs

In the past kids have immediately associated circle graphs with percents. Instead of discouraging that instinct I wanted to allow kids to find percents, then help them convert that to degrees to make a precise circle graph. Hence the empty third column. However, my students this year didn't even think of percents. So my vague step four was lost on them. When I asked them what they needed to make a precise graph they quickly came up with a protractor, so we set up proportions and skipped straight to the angle. The rest of it went fine.

So, knowing that, how would you rephrase step four? Any other ideas to make this better or more interesting?

Confronting the Insanity

The testing pressure in my district is real. The school committee met this week to talk about handing one of our elementary schools over to a private company. The skill deficits my students have are also real. Every time we talk about parallel or perpendicular lines we have to discuss how to find slope and what slopes would show the lines are parallel/perpendicular. When we are trying to find the missing angle of a polygon they struggle both to set up and to solve equations. So, the limited Algebra practice that I naturally incorporate in Geometry class isn't sufficient. I can't send them into an Algebra heavy state test in May like this, and I can't send them into Algebra 2 like this. I'm not going to quit teaching Geometry to teach Algebra, and I'm sick of forcing Algebra into Geometry (today we were identifying quadrilaterals given the four vertices, then writing the equation of the line containing one side... because they need practice writing equations of lines).

So my co-teacher and I decided today that we will set aside some time to do Algebra, maybe at the beginning of each class (the glory of 90 minute blocks is we can do this and still have a rich Geometry lesson). In our conversation we thought a good first task would be to have kids practice matching equations to graphs, and then we can work up to writing equations and graphing. But then my last block class had kids in such different places that some kids will be ready to move on from matching after day one while others will need a lot more practice. I think this has to be individualized, each kid could get a chart listing all the skills, quiz on a topic (say 5 questions on a very specific skill) and they either test out of that skill or get assigned related practice problems. If each kid gets an individualized assignment do these have to come in a particular order or can kids choose? Which skills have prerequisites? Maybe I can have kids write practice problems after they master a skill so I'm not making a million?

I hate being so skill/test focused so I'm also hoping to mix in Fawn's awesome visual patterns and estimation problems. Possible plan: each week every student must complete 2 skill assessments, a pattern and an estimation problem (then I can skip estimation on my skill list!). Patterns and estimation should be written up with a partner, skill assessments must be completed individually (but feel free to discuss practice problems with a partner). If I plan 60 minute lessons then students can work on this at the end of class (plus about half the students - the ones with a math learning disability - have a second block with my co-teacher, she has been trying to set up similar things so this can carry through to the other class).

This afternoon I set out to see what skills are important according to the state test. And came up with a list (below) that I can work with. I'm really hoping part of this matches your curriculum, that you do SBG and you have assessments and skill practice sheets that you can share with me. Please??

Algebra:
Simplify algebraic expression (order of operations, exponent rules, factor/distribute)

Solving equations (rate, linear, one quadratic-multiple choice)
Solving system of linear equations, absolute value inequality

Writing linear equation, absolute value inequality
Writing and graphing inequalities (one variable)

Number Sense
Estimate percent, square root, with data (total, average, difference)
Scientific Notation

Geometry:
Angles of triangles, parallel lines with transversal, parallelogram
Similar and congruent triangles
Pythagorean Theorem
Transformations on coordinate plane

Data
Mean, Median, Mode, Range
Box and Whisker Plot, Scatter plot, Line Plot, Circle Graph
Probability

Measurement:
Area, Surface Area, Volume

I have a better sense of what I'm hoping to achieve after writing this post, but I still don't know how best to organize this all. It's overwhelming because it seems like an entire Algebra class running on top of my Geometry course. But I'm already overwhelmed so I'd rather be overwhelmed with a purpose and a plan than continue throwing up my hands in despair. Advice greatly appreciated!

Transformations

Mind Dump!

My next unit in geometry is on transformations. I'm going to lay out all my options and then organize them via blog post, just for fun.

As a geometry team at school we got together and decided these were the important ideas:

• translate
• rotate about the origin multiples of 90 degrees
• reflect over a variety of horizontal and vertical lines, not just the axes
• wait to dilate until we get to similarity, for now rigid transformations only
• use the phrase rigid transformations
• use the prime notation P -> P'
• incorporate language about congruence, including corresponding parts
• one of our textbooks emphasizes coordinate rules, we will not emphasize them, in fact we will encourage students to physically rotate the paper rather than memorize a rule

Last year I had kids draw block letters on the coordinate plane and then perform two transformations in different orders (first translate then reflect vs. first reflect then translate) and compare. The worksheet is not getting along with scribd or I'd share.

This year I think I will have students do one example of each type (translate, rotate, reflect) individually or in pairs and then take notes on the basic definitions as a whole class activity (most kids are familiar with the terms but I have a crew of English Language Learners, definitions are important). From there we will break into stations.

1. transformation mini golf (the game link isn't working at the moment, I hope it's back soon!)
2. draw your own image and then transform it all three ways
3. basic practice problems 
4. more advanced practice problems (last year's sheet? MCAS problems?)
5. something with transparencies, because we have them and they're cool

And that's my outline. We are going to spend about a week on this. What have you done in the past? I know other people have different mini golf options - I think I have some links saved at school but I am currently at home.

Properties of Triangles

Last year when we started talking about triangles one of my colleagues did the relationship between angles and sides first (the largest angle is opposite the longest side etc.) and found great success.  The angle properties of isosceles and equilateral triangles easily follow the definition, the hypotenuse is obviously the longest side of a right triangle, the impossibility of an obtuse equilateral triangle is quickly apparent. So I wrote down that I would do that first this year. But then I forgot the definition of first, I had students define types of triangles and try to draw obtuse equilateral triangles before discovering this relationship. I know now, first means first. At the very beginning. As soon as we say the word triangle, we should be doing the exploration relating sides to angles.

A fun way to transition from lines to triangles is this activity from Mr. Stadel. I retyped it to get the instructions and diagram on one page, it's exactly the same as his though.

Matching Angles.doc

With just a single extra line, you can prove that the angles of a triangle add up to 180:

http://en.wikibooks.org/wiki/Trigonometry/Proof:_Angles_sum_to_180

Now we've mentioned the word triangle, time to explore sides and angles! This sheet is adapted from one in my textbook. I edited it with my colleagues. It was awesome. Kids sat quietly cutting out their side lengths, trying triangles and figuring out what worked. They weren't just quiet because they were cutting and tracing, there was thinking involved in the process. They were noticing and wondering, because, as one student declared, "You always ask us that!" She was complaining that I asked her to write something down when she finished, but what a great complaint 'You always want to know what we think!' I smiled, I have no idea what thoughts ran through her head, but then she commenced recording her ideas.

Triangle Inequalities.pdf

I'd have a picture of graph paper rulers, but I forgot to take one. Might remember tomorrow. Idea: cut out rectangles 1 box wide in the length needed. Compasses, wooden rulers and plastic rulers all work as well, but compasses are tough to hold steady and most rulers are long and thus unwieldy. We used graph paper just a bit larger than the standard size. Give kids one full sheet to draw on and a quarter sheet to cut from and they're good to go.

Goals of this activity: figure out what side lengths make a triangle (triangle inequality) and discover the angle-side relationship. Having the side lengths in decreasing order would make the second discovery more obvious. I chose to have kids work for it a bit, but am open to change. This page was typeset in LaTeX, if you want the original file so you can edit it please let me know.

My next step would have been to define all the words that we use to classify triangles, but since I already did that it will be to have kids work through stations practicing and applying the rules they discovered. I'm unreasonably excited about the coupon holder I got at Staples for keeping all of my station activities in order. Organizers are such fun!

Geometry Curriculum

I'm gearing up for my fourth year of teaching geometry! Having taught 11 sections over the past three years, you'd think I'd have this thing down. But that's the fun of teaching, I get to reimagine the course every year. Since I'm having a hard time getting excited about planning for this year, I'm continuing to analyze what I've done in the past. An awesome feature of doing (sorta) Standards Based Grading last year is that I can scan through my online gradebook and see exactly what topics I taught and in what order. Or at least, that's how it worked in most cases. It turned out I didn't really do quizzes or tests fourth quarter, no wonder one group was complaining so much about all the projects they had to do! Granted, we did most of the work in class so it shouldn't have been too overwhelming, but I hadn't realized how drastically I changed the class at the end of the year.

Geometry 12 13

An explanation of the categories:

Investigations: tasks that students spend class time working on and sometimes finish for homework (I will be teaching all Fundamentals of Geometry this year so homework is limited). Basically they're interesting problems or projects that I consider worthy of grading.

Standards: at the beginning of most classes I give a 3 question quiz on one recent standard. Every 2-4 standards there is a test. So each standard is assessed twice; old standards only cycle back in the way that you need to know properties of an isosceles triangle to determine something about right isosceles triangles. Students can retake quizzes and sections of a test throughout the quarter.

You can see all the topics at once!
And if you want to know what ASA means,
just flip up the cards in the way.

The "T" in the corner is for Top.
Helps kids figure out how to flip since
they write on the card before taping.

Vocabulary/Flappers: Each word/phrase is the title of an index card, definitions and examples (including diagrams!) fill the rest of the card, then it gets taped onto card stock so you can see the title of every card. This was the only formal note taking we did and it worked beautifully. An entire year's notes on a single (double sided) page - easy to refer to! Words in parentheses were discussed and used frequently, most appeared on a flapper but they were not the title of a flapper.

The plan that I wrote last summer is organized by unit, which is a more logical way to lay things out for any purpose other than analysis. I did a decent job of sticking with the plan and most of my standards line up though they're named differently. (I'm glad I did all of that work last summer and then forgot about the second document by mid-September! At least I used the first one all year, it took the most effort.)






Goals:

• Don't answer questions until flappers (or the appropriate reference sheet(s) in third quarter) are out on their desk.
• Be more focused about organizing binders (we did a great job with flappers, now I'd like to move on to keeping tests/quizzes and a chart of standards so they know what their strengths are and what to retake)
• Give some tests and quizzes in fourth quarter and don't collect every assignment (end of year slacking will happen, there's no reason to punish yourself by threatening them with grades)
• Make a comprehensive vocabulary list and talk to the English department about how we can support each other in vocabulary. (Did I miss any words?)
• Reformat the intro to triangles exploration so they learn "Angle-Side Relationship" at the very beginning (is there a better title for "longest side is opposite largest angle"?)
• Have fun!