1 year anniversary

I've been sick with a flu type virus for the last week and a half or so.  I've been in a torpor- not reading blog posts, not thinking about teaching, barely dragging myself to school and home and sometimes not even managing that.  Sadly this sick torpor couldn't have come at a worse time.  I spent days planning a spectacular pi-day (but it was supposed to extend over the week with various competitions) and I've been absent for most of it.  My husband and I were supposed to have a getaway weekend this weekend that we've been planning for a year, a YEAR! And now all I can do at this awesome resort is whimper and sniffle.  Finally, this week marked the first anniversary of the earthquake and tsunami in Japan.  If I hadn't been in a congestion induced coma I would have thought to do something at school about this.  Fortunately my boss thought of it first and sent us all packing yesterday on a field trip to visit a local university that had a talk about the earthquake and tsunami.  It was kind of a terrible talk, but it inspired me to share some of my thoughts about this anniversary.  I wish I'd been able to get my act together to talk with my students about this stuff, but maybe it's better I talk about it somewhere than nowhere.

You may wonder why I would talk to my students about this at all.  My school was established on the philosophies of Japanese education 15 years ago and all our students study Japanese for two hours a day- so the earthquake and tsunami already would have caused (excuse me but I must) waves at my school.  But a year ago I was sitting in my classroom on a grading day when I heard the news that an earthquake, then tsunami had struck Tohoku, Japan- a region I lived in for two years.  I don't know how explain my connection to the region or it's people in a non cheesy way- but I studied abroad in Morioka, Iwate when I was 19 and fell in love with my host family and the region.  For the last 8 years my host family and I have written letters to each other monthly- have bi-monthly phone calls and send each other goodies from our respective countries.  I was so torn when I left Morioka as a student that as soon as I graduated with my MAT, I applied and was accepted into the JET Programme (a program to bring native English speakers into Japanese schools) and I requested to be placed in Iwate near my host family.  For a year I worked in a town an hour away from Morioka and took a bus into Morioka every weekend to see my host family.  They took me on trips throughout Japan, took care of me when I was sick, and christened me with their last name because they said I was their American daughter.  They're alright- this is not a build up to a "now they're gone and I'm heart broken" story.  But I had a very difficult weekend a year ago when I saw video after video of the town my host-sister lived and worked in being swept away.  When I finally made contact with my host parents 3 days after the tsunami hit- my host sister still hadn't been located.  You'll never guess what my host mom kept telling me throughout this phone call and subsequent phone calls throughout the week as her daughter remained missing- "thank you so much for caring from so far away."  While her daughter was missing and her prefecture was in ruins (no power, water or food), my host mom was thanking me for caring.  The speaker we went to see for our field-trip yesterday went to Japan for a few weeks last summer to help with clean-up and he characterized the Japanese people he met as "respectful".  He had no experience with Japan before he went on this relief effort, and he's right, they are respectful, but this is such a small word to characterize how the Japanese have responded to such a massive cataclysm.

About 5 or 6 days after the tsunami hit I received word from my host mother that my host sister was ok.  She is also a teacher and she was at school with her students when the earthquake hit.  The school was up pretty high on a cliff overlooking the water, and because of its altitude, it had been designated as the place the towns people should flee to in the event of a tsunami.  The teachers of the school, when they saw the swelling water, decided that their students weren't high enough and they collected all the students in the field and evacuated the students into the hills while the towns people came in behind them fleeing to the school.  The teachers were wise and the towns people (including many parents of the students) were all killed when water flooded the school.   My students knew about my connections with Tohoku and after the tsunami, we spent months raising money for the students of my host-sister's school.  Every student in our school wrote two letters, (there are 88 of us and 160 of them) decorated manila envelopes, and collected toys to put in the envelopes for students of the school.  The money we raised went to buying games and gym equipment for the students at the school now living in shelters.  Last summer my husband and I went to Japan to visit my host family and personally delivered the letters and gym equipment to my host sister's principle.  I'm glad we did something, but I'm so sad we didn't do more.   As we drove around in Japan this past summer visiting my host family, the country seemed almost unchanged from my previous visits.  The Japanese people I met were still very respectful and polite, they still were or pretended to be impressed by my Japanese, they laughed and joked and worked as hard as ever.  They spoke very matter-of-factly about the earthquake and tsunami, often without discernible bitterness. I felt like a clown- bringing these letters and these donations to a school as a great charitable act.  It was such a small thing compared to the mountains each individual Japanese person has moved to push themselves and their country forward.  Underneath their smiles, politeness, respect, and humor lay deep wells of emotion that they control and harness for collective good.  There's a word for this in Japanese- "gaman"- it loosely translates to perseverance- but it is closer to the word "stoic" I think.  I don't really know how to put it in words- but I know most of the people I saw while I was on my trip were suffering deep personal  losses.  Their whole worlds had been turned upside down.  They lost homes, parents, children, spouses and in some cases- their entire neighborhood or town had been washed away.  But they hadn't fallen to pieces.  They hadn't turned to bitterness, over indulgence, listlessness, cynicism, or regret.  I'm sure they felt many of these things but they didn't show any of it.  They rebuilt, they welcomed tourists, they thanked us whole heartedly for the small support we were able to offer.  They gave us tea and let us take pictures of their ruined lives.  There are many things about Japanese culture and society that I disagreed with, or made me feel uncomfortable- of course there were, I am from entirely opposite cultural values- but I'm proud that my host family always said I have a Japanese heart because I can't think of a better heart to have.

I haven't really shared these pictures with anyone, but I think I want to now.  These are pictures my husband and I took when we were vising my host family this past summer.  Maybe you noticed above, but I feel pretty sheepish for even having taken these pictures.  It seemed like such an awful thing to do-fat American tourist snapping shots of ruined Japanese towns while Japanese residents watch on from shelters.  I felt compelled to take these pictures though because I was representing my school and I knew my students back home would want a share of my experiences because they had worked so hard to produce the letters and toys that I was delivering.  I did not end up sharing these pictures with my students though.  There never seemed like a good time to do it and it all just seemed too sad to bring up again.

Here's a view of where the town my host sister lived in used to be:

Here are some various sights throughout the town.  The "OK" spray painted onto buildings meant that it's OK for the buildings to be demolished- anything inside that was salvageable has been taken out already.

This is one of the most striking sights to me.  The boat is upside-down on a tsunami wall.  These walls were designed to withstand tsunamis produced by earthquakes up to 8.5 in magnitude.  

This is my host sister's car sitting outside the school where many of the towns people died.  

Here's the school itself that was hit by the tsunami.  When we first pulled in this mural- of children playing in the sea- was the first thing we saw:

Here are a few pictures around the school.  

Here's a view from the school out to the ocean

Finally, here's a picture from the new, temporary school.  They're using an old office building to house two displaced schools.  They won't be able to rebuild new schools for about 10 years according to my host sister. 

Don't Break the Quotient Rule or Our Friendly Giant will Stick You Between His Couch Cushions that Never get Vacuumed!

I've been meaning to put these up for a while.  My pre-algebra students finished their exponents brochures (description here) and they're kind of awesome so I thought I would post them.  Unfortunately, they did not do as well on the test following the brochures as I would have liked (80% average which is ok, but I wanted better).  I think they performed poorly because even though the brochures helped them cement the basic rules, it didn't help them review how to tackle the different types of problems they might see.  I was trying to avoid doing a drill and kill review worksheet by doing a project instead, but since the test had drill and kill problems, the only way to really prepare them for the test was with a worksheet.  So we're going to spend another few days reviewing exponents and we'll have a retake in a week.
I already started the review.  With our first extra day on exponents, I made a fake test for them that I had "taken" by compiling all the mistakes they made on their first test.  I showed them how I grade tests and then asked them to grade "my" test without an answer key.  I tried to make it a little bit fun in that I said I would award a prize to the student who's final grade was closest to the grade I would have given this test.  Every single student graded the test to within two or three points of the grade I would have given it.  They also found all the mistakes and discussed how frustrating it was when I didn't show my work or circle my answer.  It actually turned into a pretty fun activity because they got to scold me and they were having great discussions about which errors constituted arithmetic errors (which is only -1/2 a point) and which errors constituted understanding errors (which is -1 point).  So clearly they know the material well enough to recognize good work from bad work.  I just need to get them to recognize their own good work and bad work.  When I was wondering aloud about how the same student who got a 65% on his own test could have identified and corrected 100% of my mistakes the very next day, one of our Japanese teachers mentioned that it's easier to understand a language that you're studying than it is to speak it.  So I need to give them more speaking practice?  Maybe I've been focusing too much on error correction.  I decided to make one of my teaching goals this year to help kids learn to recognize their own errors and I guess I went a little overboard.  I have a nice, long drill and kill practice test lined up for them next.
Anyway, without further ado, here are my students' "exponent planet" brochures.

Front of brochure

Inside of brochure

Here's another one 

Front

Inside (The pictures are adorable)

And just one more because I can't resist the math puns this student used

Systems of Equations Scavenger Hunt

My algebra 1 students are coming to the end of their unit on systems of equations.  So far, they've done so much better with it than my students last year that I've been completely blown away.  They even had very little difficulty with the word problems which still has me reeling.  I set up a ladder game type activity where they did problems in pairs and then came to check answers with me before getting a new problem to work on.  I told them I'd award jolly ranchers (I know, I know.  Bribery is not the solution to student engagement, but it just works so well...) to pairs that got the answer on the very first try.  I thought I wouldn't really be handing out any candy because they would struggle so much with the word problems that they'd never be able to get the right answers the first time.  I went broke.  They tore through my stash of candies.  All students got through the 4 problems that were part of the game and about half got through the additional 9 problems that were on the homework all in the space of one class period!

So this week I've been trying to come up with a summative assessment that really challenges my students.  I'm a little bit bored of tests and I want to try something related to the real world.  I spent an hour earlier this week trolling around the internet hoping to find some good, real world application, performance assessment type activity, or worksheet or project but I didn't see anything I liked.  There are some activities out there on supply and demand, but my students I think are more intrinsically motivated to do math than to learn about economics, so a supply and demand activity that's unconnected to their direct experience seems contrived, and less engaging to my students than pure math would be.  Maybe I could fabricate some scenario when they'd be interested in figuring out a supply and demand problem?  I think that this would be a really cool way to go with a summative assessment for this unit, but I can't think of anything practical, doable and a manageable amount of prep.  I played with the idea of a Settler's of Catan type game (where students trade for resources to build stuff) but this is WAY more prep than I can manage right now.  I'm really torn because after reading so many blogs and articles that stress the importance of showing students how their math knowledge can be used practically in the real world, I really want to plan projects and simulations into my classes but I just don't know how to make it genuine.  I could try something like what dy/dan does and find a cool and interesting real life example, but I haven't found one yet and I don't have the time to keep looking.  I played with other ideas like taking a map of a town or city and plotting the paths of cars and seeing where cars may crash or where the road would be the most worn down.  This seemed doable, but again, contrived and impractical.

Because I ran out of time to work on this, I made a game that's maybe a bit of a compromise.  I made a map of our school and superimposed a coordinate grid on it.  I'll hide envelopes in 9 different locations throughout the school.  Each envelope will have copies of problems in them.  I'll hand each pair of students a system of equations problem and they need to solve it, graph the solution on their map, and go to that location to find another problem.  That problem will send them somewhere else in the school.  This is just a variation of a test, but at least it gets the kids moving around and potentially gives them a spacial understanding of what systems of equations mean.  It only took me about three or four hours to prep this game (uggg...).  And it's not really what I wanted, but I think it will be pretty fun.

Here's the map.  I'm especially proud of this.  It will be really useful to have when we're doing any graphing activity in any of my math classes.

System of Equations Scavenger Hunt Map of SJS With Cartesian Grid

Here are the scavenger hunt problems I'll hide in different places around the school:

System of Equations Scavenger Hunt Cards
Here is the worksheets students will fill out as they go around the school

System of Equations Scavenger Hunt Handout
Finally, here's the answer key.  Of course, none of this is of any use to someone who doesn't work at my school and it won't be of any use to me when I leave this school (which is probably going to be soon), but I did put a lot of time and thought into it so I felt the need to share.

System of Equations Scavenger Hunt Key

Don't Break the Product Rule or you'll be Slathered in Butter and Grilled!

My Pre-algebra class is reaching the end of their exponent unit.  I blogged about this unit before, but it was one of my first posts so I'm sorry about the roughness (I was also a little scribd happy because I was excited by how easy it was to throw documents right into my post.  It's magical!) I still feel like this unit is really important because really internalizing the exponent rules makes for a much smoother transition into algebra 2 and beyond, but after reading around a lot on other people's blogs, I'm not sure that stomping along through all the rules in order is the best way to teach them.  I also am aware that in the real world students will never have to simplify these ridiculous exponent problems. Though I still think that understanding these rules is necessary in creating a foundation for high school math and is a good way to introduce the logical system of math, I'm a little uncomfortable with how hard it is to tie them to the real world.  I ended the unit with exponential growth and decay and scientific notation which use the exponent rules in context, but I still wish I had a more concrete way to make these rules relevant to students.

I've been trying to think of a way to review what we learned throughout the whole unit.  Last year, I had the students just do a poster project where they had to neatly and creatively demonstrate all the rules, but I think that was just a desperate attempt on my part to have them review the material without adding a whole bunch more prep work on my part because I was swamped.  This year I created this review activity for them:

Exponent Unit 'Going on Vacation' Review Project After reading around so many blogs and seeing what the larger teaching community is doing, I've realized that even the activities I'm most proud of are lacking the real world context that has been stressed by so many other bloggers.  I've stressed teaching the logic of mathematics to my students because that is what is beautiful about math to me, but my students probably need more context driven activities and examples.  The problem is that my education in mathematics has been entirely traditional (i.e. contextless) and I don't think about applications and I don't interpret the "real world" through math.  I don't know how to see math in the world around me.  Yet I guess, just as we tend to repeat our parent's mistakes, it's so easy to teach the way I was taught and to focus strictly on logic.  I've realized that this is a grave deficiency in my teaching that I need to learn to correct, but changing the way I think about mathematics is going take a lot of time.  At least I'm pretty good at making fun and silly math assignments, even if they're not tied to context.  Two months ago when I came up with the idea for this project I was pretty proud of it.  Now I realize that it doesn't give students any deeper insights into math.  It will be silly and engaging I think, but I need to do better.

Gifted and Talented

I was listening again to an old Radiolab episode and though I know it's a bit out of date and the conversation about this episode has probably long dried up, it just bothered me so much that I felt like I needed to rant about it somewhere.  My husband says that's what blogs are for, so here's the episode:

In it, Malcolm Gladwell talks about why he doesn't like gifted and talented education.  I've been entrenched in the gifted and talented debate in schools for my entire life because my older brother was identified as exceptionally gifted.  My younger brother and I were also identified as gifted, but I had a severe deficiency in reading that helped me integrate more into a public school environment because I wasn't yards ahead of my classmates.  My older brother though, was exceptional all around.  My parents trusted to the public school system to meet his needs, but like many very bright students he was bored.  So bored in fact that when the classes started to get legitimately challenging, he had trouble keeping up as he had never learned study skills because he had never needed to study.  My mother, throughout my brother's school career campaigned for gifted and talented education because the needs of her child were not met.  So I guess this all makes me ridiculously biased, but as a teacher, it has helped me work with my students because it taught me a very valuable lesson- school has to meet students where they are and it has to set expectations tailored to each student's abilities.

 Maybe I'll deal point by point with my disagreements with both Robert Krulwich and Malcolm Gladwell's comments.  I love Radiolab and Robert Krulwich, but I feel that in this instance, they didn't really know what they were talking about.

About a minute into the show, Krulwich says, "As we all know, in America, there's a real hunt on from a very early age to find the gifted and talented children.  And we have programs all over the country trying to identify exceptional kids."  Perhaps it's just Oregon, but here, Gifted and Talented programs have all but been eliminated from the public school system.  My mother has been fighting for the last 25 years to force the Oregon government to adhere to laws about TAG (Talented and Gifted) students in our state because these programs keep getting cut despite regulation meant to ensure that TAG students' needs are met.  She and many other TAG parents sued the Portland school district for noncompliance and won.  TAG programs are just another expense that school districts are reluctant to pay for, especially once No Child Left Behind was passed because NCLB forced schools to expend resources for the students at the bottom of the achievement spectrum, students at the top could be ignored.  The thought that there's a hunt out there to help these bright students succeed is laughable.  Portland just tried to close the highest performing school in the district because it was a "brain drain". They thought that those students, when redistributed among the other high schools, would help those high schools meet AYP. I know that perhaps he meant any gifted and talented students- those gifted at sports, art, or music.  Not necessarily those who are identified as Gifted and Talented academically by state tests.  These academically gifted students though, because their needs are not being met, make fewer and fewer gains as they move up through school and often end up having very poor grades or dropping out because their entire experience with school has been boring and frustrating.  Unfortunately, I have a terrible memory, and this is a rant, not an essay or an article so I won't dredge up statistics for you.  My mom could.  But how often have we teachers bemoaned the poor grades of our brightest students?  I know I and my colleagues have.

About a minute and a half into the show, Malcolm Gladwell, after Krulwich comments that Gladwell hates Gifted and Talented education says "It's ridiculous... We identify a child and call that child gifted because of their performance at the age of, whatever, nine or ten or eleven years old.  Why do we care particularly how well a child performs at nine or ten or eleven years old?  They're nine or ten or eleven.  They're a good 25 years from making any kind of substantial contribution to the world.  Why don't we wait?  What's the hurry?... So one child learns to read at 4 and one child learns to read at 2 and a half. So what?  Why does it matter?  Are the things that are being read between 2 and a half and four are of such incalculable..." And here's where Krulwich cuts in.  I get to what Krulwich's response is in a minute.  I can't even begin to express how much Gladwell misses the point with this comment.  He thinks that we identify early readers because we're interested in finding the next American author?  His comment makes it seem that education is about grooming students to serve society later in life.  This is an entirely valid perspective on education at a macro level.  Yes, education is about raising a new generation of thoughtful, intelligent and creative people to carry our civilization into the future.  But I as an educator am not interested in creating the next Obama, the next Gladwell or the next Bill Gates.  I am interested in helping my students learn to love learning.  I am interested in helping them learn to navigate the world so that they can explore and work towards goals that will make them happy.  I'm interested in helping them become happy, as well as successful human beings.  I am interested in the micro.  Gifted education is not about us selfishly selecting the brightest students to serve society in 30 years ala Ender's Game (If you haven't read this book, you should).  Gifted education is about helping students navigate school successfully without getting bullied and without being bored.  Gifted education is about challenging students so that they don't get lazy, as bright students are apt to do, so that they learn important study skills, expand their knowledge, interact positively with the world around them rather than disengaging as soon as they can no longer coast by.

Now to Krulwich's response: "No no no, it's just a normal parent's response to 'oh, if he's reading at two and a half, think of the things he'll do..' and it's just an extrapolation."  This response makes it sound like parents' only interest in their child's early skill is for boasting purposes or to satisfy a parent's pride.  What about those parents who recognize that their student will go to school and will be told by a friend, or a teacher "you're not supposed to read yet."  Or "it's nice that you can read, but the other children can't yet so you'll just need to sit here until they catch up."  These are comments that are made to young children when they can do things other students can't and it cuts down their morale, makes them feel freakish, tells them that what they do naturally is not ok, or that school is about sitting around being bored.  Parents become frustrated when their children are unhappy, and parents of gifted students are very frustrated.  My mother gets calls and e-mails daily from parents torn about what to do when their children come home from school demoralized, unmotivated and sad. (To make another literature reference- Wrinkle in Time has a great example of how gifted students can get treated at school.)

Now I know that this is not what Gladwell and Krulwich were focusing on.  There are parents out there that push their children to succeed out of selfish motives.  In many cases, there's no need to separate high performing students from average students.  But there are students out there who have special needs- just like deaf students, or a students with dyslexia.  Exceptionally gifted students need special programs with teachers who understand them in order to become successful, well adjusted adults.  I had dyslexia and I was accelerated in math.  Both of my needs were met.  I was challenged in both reading and math and being challenged taught me the beauty that could be found through learning and education.  I have a good work ethic because I was constantly challenged to keep trying, keep pushing, keep engaging with the world.  All students need this challenge, all students should have programs and teachers that work with them from where they are to where they want to go.  Even naturally academically gifted students need guidance and support. Krulwich and Gladwell have forgotten that education is also about students, not just about society at large.

Perfect Squares and Perfect Cubes

My algebra 2 students are starting their unit on radicals.  This unit includes a review of square root arithmetic, the quadratic formula, higher order roots and fractional exponents.  It's kind of a dry unit.  I was trying to think of a way to spice it up and also, I was thinking about how in years past, my high school students have not been able to recognize perfect squares or perfect cubes at all because elementary schools around here don't emphasize the multiplication tables.  Then I remembered a comment on a blog (I think it was f(t) but I can't find the actual comment) from a guy who said he got students to recognize perfect squares by coding the alphabet with perfect squares from 1^2 through 26^2 and putting messages up on his board in code.  I thought this was a pretty awesome idea, so I went through and coded messages for my students for each day of the unit.  Because they're in algebra 2 though, I went beyond perfect squares and made more interesting codes.  I'm just awarding candy to those who solve the codes, but I'm also thinking about making it a competition.  The person who decodes the most messages correctly gets some prize.  I did my first one yesterday and my second today and so far, my students are really into it.  I thought I'd share the coded messages here for anyone who was interested.  I got all the messages by googling around for math jokes or math quotes.

Day 1:

“169-1-400-64  81-361  324-1-16-81-9-1-144”

Day 2:

81-400  81-361  196-225-400  400-64-25  100-225-4  225-36  169-1-400-64-25-169-1-400-81-9-1-196-361  400-225  16-225

9-225-324-324-25-9-400  1-324-81-400-64-169-25-400-81-9.   81-400  81-361  400-64-25  100-225-4  225-36  4-1-196-121 

1-9-9-225-441-196-400-1-196-400-361

Day 3:

676  196-676-49-361-484-196-676-49-324-676-169 324-64 676 529-484-25-324-576-484 441-144-81  49-36-81-169-324-169-400  576-144-441-441-484-484  324-169-49-144  49-361-484-144-81-484-196-64

Day 4:

676-289-64-361-49-144-16-361-64-4  64-324  1-16-64-169-36  676-1-121-16  361-196  4-196-400-169-361  400-225  361-196  361-484-16-169-361-576  484-64-361-49-196-400-361  361-676-100-64-169-36  196-25-25  576-196-400-289  324-49-196-16-324

Day 5:

4-144-1-9-121  64-225-144-25-361  324-25-361-441-144-400  36-324-225-169  49-225-16  16-81-484-81-16-81-196-49  400-64-25  441-196-81-484-25-324-361-25  4-625  676-25-324-225

Day 6:

441-324-25-484  144-36-49  144-441  441-144-36-81  121-484-144-121-225-484  361-676-25-484  49-81-144-36-625-225-484  16-324-49-361  441-81-676-576-49-324-144-169-64

Day 7:

3375-1728-64  2197-1-8000-512-125-2197-1-8000-729-27-729-1-2744-6859  2477-125-10648-125-5832  64-729-125  8000-512-125-15625  1000-9261-6859-8000  1728-3375-6859-125  6859-3375-2197-125  3375-216  8000-512-125-729-5832  216-9261-2744-27-8000-729-3375-2744-6859

Day 8:

8191-2-2187-256-32-8192-2-2187-512-8-729  512-729  27-243-9-19683-512-3-128  2187-256-32  8192-9-729-2187  9-4-19683-512-9-6561-729  2187-256-512-3-128  512-3  2197-256-32  4096-32-2-729-2187  9-4-19683-512-9-3561-729  59049-2-531441

Day 9:

3  1296-7776-3-7776-256-7776-256-27-256-3-625  27-3-625  64-3-49-243  64-256-1296  64-243-3-81  256-625  3-625  3125-49-243-625  3-625-81  64-256-1296  4-243-243-7776  256-625  256-27-243  3-625-81  64-243  343-256-25-25  1296-3-16807   7776-64-3-7776 3125-625  3-49-243-216-3-16-243  64-243  4-243-243-25-1296   4-256-625-243

I hope my students learn to recognize perfect squares and perfect cubes after doing these codes.  At the very least they're fun and they don't take away any class time because I'll pass them out on slips of paper at the end of class and students will have to decode in their own time.

If you want an answer key, just leave me a comment.

Parallel Lines and Transversals

I wanted to share this document I made because I used this lesson on Tuesday with my geometry class and it worked really nicely.  I did it with them on a doc camera and shared their answers over the doc camera as well.  We all really enjoyed especially the last problem which was written about a student.  He really enjoyed the problem even though it poked fun at him.  I did a terrible job after this lesson though with reinforcing all the angle relationships and their names.  I just went over all the vocab- alternate interior/exterior etc. and had them do problems out of the book.  In my defense, I just didn't have time to do anything more exciting.  But at least we had a good intro to the topic I think.

The pictures came from  world.mitrasites.com  and bookbuilder.cast.org.   By the way, I'm still pretty new to this so is it best to cite pictures as I did above, in fine print below the picture, or should I try to restrict myself to only using pictures I myself have taken?
  Parallel Lines and Transversals Worksheet