Category Archives: Algebra II

Common Core Algebra II Exam – First Thoughts – by Kirk

I don’t know about anyone else, but it was a long, long day for me. We had 475 students taking the Common Core Algebra II Regents exam for the first time at Arlington High School. The exam started at 8:00 a.m., but I was there at 6:30 along with Dawn Orlik, the head proctor of the exam, to begin setting up for it.

All 475 students were in one giant gymnasium for us. The exam actually went off without a hitch, which was fortunate given the time pressure we were under. Ten amazing Arlington Math teachers proctored the exam to all of these worried students, many of whom stayed the entire three hours.

And then, we went into overdrive mode!!! Our entire department, 23 math teachers, sat and graded from 11:30 until 3:00 in the afternoon. Organizing a large grading process like this is an undertaking all by itself, but I won’t bore you with the details. I myself spent the first half of the grading on problem #37, the 6 pointer!!! Then, I transitioned to the first four 2 point problems, #25, 26, 27, and 28. I also spent some time on the disastrous #33. So, those are the questions I am most comfortable discussing, although I did work through the entire Part II, III, and IV section and analyzed the multiple choice.

And then I didn’t bring a single exam home with me. So, now I have to give you my analysis by complete memory, which will be faulty at best. So, I’m going to give some bullet points until I can carve up an exam and have actual images of problems. Here we go:

I thought the test was fair and I overall liked the questions and the content.

I thought the multiple choice was a bit harder than I expected and the free response was a bit easier than I expected.

The multiple choice had too many problems in it where they tried to be “tricky” and ended up testing aptitude more than content. These included the one where they gave them the inverse and asked them for the original and the awful one with completing the square with silly leading coefficients of 4.

I loved the six point problem. I graded that one and students did really well on it. I liked the fact that students had to deal with a given model, then use a graph of that model to answer realistic questions.

I generally liked the free response. It made me sad that the hardest one seemed to be the systems problem with the circle. So straightforward. So mechanical. So botched!

VERY LITTLE TRIG. In fact, only four total trig problems and three of them were sinusoidal modeling. I was VERY annoyed by the two point trig graphing problem. Many students went 0 for 2 even though they had the correct midline and amplitude BUT had graphed more than one cycle plus had not labeled their period correctly. But, that’s the issue with trying to package a 3 or 4 point problem into 2 points. No tangent, cotangent, secant, etc…

I generally liked the statistics questions. A straightforward normal distribution in multiple choice and then two simulation questions, both on proportions, which I thought was a bit strange. I liked the controlled experiment question, but could it have killed them to tell us what mystery ingredient X was supposed to do (fight cavities, brighten teeth, better taste)? Come on, invest a little in your question writing.

I liked the rational algebra and polynomial work. It wasn’t too bad and stressed mostly important things. I graded the “is (x-5) a factor…” question and almost all kids got it correct. I do wonder about the use of limit notation in that multiple choice problem on polynomials, but we could easily incorporate that into lessons. It would be a nice tool to use. Even that weird proof question was on the easier side and lent itself to many different approaches.

Nothing on mortgages or Newton’s Law of Cooling. Yay! I also liked the Geometric Series question. It was fair and a good application of the formula. All in all, the sequence/series questions were challenging, as they always are, but not particularly tricky.

All in all, I think we felt pretty good about the test. I hope it helps put to bed the idea that this course still has a lot of trig in it. This is definitely NOT Common Core Algebra II and Trig. I think that a lot of precalculus courses will not need to incorporate a lot more traditional trigonometry.

I applaud NYSED for creating a test that was fair, stuck pretty much to what we knew it would, and didn’t go off the standards in any serious way. I’m hopeful they will get us the curve before their end date of June 23rd. My bet is June 22nd, if they hold true to form. I hope the exam went well for all of you out there, whether you were a student or teacher taking the test.

Common Core Algebra II Review Videos – by Kris Williamson

Kris Williamson, from Newfield, has been supplying us with Common Core Algebra II Review videos based off of the unit reviews that I posted to the site a few months ago. We’ve been linking to them as he has supplied them, but now we have the links to all the videos in one post. Just in time for the June 1st exam.

Thanks Kris!!!

Unit 1: Algebraic Essentials Review
https://www.youtube.com/watch?v=yXY4tAMXxRo

Unit 2: Functions as the Cornerstone of Algebra
https://www.youtube.com/watch?v=fQDLWq1qgoU

Unit 3: Linear Functions
https://www.youtube.com/watch?v=142RKxiAlCA

Unit 4: Exponential and Logarithm Functions (Video 1/2)
https://www.youtube.com/watch?v=CVGwJDWjxi0

Unit 4: Exponential and Logarithm Functions (Video 2/2)
https://www.youtube.com/watch?v=Vdni7pewUB4

Unit 5: Sequences and Series (Video 1/2)
https://www.youtube.com/watch?v=GY1psv-Qjr8

Unit 5: Sequences and Series (Video 2/2)
https://www.youtube.com/watch?v=zTh40pGhQk4

Unit 6: Quadratic Functions and Their Algebra (Video 1/2)
https://www.youtube.com/watch?v=YefiQibzbG8

Unit 6: Quadratic Functions and Their Algebra (Video 2/2)
https://www.youtube.com/watch?v=g1QEU1LK3rE

Unit 7: Transformations of Functions
https://www.youtube.com/watch?v=Z5RLIIAX-KQ

Unit 8: Radical Functions and the Quadratic Formula
https://www.youtube.com/watch?v=sCbcI0I8wBU

Unit 9: Complex Numbers
https://www.youtube.com/watch?v=CVjCSaYzhqQ

Unit 10: Polynomial Functions (Video 1/2)
https://www.youtube.com/watch?v=uN_3nwl-3Vw

Unit 10: Polynomial Functions (Video 2/2)
https://www.youtube.com/watch?v=pxxDZCZaFlg

Unit 11: The Circular Functions (Video 1/2)
https://www.youtube.com/watch?v=wIdt5uflfcs

Unit 11: The Circular Functions (Video 2/2)
https://www.youtube.com/watch?v=Lmhzxn7NSZA

Unit 12: Probability
https://www.youtube.com/watch?v=tlpI8iDTLQk

Unit 13: Statistics (Video 1/2)
https://www.youtube.com/watch?v=OjkSzs2GcHg

Unit 13: Statistics (Video 2/2)
https://www.youtube.com/watch?v=Bl4dRt5E_Y4

Common Core Algebra II Review Videos – by Kris Williamson

A few weeks back, I had a post that shared a bunch of great videos that Kris Williamson, from Newfield CSD, that were made based on the CC Alg II Reviews that I posted previously. At the time, Kris had the first five units done, now he has 10 out of 13. I’m posting them here along with the pdf slides that he used to go along with them. Please share these links with your students if they need extra review.

Thanks Kris for sharing all of your hard work. This will benefit many students who want to give that extra bit of effort heading into June 1st. Here are the links:

Unit 1: Algebraic Essentials Review
https://www.youtube.com/watch?v=yXY4tAMXxRo

Unit 2: Functions as the Cornerstone of Algebra
https://www.youtube.com/watch?v=fQDLWq1qgoU

Unit 3: Linear Functions
https://www.youtube.com/watch?v=142RKxiAlCA

Unit 4: Exponential and Logarithm Functions (Video 1/2)
https://www.youtube.com/watch?v=CVGwJDWjxi0

Unit 4: Exponential and Logarithm Functions (Video 2/2)
https://www.youtube.com/watch?v=Vdni7pewUB4

Unit 5: Sequences and Series (Video 1/2)
https://www.youtube.com/watch?v=GY1psv-Qjr8

Unit 5: Sequences and Series (Video 2/2)
https://www.youtube.com/watch?v=zTh40pGhQk4

Unit 6: Quadratic Functions and Their Algebra (Video 1/2)
https://www.youtube.com/watch?v=YefiQibzbG8

Unit 6: Quadratic Functions and Their Algebra (Video 2/2)
https://www.youtube.com/watch?v=g1QEU1LK3rE

Unit 7: Transformations of Functions
https://www.youtube.com/watch?v=Z5RLIIAX-KQ

Unit 8: Radical Functions and the Quadratic Formula
https://www.youtube.com/watch?v=sCbcI0I8wBU

Unit 9: Complex Numbers
https://www.youtube.com/watch?v=CVjCSaYzhqQ

Unit 10: Polynomial Functions (Video 1/2)
https://www.youtube.com/watch?v=uN_3nwl-3Vw

And here are the pdf’s of the reviews:

Unit01-AlgebraicEssentials

Unit02-FunctionsAsCornerstonestoAlgebra

Unit03-LinearFunctions

Unit04-ExponentialLogarithmicFunctions

Unit05-SequencesSeries

Unit06-QuadraticFunctionsAndTheirAlgebra

Unit07-TransformationsOfFunctions

Unit08-RadicalsQuadFormula

Unit09-ComplexNumbers

Unit10-PolynomialRationalFunctions

Huge Common Core Algebra II Reference Packet – by Elaine Perkins

Elaine Perkins of Elmira High School, upstate in the Southern Tier,  just sent us an awesome synopsis of the material in Common Core Algebra II. This isn’t a review set of problems, but a comprehensive listing of all of the content with important formulas and other ideas.

Elaine was kind enough to supply the document, which must have taken her weeks to create. She gave it to use in MS Word. So, teachers can decide if they want to use it en masse or if they want to tailor it to the material that they taught in their course. It’s simply that comprehensive. Here it is:

Algebra II Common Core Review

Thanks Elaine. This comes at a great time as we get closer and closer to June 1st.

Common Core Algebra II Review Videos – by Kris Williamson

Kris Williamson, from the Newfield Central School District, has given us a great contribution heading into the homestretch in Common Core Algebra II. He essentially took the Unit Reviews that I created, sliced and diced them, and created videos of all the reviews. Kris is relatively new to the video world, but he’s done a really nice job. I’m going to let him explain it to you…

“Hello! I am using Kirk’s Algebra 2 Lessons in Newfield, and have created some unit review videos for my students. We are just beginning the probability unit and will end up having about 4-5 days of review after Statistics, so at Newfield we have been providing students with unit reviews this last month while they learn the last two units. A big thank you to Kirk for sharing his unit reviews!

I’m just getting used to creating my own videos, so they aren’t the best quality, but I’ve tried to hit upon the big points in each unit. I invite you to use them as you wish. I have been able to create videos for units 1-4 so far, and Kirk has graciously let me share them with you all. I hope that you find them helpful and useful these last few days before our June 1 Regents!

In my videos, I recap the lessons in the unit, share the Common Core Standards briefly, then I provide topical reviews, and lastly I sum up the unit by indicating all the important things students should be able to do from the unit. Unfortunately, school blocks YouTube, so I have been using zaption.com to share them with my students. Feel free to use them as you wish!

Unit 1: Algebraic Essentials
https://www.youtube.com/watch?v=yXY4tAMXxRo

Unit 2: Functions as the Cornerstones of Algebra
https://www.youtube.com/watch?v=fQDLWq1qgoU

Unit 3: Linear Functions and Their Algebra
https://www.youtube.com/watch?v=142RKxiAlCA

Unit 4: Exponential and Logarithmic Functions Video 1/2
https://www.youtube.com/watch?v=CVGwJDWjxi0

Unit 5: Exponential and Logarithmic Functions Video 2/2
https://www.youtube.com/watch?v=Vdni7pewUB4

These are just the first five units. Kris will be sharing the others as he makes them. He also contributed the blank slides that go with the videos. Here are those as well (but only for the first four units – still need #5). Thanks Kris!!!!

Unit01-AlgebraicEssentials

Unit02-FunctionsAsCornerstonestoAlgebra

Unit03-LinearFunctions

Unit04-ExponentialLogarithmicFunctions

New Functionality in Statistical Simulators – CC Alg II – by Kirk

Recently my good friend Joey Shavelle of Jolo Studios put some additional functionality into two of our statistical simulators. For the sample mean simulator he added code that calculates the mean of the means and the standard deviation of the means. Here’s what it looks like when I had a population mean of 127, a standard deviation of 32, and a sample size of 50 with 1000 simulations:

Mean of Means with Histogram

(Click on the image to see it more clearly.)

He also added the same code to the sample proportion simulator that calculated the mean sample proportion and the standard deviation of the sample proportions. Here’s a shot of what it looked like when the population p was 0.3, a sample of size 100 was used, with 1000 simulations:

Proportion Means

So, check them out and use this new functionality in our lessons that require it if you haven’t already finished statistics at this point.

New CC Algebra II Statistics Videos – by Kirk

A few weeks back, I published three new lessons to go with our final unit in Common Core Algebra II – Statistics. These lessons extended the work that we began in inferential statistics by introducing more formal ways of finding variation within sample statistics, such as sample means and proportions. By continuing to use simulation, we introduce students to the Central Limit Theorem for both sample means and sample proportions and extend this to a final lesson on margin of error. All of the lessons and their answer keys can be found at this post:

Simulation and Inferential Statistics – by Kirk

It took awhile, but I finally found time this weekend to record the videos for these lessons. I hope the videos are in time to help both students and teachers with this challenging material. I believe that these lessons will help students understand inferential statistics and how the results of the simulations can be extended to standard methods within statistics. Here are the YouTube links to the three new videos:

CC Alg II.Unit #13.Lesson #8.The Distribution of Sample Means

CC Alg II.Unit #13.Lesson #9.The Distribution of Sample Proportions

CC Alg II.Unit #13.Lesson #10.Margin of Error

I know that the numbering system here is in conflict with our final two lessons on regression. We will be re-numbering those last two lessons on regression as Lessons #11 and #12 respectively (as well as figuring out where they really make sense in the overall text).

 

Google Sheets for CC Alg II Statistical Simulation – by Kirk

As many of you know eMathInstruction offers online apps for performing the statistical simulations required in Common Core Algebra II. You can find links to those simulators at this post of mine:

Statistical Simulations and Inferential Statistics – by Kirk

The simulators work like a charm and I explain how to use them as well as how to use our TI based calculator programs in this YouTube video:

YouTube – Statistical Simulation Programs from eMathInstruction

One of the things I mention in the video is using Google Sheets to analyze the results of the simulations. Google Sheets is super convenient to use given that everyone has access to it. I also find it is much, much easier to use than Excel in terms of creating distribution histograms of the simulation results. Here’s a distribution of 1000 sample means that I created just now:

Histogram

I show how to create these in the video. But, now I’ve created three Google Sheets for the simulators that people can just copy and paste their data into. I’ll give a links in a second. One of the nice things is that it will report the mean of the results, the standard deviation of the results, the 2.5% percentile, the 5% percentile, the 95% percentile, and the 97.5% percentile (hopefully you know why these are good to have). Here’s what a screen shot looks like:

NormSamp Screen

As always, click on the images to see them much more clearly. Then hit the back arrow to come back to the article.

Anyhow, I’m going to supply links to all of these sheets. Now, and this is key, we don’t want to make it so that everyone is sharing and editing the same Sheet. So, I’m going to make the links “View Only.” All you have to do to have them all for yourself is make a copy:

Screenshot 2016-04-19 20.43.00

Then, give it a cool new name and, as all things Google, it will magically now be in your Google Drive (assuming you have a Google account).

Screenshot 2016-04-19 20.44.01

O.k. Here are the links to all of the Google Sheets:

Normal Sample (NORMSAMP) Google Sheet

Proportion Simulator (PSIMUL) Google Sheet

Difference in Sample Means (MEANCOMP) Google Sheet

Have fun with them. If you are going to share the links with students, consider making copies on your own drive and then sharing them from there.

CC Alg II – Polynomial Functions and Their Graphs – by Brett Widman

Our long time friend Brett Widman from West Seneca sent us a great review packet on polynomial functions and their corresponding graphs. We have all seen in the sample problems the clear emphasis on the connection between the equations of these polynomials and their zeros. Brett’s problem set does a great job reviewing this. Here it is!!!

Algebra 2 Writing Polynomial Equations

Thanks Brett for this really great contribution.

Cubic