Category Archives: Algebra I

Common Core Algebra I – Regression and Residuals Reviews – by Brett Widman

Brett Widman, who contributes a lot of material here on the Forum, has given us another great resource. He created a Common Core Algebra I Review on Regression and Residuals. We don’t have much time left with the kids, but if they are confused about this fairly complex topic (especially for Algebra I students), check out Brett’s review.

As always, thanks Brett!

Algebra 1 WU Regression and Residuals

Common Core Algebra I Regents Exam M.C. Review Videos – by Kevin Dorey

A new contributor to the site, Kevin Dorey of Pittsford Central Schools, just sent us a great email. He has been making videos that go through ALL of the multiple choice problems from previous Common Core Algebra I exams.

This is a truly great resource to share with those students who are willing to go the extra mile to try to master the material on this upcoming exam. I want to really thank Kevin for making these and choosing to share them on the eMath Forum. Here are the links:

June 2014





August 2014




January 2015




June 2015




1-Var Stat Summary on the TI-83/84 – by Aimee Campagnola

Our good friend Aimee Campagnola from the Three Villages School District gave us a hand little reference for the one variable statistics option on the TI-83/84, especially how it links in to creating box plots.

Calc Instructions

As we all know, these button pushing procedures on the calculators can be tough for kids to remember, and this is a great guide on how to do it. She sent both the filled in version and a blank version, in case teachers wanted to use it as a classroom activity. Here they are:

1 Variable Statistics and Box Plots Blank

1 Variable Statistics and Box Plots

Thank you for sharing Aimee!!!

CC Algebra I and II Calculator Guide – by Michael Desharnais and Giselle Ferraro

So, Michael Desharnais and Giselle Ferraro from Port Chester Schools put together a simply amazing graphing calculator resource guide for the Common Core Algebra exams in New York state. Granted, it is mostly geared towards CC Algebra I calculator skills, but I wouldn’t hesitate for a second to give it to my CC Algebra II kids as well.

It has everything in terms of button by button directions. Here’s a good example from statistics. Click on the image to see it more clearly.

Calculator Guide #2

They’ve even included a really handy Table of Contents. Anyhow, this thing has it all. Consider printing it out or posting it to your website. I’ve included both a Word version here and a pdf version.

Thanks so much Giselle and Michael. Much appreciated!!!

Common Core Regents Exam Calculator Skills

Common Core Regents Exam Calculator Skills


Intervals of Increase and Decrease – Put to Bed!

Ever since I wrote Common Core Algebra I, I’ve had numerous discussions with math teachers around the country who weighed in on the subject of intervals of increase and decrease of functions. In my answer key, I often include the x-coordinates of the turning points in the intervals, although I’m not entirely consistent on it.

Now, I’ve had folks email me clearly upset that I’ve included the x-values of turning points in the intervals. Their logic is sound. If an x-value of a turning point is included in an interval of increase, then it would also be included in an interval of decrease. Thus making the function both increase and decrease there.

So, their point was not lost on me. In fact, my inconsistency on the convention is actually consistent with the fact that the turning point x-values can either be included or excluded. I wrote a really lengthy post on why and won’t go into it at depth. If you want to read it, here’s the link:

Unit #3.Lesson #4.Intervals of Increase and Decrease

Ultimately, my belief has always been that the Regents would take either convention. Of course, they never saw fit to clarify this point in their CC Algebra I standards clarifications. More on that later.

For all intents and purposes, though, the debate is now over because the January 2016 Common Core Algebra I exam had a free response question on it that included the question of a decreasing interval. Up to this point, we had only seen multiple choice questions and this issue didn’t arise in those.

Here’s Question #33 from this January. It’s a standard projectile motion question that we love so much. I like the question quite a bit because students can explore it on their calculator. Click on the image to see it better.

Q33 - 1-29-16, 9-40 AM

It’s the second part of the problem, where students are asked for the interval of time over which the height is decreasing. By the way, using whatever technique, the time to the peak height is 2 seconds.

Now, the first thing I looked at was the standard state rubric. My heart was beating. Would it state that we had to take off one point for the student including the 2 in the decreasing interval? What about 5 seconds when it hits the ground? Is the height decreasing there? Here’s the state rubric. Again, click to see clearer.

Q33 Rubric - 1-29-16, 10-02 AM

Wow! They didn’t even state the interval. Notice, they are quite clear on the time to the maximum being 2. But, they aren’t even willing to state what the “correct” interval is. Just that the student has given a “correct” interval with an explanation written.

O.k. Well, those of you who have graded these tests before now know where we turn to – the sample responses. For those of you not in the business, these are copious amounts of sample responses with the appropriate points given for the work shown. Here’s the first full credit response (4 points) they gave us:

Q33 Sample 1 - 1-29-16, 7-45 AM

Oh, snap! The first full credit solution includes the turning point (and the endpoint). Most importantly, it states unequivocally that the student has a complete and correct response. Thus, that settles it for good. The turning points must be included for full credit.

Oh, wait. Here’s the second full credit solution they provide:

Q33 Sample 2 - 1-29-16, 7-46 AM

Wait, now they’re accepting 2<t<5 for full credit. Huh?!? I guess the turning point x-coordinate doesn’t have to be included.

O.k. Enough with the sarcasm. What these two responses now give us is clarity! A student can include the x-coordinate of the turning point in an interval of decrease or increase (as seen from our first sample response) or they can leave it off (as seen from our second sample response).

So, I’m glad I won’t have to address that ever again. I’ll just refer to this post.

On a much larger picture, though, this illustrates something I have been very troubled by in the current transition to Common Core in New York – the lack of specificity in the standards. Seriously! There are so many issues out there that are completely predictable that need clarity like this one. The good and overworked folks up in Albany could just make a list to give us the clarity we need to create robust and consistent curriculum. Instead they try to get the word out in random forums, emails, and footnotes. I don’t know if this is some sort of philosophical change up there, but it makes is harder to do our jobs well.

I hope that with a new head of the Regents and change in the wind, they will consider making their expectations clear to us in very fine grained detail. This will allow us to craft our own stories of rich mathematics and not get bogged down in trying to answer questions about what will be on the exam and what will not.

SMART Notebook Files for CC Alg I from Julie Merana-Spanarelli (Unit 4, 5 and 6)

Last year Julie Merana-Spanarelli submitted tons of great stuff to us, mostly SMART Notebook files for Common Core Algebra I. She has made pretty much a file for every eMathInstruction Common Core Algebra I lesson. We’ve posted the first three units earlier this year. Here are Units 4, 5, and 6. Thanks Julie!!!

Unit 4

CCAlg1-U4L1-Proportional Relationships

CCAlg1-U4L2 Unit Conversions

CCAlg1-U4L3-Nonproportional-Linear-Relationships Smart

CCAlg1-U4L4-More-Work-Graphing-Linear-Functions-Lines Smart

CCAlg1-U4L5-Writing-Equations-in-Slope-Intercept-Form Smart

CCAlg1-U4L6-Modeling-with-Linear-Functions Smart

CCAlg1-U4L7-More-Linear-Modeling Smart

CCAlg1-U4L8-Strange-Lines-Vertical-and-Horizontal Smart

CCAlg1-U4L9-Absolute-Value-and-Step-Functions Smart

CCAlg1-U4L10-The-Truth-About-Graphs Smart

CCAlg1-U4L11-Graphs-of-Linear-Inequalities Smart

CCAlg1-U4L12 Introduction to Sequences

CCAlg1-U4L13-Arithmetic-Sequences Smart

Unit 5

CCAlg1-U5L1-Solutions-to-Systems-and-Solving-by-Graphing Smart

CCAlg1-U5L2-Solving-Systems-by-Substitution Smart

CCAlg1-U5L3-Properties-of-Systems-and-Their-Solutions Smart

CCAlg1-U5L4-The-Elimination-Method Smart

CCAlg1-U5L5-Modeling-with-Systems-of-Equations Smart

CCAlg1-U5L6-Solving-Equations-Graphically Smart

CCAlg1-U5L7-Solving-Systems-of-Inequalities Smart

CCAlg1-U5L8-Modeling-with-Systems-of-Inequalities Smart

Unit 6

CCAlg1-U6L1-Simplifying-Expressions-Involving-Exponents Smart

CCAlg1-U6L2-Zero-and-Negative-Exponents Smart

CCAlg1-U6L3-Exponential-Growth Smart

CCAlg1-U6L4-Introduction-to-Exponential-Functions Smart

CCAlg1-U6L5-Percent-Review Smart

CCAlg1-U6L6-Percent-Increase-and-Percent-Decrease Smart

CCAlg1-U6L8-Linear-Versus-Exponential Smart

CCAlg1-U6L9-Geometric-Sequences Smart

History of CC Algebra I Questions – by Christopher Desmond and the Nottingham Teachers

So, Chris Desmond from Nottingham High School in Syracuse, where I did my formative student teaching many, many years ago, has done an analysis with his fellow teachers of all of the questions that have been asked so far on the Common Core Algebra I exam given in New York state.

Their analysis comes in the form of an Excel spreadsheet where they’ve analyzed the percent of questions that are tied to a given CC Standard. What is really, really amazing is that they also include snips of all of the questions themselves. This is really amazing work and would be great for any school to analyze at this point.

The Excel file is large because of all of the graphics that are embedded in it. Here is is:

History of Common Core Assessment – Algebra

SMART Notebook Files for CC Algebra I – Units 1 through 3 – by Julie Merana Spanarelli

Well, we’ve gotten a wonderful end of summer present from our friend Julie Merana-Spanarelli from Central Islip CSD.

She sent us SMART Notebook files for all of the eMathInstruction lessons for Common Core Algebra I. I’m only posting Units 1 through 3 here, but will post more as the school year progresses. Thanks Julie for this great beginning of the year contribution.

CCAlg1-U1L1-Rates-Patterns-and-Problem-Solving Smart

CCAlg1-U1L2-Variables-and-Expressions Smart

CCAlg1-U1L3-The-Commutative-and-Associative-Properties Smart

CCAlg1-U1L4-The-Distributive-Property Smart

CCAlg1-U1L5-Equivalent-Expressions Smart

CCAlg1-U1L6-Seeing-Structure-in-Expressions Smart

CCAlg1-U1L7-Exponents-as-Repeated-Multiplication Smart

CCAlg1-U1L8-More-Complex-Equivalency Smart

CCAlg1-U1L9-More-Structure-Work Smart

CCAlg1-U1L10-Translating-English-to-Algebra Smart

CCAlg1-U1L11-Algebraic-Puzzles Smart

CCAlg1-U2L1-Equations-and-Their-Solutions Smart

CCAlg1-U2L2-Seeing-Structure-to-Solve-Equations Smart

CCAlg1-U2L3-A-Linear-Equation-Solving-Review Smart

CCAlg1-U2L4-Justifying-Steps-in-Solving-an-Equation Smart

CCAlg1-U2L5-Linear-Word-Problems Smart

CCAlg1-U2L6-More-Linear-Equations-and-Consecutive-Integers-Games Smart

CCAlg1-U2L7-Solving-Linear-Equations-with-Unspecified-Constants Smart

CCAlg1-U2L8-Inequalities Smart

CCAlg1-U2L9-Solving-Linear-Inequalities Smart

CCAlg1-U2L10-Compound-Inequalities Smart

CCAlg1-U2L11-More-Work-with-Compound-Inequalities Smart

CCAlg1-U2L12-Interval-Notation Smart

CCAlg1-U2L13-Modeling-with-Inequalities Smart

CCAlg1-U3L1-Introduction-to-Functions Smart

CCAlg1-U3L2-Function-Notation Smart

CCAlg1-U3L3-Graphs-of-Functions Smart

CCAlg1-U3L4-Graphical-Features Smart

CCAlg1-U3L5-Exploring-Functions-Using-the-Graphing-Calculator Smart