In September of 2022, schools around New York State will transition from the Common Core State Standards version of Algebra I to the New York State Next Generation Mathematics Learning Standards version of Algebra I. Although this transition could certainly be viewed skeptically as simply a rebranding of the Common Core standards, there are important changes that have been made to the course. At eMATHinstruction, we will be publishing N-Gen Math Algebra I to align to the Next Generation standards, but have also included material to make the text more broadly align to the CCSS. In this post, I’d like to discuss some of the changes to the CCSS that will occur with the Next Generation standards and how we will be treating them in our new text. As longtime readers of eMATHinstruction posts also know, I will add my own color commentary here and there.
The New York State Education Department (of NYSED) has a fantastic website devoted to the Next Generation Learning Standards for Mathematics. There are lots of helpful links on timelines, testing, and curriculum. Here are some vital links from this page:
New York State Next Generation Mathematics Learning Standards (nysed.gov)
Mathematics Learning Standards | New York State Education Department (nysed.gov)
New York State Next Generation Mathematics Learning Standards Algebra I Snapshot (nysed.gov)
New York State Next Generation Mathematics Learning Standards Algebra I Crosswalk (nysed.gov)
The first link takes you to the official Next Generation Math Standards in all of their glory. The second link is to the launching page for all of the math standards, a good place to start if you want to find information for all grade levels. The last two links are what we will concentrate on in this post. The Snapshot gives a very quick summary of the new standards, standards that have been removed, and standards that have their content clarified.
We can see immediately that one new addition to Next Generation Algebra I is the topic of linear-quadratic systems (AI-A.REI.7a). This is, of course, a topic that many Algebra I teachers have taught for years to their students. I love that the topic has been placed in Algebra I as it reinforces many of the big ideas and skills students have seen while still keeping the math relatively simple. We have included a lesson on this in our new N-Gen Math Algebra I text.
Besides that topic, the only other new content added in Next Generation Algebra I is the skill of rationalizing a denominator. I was sincerely hoping when New York State redid its standards it might throw out the idiotic rational/irrational number hooey that is contained in Algebra I, but I was sorely disappointed. Not only did they not get rid of it, they upped the ante by including rationalizing the denominator and even more rules about how rational and irrational numbers combine using operations to form either rational or irrational numbers. This is where the Crosswalk link above is helpful in really understanding how the CCSS math standards compare to the NGLS:
So, in addition to knowing the ever-so-useful fact that the sum of a rational and an irrational number is always irrational, they now must know that the sum and product of two irrational numbers could be either rational or irrational. I just cannot fathom in what universe it is important for a student who is taking Algebra I (typically kids who are as young as 13 and 14 years old) to know that when we have the following sum:
its result is a number whose decimal representation does not terminate nor repeat. I honestly think our entire treatment of irrational versus rational numbers makes no sense whatsoever. Irrational numbers are one of the amazing mysteries of mathematics. Yet, what we ask kids to know about them gets at none of the magic behind these numbers. More on that in a future post.
The Snapshot link above also gives us topics or standards that have been removed from Algebra I.
Some of these have been removed due to the ambiguous nature of the standard (yes, I’m talking about you N-Q.2). Others have been removed because there was redundancy in the standards (A-SSE.3a and S-ID.6c are good examples). And some have been removed because they are developmentally inappropriate for this level. A great example of this is the removal of residuals as a topic in Algebra I. I’ll be honest, after having taken three college courses in statistics, I had never even heard of residuals before the Common Core Standards, so I’m not crying to see them go. One interesting note in these subtractions is the fact that completing the square will not be used to transform a quadratic function into its vertex form. It will only be used as an algebraic procedure to solve quadratic equations. I’m a bit ambivalent to this particular change. I love the fact that they are keeping completing the square as a technique for solving quadratic equations, but am a bit sad to see it go in terms of the vertex form of a quadratic function. Be careful as some subtractions are not contained in this section of the Snapshot. A good example of this is the removal of the cube root function (a topic that has been moved to Algebra II).
Besides these additions and subtractions, the rest of the Snapshot and Crosswalk links above give us more specificity about particular standards. As a math teacher and curriculum creator, I very much appreciate this. One of the oddities of the Common Core Standards at the high school level was always the fact that they were not broken into courses the way the standards were at the K-8 level. Although I can appreciate the fact that the writers on the CCSS wanted to give states flexibility in terms of how the standards were arranged in courses at the high school level, it meant that the depth to which certain standards should be taught was not as clear as it should be. This is not a minor issue! Seemingly small changes in how generally a topic should be taught can transform it from being a one lesson topic to a one week topic.
A great example of exactly this is the topic of factoring. In the CCSS the topic is fairly vague, urging students to “use the structure of an expression to identify ways to rewrite it.” I do love the idea of students recognizing structure in expressions, but a standard that is this vague leaves a lot of guessing to be done by the teacher in terms of how deeply to pursue it. The Next Generation Standards clean this issue up nicely:
Notice now that specific types of factoring are required with other noted types being excluded, such as factoring by grouping and the sum and difference of perfect cubes. Additionally, we now see that students will only need to factor trinomials whose leading coefficient is equal to 1. Of course, this single additional specification takes a topic that might warrant four class days and cuts it down to one or two. In our own Next Generation aligned Algebra I text, we concentrate on factoring trinomials with lead coefficients equal to one, but also include one lesson on more challenging trinomials. Another nice example of where the Next Generation Standards become more specific is when using completing the square to solve a quadratic equation. In this case, they now specify that the leading coefficient of the equation will be equal to one and the linear coefficient will be an even number.
I love the fact that New York considered these two factors when specifying what kids will need to be able to do with completing the square, a procedure that is certainly within their ability to grasp. Students at the Algebra I level are being flooded with a level of abstraction and technique that is far above anything they experienced in grades K through 8. As every math teacher knows, lots of students enter 9th grade with only a tenuous grip on fractions. When we have them learn a process such as completing the square, we do not want their deficiencies with a topic like fractions to hinder their acquisition of an important new skill.
Another important change to the standards involves sequences. In the Next Generation version of the standards, sequence rules will only be given in explicit form, i.e. no recursion, and will only be given using subscript notation, not function notation.
This again seems like a minor change, but now teachers can feel free to bypass the enormous amount of confusion that occurs from the recursive definition of a sequence. My feelings are a little more mixed on getting rid of function notation for sequences, as I thought this was a good way to connect them to the overall concept of a function. I can live with that change, and our new N-Gen Math Algebra I will only include subscript notation when working with sequences.
One area with both subtraction and more specificity is the topic of transformations of functions.
Notice that the horizontal stretch/compression of a function has been removed. I very much support that move. I think the horizontal stretch and compression is far too difficult to understand at the Algebra I level, even if its basic mechanics can be mastered. I also like how very specific the standard now is in terms of what students should be able to do regarding the value of the transformation constant k.
The topic of statistics, even though amazingly out of place in this course, remains mostly unchanged. We have already mentioned that the topic of residuals is being removed. As well, the only type of regression that is studied is linear, with both exponential and quadratic pushed off until Algebra II. Finally, and I think this is an interesting one, a change in terms of what types of data sets kids will work with:
Notice that the Next Generation standards are very clear that data sets, when given, will be sample data sets only. There will be no population data sets. As such, only the sample standard deviation will ever be used. I love this for multiple reasons. One, it means we don’t have to constantly debate whether we should use the population or sample standard deviation. More importantly, though, it allows us to focus on what statistics is most often used for, making statements about populations based on samples. We are very happy with our new statistics unit in N-Gen Math Algebra I. We revolve this unit around using tools from statistics to answer statistical questions, including how we can compare samples statistics to make inferences about populations.
One final note I’d like to mention is on the topic of exponent properties. Algebra courses, both Algebra I and Algebra II, often concentrate on exponent properties in order simplify expressions that might look something like this:
The Common Core Standards, and to a greater extent the Next Generation Standards, are more interested in using these properties to transform exponential expressions where the base is a constant into equivalent exponential expressions.
Notice that now students will have to work with exponential expressions whose exponents could be linear expressions (with integer coefficients). In other words, they might now have to solve problems such as:
This type of manipulation is included now because in Algebra II we want a student to be able to take a function like:
And rewrite it like this:
This type of manipulation allows us to transform an exponential function that is written in terms of doubling time (8 years) and to one in terms of annual percent increase (9.05%). This is an important skill and an important shift from how we have traditionally thought of using exponent properties. That’s not to say that a problem like the following won’t show up on a state assessment:
The New York State Next Generation Mathematics Learning Standards are based off of the Common Core State Standard. All public schools in New York will be aligning their Algebra I courses to this set of standards in the fall of 2022. Schools outside of New York who use the CCSS as the basis of their high school courses can have confidence that eMATHinstruction’s N-Gen Math Algebra I text will closely align to the CCSS, with some important changes as discussed above. Although the Next Generation Algebra I standards are certainly different from the CCSS, it is mostly in specificity and not content that they differ.
