The EngageNY Lessons for Mathematics grades 6-11 are an excellent interpretation of the Common Core State Standards and the Standards for Mathematical Practice. But often the lessons are dense, progress in complexity too quickly, or assume [a lot!] of prior knowledge.
So how do you customize the lesson to make it accessible for students without compromising the rigor?
In this two-part blog, I will share a lesson planning process that has helped me to remix lessons to make them a hit for teachers and students alike.
Below is an overview of the process. Part 1 will cover the first two steps, and Part 2 will cover the remaining three steps.
Yes — we all know… DO THE MATH! But the reality is that when pressed for time, it is easy to skip over this step. Solving the problems on the student version and reading the entire Teacher Version of the Lesson will give you an understanding of the progression of learning and how it relates to the student outcome(s) and standard(s). It will also help you to refresh your memory so that you are attuned to the demands of the work and the potential student misconceptions.
Step 2: Identify the standard(s).
The standard for Lesson 8 is:
A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
In this lesson, students first familiarize themselves with the concept of base and generalize this to polynomials in base x. Here we see a connection between integers and polynomials by looking at their basic composition: integers are based on groupings of 10 (base 10 system), while polynomials are based on a generalized grouping, let’s say x (base x system). See Exercise 1, part b and Exercise 2.
I found this connection to be profound because we are alluding to the fact that although polynomials look crazy and weird to students, their composition is very similar to that of integers.
Another idea directly related to the standard is to understand what we are allowed to “do” with the “stuff” in each system. Let’s consider integers first. If we add/subtract or multiply integers, we know that we will get another integer. But when we divide integers, we introduce a new system, or number set — the set of rational numbers.
Again — profound!
The same is true for polynomials. We can add/subtract or multiply polynomials, and we will create another polynomial. This can be observed in the definition of a polynomial expression in Lesson 8:
This definition is saying: given a number or a variable, if we add or multiply these number and/or variables, we will create another polynomial. Thus, the system of polynomials is similar to that of the integers because not only are their compositions similar, their systems are “closed” or defined by the things we’re allowed to “do” within the system (aka — you can add/subtract or multiply things within the system and get a result that still fits the requirements of the system).
Aha! THIS is what A.APR.1 means!
Math is usually thought of as a series of unconnected topics, but this standard is calling out that even though we are working with a different system, it functions similarly to a more familiar one, so we don’t need to learn new things, we just need to connect ideas to make sense of them.
Now I pause and consider how students will uptake this information, if at all. What do I need to be sure to emphasize and what additional opportunities do I need to provide for students to absorb and process this idea? I see this has as the first opportunity to customize the lesson and to make this connection more explicit.
The lesson then proceeds to cover a ton of vocabulary with few opportunities for students to progress from practicing identification to creating or explaining. This would also be an opportunity to customize the lesson for explicit learning and practice around vocabulary.
Finally, students find the sum or difference of polynomial expressions in Exercise 4:
In part a, we again see the connection between combining integer like terms (hundreds, tens, ones) and combining polynomial like terms (x³, x², x, constant).
When I was a middle and high school student, I did not learn math to the depth of the Common Core State Standards. If I expect my students to master the standards, I too, must be persistently studying to become well-versed. Engaging in the first two steps of this process has been incredibly helpful to doing just that.
On Thursday this week, I’ll post the second part of this blog detailing the final three steps of this remix.
Tell us what you think of this breakdown of engage, or share your own thoughts about improving student accessibility of the lessons in the comments below.
See you Thursday!
Leah is an AUSL Curriculum and Development Coach for HS Math. She previously taught Intrinsic Charter School, Muchin College Prep, and ACT Charter School and received her M.Ed. in Learning and Teaching and Mathematics Instructional Leadership from Harvard Graduate School of Education.