We know, you thought that we advertised ourselves as being able to help you plan for your algebra instruction. You’re right. We did. And regardless of all of the elementary content you see on our website, we would argue that we are.
Differentiation. Personalized Learning. Enrichment. Intervention. We get it. As a teacher, we are expected to be everything to everyone. The expectations (as well as the educational jargon) can be a heavy weight to carry around.
This is where we hope to help. While understanding that there isn’t a silver bullet that will magically lift these expectations off of your back, we hope to be able to lighten the load by providing you with resources to help differentiate for the needs of the students who show up in your classroom each day with varied and vast mathematical experiences and background understandings.
We started this month by identifying some common misunderstandings related to CCSS.Math.Content.HSA.SSE.A.1 and CCSS.Math.Content.HSA.SSE.A.2 (seeing structure in expressions). During the second week, we worked backwards from an A.2 example problem [3(m - 3) + 2m - 1 + 6] and identified the elementary building blocks that were intertwined in the process of simplification. During the third week, we built on the elementary building blocks and identified some foundational topics taught in middle school that lead to fluency in simplifying expressions.
Each week, we’ve highlighted five correlated skills/understandings (what we’ve been referring to as the building blocks of understanding). For each correlated skill, we’ve found an instructional video that explains whY, and we’ve also identified instructional resources that you can utilize in a small group when planning for intervention, or with the whole class if you’ve identified a gap in understanding of the building blocks.
Differentiation is whY we’re decomposing these algebra standards to their elementary and middle school roots. Pre-assessment is how you can effectively identify which of the foundational skills you need to return to before pushing ahead in the algebra curriculum.
We advocate for the use of open-ended pre-assessments. By leaving the questions open-ended, it allows us the space to identify fundamental misunderstandings within student solutions. For example, one of the questions on our ‘Seeing Structure in Expressions’ pre-assessment asks students to simplify 5 + 3 (m - 3) + 2m - 1. Each time I give a pre-assessment in my classroom, I scan student papers for their responses to each question one at a time. I don’t mark-up the papers, but simply sort the papers into groups based on similarities in their solutions. As I was looking over this particular pre-assessment question, I noticed that some of my students were adding 5 and 3 before they distributed (ending up with something close to 8m - 24 + 2m - 1 before combining like terms). I was then able to sort my students into two groups. With one group, I planned to help them use algebra tiles to identify why 8 (m - 3) is not the same as “five plus three groups of (m - 3).” When we worked in this small group, I remember that the use of the algebra tiles led to a great discussion about the order of operations and why it makes sense that we would have to distribute before we add in this expression. With the students in my second group, I planned to push their thinking to show the distribution of a variable by an expression. This intervention and enrichment were both made possible because students had prior experiences using area models to multiply.
This is how it all ties together. Understanding of multiplication as a representation of equal groups (and visualizing these products with an area model) is first taught in third grade (CCSS.Math.Content.3.OA.A.2). The concept of working within grouping symbols following the order of operations is first introduced in fifth grade (CCSS.Math.Content.5.OA.A.1) and then revisited in relevant contexts each year thereafter (check out the ‘Four 4s’ activity in our week 3 resources). The distributive property is first introduced in third grade (CCSS.Math.Content.3.OA.B.5) as a way to break down a product into its friendly factors. Computation with negative integers is taught in seventh grade CCSS.Math.Content.7.NS.A.1 and CCSS.Math.Content.7.NS.A.2, but understanding of negative rational numbers is solidified in sixth grade. We could go on, but we think you get our point. All of these understandings are essential to being able to simplify 5 + 3 (m - 3) + 2m - 1.
So, yes, we’ve highlighted a kindergarten standard on a blog that has marketed itself as a place for resources for algebra teachers. But, we hope you can use the resources we’ve compiled to help your students build on their past experiences and fill in any gaps they may have in the understanding of the building blocks.
If you’re interested in the ‘Seeing Structure in Expressions’ pre-assessment we’ve described above, we’d love to share it with you! It is attached as a PDF here: