Welcome back to the Unfinished Learning in Middle School Math series, in which math educators Chrissy Allison and Becca Varon illustrate how to make some of the trickiest standards in grades 6-8 accessible for all students. Over the course of six blog posts, we’ve provided concrete examples of how math educators can address unfinished learning within the context of grade-level lessons, which in the long term will help prevent an entrenched pattern of over-remediation and below-grade-level teaching. You can read our introductory post here. In this final post, we will explore ways to “bridge the gap” with the 8th-grade standards 8.EE.B.5 and 8.EE.B.6. 

Let’s take a look at these two impactful standards:

8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

For our final post in this series, we knew we had to tackle slope. This topic pulls together countless threads of prior learning from grades K-8. The intention is that the study of linear relationships in 8th grade should weave those threads into a parachute that will allow them to jump fearlessly into high school, where they’ll apply that knowledge to other types of functions. However, in practice, this unit is often a major source of struggle. Some of those threads of prior knowledge are inevitably missing or frayed, and it can be tough to know how to meet students’ needs in the moment. As a result, data shows that students often depart 8th grade missing this key foundation for Algebra I.

Let’s take a moment to study and understand two of the standards that are essential to building students’ knowledge of how to use slope:

In order to start to see all of those threads of prior knowledge — and to wrap our heads around the “slope triangles” in 8.EE.B.6 — let’s solve this task from Illustrative Mathematics.

As you look to diagnose students’ prerequisite understanding, here are a couple of things we noticed as we solved this task:

• There’s a lot of geometry involved. And not just the 8th-grade standards addressing similarity, translations, and dilations (8.G.A), which are typically taught soon before this learning and have several of their own prerequisites. Students will also need some comfort with finding side lengths of polygons graphed on coordinate planes (6.G.A.3), especially in future tasks where a visual isn’t provided for them. Unfortunately, we know from experience that geometry standards are under-taught before 8th grade. Since they aren’t in the K-7 Major Work of the Grade, they’re often the first to go when teachers make hard decisions about pacing. 
• Proportional reasoning is at the heart of this task. Students need to see that the line rises 2 units for every horizontal increase of 3 units, no matter how large the triangle gets. This builds on an array of skills that stretch back to equivalent fractions and multiplicative reasoning in elementary school.

In a prior blog post for ANet, we wrote about some general teaching methods that will boost student understanding of linear relationships. All of those suggestions, like using visual patterns and encouraging students to discuss the connections between representations, still apply. 

Here are a couple of additional ideas for ways to take action in supporting students with specific threads of unfinished learning:

1. Use slope triangle manipulatives. There are some excellent tools out there, like this Desmos activity, that allow students to experiment with translating and dilating slope triangles. Supplement these grade-level activities with questions that probe students to think deeply about concepts that may still be unclear to them. For example, if your students have unfinished learning about polygons graphed in the coordinate plane, ask them to predict the coordinates for triangles with different side lengths.

2. Bring in real-world contexts. Tangible scenarios make things click sometimes. For example, check out this lesson in the IM curriculum that investigates proportionality between the shadows cast by different objects. We can totally imagine students having an aha moment about how shadows “match” their objects in predictable (proportional) ways. Or, your students might enjoy experimenting with ways to safely land a plane using equations. Instead of waiting until the culmination of your unit to decide whether students are “ready” for real-world activities, try using them at the beginning to generate curiosity and stories that you can refer back to.

We hope you found this blog series helpful, and we’re cheering you on as you tackle the hard and critical work of supporting your students. We’d love to keep the conversation going in the comments by hearing from you! Please share:

• What approaches have you used to “bridge the gap” to support students with slope?
• How have you helped students find success with linear relationships?

Finally, don’t forget to check out other posts in the Unfinished Learning in Middle School Math series for additional strategies, examples, and ideas.