What might be happening when 8th and 9th graders get stuck with linear functions, and how can we address their unfinished learning? Build your pedagogical content knowledge for teaching or feedback with these three tips.
Eighth grade provides the thrilling opportunity for students to make the leap from proportional relationships, y = mx, to linear ones, y = mx + b. (This equation reveals the slope of the line, m, and the y-intercept, b.) As you can see in Student Achievement Partners’ Focus Documents, linear relationships are not only included in the major work of the grade, they’re singled out as the most critical area of the year:
But, despite the significant time most curricula spend on this topic, many students never fully grasp how the same function can be shown in different ways: with verbal descriptions, tables, graphs, and equations. For example, ANet’s interim assessment data shows that most 8th graders can’t yet visualize that when the graph of a linear function includes the point (0, 2), that’s the same thing as having an “initial value” of 2.
Not only is this an important requirement of standards like 8.F.A.2, it’s the critical understanding that will allow students to apply function thinking to their lives—like visualizing why a phone plan with a high initial buy-in but a low monthly rate may be a better deal over time.
How can we make sure our students leave eighth grade primed for success in Algebra I, which will quickly branch out from linear relationships to exponential and quadratic ones? Based on some trends we see across grades, we have a few suggestions.
Eighth graders are not too old for concrete and pictorial representations! If students are struggling to figure out the meaning of the “m” or the “b” in y = mx + b, spending a few minutes a day with a routine like this one could make a big difference. Fawn Nguyen’s Visual Patterns site is a great resource for these.
Can you figure out how many shapes would be in…
Visual pattern routines are also great for formatively assessing students’ unfinished learning (read on!).
Students have been preparing for functions since elementary school, where they generated patterns (4.OA.C.5) and graphed them on coordinate planes (5.OA.B.3). But the most critical missed connections often come from 6th and 7th grades. Based on our interim data, the following prerequisite standards are often trouble spots:
| PREREQUISITE STANDARDS | EVIDENCE OF UNFINISHED LEARNING |
|---|---|
| 6.RP.A: Understand ratio concepts and use ratio reasoning to solve problems. 6.EE.C: Represent and analyze quantitative relationships between dependent and independent variables. |
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| 7.EE.B: Solve real-life and mathematical problems using numerical and algebraic expressions and equations. |
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| 7.RP.A: Analyze proportional relationships and use them to solve real-world and mathematical problems. 8.EE.B: Understand the connections between proportional relationships, lines, and linear equations. |
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Teachers can address these areas with targeted support.
Does your curriculum lack explicit structures for student-to-student discussion? As described in this post, the Math Language Routines are a universal support to get students talking and processing more effectively. Math Language Routine 7 fits especially well into lessons that ask students to compare verbal descriptions, tables, graphs, and equations. It’s a great way to push students’ meta-awareness of the key information that each of those representations reveals, and to build the flexibility they’ll need to be successful in high school.
The most important thing you can do to support your students is to ensure your materials line up with their needs. Students who fall behind in math need support that removes barriers without watering down grade-appropriate content. We hope this post helps you get students talking about and representing functions in a more accessible way.