Nix the Tricks Book Study – Chapter 6: Equations and Inequalities

This post is part of an ongoing book study series on Nix the Tricks by Tina Cardone. Each week, a group of math teachers meets to reflect on one chapter of the book and discuss how we can move away from shortcuts and toward true mathematical understanding. You can find our reflections from earlier chapters and follow along as we continue exploring.

Chapter 6, “Equations and Inequalities,” took us into one of the most common danger zones for student misconceptions. This is the chapter where many of the “classic” algebra tricks show up: move the 3 to the other side and “change the sign,” “cross off” matching terms, or use the “log circle” as a shortcut to solving exponential equations.

Our discussion centered around whether the fixes offered in this chapter are realistic for our students, and how our own language and instructional habits can inadvertently cement tricks instead of understanding.

Thoughts on the Fixes

While many of us agreed with the fixes presented, especially the emphasis on structure, inverse operations, and precise language, we also acknowledged some real-world challenges.

A few reflections stood out:

• Some tricks are overly simplistic, but their replacements can feel overwhelming to students encountering the idea for the first time.
• The example that came up most was the log circle trick. The fix, understanding logarithms conceptually, is absolutely correct, but expecting CP Algebra 2 students to fully internalize a deep conceptual model of logarithms during their first exposure may not be realistic.
• Vocabulary matters. Many students do not know what a term is versus a factor, which causes confusion when solving.
• Avoiding the word cancel can shift the tone of the whole lesson. Saying “this simplifies to 1” or “this evaluates to 0” builds meaning.

We agreed that even if the fixes feel ambitious, the underlying message is crucial: students need to understand why an equation-solving move works, not just memorize a mechanical step.

Instructional Shifts We Plan to Make

Several teachers identified small yet powerful changes they want to bring into their instruction:

• Use more precise language. A great example that came up was defining absolute value as “distance from zero,” not “make it positive.”
• Intentionally avoid shortcuts in early instruction. Formal algebraic moves should be introduced through patterns, reasoning, and structure.
• Lean into consistent definitions. If students don’t know what “term,” “coefficient,” or “factor” means, solving becomes guesswork.

Even small adjustments in the way we talk about math can dramatically shift student understanding.

How Tricks Impact Future Learning

When we look back at our own teaching, it’s clear that tricks don’t just affect the current unit; they shape how students approach math later on.
A few big takeaways from our conversation:

• Teaching tricks teaches students that there’s only one right way to solve a problem.
• Students who learn shortcuts often struggle to adapt when the structure changes even slightly.
• When students learn multiple methods, they develop flexibility, a critical skill for algebra and beyond.
• Showing multiple approaches also normalizes mathematical thinking rather than memorized steps.

One teacher noted that they intentionally model solving equations in several ways, isolating variables using inverse operations, graphing, or even using tables, so students see algebra as a landscape of choices, not a fixed set of moves.
This is connected to a bigger philosophical question from the group: Do we need to pick just one method for students?  Our collective answer: Definitely not.

Big Takeaways

• Fixes that emphasize understanding are vital, even if they take more instructional time.
• Students need precise language to make sense of algebraic structures.
• Avoiding words like “cancel” prevents conceptual shortcuts.
• Showing multiple methods strengthens mathematical reasoning and transfer.
• Tricks may feel helpful in the moment, but they often hinder future learning.

Next time, we’ll move on to Chapter 7: Graphing, where we’ll explore common shortcuts, misconceptions about slope and intercepts, and ways to help students visualize relationships with meaning.