Nix the Tricks Book Study - Chapter 3: Proportional Reasoning
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 Chapter 1: Introduction and Chapter 2: Operations in previous posts and follow along as we continue through the book.
Chapter 3, “Proportional Reasoning,” was one of the heaviest sections we’ve read so far. It’s packed with tricks that almost every math teacher has encountered at some point: Butterfly Method, Flip and Multiply, Cross Multiply, and the Formula Triangle, to name a few. These are the kinds of tricks that often sneak into students’ repertoires long before they truly understand what a proportion or ratio represents.
As we discussed this chapter, one central question emerged: “Do students ever really understand the math behind these shortcuts, or do they lose the reasoning once the trick takes over?”
Rethinking “Multiply by the Reciprocal”
One example that generated a lot of discussion was Flip and Multiply. The book reminds us that “multiply by the reciprocal” isn’t really a definition of division; it’s an outcome that makes sense after students understand what division means. If we start with the shortcut, students may get correct answers for a while, but never develop a deep sense of why it works.
In our conversation, several teachers noted how quickly students latch onto “flip the second fraction” without understanding why. Later, when they hit algebraic fractions or complex rational expressions, that same lack of reasoning resurfaces. They may remember the motion, but not the meaning.
The chapter’s fix, helping students reason through division of fractions using models, common denominators, and equivalent ratios, feels slower at first but pays off in long-term understanding. We agreed that students who see why the reciprocal works are better equipped to reconstruct the process, even if they forget the rule.
The Problem with Cross-Multiplying Everything
Another recurring theme was the overuse of cross multiplication. It’s one of those procedures that students apply everywhere, fraction multiplication, division, proportions, and even equations, where it doesn’t belong. The phrase itself becomes a stand-in for thinking.
We talked about reframing proportional reasoning not as a method, but as a relationship. When students see proportions as equivalent ratios or as direct variation, they can make sense of what’s happening numerically and visually. Instead of memorizing a diagonal trick, they’re recognizing structure, what the book calls “legal algebra.”
Some teachers shared that they now use tables, double number lines, or unit rate reasoning before ever showing algebraic proportions. This shift helps students build the intuition that multiplication and division are two sides of the same relationship, rather than two separate tricks.
Why This Chapter Matters
This chapter pushed us to reflect on the long-term effects of teaching tricks that work in the short term. When students rely on memorized steps without conceptual grounding, those gaps show up later, especially in algebra, when proportional reasoning connects to slope, similarity, and function relationships.
The underlying message of this chapter was clear: Proportional reasoning isn’t a single skill; it’s a way of thinking. Students need to understand the multiplicative relationships that tie together ratios, rates, and scaling. Tricks might speed up early success, but they can rob students of the opportunity to reason deeply.
Looking Ahead
Next time, we’ll move into Chapter 4: Geometry and Measurement, where we’ll unpack common misconceptions about area, perimeter, and angle relationships, and how to replace formula memorization with meaning.
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