Nix the Tricks Book Study – Chapter 4: Geometry and Measurement

 

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 prior chapters in previous posts, and follow along as we continue through the book.

Chapter 4, “Geometry and Measurement,” took us into one of the most visual and language-rich areas of mathematics. For many of us, geometry is where we first learned to “see” math, but it’s also a place where shortcuts and memorized formulas can overshadow the reasoning behind them.

Our discussion this week revolved around three major themes: definitions, sense-making, and vocabulary.

Rethinking Tricks and Formulas

When reflecting on the “tricks” in this chapter, several teachers admitted they’ve used them in the past, especially when teaching CP (College Prep) Geometry classes. It’s easy to fall back on formula lists and quick rules when students struggle with abstract reasoning or vocabulary.

But this chapter was a reminder that when we focus too much on memorization, students lose the opportunity to derive formulas and make sense of geometric relationships. One participant shared that they loved this chapter because it affirmed that the conceptual approach used in Honors Geometry, where students discover relationships through exploration, is exactly what we should strive for at every level.

The conversation turned to a few specific questions that reveal common points of confusion:

  • Do students truly know the difference between a rectangle, a square, and a rhombus?

  • How consistent are we about definitions of trapezoids and kites, especially when textbooks or standards vary?

  • Can we skip the endless list of surface area and volume formulas and instead focus on deriving them from a few key relationships?

The consensus was clear: fewer formulas, more reasoning. When students uncover why a formula works, they don’t have to memorize dozens; they remember one idea that can be applied flexibly.

Shifting Instructional Practice

Many of us shared ideas for making geometry instruction more meaningful in the future. Some highlights included:

  • Using more precise language and reinforcing vocabulary through everyday use, not just at test time.

  • Focusing on discovery and pattern recognition, allowing students to notice relationships before formalizing them into definitions or formulas.

  • Providing repeated exposure to big ideas rather than covering them once and moving on.

One teacher shared that if they teach CP Geometry again, they plan to bring in the same level of conceptual discussion used in Honors classes, helping students understand why area and volume formulas work rather than just applying them.

Teaching Vocabulary with Meaning

We also spent time discussing how to teach vocabulary effectively, since precise language is at the heart of geometry. Some strategies we shared included:

  • “Say what you mean.” Model clear, accurate mathematical language and expect the same from students.

  • Start with examples before definitions. Let students explore, categorize, and then generalize.

  • Dissect definitions. Write the formal definition, then analyze it piece by piece using diagrams or examples.

  • Connect definitions to theorems. Emphasize that definitions are biconditional statements, while theorems build upon them. For instance, understanding the difference between the definition of a midpoint and the midpoint theorem helps students appreciate the precision of geometric reasoning.

These conversations reminded us that vocabulary instruction in math isn’t about memorizing words; it’s about developing a shared language for thinking logically and communicating reasoning.

Big Takeaways

  • Prioritize reasoning over recall: fewer formulas, more discovery.

  • Use consistent, precise language. Clarity builds confidence and accuracy.

  • Explore before defining. Let students make sense of patterns before formalizing them.

  • Teach definitions as tools, not trivia. Understanding “what it says and why” helps students connect concepts to theorems.

  • Geometry is sense-making. Visual models and verbal reasoning go hand in hand.

Next time, we’ll explore Chapter 5: Algebra, where we’ll reflect on how to balance procedural fluency with conceptual understanding and help students see algebra as a tool for thinking, not just symbol manipulation.

Comments

Post a Comment

Popular posts from this blog

My Favorite: Doceri

Image
  My favorite classroom tool is Doceri.  It does cost $30, but I use this iPad app every day, so it was well worth the cost.  You need a computer (laptop/desktop), wireless network, and projector to use Doceri.  It basically turns your computer into a whiteboard.  You can control your computer and any files remotely from the iPad.  I use this feature for PowerPoints during class.  You can eccentrically take a screen shot of whatever is on your screen (or use a number of included backgrounds such as graph paper) and then write on that image.  I use this feature most often.  I will make a PowerPoint presentation of my notes each day and then use Doceri to annotate each slide.  You have a ton of pen style/color choices so you can color code and highlight important parts of your class notes.  Doceri can save each annotated image and you can email it to yourself in PDF form.  This can become a document to post on your cla...
Read more

Practice Structures: Error Analysis Revisited

Image
Last summer I wrote about using error analysis as one of my favorite ways to practice math skills.  Since then I've found a few more resources and created a few more that I'd like to share.  In general, error analysis activities involve purposefully making common mistakes, then having students try to find them.  Here is a link to the original post if you'd like to see it.  In that post I shared an example for the segment addition postulate and midpoint as well as a second example for multiplying polynomials . Here are some other examples that I found.  I did not create these. Grade 2: Adding and Subtracting up to 3 digits. Grade 8: Solve Linear Equations. Algebra 1: Graphing Linear Inequalities in two Variables
Read more

One Good Thing

Image
This past week was the last week of classes for this semester.  Some of my classes are full year classes while others are semester courses.  Either way, this seemed like a good time to make some changes for the new semester and to allow students to give some feedback about those changes. I gave my students a quick exit ticket on our last day of class.  I asked them to think back on the year so far and to tell me one thing that they liked about our class that they would like to keep the same for the new semester.  Then I asked for one thing that they would like to change for the new semester. Now, I must preface their responses by saying that I teach in a very non-traditional classroom.  I use hybrid learning.  You can read more about it on the hybrid page of my blog.  Basically it involves students rotating through various stations throughout the lesson.  Students have more freedom than they have ever had before and some of them fail mi...
Read more