Perhaps you have seen the diamond paper graphical organizer.  Teachers can pose a problem or question inside the diamond and ask students to justify or explain their work in different ways in the four rectangles.  Here’s a version that I created for my Honors Pre-Calculus class: I learned about “diamond paper” from Jo Boaler — I either read about it in her book Mathematical Mindsets or saw it in her online course.  When I attended one of Jo’s workshops, I worked through a diamond paper problem from the perspective of a student and was immediately hooked.  I often ask students to look at a problem graphically, analytically, numerically, and in written form, but this graphical organizer furthers the discussion by visually organizing their work.

For these activities, my students work in groups to solve the problem in the center.  They record their work in Microsoft OneNote.  When discussing their results, I either project a copy of work from one group or ask each group to use markers to fill in one of the sections on a whiteboard.  We discuss and annotate the work as we go.  I take a picture at the end to recap our discussion.

I created the “diamond paper” activity shown in the picture at the beginning of this post to introduce inverse trigonometric functions.  My hope was that students would:

• Recall AND connect the different methods to find an angle whose sine is 1/2.
• Discuss how many solutions to provide.

Here is a picture of student work that we reviewed in class: Some groups found one solution for the equation, some found two solutions, and some found many solutions.  This led to a discussion about how to restrict the domain of sine so that it has an inverse.  Students unanimously agreed that the toughest part of the diamond paper to complete was the section that asked them to explain with words.  We had a productive discussion about the difference between explaining their method in words versus explaining their solution in words.

One of the unexpected outcomes was a discussion of the various graphs that students made for the “justify with a graph” section.  Some students graphed y = sin x and y = 1/2 in various software (see an example from Mathematica below): Some students graphed sin x = 1/2 in software (see an example from Desmos below): The students who created this graph were perplexed as to why the graph did not look like a typical sine wave.  Students shared their thoughts about how to interpret this graph and explained why its shape was not sinusoidal.

I created several versions of the activity below for the topic of solving trigonometric equations.  Students were especially excited to see the connection between the solutions of this equation and the x-intercepts of the graph.  We spent a substantial amount of time on the “justify in words” section, again discussing the difference between describing a method and justifying a solution.

Diamond paper is perfect for calculus as well.  I use it to help students see connections in calculus topics like continuity.  Here’s an example of work from a group of my students a few years ago.  As an aside, the graph in this picture was made using FluidMath graphing software.  I prefer to use FluidMath when working with rational functions and continuity because it shows holes in the function, as you can see if you look closely at the graph below. As a bonus, diamond paper is easy to make.  I created a template in PowerPoint so that I can simply change the words for new activities.  Here’s a blank graphical organizer (diamond paper) to get you started.