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## Using Covariation Reasoning to Support Mathematical Modeling

Table representations of functions allow students to compare rows as well as values in the same row.

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## An Excel–lent Card Trick

Card tricks based on mathematical principles can be a great way to get students interested in exploring some important mathematical ideas. Bonomo (2008) describes several variations of a card trick that rely on nested floor functions, but these generally go beyond the reach of beginning algebra students. However, a simple spreadsheet implementation shows students why the card trick works and allows them to explore several variations. As an added bonus, students are introduced to composite functions, the floor function, and iteration, and they learn how to use formulas and the INT function in Microsoft Excel. The depth of the mathematical explanation can be varied according to students' background.

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## Investigating Functions with a Ferris Wheel

A computer activity helps students make sense of relationships between quantities.

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## Are We Contributing to Our Students' Mistakes?

Teachers who have been in the classroom for a while know that year after year we see students make the same mistakes that previous students made. To prevent these errors, we try a variety of techniques, but it seems that no matter what we do, they reappear in students' work time and time again. Are we, despite our best efforts, contributing to these mistakes? I believe that we do, indeed, use some practices that inadvertently undermine our best intentions.

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## Using Multiple Representations to Teach Composition of Functions

Experience with multiple representations fosters students' robust understanding of what functions are, how they behave, and how they can be composed.

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## Building Sinusoids

A measurement-based activity can help students struggling to understand trigonometric functions.

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## Kick-Starting Discussions with the Flipped Classroom

A four-phase process and three principles for building a mathematics learning community use rich discussion of student work.

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## Students as Mathematics Consultants

Five problems—relating to gas mileage, the national debt, store sales, shipping costs, and fish population—require students to use functions to connect mathematics to the real world.

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## Algebra Aerobics

A kinesthetic approach to developing ideas of function transformations can get students physically and intellectually involved.

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## The Common Core and Inverse Functions

As we design curriculum programs based on CCSSM, we need to be careful when we consider the inclusion of some “nonessential” standards.