## From What iS? to What iF?

To open opportunities for students’ math reasoning, change the questions. Instead of What iS? Ask What iF? The questions look similar. Yet they imply very different responses. What iS? “Give an answer” What iF? “Consider the possibilities” Too often, students experience math as a pursuit of “What iS?” rather than an exploration of “What iF?“ …

## The Same Situation. Two Different Graphs.

Students have many opportunities to use different types of representations to show the same relationship between variables (e.g., graphs, tables, equations). Students also benefit from opportunities to use different forms of the same type of representation (two different looking graphs) to show the same relationship between variables. Wonder how that can be? Check out this …

## #HowGraphsWork

I have been thinking hard about how students make sense of graphs. In my April 17 Global Math Department webinar, we’ll explore ways to help students see #HowGraphsWork I hope many are able to join us. In case you aren’t able to make it, or if you would like to access resources after the webinar, …

## Increases can increase? Learn what students think

Sea levels aren’t just rising. They’re rising FASTER. https://www.cnn.com/2018/02/12/world/sea-level-rise-accelerating/index.html Yet how do students come to make sense of variation in change? How do “increasing” increases become things for students? In a March 2018 episode of the Math Ed Podcast, I talked with Sam Otten (@ottensam) about an article I co-authored with Evan McClintock. I share …

## Give students opportunities to make sense of varying increases

2018 began with news articles about varying increases: U.S. Private Payrolls Growth Accelerates; Jobless Claims Up UK Productivity Growth Hits Six-Year High After Weakest Decade Since 1820s Euro Zone Factory Growth Surges to Record; More Uneven in Asia What kinds of opportunities help students to make sense of accelerating growth? In our recent research article, …

## Make Graphs about Relationships with Cannon Man

In math classes, students work with graphs. A LOT. Yet, what do students think graphs are? Why might students sketch or use graphs? A powerful way for students to think about graphs: As relationships between “things” that can change Together with Dan Meyer and the team at Desmos, I developed activities, “Techtivities” to provide students …

## Are transformations of functions giving your students trouble? Try a covariation approach.

Consider this problem: Sketch a graph of a function y=f(x). Now sketch a graph of y=f(ax) for some constant value a>1. Reflect. What kind of graph did you sketch for y=f(x)? How did you decide what to sketch? Did you choose a particular value for a? How does your graph of y=f(ax) compare to your graph of y=f(x)? What is …

## Investigating Functions with a Ferris Wheel. Part 3: Exploring Distance and Width

Here are some tips for using the Ferris Wheel Distance-Width Interactive with students. The format is parallel to Investigating Functions with a Ferris Wheel: Part 2. I suggest using the Ferris Wheel Distance-Width Interactive after students have explored the Ferris Wheel Distance-Height Interactive. I introduced these interactives in Investigating Functions with a Ferris Wheel: Part 1. Explore changing distance and width: Ferris wheel animation Click Hide Width, Hide …

## Investigating Functions with a Ferris Wheel. Part 2: Exploring Distance and Height

In Investigating Functions with a Ferris Wheel: Part 1 I shared two Web Interactives. Here are some tips to for using the Ferris Wheel Distance-Height Interactive with students. Explore changing distance and height: Ferris wheel animation Click Hide Height, Hide Distance, Hide Point, and Hide Trace. Press Animate Point. Questions for students: For a car beginning at start and moving once around the wheel, …

## Investigating Functions with a Ferris Wheel: Part 1

Investigating Functions With A Ferris Wheel coauthored with Peter Hornbein (@phornbein1) and Sumbal Azeem, appeared in the December 2016/January 2017 issue of NCTM’s Mathematics Teacher journal. This winter, Max Ray-Riek (@maxmathforum) and Annie Fetter (@MFAnnie) converted my Geometer’s Sketchpad sketches into Web Sketchpad Activities, freely available through NCTM Illuminations: Investigating Functions with a Ferris Wheel: Distance vs. …