# 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 similar? What is different?

## It is no secret that students have difficulty making sense of transformations of functions. A covariation approach can help. Here’s how.

A graph of y=f(x) represents a relationship between two “things” that can change: “y” and “x.”

## Think about a graph of y=f(x) as representing a relationship between two “things” that can change: “y” and “x“

The notation y=f(x) means that y is a function of x. Functions are special kinds of relationships between two “things” that can change. When y is a function of x, as the values of x change, the values of y change in predictable ways.

## A graph of y=f(ax) represents a relationship between two “things” that can change: “y” and “ax“

The notation y=f(ax) means that y is a function of ax. When y is a function of ax, as the values of ax change, the values of y change in predictable ways. Because a is some constant value greater than one, does not change. Therefore, the changing values of ax depend on changes in the values of x.

## Working from a relationship between “y” and “x,” students can make sense of a relationship between”y” and “ax.”

For a function y=f(x), as the values of x change, the values of y change along with them.

For a function y=f(ax), the values of x change by a factor of “a.” This means that “1/ath of the change in x will yield the same amount of change in y as for y=f(x).

## Using relationships between changing “things” can help students to make sense of graphs of transformations of functions.

For example, sketch a graph y=f(x).

Now sketch a new graph with this constraint: “1/2 the change in x will produce the same change in y as for the original function.”

## Reflect.

What kind of graph did you sketch for y=f(x)? How did you decide what to sketch? How does your new graph compare to your graph of y=f(x)? What is similar? What is different?

# Let’s share our ideas to grow them

## What might happen if we start sharing our ideas to grow them?

I think about new ideas as living organisms rather than completed products. As I share ideas, I grow them.

I enjoy having written products, because they serve as records of my thinking at particular moments in time. And written products help me to learn how ideas have grown and continue to grow.

## Here is one of my very first pieces of writing that I shared with a broader community

I describe a lesson that I submitted for my PAEMST application.

Using Data and Linguine to Discover the Triangle Inequality_PCTM_2003

## Forging ahead with new ideas

In my current work, I design online tasks to provide students’ opportunities to engage in mathematical reasoning about difficult to learn concepts such as function and rate. In Why is it so hard for students to make sense of rate? I share where my ideas have come and are going.

# Graph Makeover: It’s about time

Think back to a time when you encountered a “real world” graph in a math class. What was on the horizontal axis?

### Probably TIME.

Graphs represent relationships between two things. When working with graphs, it is important for students to form and interpret relationships between TWO things that can change. Yet, if one of those things is passing time, students might be thinking about only ONE thing changing.

### Want to provide students opportunities to think about TWO things that can change?

Have students interpret graphs that represent two changing lengths.

### Wonder how to get started?

Try these Ferris Wheel Interactives on NCTM Illuminations.

It’s about time that students have opportunities to encounter “real world” graphs representing things other than time.

# Use Tech to Broaden Students’ Opportunities for Math Reasoning

Subtitle: The reason I’m giving this talk at #NCTM2017.

Think of technology as “playground equipment” that teachers can use to create online “learning playgrounds” for students.

By using different kinds of equipment, we can broaden students’ opportunities to engage in mathematical reasoning.

If you subscribe to NCTM’s Mathematics Teacher journal, you can read more here.

# 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 Distance, Hide Point, and Hide Trace.
• Press Animate Point.
• Questions for students: For a car beginning at start and moving once around the wheel, (1) How is its distance from start changing? (2) How is its width from the center (horizontal distance) changing?
• Teaching Tip: Have students use their fingers to trace along the Ferris wheel to show the distance and width. [Students might think the literal words ‘distance’ and ‘width’ are changing. Focus their attention on the lengths.]

## Explore changing distance: Animation & Dynamic Segments

• Drag the active point (Ferris wheel car) to the right side of the wheel. Click Show Distance.

• Before pressing Animate Point, ask students to predict how the dynamic distance segment would change as the car moves once around the wheel.
• Once students make predictions, press Animate Point. The Ferris wheel animation and dynamic distance segment will move together.
• Teaching Tips:
• Have students use their fingers to show how the dynamic distance segment will change.
• Students might be surprised that the dynamic segment stays on the horizontal axis, because they may not have seen many graphs with points only on an axis.
• Students might think that the dynamic segment for distance has to be the same length as the actual distance around the wheel. Allow students to investigate why this does not need to be the case.
• If students have already worked with the Distance-Height Interactive, ask them if the distance will change in the same way. [The distance does change in the same way, but students might think it would be different because it is a new situation.]

## Explore changing width: Animation & Dynamic Segments

• Drag the active point (Ferris wheel car) to the right side of the wheel. Click Show Width. Click Hide Distance.

• Before pressing Animate Point, ask students to predict how the dynamic width segment would change as the car moves once around the wheel.
• Once students make predictions, press Animate Point. The Ferris wheel animation and dynamic width segment will move together.
• Teaching Tips:
• See the Teaching Tips for distance. Apply those Teaching Tips for width.
• If students have already worked with the Distance/Height Interactive, ask them to compare how the width and height change. [The width segment changes direction twice, but the height segment changes direction only once.] Ask students to use the Ferris wheel situation to explain why this is the case.

### Explore changing distance and width: Animation & Dynamic Segments

• Drag the active point (Ferris wheel car) to the right side of the wheel. Click Show Width. Click Show Distance.

• Before pressing Animate Point, ask students to predict how the dynamic distance and width segments would change together as the car moves once around the wheel.
• Once students make predictions, press Animate Point. The Ferris wheel animation and dynamic distance and width segments will move together.
• Teaching Tips:
• Ask students if changing the speed of the Ferris wheel would affect the dynamic distance and width segments. [The motion would occur faster or slower, but the dynamic width and distance segments would still change in the same way.]
• Ask students to compare and contrast the ways in which the dynamic distance and width segments change. [The width segment changes direction twice. As the car is moving around the Ferris wheel, the dynamic width segment increases and decreases, until it reaches the top of the wheel, then increases and decreases until it returns to the bottom of the wheel. The increases and decreases in the width segment are faster or slower depending on where the car is on the wheel. The distance segment only increases, and it increases at a constant rate.]

## Want more?

In upcoming blog posts, I’ll be sharing more ideas for using these Web Sketchpad activities.

## What do you think?

How have you used these Web Sketchpad activities with your students? Let me know in the comments, or let me know on Twitter @HthrLynnJ.

# I study students’ mathematical reasoning, and I take real breaks.

This week I revisited an advice column, Workload Survival Guide for Academics, which I came across last year.

Professor Andrew Oswald identified a price that comes along with the privilege of being a university faculty member: No clearly defined leisure time.

# 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, (1) How is its distance from start changing? (2) How is its height from the ground changing?
• Teaching Tip: Have students use their fingers to trace along the Ferris wheel to show the distance and height. [Students might think the literal words ‘distance’ and ‘height’ are changing. Focus their attention on lengths.]

## Explore changing distance: Animation & Dynamic Segments

• Drag the active point (Ferris wheel car) nearer to Start. Click Show Distance.

• Before pressing Animate Point, ask students to predict how the dynamic distance segment would change as the car moves once around the wheel.
• Once students make predictions, press Animate Point. The Ferris wheel animation and dynamic distance segment will move together.
• Teaching Tips:
• Have students use their fingers to show how the dynamic distance segment will change.
• Students might be surprised that the dynamic segment stays on the horizontal axis, because they may not have seen many graphs with points only on an axis.
• Students might think that the dynamic segment for distance has to be the same length as the actual distance around the wheel. Allow students to investigate why this does not need to be the case.

## Explore changing height: Animation & Dynamic Segments

• Drag the active point (Ferris wheel car) nearer to Start. Click Show Height. Click Hide Distance.

• Before pressing Animate Point, ask students to predict how the dynamic height segment would change as the car moves once around the wheel.
• Once students make predictions, press Animate Point. The Ferris wheel animation and dynamic height segment will move together.
• Teaching Tips: See the Teaching Tips for distance. Apply those Teaching Tips for height.

## Explore changing distance and height: Animation & Dynamic Segments

• Drag the active point (Ferris wheel car) nearer to Start. Click Show Height. Click Show Distance.

• Before pressing Animate Point, ask students to predict how the dynamic height and distance segments would change together as the car moves once around the wheel.
• Once students make predictions, press Animate Point. The Ferris wheel animation and dynamic height and distance segments will move together.
• Teaching Tips:
• Ask students if changing the speed of the Ferris wheel would affect the dynamic height and distance segments. [The motion would occur faster or slower, but the dynamic height and distance segments would still change in the same way.]
• Ask students to compare and contrast the ways in which the dynamic height and distance segments change. [The height segment changes direction. As the car is moving around the Ferris wheel, the dynamic height segment increases and decreases faster or slower depending on where the car is on the wheel; the distance segment only increases, and it increases at a constant rate.]

## Want more?

In upcoming blog posts, I’ll be sharing more ideas for using these Web Sketchpad activities.

## What do you think?

How have you used these Web Sketchpad activities with your students? Let me know in the comments, or let me know on Twitter @HthrLynnJ.

# 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.

### A few quick notes to get started:

• Click Animate Point to move the car around the Ferris wheel.
• Click the action buttons to show/hide features and move between pages.
• Drag the Active Point (the car on the Ferris wheel) to control the animation.

### Who might use these activities?

I used these activities with 9th grade students in Algebra 1, but they could be appropriate for students with different kinds of mathematical experience.

### Want more?

In upcoming blog posts, I’ll be sharing ideas for using these Web Sketchpad activities.

### What do you think?

How have you used these Web Sketchpad activities with your students? Let me know in the comments, or let me know on Twitter @HthrLynnJ.

# Narrating students’ mathematical reasoning

### When I narrate students’ mathematical reasoning, I engage in storytelling.

I work to analyze what students say and do to better understand their perspectives.

I have the great privilege of learning from students who are willing to talk with me about their work to solve mathematics problems.

This weekend’s stories remind me of the responsibility that comes with that privilege.

### To read one of my analyses of students’ mathematical reasoning, click the link below:

Johnson, H. L. (2015, July). Task design: Fostering secondary students’ shifts from variational to covariational reasoning. In Beswick, K., Muir, T., & Wells, J. (Eds.) Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 129-136). Hobart, Tasmania: University of Tasmania

# Reaching a milestone, Allowing for introspection, Forging new possibilities

### Reaching a milestone

In 2016 I reached a milestone that I have been working toward for more than a decade. I earned tenure as a faculty member at a research intensive university. In 2005, when I shifted from high school classroom teacher to full time graduate student, I had only glimmers of notions of the journey that would lie ahead.

### Allowing for introspection

I have been working so fiercely for so long that it feels somewhat unsettling to entertain the possibility of a pace less relentless. I am working to carve out more spaces for noticing, reflecting, and growing. I am learning that embracing my fervent passion for fostering students’ mathematical reasoning need not come at the expense of my well being.

### Forging new possibilities

In late summer/early fall, I began to share my thinking and learning more publicly through my Twitter account, @HthrLynnJ. I started this blog. I intended to include more posts. I am giving myself space to be okay with my current number of blog posts. I am looking forward to new possibilities yet to come.

Taking in a view on the Trading Post trail at Red Rocks Park on the eve of NYE