# Blog

## 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

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

## How would you use mental math to solve 36×25?

Last week I asked #MTBoS followers how they would use mental math to solve 36×25.

Below are a few responses that I received:

Having students use mental math to solve multiplication problems such as 36×25 can provide them with opportunities to informally use mathematical properties that are fundamental to algebraic reasoning.

For instance,

When students talk about their mathematical thinking, they have opportunities to clarify their ideas and to make meaning from symbol sentences. Tina, Annie, and Max all mentioned how they interpreted the symbol sentences they wrote. I think it is especially interesting how Tina said that the first two sentences were how she “thought it through.”

When I investigate students’ reasoning in my own research, I have learned that students may use the same written representation, but think about it very differently. The selection of Tweets that I included here is only a sampling of the interesting ways that #MTBoS followers thought about 36×25.

I wonder what might have changed if I had asked #MTBoS followers how they would use mental math to solve 25×36.

This semester, I am teaching a fully online class – Expanding conceptions of algebra— #MTED5622. In #MTED5622 we are investigating how students in grades K-12 engage in Algebraic Reasoning. One of the resources we are using is from NCTM’s Essential Understanding Series –Developing Essential Understanding of Algebraic Thinking Grades 3-5. This fall, I’ll be working to include blog posts focused on my work in teaching this course.

## Why is it so hard for students to make sense of rate?

Earlier this year, I searched for some newspaper headlines that dealt with rates. Here are a few that I came across:

• “Colorado unemployment rate among 10 lowest in the country”
• “Unintended pregnancy rate in U.S. is high, but falling”
• “Oil ends sharply higher. Logs 10% weekly gain as output draws focus”

Many of the headlines talked about rates, and not just rates, but varying rates. When I think about these headlines, I wonder how students make sense of varying rates. For example, what might a student think it means for a rate to be affected, for a rate to be low, or for a rate to be high, but falling? Furthermore, what does it mean for something to end “sharply” higher? Are there other kinds of ways to end higher? For example, what might ending “gradually” higher be like?

In my research, I investigate how students make sense of change, and more specifically, I study how students make sense of variation in change. Put another way, I want to know how—from a student’s perspective—an “increase” (or “decrease”) can be a thing that can vary. When I interact with students, I work to design learning experiences that can provide them opportunities to investigate different kinds of increases (and decreases).

When it comes to rate, I have found that it matters how students form and interpret relationships between quantities

If we want students to think about rate as something that is capable of varying, we should help students coordinate change in two different quantities, such as the height and volume of liquid in a filling bottle. Specifically, we should provide opportunities for students to think about one quantity as continuing to change while another quantity is changing along with it. For example, students could think about how the height of the liquid in a filling bottle continues to change while the volume of liquid in the filling bottle changes along with it.

I developed a framework that explains how students’ different ways of forming relationships between quantities can impact how they think about rate. I wrote about this framework in a 2015 article published in the journal Mathematical Thinking and Learning. The published version of the article is available here: http://www.tandfonline.com/doi/pdf/10.1080/10986065.2015.981946

Here is the full citation for the article (APA 6th)

Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Mathematical Thinking and Learning, 17(1), 64-90.

Here is the accepted manuscript of the article that you can download:

HLJohnson_QuantRatioRate_MTL