#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, I included links in this space.

Slides from the webinar

180417_HowGraphsWork_GlobalMath_HthrLynnJ

webinar video

Open Access Online activities

Desmos Activities: Cannon Man, Toy Car, and Ferris Wheel

NCTM Illuminations Ferris Wheel Interactive

a Blog post and an Article 

Steve Phelps’ (@giohio) Desmos Sketches

Isosceles Triangle   

Isosceles Triangle v4 

New York Times Column: What’s Going on in This Graph?

(Graphs selected in partnership with Sharon Hessney, the Statistics Content Director at Mass Insight Education)

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 results of the study and offer insights into our research process.

We found that students who discerned variation in increases also reasoned about attributes as being capable of varying and possible to measure.

Students’ willingness to share their thinking is key to our research. Learn how we position students as experts when conducting math interviews.

Our article is available open access:

Johnson, H. L & McClintock, E. (2018). A link between students’ discernment of variation in unidirectional change and their use of quantitative variational reasoning. Educational Studies in Mathematics. 97(3), 299-316. doi: 10.1007/s10649-017-9799-7

I also talked about this article in an earlier blog post:

Give students opportunities to make sense of varying increases

Give students opportunities to make sense of varying increases

2018 began with news articles about varying increases:

In our recent research article, we report on a study identifying learning conditions that help early secondary students to “discern variation in unidirectional change.” Or put another way, make sense of accelerating growth.

We identify two conditions:

Provide students opportunities to conceive of attributes as capable of varying and possible to measure.

Possible to Measure

How students think about the attributes matters. Attributes such as length can be easier for students to conceive of measuring, than attributes such as volume or time.

capable of varying

Give students opportunities to make sense of change as happening. Too often, students only have opportunities to think about how much change has accrued.

Want more? Click the link to read our article.

Johnson, Heather Lynn, and Evan McClintock. “A Link between Students’ Discernment of Variation in Unidirectional Change and Their Use of Quantitative Variational Reasoning.” Educational Studies in Mathematics, Springer Netherlands, Jan. 2018, pp. 1–18.

Want to try this with students?

Use one of the Techtivities we developed in collaboration with Dan Meyer and the team at Desmos.

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 opportunities to think about graphs as representing relationships.

In this audio clip from a recent presentation, I talk through one of the Techtivities, the Cannon Man:

Want to find out more about how we’re using the Techtivities? See our ITSCoRe project website.

Tried one of the Techtivities? Have questions about when or how the Techtivities? I would enjoy hearing and responding to your comments.

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.Screen Shot 2017-04-01 at 11.49.22 AM

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.Screen Shot 2017-03-30 at 10.56.10 AM
  • 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.

Screen Shot 2017-03-30 at 11.04.06 AM.png

  • 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.

Screen Shot 2017-03-30 at 11.04.35 AM.png

  • 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.

Screen Shot 2017-03-30 at 11.11.12 AM.png

  • 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.

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.Screen Shot 2017-03-13 at 3.07.58 PM.png
  • 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.

Screen Shot 2017-03-13 at 3.18.34 PM.png

  • 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.

Screen Shot 2017-03-13 at 3.36.48 PM.png

  • 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.

Screen Shot 2017-03-13 at 3.49.04 PM

  • 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.

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. Height

Screen Shot 2017-03-06 at 11.22.45 AM

Investigating Functions with a Ferris Wheel: Distance vs. Width

Screen Shot 2017-03-06 at 11.28.07 AM.png

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.