This case study is an investigation into how students make sense of speed. In other words, how do they quantify speed? Speed is a ratio, meaning that it is quantified by the relationship between two other quantities - distance and time. This is complex, and for many students, quantifying speed poses a unique challenge.

You will watch videos and analyze student thinking from a research project your instructor is a part of called IDReAM. (If you're interested, you can read more about the IDReAM project at http://www.indiana.edu/~idream/ (Links to an external site.)) In this project, the lead researcher, Amy Hackenberg, co-taught a unit with a 7th grade mathematics teacher. We will focus on three days of the unit, in which the students were working with an applet to investigate speed. You will work on the tasks yourself, watch videos of student work, and answer questions about students' thinking as well as questions about the choices made by the teachers.

At our presentation at AMTE on February 8, 2020, we used fewer videos and shorter versions, which we've also made available to you. Find them here.

Click here for a transcript of the video clips.

Part 1: Introduction

What you need to do for Part 1:

First, visit https://www.geogebra.org/m/hc3ecnnt (Links to an external site.) and complete the same tasks the students were asked to complete.

NOTE: This applet will work best if you use it from a tablet, laptop, or desktop. (It is often glitchy when you access it with a phone.)

Make the red car go slower than the silver car when both cars travel the same distance of 15 miles, and the silver car travels for 6 minutes.
Make the red car go slower than the silver car when the two cars travel the same number of miles. Can you think of a "rule" to use that would always make the red car go slower when both cars travel the same distance?
Make the red car go slower than the silver car when both cars travel for the same amount of time, 6 minutes, and the silver car travels 15 miles.
Make the red car go slower than the silver car when the two cars travel the same number of minutes. Can you think of a "rule" to use that would always make the red car go slower when both cars travel for the same amount of time?
Make the red car go the same speed as the silver car (with a different pair of numbers) if the silver car travels 18 miles in 3 minutes. Draw a picture to help you justify why the two distance time pairs produce the same speed.
Make the red car go the same speed as the silver car (with a different pair of numbers) if the silver car travels 15 miles in 6 minutes. Draw a picture to help you justify why the two distance time pairs produce the same speed.
Make the red car go the same speed as the silver car (with a different pair of numbers) if the silver car travels 15 miles in 9 minutes. Draw a picture to help you justify why the two distance time pairs produce the same speed.

Click here if you'd like to see the task sheet that we gave to students.

Second, read the Quantifying Speed Introduction. This gives you a general overview of the investigation as well as providing important information about the classroom and setting for the project.

Part 2: Sara & Lisa

For Part 2, you will be looking closely at Sara & Lisa as they work through the speed tasks. (Note that there is another student, Ivan, in their group, who does not participate much and is off-camera most of the time.)

There are 3 video clips to watch, with questions to consider. You can also see a full list of the questions by clicking here for a .pdf version.

(For shorter videos, click here.)

Sara and Lisa are working on the following task: One of the cars is travelling 15 miles in 6 minutes. Can you make the other car travel at the same speed but with a different number of miles and minutes?

First, watch Clip 1:

Sara&Lisa Clip 1 - Breakthrough.mp4

(Click here for a transcript of the video clips.)

Questions to consider:

Lisa's first suggestion is to try 14 miles in 5 minutes. Why do you think this makes sense to her?
After trying 14 miles in 5 minutes and 15.1 miles in 6.1 minutes, Sara claims “it’s impossible.” Then she has a “breakthrough” about doubling. What do you think Sara understands about doubling, initially? What evidence is there that Lisa understands Sara’s idea? What about Ivan?

Next, watch Clip 2.

Sara&Lisa Clip 2 - Doubling.mp4

Questions to consider:

What do you think Sara and Lisa understand about speed?
Why does Amy ask them to draw a picture? What do you think Lisa understands about her picture with regard to justifying why 30 miles in 12 min and 15 miles in 6 min are the same speed?

Then, watch Clip 3.

Sara&Lisa Clip 3.mov

Questions to consider:

Once they determine that 7.5 miles in 3 min is the same speed as 15 miles in 6 min, why do you think the girls talk about subtracting 1 and other subtraction ideas, like “7.5 miles less”?
What epiphanies did the girls have during this part of class? Are you surprised by any of them? What do you think their work means about their understanding of speed? Of proportional relationships more generally?

Here are the pictures Lisa drew as she was working on the task:

Lisa Picture 1.png

Lisa Picture 2.jpg

Part 3: Emily & her group

For Part 3, you will be looking closely at Emily and her group as they work through the speed tasks. (Focus on Emily, although Kendra and Lena are also working on the task in these clips.)

There are 2 video clips to watch, with questions to consider. You can also see a full list of the questions by clicking here for a pdf version.

(For shorter video clip, click here.)

Emily, Lena, and Kendra are working on the following task: One of the cars is travelling 18 miles in 3 minutes. Can you make the other car travel at the same speed but with a different number of miles and minutes?

First, watch Clip 1.

Emily Clip 1.mp4

(Click here for a transcript of the video clips.)

Questions to Consider:

Emily initially suggested 9 and 6, and she also tried 18 and 6 and 18 and 2. It’s possible that the she was “just guessing,” but usually there is some logic behind students’ suggestions. What logic can you find behind these initial suggestions?
Emily is visibly very excited when she runs the race where the red car goes 36 miles in 6 min. She seems a little subdued when the red car travels beyond the blue car and asks Lena if it matters that it goes farther. What do you think she understands about speed at this point?

Next, watch Clip 2.

Emily Clip 2.mp4

Questions to Consider (For these questions, take a look at Emily's pictures, found below the questions.)

How did Amy interact with Emily when she did not know how to draw a picture? Did Amy tell her what to draw?
In Emily’s second picture, the segments representing the journeys are basically identical. Why do you think she drew them that way, after the talk about doubling the quantities?
In Emily’s third picture, she extends the segment representing the 36 mile-6 min journey, but it is not exactly twice as big as the segment representing the 18 mile-3 min journey. She says it’s not drawn to scale. And then she shows two 18 mile-3 min journeys making up the 36 mile-6 min journey. What do you think she understands about making the cars go the same speed at this point?

Emily's first picture:

Emily picture 1.png

Emily's second picture:

Emily picture 2.png

Emily's third picture:

Emily picture 3.png

Part 4: Joanna & her group (and comparing groups)

For Part 4, you will be looking closely at Joanna and her group as they work through the speed tasks.

There are 2 audio clips to listen to, with questions to consider. There are also some questions to consider comparing the various groups. You can also see a full list of the questions by clicking here for a pdf version.

(For shorter audio clips, click here.)

Joanna, Mark, Brianne, and Jenni are working on the following task: One of the cars is travelling 15 miles in 9 minutes. Can you make the other car travel at the same speed but with a different number of miles and minutes?

First, listen to Clip 1.

Joanna Clip 1.mp3

(Click here for a transcript of the clips.)

Questions to Consider:

How is Joanna solving this problem?
How is Mark thinking about creating same speeds?

Then, listen to Clip 2.

Joanna Clip 2.mp3

Questions to Consider:

How does Joanna’s picture (see below) justify that the distance time pair of 5 miles in 3 minutes produces the same speed as 15 miles in 9 minutes?
What teacher move does Rebecca make with Mark and Jenni in the group? Why does she make this move?

Joanna's picture:

Joanna's Picture.png

Mark's picture:

Mark's picture.png

Brianne's picture:

Brianne's picture.png

Next, consider the following questions comparing/contrasting the three groups and their work.

What similarities and differences do you notice about the thinking across the three groups of students?
What appeared to support progress for Lisa and Sara? For Emily?

Part 5: Whole Class Discussion

On Day 13 of the unit, after all the students had significant time to work on the same speed task, the whole class came together for a discussion about what they had learned. For part 5, you will watch a video clip of this discussion, and consider some questions about it. You will also look at a summary table of the groups' work and read through some takeaway messages for the unit. You can see a full list of questions for Part 5 by clicking here.

First, watch a clip of the whole group discussion.

Whole Group Discussion & Presentations.mp4

Questions to consider:

Students presented the numbers they worked on. What are possible disadvantages and advantages of having students present different problems/numbers?
What do you think Lisa and Sara understand about same speeds? Emma? Joanna?
What other questions could Amy have asked to help students articulate how their pictures justified same speeds?

Then, take a look at the following summary chart of the three groups' work:

Summary Chart.png

Next, consider the following take away messages for the case study:

We want all students to make progress where they are; tiered tasks with some purposeful changes can support that. Note that this was basically a Choice Problem where we (the researchers and teacher) made the choices for what numbers groups of students would work on.
From each group we learned more about how students think about ratios. It was formative assessment for us, big time. This helps us in the moment and in the future.
We actually are not expecting all students to think like Joanna right away, where she determined the smallest whole number pair of quantities that would work, and she saw this pretty quickly. People are often surprised at this, because Joanna’s thinking is a powerful and a worthwhile learning goal.
Doubling each quantity, multiplying each by other whole numbers, and then dividing each by whole numbers is important, legitimate work for many middle school students when working with ratios. People are often surprised at this too.

Our work has been published in the February, 2020 issue of Mathematics Teacher: Learning and Teaching PK-12 (NCTM's new journal). Visit https://pubs.nctm.org/view/journals/mtlt/113/2/article-p124.xml for more information!

MTLT Article Feb 2020.png