# Ely, Nevada and its surrounding area

I discovered Ely, Nevada’s homepage recently and particularly enjoyed it: Ely, Nevada Home Page., but especially for the sentence (near the bottom of the page): “Centrally located in the intermountain west, Ely and the surrounding area contains a population of over 6 million people within a 500 mile radius.”

I loved the humor and creativity evident in that sentence. And then I thought, “What questions could I ask to make sense of this statement? How could I share this statement with a math class?” I considered the questions:

• In order to engage in answering any of the questions below, what tools can I use?
• Is there an online tool that, given a map, would create the circle for me? Is there a tool that would add up the populations for me?
• Creating the circle (if using the map, figuring out scales, etc. is important):
• Get a map of US (either a folded map or print out a map from Google Maps) and a compass, measure the size of “500 miles” and use the compass to create a circle
• Get a digital map of US and use GeoGebra to draw a circle (depending on how precise I want)
• Finding populations:
• How precise should I be?
• Should I choose the largest cities, and look up the metro populations? What is my “cut-off point” for city size?
• How far would I be under-estimating, if I don’t count all of the smaller towns and cities?
• What does “surrounding area” really mean?
• If I draw my intuitive sense of what “surrounding area” means on a map, how big of an area does it include? Do my classmates have similar or different ideas about the size of “surrounding area”?
• How big of a radius should I use to include the whole US? Where would I center it? How much of Canada, Mexico, or other countries would be included?
• How big is a region with a 500-mile radius?
• What cities / metro areas are included in the 500-mile radius circle centered at Ely?
• Could they have chosen a smaller radius and still included an impressive population?
• Are there particular cities they wanted to include that inspired their choice of 500 miles?
• Where I am now (in East Lansing, MI):
• If I created a circular region of radius 500 miles centered on East Lansing
• How many people would be contained in that region?
• What cities would be contained in that region?
• If I wanted to include a population of 6 million people, how large should the radius be, if I centered the circle on East Lansing?

I will try to engage in this task when I have more time…

In order to engage in answering any of the questions below, what tools can I use?

• Is there an online tool that, given a map, would create the circle for me?
• Is there a tool that would add up the populations for me?
• Finding populations:
• How precise should I be?
• Should I choose the largest cities, and look up the metro populations? What is my “cut-off point” for city size?
• How far would I be under-estimating, if I don’t count all of the smaller towns and cities?

How big is a region with a 500-mile radius?

• What cities / metro areas are included in the 500-mile radius circle centered at Ely? The map shows some large cities are: Salt Lake, Ogden, Provo, Las Vegas, Reno, Phoenix, Los Angeles, San Jose, San Francisco, Fresno, Sacramento, San Diego, Tijuana (Mexico), Boise, Idaho Falls
• I’m going to estimate the population. I notice that all of Nevada (2.839 million) and Utah (2.943 million) are contained. So that’s already about 6 million people! Most of California is contained, except for a tiny piece in the northwest that includes Crescent and Eureka, and California has population 38.8 million – let’s be conservative and knock off about 8.8 million for that tiny corner and we still have 36 million contained. The Phoenix metro area adds about 3 million. Tijuana adds about 1.483 million. Boise (0.616) and Idaho Falls (0.130) add a little less than a million. I could keep adding on, but a low estimate is still over 40 million.
• I found that the same site I used to create the circle also will estimate population within the radius, and it says “The estimated population in the defined area is 49,997,320.”
• Could they have chosen a smaller radius and still included an impressive population?
• I’m going to guesstimate and say 250-300 mile radius would probably still include 6 million people.
• Let me test my conjecture. (Note that I needed to reset the site before changing the radius, or it told me the same population as above.)  I entered 250 miles and was told “The estimated population in the defined area is 4,331,681.”  300 miles was closer with 5,438,487, and 315 was slightly more than 6 million with 6,396,013.
• Are there particular cities they wanted to include that inspired their choice of 500 miles?
• I’m guessing it is cool to say that San Fran and other California cities are “in the surrounding area.”

Where I am now (in East Lansing, MI):

If I created a circular region of radius 500 miles centered on East Lansing

• How many people would be contained in that region? Well, it includes all of Michigan (9.91), Wisconsin (5.758), Illinois (12.88), Indiana (6.597), Ohio (11.59), West Virginia (1.85), Washington, D.C. (0.658), and Kentucky (4.413). So, as a low estimate, a 500-mile region would include a population of at least 53.656 million.
• I was much, much too low! With the population finder, I found  the estimated population in the defined area is 101,594,400
• If I wanted to include a population of 6 million people, how large should the radius be, if I centered the circle on East Lansing? I conjecture 90 miles. Metro areas for Detroit (3.734), Lansing (0.464), Grand Rapids-Muskegon-Holland (1.321), Kalamazoo (0.326), and Flint (0.425) would be just over 6 million.

I’m really under-estimating though, because I choose to balance time over precision (time of looking up populations), but the radius around East Lansing of 90 miles would definitely include 6 million people. Using the population tool, I found that 75 miles radius would include 6,795,008 people.

# Fraction Multiplication and Division Using Virtual Measurement Models

Session Presented at the National Council of Supervisors of Mathematics (NCSM) annual conference in April, 2015, in Boston, MA. Because it was NCSM, our participants were professional development facilitators and other teacher educators. We presented a similar session in February, 2015 at regional practitioner conference – we deliberately changed very little between the two sessions, so that we could talk to the teacher educators about what we had seen in the teachers’ interaction with manipulatives in the session

In this session we explored various web-based applets created to support conceptual understanding of fraction, multiplication, and division using spatial measurement models (length, area, volume). We also discussed how these applets can be used in professional development or classroom settings with students.  (View Session Site; Download Session Hand-out and Session Slides)

Overview of session:

We hoped to stimulate discussion focused on the complex interactions of:

• Students’ understanding of mathematical ideas, such as: measurement models, number lines, and fraction operations
• Relationships between: quantities and numbers / actions and operations
• Benefits and limitations of instructional models: manipulatives / applets

Exploration of physical manipulatives:

We provided participants with several physical tools, including rubber bands, linking cubes, WikiStix, and graph paper. We asked them to use the tools to make sense of representing fraction multiplication with discrete quantities and with continuous quantities.

Exploration of virtual manipulatives:

We asked the participants to interact with virtual manipulatives (http://tinyurl.com/STEM-NCSM2015) that we had created in an attempt to explore discrete versus continuous quantities in multiplication and division of fractions.

We created the applets specifically to support thinking about measurement dynamically and continuously because research has shown students see measurement as static and discrete most often, and yet dynamic and continuous experiences can support students develop flexible understandings of multiplication and division that may support them in understanding later covariational relationships.

Final discussion:

We asked participants to talk about: How can focusing on these ideas support students’ and teachers’ thinking about measurement and fractions? When are virtual manipulatives appropriate? When are physical manipulatives appropriate?

Stehr, E.M., Satyam, V.R., Smith, J.P., & Gilbertson, N. (2015, April). Fraction multiplication and division using virtual measurement models. Presentation to the 2015 Annual Meeting of the National Council of Supervisors of Mathematics, Boston, MA.

# Modeling 5 Practices – Monitoring

Attachment: Measurement Applets – Monitoring Sheet

In Spring 2015, I taught one section of a mathematics methods course for senior preservice K-8 teachers (PSTs). The seniors spent about 5 hours each week in a classroom – most were in the same classroom all year. In two large assignments, they created high-level tasks, wrote lesson plans, and then implemented their lesson plans. Throughout the process they used, as resources: Van de Walle (2013), Smith, Bill, and Hughes (2008), and Stein, Engle, Smith, and Hughes (2008). One requirement of the lesson plan was that it explained how they would incorporate the “5 practices” to lead a productive mathematics discussion.

My seniors struggled with anticipating and monitoring, so I developed an in-class task where they could experience the monitoring, selecting, sequencing, and connecting practices. I did the “anticipating” and created monitoring sheets. The seniors engaged in five tasks in pairs, using applets from https://www.msu.edu/~stemproj/ (links are found in the attached monitoring sheets) on the classroom computers. They were in five groups, so for each task one group monitored their classmates and four groups engaged in the task.

After the five tasks, the seniors had ten minutes to talk in groups about how they would select and sequence the strategies they had seen to support a particular measurement learning goal. Then each group led a (unfortunately too-brief) five-minute discussion to support their classmates in making connections.

The applets were designed by the Strengthening Tomorrow’s Education in Measurement research project to support K-8 students in confronting misconceptions about particular aspects of measurement. I helped design the applets, and (for most of them) wrote the code to create the applets. My seniors thus were able to talk deeply about measurement strategies as well as to experience the decision-making process around the 5 practices.

References

Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a Lesson: Successfully Implementing High-Level Tasks. Mathematics Teaching in the Middle School, 14(3), 132–138.
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.
Van de Walle, J. A., Karp, K. S., Lovin, L. H., & Bay-Williams, J. M. (2013). Teaching student-centered mathematics: Developmentally appropriate instruction for grades Prek–2 (Vol. 1). Boston, MA: Pearson.
Van de Walle, J. A., Karp, K. S., Lovin, L. H., & Bay-Williams, J. M. (2013). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Boston, MA: Pearson.

# Fraction Multiplication and Division using Virtual Measurement Models

Session Presented at the Math in Action annual conference in February, 2015, at Grand Valley State University in Allendale, Michigan. We presented a similar session in April, 2015 at a national conference (National Council of Supervisors of Mathematics) – we deliberately changed very little between the two sessions, so that we could talk to the teacher educators about what we had seen in the teachers’ interactions with manipulatives in the session

In this session we explored various web-based applets created to support conceptual understanding of fraction, multiplication, and division using spatial measurement models (length, area, volume). We also discussed how these applets can be used in professional development or classroom settings with students.  (Visit Session Site; Download Session Slides and Session Hand-Out)

Overview of session:

We hoped to stimulate discussion focused on the complex interactions of:

• Students’ understanding of mathematical ideas, such as: measurement models, number lines, and fraction operations
• Relationships between: quantities and numbers / actions and operations
• Benefits and limitations of instructional models: manipulatives / applets

Exploration of physical manipulatives:

We provided participants with several physical tools, including rubber bands, linking cubes, WikiStix, and graph paper. We asked them to use the tools to make sense of representing fraction multiplication with discrete quantities and with continuous quantities.

Exploration of virtual manipulatives:

We asked the participants to interact with virtual manipulatives (http://tinyurl.com/STEM-MIA2015) that we had created in an attempt to explore discrete versus continuous quantities in multiplication and division of fractions.

We created the applets specifically to support thinking about measurement dynamically and continuously because research has shown students see measurement as static and discrete most often, and yet dynamic and continuous experiences can support students develop flexible understandings of multiplication and division that may support them in understanding later covariational relationships.

Final discussion:

We asked participants to talk about: How can focusing on these ideas support students’ and teachers’ thinking about measurement and fractions? When are virtual manipulatives appropriate? When are physical manipulatives appropriate?

Stehr, E.M. & Gilbertson, N. (2015, February). Fraction multiplication and division using virtual measurement models. Paper presented at the Math in Action Conference, Grand Valley State University, Allendale, MI.

# Creating Technologies to Support Measurement

I worked with a student teacher who had been an undergraduate research assistant for the Strengthening Tomorrow’s Education in Measurement project, and was a recent graduate of Michigan State’s Teacher Education program. She is currently interning in a nearby classroom. She agreed, along with her mentor teacher, to design and implement a lesson with my support. Two redesigned applets resulted, and a third is possible. The link below connects to her lesson.

The student teacher had taught a lesson that incorporated van de Walle’s Crooked Path and Broken Ruler activities two weeks before the technology lesson (Van de Walle, 2013). She created crooked paths in her classroom using masking tape on the floor and on some tables. Students measured them, moving from path to path as they completed each measurement.

The student teacher felt that the STEM Jagged Path applets (see link above) would support the learning goals of her previous lesson by encouraging students to practice skills they had developed as well as deepening their earlier thinking. She asked all students to explore Activity 1, using Activity 3 as an extension for students who finished quickly.

The student teacher and I talked above another applet that could support students’ thinking. She said that her mentor teacher worked hard to encourage her students to confront misconceptions and change their thinking. We designed and created a third applet (shown in Activity 2, using the site link above) that would show the student two “wrong” ways to measure. We asked the students to explain what they thought was the thinking behind each method, and to explain which method they thought was correct – or if neither was correct, to explain what a correct method would be.

# Fractions as Lengths

Session presented at Math in Action annual conference February 22, 2014 at Grand Valley State University in Allendale, MI.

The main take-away from the session was that: If students are going to understand the number line in rich and meaningful ways, they should understand how numbers represent accumulated quantities of lengths. To support our participants in their understanding and future teaching, we created opportunities for them to explore connections between fraction operations and fraction representations using a length model.

Connections between measurement models and fraction operations

We reminded teachers that often an area model is used to represent fractions and fraction operations. We also brought an example of a word problem that connected fraction operations to a length model (as represented by lengths of string).

We asked participants to represent fractions as numbers, and then operations on fractions, using sentence strips (as a length model).

Stehr, E.M., Gilbertson, N., & Clark, D. (2014, February) Fractions as Lengths. Paper presented at the Math in Action Conference, Grand Valley State University, Allendale, MI.

# Filling the space in length measurement

Session presented at the Michigan Council of Teachers of Mathematics annual conference in Traverse City, MI. In this session we provided a variety of tasks to enhance student learning of length measure. We examined the meaning of length as an attribute and ways of quantifying length. We explored length measurement with a variety of tasks that can push us (and our students) to think about aspects of measurement, such as filling a space without leaving gaps or overlaps.

In this session, we provided stations, each with a different measurement task and set of physical tools. Tasks were created based on tasks designed by Strengthening Tomorrow’s Education in Measurement project, based on articles from Teaching Children Mathematics, or based on observed tasks from methods courses. Participants engaged in two or more stations, tracking the tools and ideas they used, and we finished with a discussion. The hand-outs and a brief description are provided below:

• Station01-Comparing Units – An illustration of three “wrong” strategies for measuring and comparing the lengths of paths is given. Students are asked to explain why or why not they agree with the measurement. Students could also be asked to explain the possible thinking behind each of the three strategies, and why it would or would not “work.”
• Station02-Measuring Paths – Fourth paths are given. Each path is partitioned in different ways. Students are asked to compare lengths of paths, to explain their reasoning, and how they chose measurement units.
• Station03-Measuring Height – Students in pairs compare their heights and indicate who is taller. They use non-standard units of measure to prove they are correct by measuring each other and comparing responses. They should each measure each other with two tools.
• Station04-Measuring Tools – Students choose an object to measure, and use a non-standard unit of measurement to measure the object. They write or draw their strategy and then write or draw their understanding of the meaning of length.
• Station05-Popsicle Comparison – Students measure two objects, each with a longer popsicle stick and shorter popsicle stick. The students then discuss any ideas they have or anything they notice about the measurements.
• Station06-Strange Rulers – In order to make sense of the important features of standard rulers, students use several “strange rulers” to measure a paperclip and a piece of string and make sense of which rulers “work” and which don’t.

Stehr, E.M., Gönülateş, F., & Nimtz, J.L. (2013, August) Filling the space in length measurement. Paper presented at the Michigan Council of Teachers of Mathematics Conference, Traverse City, MI.