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.

Supporting spatial measurement tasks with technology

Session presented at the Michigan Council of Teachers of Mathematics annual conference in Traverse City, MI.

Download slides here.

In this hands-on session, we introduced several sample tasks that included use of computer applets to strengthen understanding of length and area concepts. Many students struggle with length and area, using only pen and paper or physical manipulatives. In this session, we provided materials to support their learning (that will be available for use during and after the session).

We asked participants to use their devices to work through one sample lesson (Area of Rectangular Regions) including Launch / Explore / Summarize components. We gave participants two sample lessons that they could in their classrooms, along with applets designed and created by Strengthening Tomorrow’s Education in Measurement project.

Stehr, E.M., Gönülateş, F., & Siebers, K. (2013, August) Supporting spatial measurement tasks with technology. Paper presented at the Michigan Council of Teachers of Mathematics Conference, Traverse City, MI.

GeoGebra – Malfatti’s Problem

In Axiomatic Geometry, I and a partner explored Malfatti’s problem (see references below). I created a GeoGebra worksheet to support the presentation we later gave. I share some details here.

GeoGebra-Malfatti-02

Malfatti originally asked this question is 1803 and the question, “How to arrange in a given triangle three non-overlapping circles of greatest area?” remained open until 1994.

GeoGebra-Malfatti-03

As far back as 1384, in a manuscript of Italian mathematician Gilio de Cecco da Montepulciano a problem is presented of how to construct 3 circles, inside a triangle, that are tangent to each other and tangent to 2 edges of triangle. The same problem was posed by 18th century Japanese mathematician Ajima Naonubu. In 1803, Malfatti refined the problem to include minimizing the amount of area in the triangle “wasted”, i.e., constructing the largest circles possible. In 1826, Steiner (that sly old fox) developed a construction of three circles, tangent to each other and touching 2 edges of the triangle.  This was a solution to the problems posed by Montepulciano and Naonubu, but not Malfatti.  Stiener demonstrated 3 circles in a triangle but did not minimize the area “wasted”. In 1930, Lob and Richmond showed Malfatti’s circles do not produce the largest circles (i.e., minimal waste).  Optimal circles are obtained using greedy algorithm. In 1967, Goldberg showed that the Mafatti circles will never produce greatest area and the Lob and Richmond ‘greedy algorithm’ always produces the largest total area.  In 1994, Zalgaller and Los classified all the different ways 3 circles can be arranged in a triangle and developed a formula for determining which method of arranging the circles is optimal for a given triangle.  They also proved that there are 14 combinations of circles in a triangle that will yield optimal area.

GeoGebra-Malfatti-04

Essentially, if there is room for a circle to ‘grow’ then the system is NOT rigid.  The two examples on the left are rigid, the one on the right is not, the middle circle could change radius or ‘grow’.

GeoGebra-Malfatti-05\

Zalgaller and Los proved the result systematically, reducing all possibilities to 14 combinatorially, non-identical, rigid configurations of three circles in a triangle. See cases below.

GeoGebra-Malfatti-06

Malfatti’s Marble Problem asks:

  • How can we arrange in a given triangle three non-overlapping circles of greatest area?
  • For an Isosceles triangle, the first two circles are drawn as K1 and K2. Then we choose K3a or K3b depending on the relationship between sin(alpha) and tan(beta/2).
  • Play around with the triangle and tell me –
    • Do we use K3a or K3b when sin(alpha) > tan(beta/2)?
    • Do we use K3a or K3b when sin(alpha) < tan(beta/2)? W
    • hat do we do when sin(alpha) > tan(beta/2)?
    • Also, are the three circles always symmetric for an Isosceles triangle?

GeoGebra-Malfatti-01

References

Andreatta, M, Bezdek, A, & Boronski, J (2010) The Problem of Malfatti: Two Centuries of Debate. The Mathematical Intelligencer.

Coolidge, J. L. (1923). Some Unsolved Problems in Solid Geometry. The American Mathematical Monthly, 30(4), 174-180.

Goldberg, M. (1967). On the original Malfatti problem. Mathematics Magazine, 40(5), 241-247.

Zalgaller, V. A., & Los’, G. A. (1994). The Solution Of Malfatti’s Problem. Journal Of Mathematical Sciences, 72(4), 3163-3177.

Presentation Reference

Humes, W.R. & Stehr, E.M. (2013). Malfatti’s Marble Problem. Presentation prepared for final project in D.S. Durosoy’s MTH 432: Axiomatic Geometry course.