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.