1. 302B Technology-Based Lessons “When technology tools are available, students can focus on decision-making, reflection, reasoning, and problem-solving” (NCTM, 2000, p.24). National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Authors.

2. Type 1 Lessons* The instructions require that students identify relationships and make conjectures. The instructions specify both what is to be investigated and how it is to be investigated. Students move sequentially through the instructions with little or no decision-making required. If the lesson is implemented successfully, students will have used similar methods and made similar conjectures. *McGraw, R., & Grant, M. (2005). Investigating mathematics with technology: Lesson structures that encourage a range of methods and solutions. In W. Masalski (Ed.), Technology-supported mathematics learning environments. Reston, VA: National Council of Teachers of Mathematics.

3. Type 1 Lessons [Students are given instructions for constructing a parallelogram and measuring lengths and angles.] Measure the lengths of the sides of the parallelogram. Measure each angle of the parallelogram. Study the side lengths and opposite and adjacent angle measures as you change the size and shape of the parallelogram. Make several conjectures about the relationships among side lengths and among angle measures of parallelograms.

4. Type 2 Lessons* The instructions require that students identify relationships and make conjectures. Students play a role in determining what to investigate and how to investigate it. Mathematical reasoning and decision-making are required throughout the lesson. If the lesson is implemented successfully, students will have used a variety of methods and made a variety of conjectures. *McGraw, R., & Grant, M. (2005). Investigating mathematics with technology: Lesson structures that encourage a range of methods and solutions. In W. Masalski (Ed.), Technology-supported mathematics learning environments. Reston, VA: National Council of Teachers of Mathematics.

5. Type 2 Lessons [Students have previously constructed and saved various quadrilaterals.] Select a quadrilateral. Drag vertices and sides to change the size and shape of the quadrilateral. What conjectures can you make about the properties of the quadrilateral? Feel free to construct midpoints, diagonals, and parallel and perpendicular lines; you could also measure lengths, angles, or areas. Do your conjectures seem to hold for other quadrilaterals? Why or why not?

6. Goals for 302B Lesson(s) Internal representations of quadrilaterals Knowledge of characteristics of quadrilaterals Ability to investigate shapes with computer software Ability to make and test conjectures Necessary and sufficient conditions for definitions Class inclusion within the quadrilateral family

7. The Lesson Students work in computer lab w/ handout Whole-class discussion Develop a list of the characteristics of each quadrilateral Select defining characteristics from among those listed Discuss class inclusion

8. Investigating Implementation Pre- and post-tests Observations of implementation Instructor feedback

9. Tools for Analyzing Instruction Analyzing the “type” of lesson Analyzing the content and structure of teacher-student and student-student discourse

10. Questions for Reflection* How much time do students spend trying to understand and follow directions versus trying to identify patterns and find relationships? How much time do students spend reasoning about the mathematics versus trying to get the technology to work? How much variation is there in the conjectures that students make? How much time do students spend testing, refining, and attempting to prove their conjectures? How much time do students spend waiting for the teacher to help them with the technology? How much time does the teacher spend discussing mathematics and methods for investigating mathematics with students versus helping students interpret and follow directions? *McGraw, R., & Grant, M. (2005). Investigating mathematics with technology: Lesson structures that encourage a range of methods and solutions. In W. Masalski (Ed.), Technology-supported mathematics learning environments. Reston, VA: National Council of Teachers of Mathematics.