Using Clicker Items to Deepen Understanding of Measurement Concepts

1. Using Clicker Items to • Deepen Understanding of Measurement Concepts • Foster Desirable Habits of Mind

2. Logging In Procedure • 1. Turn-on your clicker • 2. Wait until it says “Enter Student ID”(Enter your 5-digit ID) • 3. The screen should display “ANS”

3. Suppose p kilometers is equal to q feet, where p and q are positive numbers. • Which statement is correct? • p > q • p <q • p = q • None of the above Item 1

4. Suppose p kilometers is equal to q feet, where p and q are positive numbers. • Which statement is correct? • p > q • p <q • p = q • None of the above Revote 1

5. Suppose p kilometers is equal to q feet, where p and q are positive numbers. • Which statement is correct? • p > q • p <q • p = q • None of the above Fact: 1 km  0.62 mile; 1 mile = 5280 feet Procedure: 1 km  0.62 x 5280 feet = 3273.6 feet HoM: Explore and generalize a pattern

6. Concept: Conservation (recognizing smaller units will produce larger counts) HoM: Explore and generalize a pattern

7. Concept: Conservation (recognizing smaller units will produce larger counts) 1 wav ? wavs ? 1 arro ?arros

8. Concept: Conservation (recognizing smaller units will produce larger counts) 1 wav  3.7 wavs 1 arro  7arros Concept: Measurement involves iterating a unit

9. Concept: Conservation (recognizing smaller units will produce larger counts) 1 wav  3.7 wavs 1 arro  9.6 arros Concept: Measurement involves iterating a unit Concept: Units must be consistent Concept: Inverse relationship between the size of a unit and the numerical count

10. True or False: If the volume of a rectangular prism is known, then its surface area can be determined. Item 2

11. True or False: If the volume of a rectangular prism is known, then its surface area can be determined. Revote 2

12. True or False: If the volume of a rectangular prism is known, then its surface area can be determined. Concept: Volume = Length  Width  Height HoM: Reasoning with Change and Invariance

13. “[S]ome students may hold the misconception that if the volume of a three-dimensional shape is known, then its surface area can be determined. • This misunderstanding appears to come from an incorrect over-generalization of the very special relationship that exists for a cube.” • (NCTM, 2000, p. 242)

14. True or False: If the surface area of a sphere is known, then its volume can be determined. Item 3

15. True or False: If the surface area of a sphere is known, then its volume can be determined. Revote 3

16. True or False: If the surface area of a sphere is known, then its volume can be determined. Concept: A = 4 r2 V = 4/3 r3 HoM: Reasoning with Formulas

17. True or False: If the area of an equilateral triangle is known, then its perimeter can be determined. Item 4

18. True or False: If the area of an equilateral triangle is known, then its perimeter can be determined. Revote 4

19. True or False: If the area of an equilateral triangle is known, then its perimeter can be determined. CU: Area = ½LH = ½L [L2 – (L/2)2] 0.5 = ½L (0.75L2)0.5 = ½L (0.75)0.5 L  0.433L2 L L H L/2 L HoM: Reasoning with Relationships

20. True or False: As we increase the perimeter of a rectangle, the area increases. Item 5

21. True or False: As we increase the perimeter of a rectangle, the area increases. Revote 5

22. True or False: As we increase the perimeter of a rectangle, the area increases. HoM: Seeking causality

23. True or False: As we increase the perimeter of a rectangle, the area increases. 16 m 2 m 4 m 8 m Concept: Perimeter = 2L + 2W ;Area = LW HoM: Seeking counter-example

24. True or False: As we increase the perimeter of a rectangle, the area increases. 20 m 0.5 m 1 m 2 m 16 m 4 m 12 m 8 m Concept: Perimeter = 2L + 2W ;Area = LW HoM: Reasoning with change and invariance

25. “While mixing up the terms for area and perimeter does not necessarily indicate a deeper conceptual confusion, it is common for middle-grades students to believe there is a direct relationship between the area and the perimeter of shapes and this belief is more difficult to change. • In fact, increasing the perimeter of a shape can lead to a shape with a larger area, smaller are, or the same area.” • (Driscoll, 2007, p. 83)

26. Consider this two-dimensional figure: 4 cm 7 cm 10 cm Note: Each corner is a right angle.

27. Consider this two-dimensional figure: Which measurement can be determined? Area only Perimeter only Both area and perimeter Neither area nor perimeter 4 cm 7 cm 10 cm Note: Each corner is a right angle. Item 6

28. Consider this two-dimensional figure: Which measurement can be determined? Area only Perimeter only Both area and perimeter Neither area nor perimeter 4 cm 7 cm 10 cm Note: Each corner is a right angle. Revote 6

29. 4 cm 7 cm 10 cm HoM: Reasoning with Change and Invariance

30. Consider this two-dimensional figure: 4 m 3 m 10 m Which measurement can be determined? Area only Perimeter only Both area and perimeter Neither area nor perimeter Note: The two horizontal lines are parallel. Item 7

31. Consider this two-dimensional figure: 4 m 3 m 10 m Which measurement can be determined? Area only Perimeter only Both area and perimeter Neither area nor perimeter Note: The two horizontal lines are parallel. Revote 7

32. Consider this two-dimensional figure: 4 m 4 m 4 m 4 m 4 m HoM: Reasoning with Change and Invariance Note: The two horizontal lines are parallel.