Measurement

1. Measurement

2. Big Ideas in Measurement • The attribute to be measured determines the unit and the tool • Measurements are estimates; the more precise the tools/units, the closer one can get to the actual measure • Measurements are accurate to the extent that the appropriate units/tools are used properly

3. Big Ideas in Measurement • Perimeter/circumference and area of 2-D figures are related to surface area and volume of 3-D figures • Formulas are derived from the measures of the attributes and relationships of 2-D and 3-D figures

4. Covering and Comparing • Make a conjecture about which figures have the smallest and largest perimeters • Make a conjecture about which figures have the greatest and least areas • After recording your conjectures, find the perimeter and area of each figure using the measuring tools provided • Describe your process in words

5. Discussion Questions • What tool(s) did you use to measure? • Why did you choose a particular tool? • How did you measure the perimeter of each figure? • How did you measure the area of each figure?

6. Nonstandard Units of Measure What were some of the challenges of using nonstandard measuring tools, such as the lima beans? • Measuring around edges • Units are not always the same size • Units may leave gaps when covering regions

7. Nonstandard Units of Measure • Compare your measures for the perimeter and area of Figure B with someone who used a different tool • What problems arose when comparing these measurements?

8. Compensatory Principle When measuring, how does the size of the unit affect the number of units used?

9. Compensatory Principle The smaller the unit used to measure an attribute, the more of those units it will take • It takes many baby steps to equal one GIANT step and vice versa • This idea leads to unit conversions within and between systems

10. Compensatory Principle How can a simple activity such as the baby steps/giant steps activity help students understand the concept of unit conversions?

11. Perimeter and Area

12. Covering and Comparing • Which figures have the smallest and largest perimeter? • Which figures have the greatest and least area? • How do the areas of figures A and C relate to one another? What about the perimeters? • What do you notice about the measures of Figures A and B?

13. Figure D Area = 9 sq cm Perimeter = 12 cm Figure E Area = 4.5 sq cm Perimeter = 10.5 cm What do you notice?

14. What is Perimeter? • Total outer boundary of a figure • It is important to distinguish between the idea of perimeter as the distance around and the process used to find the perimeter

15. What is Area? 2 5 • Measurable attribute of a two-dimensional figure • Number of non-overlapping units that cover or are contained in the interior of a figure

16. The Process of Measuring Attributes, Units, and Tools

17. Process of Measuring • Select an attribute of something you want to measure • Choose an appropriate unit of measure • Determine the number of units by using a measuring tool

18. Interesting Objects • Look around the room and select an object with attributes you would like to measure • Using the handout, list all of the measurable attributes of your object • Now list the unit and tool you would use to measure each attribute

19. Attributes • What are some of the measurable attributes of the objects that were collected? • How could these attributes be classified?

20. Weight and Mass • How can mathematics teachers and teachers of science work together to help students be clear on these attributes? • To assist in discussions, an article by Anita Bowman clarifying concepts of weight and mass is included on the project CD • The article is also posted on Partners website http://partners.mrooms.org/

21. Measurement Components • Units – the type of units used to measure depends upon the attribute • Unit Iteration – the repetition of a single unit

22. Unit Iteration • Use one tube of lipstick, chap stick, or a paper clip to measure the length of your table or desk • What strategies are helpful in measuring accurately and keeping track of the number of units?

23. Standard Measuring Units and Tools • What are some problems with iterating a single unit each time as a way to measure an object? • How do these problems affect decisions on the tools with which we choose to measure?

24. Formalizing Formulas A = bh

25. Formulas What are formulas? • Formulas summarize relationships • They generalize patterns Volume is area of base x height

26. Why do we use formulas? • Formulas help us to determine indirectly some measure that may otherwise be difficult to obtain • Utility and Efficiency

27. Consider the following: • Having adequate time to teach all of the mathematical concepts from the curriculum is an issue • What is the advantage of having students derive formulas rather than just memorize them?

28. Generating Formulas How can students use their knowledge of area of a rectangle to generate formulas for the area of parallelograms, triangles, and trapezoids? 7 cm 3 cm

29. Developing Formulas “Whenever possible, students should develop and apply formulas meaningfully through investigation rather than simply memorize them.” Principles and Standards for School Mathematics, 2000, p.244

30. B A C Investigation • Identify the midpoint of the longer edge of a 3 x 5 index card • Label the card using letters indicated above • What is the area of each of these figures? •Triangle ABF •Parallelogram BCEF •Trapezoid BCDF • Rectangle ABEF D F E

31. B A C Investigation • How did you begin to think about the areas? What is the area of each figure? • What mathematical relationships do you know that helped you to find the areas? • Which area was easiest to determine? • Are there alternative ways to find the area of the figures? D F E

32. Understanding Formulas “Students are better served when they understand the formulas and their origins and can reason about the appropriateness of their choice of formulas and calculations.” • What evidence can you give to support this statement? Chapin & Johnson, 2006, p. 286

33. Applying Formulas When applying formulas, students need to know: • How to interpret the formula as it applies to the problem solving situation • The meaning of the variables/attributes • How to substitute appropriate values for the variable • How to trace units through the process

34. Paper Cylinder Task • Make 2 cylinders out of standard sheets of paper • One should have a height of 8½” and the other should have a height of 11” • Tape the seam to close the cylinders

35. Paper Cylinders • Which cylinder has the greater surface area or are the surface areas the same? • Does it make a difference if the bases are included? • Why or why not? • How does knowing the radius help you?

36. Paper Cylinders • Considering the same cylinders, which cylinder has the larger volume or are the volumes of the two cylinders the same? • How do you know? • Which would affect the volume more - adding an inch to the height or to the radius of the base of the same figure?

37. Paper Cylinders • How might a student respond to the questions about the surface areas? The volumes? • What kinds of knowledge must students possess to be able to answer these questions? • Do students have to know a formula to be able to answer the questions?

38. Common Difficulties • Common difficulties for students in applying formulas can arise from overemphasizing formulas or jumping prematurely to formulas to determine measurements • Formulas can be efficient but they can mask what is being measured

39. Common Difficulties • Formulas are definitions; certain tasks require an understanding of how formulas work • Without concrete experiences students may find it difficult to conceptualize the meaning of height, volume, and other geometric terms

40. Estimation, Precision, and Accuracy

41. Why is estimation important? • Helps students internalize measurement concepts • Helps verify if solutions are reasonable

42. Which is Which? Which target represents accuracy and which represents precision? Why?

43. What is precision? • Measure of the extent to which individual measurements of the same quantity or attribute agree • The degree to which further measurements or calculations show the same results The cluster above is considered precise since all arrows struck close to the same spot. The placement of the arrows is precise, though not necessarily accurate

44. What is accuracy? • How correctly a measurement has been made • Accuracy is the degree of veracity while precision is the degree of reproducibility • Accuracy describes the closeness of arrows to the bulls eye at the target center

45. What affects accuracy? • Person doing the measuring • Measuring tool • Continuous vs. discrete quantities — measurement of a continuous quantity (e.g. length) is always an approximation

46. What difficulties do students have with accuracy? • Interpreting the subunits on a measuring device • Iterating units inaccurately, ignoring the continuous nature of the attribute • Inaccurate measurement devices Two measurements can both be accurate, but one can be more precise

47. Two-dimensional Measurements of Three-dimensional Figures

48. Making Connections • Students need opportunities to reason about and explore how 2-D measures combine to create surface area and volume

49. Banners and Cakes You are planning a surprise birthday party for your friend. You need to create a “Happy Birthday” banner. In trying to keep the cost of the party reasonable, you need to investigate several sizes for the banner. The original size of the banner paper is 4ft by 6ft. Happy Birthday

50. Banners Happy Birthday Using square tiles, investigate how much square footage of paper you will need if you change the dimensions of the banner to: 2ft by 6ft 4ft by 12ft 2ft by 12ft 8ft by 12ft 8ft by 18ft 4ft by 18 ft