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

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? Measurable attribute of a two-dimensional figure2 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

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. A B C
Investigation D
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 F E

31. A B C
Investigation D
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? 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

51. Banners
Complete the chart of the dimensions for each banner, the factor of change from the original, and the new area
Make a conjecture about how the change in the dimensions is related to the change in the area
Happy Birthday

52. Banners
Happy Birthday
Look at the banners where only one dimension is changing. What can you say about the change in area?
What happens to the area for the banners where both dimensions change by the same factor?
How do the areas change when both dimensions change, but not by the same factor?

53. Banners
Happy Birthday
What relationship do you see between how the dimensions of the figures change and the resulting change in the area?
Does this hold true when the dimensions change by fractional factors of change? Give an example to support your answer

54. Models for Banners
The same relationship that was just developed with numbers can also be seen using an area model
Using representations such as these is important in helping students visualize relationships they may fail to grasp when presented with only an algorithm

55. Model for Banners Original figure is a 6” by 4” rectangle
New rectangle has dimensions of 6” by 10”
The new rectangle contains two and a half original rectangles

56. Cakes It’s time to order a cake
The price of the cake is determined by the number of cubic inches you purchase
Investigate how changing the dimensions affects the number of cubic inches you will have to pay for

57. Cakes The original cake is 2” by 8” by 10”
Here are the other options for cake sizes:
2” x 4” x 10” 4” x 4” x 5”
4” x 8” x 10” 4” x 16” x 20”

58. Cakes
Use the chart on the Banners and Cakes handout to record the factor of change for each dimension and the new volume
How does the relationship between dimensional change and area explored in the previous activity relate to dimensional change and volume?

59. Cakes
The relationship between the factor of change in the dimensions of a three-dimensional figure is related to the resulting change in the volume
How do the banners and cakes activities work together to help students understand the relationships between two and three-dimensional figures?

60. Big ideas
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

61. Big Ideas
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

62. Reflecting on MS Measurement
In what ways do the objectives in the measurement strand for your grade contribute to students’ overall understanding and use of measurement concepts and skills?
What strategies from this professional development can you incorporate into your teaching?

63. DPI Mathematics Staff Everly Broadway, Leanne Barefoot Robin Barbour
Carmella Fair
Chief Consultant
Donna Thomas
Mary H. Russell
Johannah Maynor
Partners for Mathematics Learning is a Mathematics-Science Partnership Project funded by the NC Department of Public Instruction. Permission is granted for the use of these materials in professional development in North Carolina Partner school districts.
Partners
for Mathematics Learning

64. PML Writers Ana Floyd Jeane Joyner Rendy King Katherine Mawhinney
Gemma Mojica
Elizabeth Murray
Wendy Rich
Catherine Stein
Please give appropriate credit to the Partners for Mathematics Learning project when using these materials. Permission is granted for their use in professional development in North Carolina Partner school districts.
Jeane Joyner, Project Director

65. PML Consultants Amanda Baucom Julia Cazin Anna Corbett Gail Cotton
Ryan Dougherty
Tery Gunter
Kathy Harris
Joyce Hodges
Karen McCain
Vicki Moss
Kayonna Pitchford
Ron Powell
Susan Riddle
Judith Rucker
Shana Runge
Kitty Rutherford
Penny Shockley
Pat Sickles
Nancy Teague
Bob Vorbroker
Jan Wessell
Carol Williams
Stacy Wozny

66. Measurement