1. Chapter 16 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.

2. Guiding Questions 1.Why should measurement be included throughout the elementary mathematics curriculum? 2.What is the measurement process, and why is it important? 3.Why is it important to develop concepts related to a unit of measurement? 4.How is measuring one attribute (e.g., length) like measuring another attribute (e.g., area)? 5.How can you use estimation to strengthen measurement skills?

3. Why Teach Measurement? It provides many applications to everyday life. It can be used to help learn other mathematics. It can be related to other areas of the school curriculum. It engages students in learning.

4. How to Teach Measurement Children must measure frequently and often, preferably on real problems rather than on textbook exercises. Children must develop meaning for common referents (e.g., centimetre, metre) and numbers in order to apply them to estimates and measurements.

5. How to Teach Measurement Children should encounter activity-oriented measurement situations by doing and experimenting rather than passively observing. The activities should encourage discussion to stimulate the refinement of ideas and concepts. Instructional planning should emphasize the important ideas of measurement that transfer or work across measurement systems.

6. The Measurement Process I. Identify the attribute by comparing objects. A.Perceptually B.Directly C.Indirectly through a reference II.Choose a unit. A.Nonstandard B.Standard

7. The Measurement Process III. Compare the object to the unit by iterating the unit. IV. Find the number of units. A.Counting B.Using instruments C.Using formulas V. Report the number of units.

8. Identifying Attributes To measure with understanding children should know what attribute they are measuring. One of your first tasks when teaching measurement is to build an understanding of measurable attributes.

9. Identifying Attributes Three types of comparisons can build understanding of attributes: – Comparing two objects perceptually (they look the same or different). – Comparing two objects directly (they are placed next to each other). – Comparing two objects indirectly (a third object is used to compare objects).

10. Identifying Attributes: Length Investigate concepts of length as it applies to length of objects, distance, perimeter, and circumference.

11. Identifying Attributes: Capacity Capacity is the attribute that tells “how much a three-dimensional container can hold.” Which container holds more seeds? Explain why you think the one you chose holds more.

12. Identifying Attributes: Weight/Mass The weight/mass of two objects are compared perceptually by lifting the two objects. Children need opportunities to compare small and heavy with light and large objects. Children should learn that to find which is heavier, they must do more than look at the object. A balance scale can be used to show which rock weighs more. Weight and mass are different: mass is the amount of substance and weight is the pull of gravity on that substance.

13. Identifying Attributes: Area Area is an attribute of plane regions that can be compared by sight (perceptually) if the difference is large enough and the shapes similar enough. The first direct comparisons children make should be made with two regions, one of which fits within the other. For example regions B and C below.

14. Identifying Attributes: Volume Volume is defined as how much three- dimensional space something takes up. This is a challenging topic and should not receive much attention before grade 4 or 5. How many blocks will fill the box?

15. Identifying Attributes: Angle If an angle is considered a turning (such as the clock hands), then even young children can compare perceptually two angles (the amount of turning). Young children can also compare angles directly by comparing the amount of space the turn would make.

16. Identifying Attributes: Temperature Before introducing reading the thermometer, you can have children compare to see which of two objects is colder (or warmer). You can also talk about things (or times) that are hot or cold.

17. Identifying Attributes: Time Time is a very abstract attribute which can be measured with events. Two attributes of events are occurrence and duration. Occurrence: You can begin describing the time of occurrence. For example, we went to the gym yesterday, we have rug time in the morning, or we collected fall leaves in October. Duration: Children can tell which of two events takes longer (duration) if their lengths are greatly different. Does it take longer to brush your teeth or read a story?

18. Choosing a Unit After children have begun to develop a firm concept of an attribute through comparison activities, it is important to help them move onto more accurate ways of describing.

19. Choosing a Unit As children gain experience with measurement, they will develop an understanding of the following: 1. A unit must remain constant. 2. A measurement must include both a number and the unit. 3.Two measurements may be easily compared if the same unit is used. 4. One unit may be more appropriate than another to measure an object. 5. There is an inverse relationship between the number of units and the size of the unit.

20. Choosing a Unit (cont.) 6.Standard units are needed to communicate effectively. 7.A smaller unit gives a more exact measurement. 8.Units may be combined or subdivided to make other units. 9.Units must match the attribute that is being measured.

21. Comparing an Object to a Unit and Finding the Number of Units Three techniques are used to find the number of the units: 1. Counting Units 2. Using Instruments – Ruler – Scaled Instrument – Clocks – Protractors 3. Formulas

22. Comparing an Object to a Unit and Finding the Number of Units 3. Using Formulas – The skill of using formulas should be developed, but not at the expense of helping students build meaning for the formulas.

23. Reporting Measurements The last step in the measurement process requires the students to tell both the number and the unit.

24. Creating Objects Given The Measurement We have concentrated on measuring a given object. Equally important is creating an object of a given measurement. An example question is “Create a rectangle that is 8 square centimetres.”

25. Comparing Measurements Equivalences As you introduce new standard units, you should relate them to others For example, if you are introducing the millimetre, you could relate it to the centimetre by showing that it is smaller and that it is one-tenth of a centimetre.

26. Comparing Measurements (cont.) Conversions To change from one unit to another, children must know the equivalence or relation between the two units. This relies on the children knowing the relative size of the units and understanding that the smaller the unit the more it takes to represent the attribute.

27. Estimating Measurements Estimating is the mental process of arriving at a measurement without the aid of measuring instruments.

28. Estimating Measurements (cont.) When including measurement in your program, make it a natural part of measurement activities. 1.Encourage children to see if they can tell about how long or heavy the object is before they measure it. 2.Look for ways to include estimating in other subject areas. Such as asking children “About how far did you jump?” 3.Plan estimating activities for their own sake or use brief ones as daily openers for several weeks throughout the year.

29. Connecting Attributes Activities involving two attributes can help children see how the attributes are related or how one attribute does not depend on the other. For example: – Area and Shape – Volume and Shape – Perimeter and Area – Volume and Surface Area

30. For Class Discussion… The following slides show some activities connected to measurement. As you engage in the activities, think about how you might use or adapt this for your own classroom. Be sure to share your thinking around these activities with your classmates.

31. Problem Stack cuisenaire rods on a table so that the longest rests on the table and others are on top in order from longest to the shortest. Find the surface area of the structure that could be painted without moving any rods.

32. One possible solution: Explanation: I put two together. Each large face is 10 by 11 or 110 square units. The front and back would add to 220 square units. Both sides add to 20 and the top is 11, so the surface area is 251 square units. We would not count the bottom since the structure is resting on this part.

33. Early Calendars Why does February have only 28 days ? Here is a calendar showing the number of days in each month during the Roman era. J F M A M J J A S O N D 31 29 31 30 31 30 31 30 31 30 31 30 Describe any patterns you see. How is this calendar different from our current calendar?

34. Early Calendars The emperor Augustus changed the calendar twice. Here is the first change. J F M A M J J A S O N D 31 28 31 30 31 30 31 31 31 30 31 30 Here is the second change. J F M A M J J A S O N D 31 28 31 30 31 30 31 30 30 31 30 31

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