Chapter 16Chapter 16

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

 

Guiding Questions1. 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?

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.

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.

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.

The Measurement Process

I. Identify the attribute by comparing objects.A. PerceptuallyB. DirectlyC. Indirectly through a reference

II.Choose a unit.A. NonstandardB. Standard

The Measurement Process

III. Compare the object to the unit by iterating the unit.

IV. Find the number of units.A. CountingB. Using instrumentsC. Using formulas

V. Report the number of units.

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.

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).

Identifying Attributes: Length

Investigate concepts of length as it applies to length of objects, distance, perimeter, and circumference.

Identifying Attributes: CapacityCapacity 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.

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.

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.

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?

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.

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.

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 thegym 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?

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.

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.

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.

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 Units2. Using Instruments– Ruler– Scaled Instrument– Clocks– Protractors

3. Formulas

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.

Reporting Measurements

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

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.”

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.

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.

Estimating Measurements

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

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.

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

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.

ProblemStack 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.

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.

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?

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 D31 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 D31 28 31 30 31 30 31 30 30 31 30 31

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Copyright © 2010 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.