CHAPTER 16 Measurement Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al. ©2012 John Wiley & Sons  

Focus Questions Why should measurement be included in school mathematics? What is the measurement process and why is it important? Why are concepts related to the unit of measurement important to develop? How is measuring one attribute (e.g., length) like measuring another attribute (e.g., area)? How can you use estimation to strengthen measurement skills? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Instructional programs from pre-k through grade 12 should enable all students to: Understand measurable attributes of objects and the units, systems, and processes of measurement. Apply appropriate techniques, tools, and formulas to determine measurements. NCTM (2000). Principles and Standards for School Mathematics. Master 16-1: The Measurement Standard Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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 the students in active learning. Master 16-2: Why Teach Measurement Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

How to Teach Measurement Children must measure frequently and often, preferably on real problems. Children must develop estimation skills with measurement in order to develop common referents. Children should encounter activity-oriented measurement situations by doing and experimenting rather than passively observing. Instructional planning should emphasize the important ideas of measurement that transfer or work across measurement systems. Master 16-3: The Measurement Process Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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 Master 16-3: The Measurement Process Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

The Measurement Process (cont’d) III. Compare the object to the unit. IV. Find the number of units. A. Counting B. Using instruments C. Using formulas V. Report the number of units. Master 16-3: The Measurement Process Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Identifying Attributes: Length Investigate concepts of length as it applies to length of objects, distance, perimeter, and circumference. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Identifying Attributes: Capacity FIGURE 16-3 Which container holds more seeds? Tell why you think the one you chose holds more Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Identifying Attributes: Weight/Mass Many activities may be set up for children to compare the weights of two objects. One challenging activity is to compare five identical containers, with lids secured and filled with different amounts of objects and to put them in order from lightest to heaviest. If the masses are not perceptually different, then this task requires multiple comparisons on the balance. FIGURE 16-4 Use a balance to show which rock weighs more. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Identifying Attributes: Area Ask the children which is different and to describe the difference. When children have some idea of conservation of area—that a region can be cut and rearranged without changing the area—this experience can be meaningful. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Identifying Attributes: Volume Challenge your students to see if they can figure out how many cubes it would take to fill a box. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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. Look at the direct comparison of these two angles: Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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; however, there are few other comparisons you can make without an instrument (thermometer). Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Identifying Attributes: Time You can begin describing the time of occurrence: we went to the gym yesterday, we have rug time in the morning, or we collected fall leaves in October. 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? Occurrence Duration Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Choosing a Unit 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Choosing a Unit (cont’d) 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Comparing an Object to a Unit and Finding the Number of Units 1. Counting Units 2. Using Instruments A. Ruler B. Clocks C. Protractors Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Comparing an Object to a Unit and Finding the Number of Units 3. Using Formulas Rectangle Parallelogram Triangle Trapezoid Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Reporting Measurements The last step in the measurement process requires the students to tell both the number and the unit. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Elementary Measurement Attributes Unit Examples Instruments Length inch, cm ruler Capacity cups, liters measuring cup Weight/Mass pounds, grams scale Area square inches ruler Volume cubic inches graduated cylinder Angle degrees protractor Temperature Celsius degrees thermometer Time minutes, days clock Master 16-4: Elementary Measurement Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Comparing Measurements Equivalences Conversions: To convert within the metric system, simply move the decimal in the same direction and count the number of moves. (i.e. 1.5 meters = 150 centimeters) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Estimating Measurements Estimating is the mental process of arriving at a measurement without the aid of measuring instruments. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Connecting Attributes Area and Shape Volume and Shape Perimeter and Area Volume and Surface Area Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Problem Stack the 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. Master 16-5: Paint Problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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. Master 16-5: Paint Problem Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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? Master 16-6: Calendars Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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. 31 28 31 30 31 30 31 30 30 31 30 31 Master 16-6: Calendars Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012