1. CHAPTER 19 Developing Measurement Concepts
Elementary and Middle School Mathematics Teaching Developmentally
Ninth Edition
Van de Walle, Karp and Bay-Williams
Developed by E. Todd Brown /Professor Emeritus University of Louisville

2. Big Ideas
Measurement involves a comparison of an attribute of an item or situation with a unit that has the same attribute.
Estimation of measures and the development of benchmarks for frequently used unit of measure help students increase their familiarity with units, preventing errors and aiding in the meaning use of measure.
Measurement instruments (e.g. rulers) group multiple units so that you do not have to iterate a single unit multiple times.
Area and volume formulas provide a method of measuring these attributes by using only measures of length.
Area, perimeter, and volume are related.

3. The Meaning and Process of Measuring
Measurement is: A number that indicates a comparison between the attribute of the object being measured and the same attribute of a given unit of measure. Measurement means that the attribute being measured is filled, covered, or matched with a unit of measure with the same attribute.

4. Measurement Instruction: A Sequence of Experiences

5. Why use Nonstandard Units?
Focus directly on the attribute being measured- irregular shapes can be measured with square tiles or circle counters.
Avoids conflicting objectives in introductory lessons- lesson on what it means to measure area.
Provide a rationale for using standard units. Standard units have more meaning after students have experienced measuring with their own collection of nonstandard units.
Examples of nonstandard for length- Giant footsteps, measuring ropes, drinking straws, connecting cubes, and paper clips.

6. Recording Measurements with Nonstandard Standard Units to find Length
Questions to ask: How did you get your measurement? Did students who measured with the same unit get the same answers? Focus on the value of lining units up end to end. Discuss what happens if you overlap units, have a gap in units, or don’t follow a straight line.

7. Three Broad Goals for Teaching Standard Units of Measure
Familiarity with the unit.
Ability to select an appropriate unit.
Knowledge of the relationship between units.

8. Goal 1. Familiarity with the unit Try this one Activity 19
Goal 1. Familiarity with the unit Try this one Activity 19.2 Personal Benchmarks
Distance around your waist, neck, wrist Height from floor to waist, floor to shoulder, floor to head Graph data and explore- Which measures are similar, about double? Can you find a personal benchmark, 1cm? 10cm?
Materials- metric measure Directions: Measure about how long: Foot Stride Hand Span Width of your finger

9. Goal 2. Ability to select an appropriate unit
Should the room be measured in inches or feet? Should the concrete blocks be weighed in grams or kilograms? To cut a piece of molding do I measure to the nearest inch?
To determine how many 8-foot pieces of molding to buy do I measure to the nearest foot? Knowing the size of the unit and the level of precision are important to meet this goal.

10. Goal 3. Knowledge of the relationship between units
Relationships between customary or metric units are conventions. Therefore students must be told and then have experiences to reinforce the conventions. Avoid mechanical rules and create conceptual, meaningful methods for conversion.
Instructionally- begin with common items and use those measures as references and benchmarks.
Doorway- 2 meters
Doorknob- 1 meter from floor
Sack of flour- 5 pounds
Paper clip- a gram

11. Role of Estimation and Approximation
Measurement estimation is a process of using mental and visual information to measure without using measurement instruments.
Focus on the attribute being measured and the measuring process
Provides an intrinsic motivation for measurement activities.
Helps develop familiarity with units.
Use of benchmarks to make an estimate promotes multiplicative reasoning.

12. Strategies for Estimating Measurement
Where in this picture do you see
Benchmarks or referents
Chunking or using subdivision
Iterate units

13. Try one Activity 19. 4 Estimation Quickie
Materials- an object like a box, pumpkin, a painting on the wall or a person. Directions- each day select a different attribute or dimension to estimate. Ask students to estimate attributes like height, circumference, weight, volume, and surface area. Review seven teaching tips on page 460 in the text

14. Common Misconceptions about Length Measurement
Measuring from the wrong end of the ruler or beginning at 1 rather than 0
Counting the hash marks rather than the space (units)
Not aligning two objects when comparing them

15. Addressing Length Misconceptions
Not aligning two objects when comparing them. Try this one Activity Longer, Shorter, Same Materials- objects of various lengths and some the same lengths Directions- Identify a “target” object and ask students to find items that are shorter, longer, and the same. Suggest they set the objects in order from shortest to longest.

16. Physical Models and Length Units
Four important principles of iterating units of length:
Units must be of equal length or you cannot iterate them by counting.
Units must align with the length being measured or a different quantity is measured.
Units must be placed without gaps or a part of the length is not measured.
Units must be placed without overlaps or the length has portions that are measured more than one time.

17. Area- a measure of two-dimensional space inside a region
Comparison activities
help students distinguish between size and shape, length and other dimensions.
Using units of area

18. Try this one Activity 19.15 Two-Piece Shapes
Materials: cut a number of rectangles of the same area about 3 inches by 5 inches
Directions- Each pair of students
will need 6 rectangles. Ask them
to fold on the diagonal and make
two identical triangles. Ask them
to rearrange the triangles into
different shapes. Only sides the
same length can be matched up.

19. Using Physical Models of Area Units
Students need multiple opportunities to “cover the surface”. Index cards Square tiles Cuisenaire rods Encourage students to wrestle with partial units.

20. Using Physical Models of Area Units
Goal- apply students’ developing concepts of multiplication to the area of rectangles. First row thought of as a single unit that is replicated to fill the rectangle. Grids are the “area rulers.” It lays out the units for you.

21. Try this one Activity 19.19 What’s the Rim
Perimeter is a length measure of the distance around a region. Perimeter- good hint Materials- identify objects to measure Tools- rulers, cash register tape, non-stretching string Measure the actual amount and record noting the unit.
Discuss-
Methods used to measure the objects
Comparison of common items measured
How they determined what measurement tool to use?

22. Developing Formulas for Area
Overemphasis on formulas with little or no conceptual background results in misconceptions
Confusing linear and square units
2. Difficulty in conceptualizing the meaning of height and base
Heights on two-dimensional figures are not always measured along an edge.

23. Areas- Rectangles, Parallelograms, Triangles and Trapezoids
Transform a parallelogram into a rectangle
Transform a triangle into a parallelogram Transform a trapezoid into a parallelogram

24. Circumference and Area of Circles
Circumference- distance around or perimeter Diameter- a line through the center joining two points on the circle Development of the formula

25. Volume and Capacity
Measures of the “size” of three-dimensional and/or the capacity of a container
Comparison activities
fill a container with a liquid and pour that amount into a comparison container
Capacity Sort – sort collection of containers into “holds more, less, or same” around a target container
Fixed Volume: Comparing Prisms –students build rectangular prisms with centimeter cubes to compare surface area
Tools for measuring volume- solid units and containers

26. Developing formulas for Volume and Common Shapes
Models, precise language, and illustrations will guide the development and understanding of the formulas.

27. Area and Volume Formulas
Parallelograms are rectangles that have been shifted to make slanted sides. Areas of both: B × h
Triangles are halves of parallelograms. Area: 1∕2 B × h

28. Area and Volume Formulas cont.
3. Trapezoids are also related to parallelograms with the same height. They have a short and a long base. Area: 1∕2 (B × h) + 1∕2 (b × h) 4. The area of the circle is found by arranging small sectors of the circle to create a parallelogram. Base is 1∕2 the circumference and height is the radius. Area: 1∕2 (2πr) (r) or πr2 5. For volumes of cylinders (and prisms) find cubes that fit on the base (same as the area of the base) and multiply by the height. Volume: B × h or area of base × height 6. Cones (and therefore pyramids) are 1∕3 the volume of the corresponding cylinder (prism).

29. Weight- measure of the pull or force of gravity on an object Mass- the amount of matter in an object and a measure of the force needed to accelerate it
Comparison activities
Hold an object in each hand and feel the pull
Pan balance (Mass) or Spring scale (Weight)
Units of weight or mass
Non-standard large paper clips, wooden blocks, metal washers, coins
Standard gram or ounce weights

30. Measuring Angles- the spread of the angle’s rays
Comparison activities
Trace one angle and place the other angle over it to compare (use angles with varied ray lengths)
Use a smaller angle (A Unit Angle) to measure a larger angle

31. Time- duration of an event from beginning to end
Clock reading—suggested approach
Begin with one-handed clock- use approximate language- “It’s about 7 o’clock”
Discuss what happens to the big hand as the little hand goes from one hour to the next
Use two clocks, one with hour hand only and one with two hands
Teach time in 5-minute intervals- point to 4 and say “it’s about 20 minutes after the hour”
Predict the reading on a digital clock when shown an analog and vice versa
Relate time after the hours to the time before the next hour.
Discuss the issue of A.M. and P.M
Relate the time after the hour to the time before the next hour.

32. Elapsed time Solving a problem involving conversion of one measure of time to another
Solving problems involving addition and subtraction of time intervals.
Empty number line
is a computational
model.

33. Money Money ideas and skills required in primary grades
Recognizing coins
Identifying and using values of coins
Counting and comparing sets of coins
Creating equivalent coin collections
Selecting coins for a given amount
Making change
Solving word problems involving money