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

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.

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.

Measurement Instruction: A Sequence of Experiences

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.

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.

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.

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

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.

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

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.

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

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

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

Addressing Length Misconceptions Not aligning two objects when comparing them. Try this one Activity 19. 6 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.

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.

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

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.

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.

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.

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?

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.

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

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

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

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

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

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).

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

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

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.

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.

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