# Standard Units of Measurement

__Units of Measurement__

Revised by Grace Ma and Jessica Jang (UCIrvine, Summer 2010)

by Rens de Bruijn, Jane Somboonsup, Jenny Banian, Jenny Nisley, Ashley Riker, and Heidi Adams (UCIrvine, August 2008)

## Contents |

## BACKGROUND

- 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 (Van de Walle, 2010).
- Units of measure are used to determine a numeric relationship, or the measurement, between what is measured and the unit.
- The conceptual prerequisites for measurement estimation fall into two categories—logical reasoning processes and knowledge of specific measurement concepts (Towers & Hunter, 2010).
- Many traditionally established instructional approaches of measurement have left students' understanding of length underdeveloped. This is because a link is never made between the procedures and concepts of measurement (Barrett & Clements, 2003).
- Instruction of measurement often starts with units, without discussing contexts and giving students the opportunity to practice measuring until later.

- National Council of Teachers of Mathematics (2000) expect children to understand units of measurement by second grade, but studies have shown that this is unrealistic (Reece & Kamii, 2001).

## STANDARDS

### Kindergarten

**1.0 Students understand the concept of time and units to measure it; they understand that objects have properties, such as length, weight, and capacity, and that comparisons may be made by referring to those properties:**- 1.1 Compare the length, weight, and capacity of objects by making direct comparisons with reference objects (e.g., note which object is shorter, longer, taller, lighter, heavier, or holds more).
- 1.2 Demonstrate an understanding of concepts of time (e.g., morning, afternoon, evening, today, yesterday, tomorrow, week, year) and tools that measure time (e.g., clock, calendar).
- 1.3 Name the days of the week.
- 1.4 Identify the time (to the nearest hour) of everyday events (e.g., lunch time is 12 o’clock; bedtime is 8 o’clock at night).

**2.0 Students identify common objects in their environment and describe the geometric features:**- 2.1 Identify and describe common geometric objects (e.g., circle, triangle, square, rectangle, cube, sphere, cone).
- 2.2 Compare familiar plane and solid objects by common attributes (e.g., position, shape, size, roundness, number of corners).

### 1st Grade

**1.0 Students use direct comparison and nonstandard units to describe the measurements of objects:**- 1.1 Compare the length, weight, and volume of two or more objects by using direct comparison or a nonstandard unit.
- 1.2 Tell time to the nearest half hour and relate time to events (e.g., before/after, shorter/longer).

**2.0 Students identify common geometric figures, classify them by common attributes, and describe their relative position or their location in space:**- 2.1 Identify, describe, and compare triangles, rectangles, squares, and circles, including the faces of three-dimensional objects.
- 2.2 Classify familiar plane and solid objects by common attributes, such as color, position, shape, size, roundness, or number of corners, and explain which attributes are being used for classification.
- 2.3 Give and follow directions about location.
- 2.4 Arrange and describe objects in space by proximity, position, and direction (e.g., near, far, below, above, up, down, behind, in front of, next to, left or right of).

### 2nd Grade

**1.0 Students understand that measurement is accomplished by identifying a unit of measure, iterating (repeating) that unit, and comparing it to the item to be measured:**- 1.1 Measure the length of objects by iterating (repeating) a nonstandard or standard unit.
- 1.2 Use different units to measure the same object and predict whether the measure will be greater or smaller when a different unit is used.
- 1.3 Measure the length of an object to the nearest inch and/or centimeter.
- 1.4 Tell time to the nearest quarter hour and know relationships of time (e.g., minutes in an hour, days in a month, weeks in a year).
- 1.5 Determine the duration of intervals of time in hours (e.g., 11:00 a.m. to 4:00 p.m.).

**2.0 Students identify and describe the attributes of common figures in the plane and of common objects in space:**- 2.1 Describe and classify plane and solid geometric shapes (e.g., circle, triangle, square, rectangle, sphere, pyramid, cube, rectangular prism) according to the number and shape of faces, edges, and vertices.
- 2.2 Put shapes together and take them apart to form other shapes (e.g., two congruent right triangles can be arranged to form a rectangle).

### 3rd Grade

**1.0 Students choose and use appropriate units and measurement tools to quantify the properties of objects:**- 1.1 Choose the appropriate tools and units (metric and U.S.) and estimate and measure the length, liquid volume, and weight/mass of given objects.
- 1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them.
- 1.3 Find the perimeter of a polygon with integer sides.
- 1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes).

**2.0 Students describe and compare the attributes of plane and solid geometric figures and use their understanding to show relationships and solve problems:**- 2.1 Identify, describe, and classify polygons (including pentagons, hexagons, and octagons).
- 2.2 Identify attributes of triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle).
- 2.3 Identify attributes of quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square).
- 2.4 Identify right angles in geometric figures or in appropriate objects and determine whether other angles are greater or less than a right angle.
- 2.5 Identify, describe, and classify common three-dimensional geometric objects (e.g., cube, rectangular solid, sphere, prism, pyramid, cone, cylinder).
- 2.6 Identify common solid objects that are the components needed to make a more complex solid object.

### 4th Grade

**1.0 Students understand perimeter and area:**- 1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m 2), square kilometer (km 2), square inch (in 2), square yard (yd2), or square mile (mi 2).
- 1.2 Recognize that rectangles that have the same area can have different perimeters.
- 1.3 Understand that rectangles that have the same perimeter can have different areas.
- 1.4 Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes.

**2.0 Students use two-dimensional coordinate grids to represent points and graph lines and simple figures:**- 2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3x and connect them by using a straight line).
- 2.2 Understand that the length of a horizontal line segment equals the difference of the x-coordinates.
- 2.3 Understand that the length of a vertical line segment equals the difference of the y- coordinates.

**3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems:**- 3.1 Identify lines that are parallel and perpendicular.
- 3.2 Identify the radius and diameter of a circle.
- 3.3 Identify congruent figures.
- 3.4 Identify figures that have bilateral and rotational symmetry.
- 3.5 Know the definitions of a right angle, an acute angle, and an obtuse angle. Under stand that 90°, 180°, 270°, and 360° are associated, respectively, with 1⁄4, 1⁄2, 3⁄4, and full turns.
- 3.6 Visualize, describe, and make models of geometric solids (e.g., prisms, pyramids) in terms of the number and shape of faces, edges, and vertices; interpret two-dimensional representations of three-dimensional objects; and draw patterns (of faces) for a solid that, when cut and folded, will make a model of the solid.
- 3.7 Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their attributes.
- 3.8 Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid).

### 5th Grade

**1.0 Students understand and compute the volumes and areas of simple objects:**- 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).
- 1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.
- 1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm 3], cubic meter [m3], cubic inch [in 3], cubic yard [yd3]) to compute the volume of rectangular solids.
- 1.4 Differentiate between, and use appropriate units of measures for, two- and three-dimensional objects (i.e., find the perimeter, area, volume).

**2.0 Students identify, describe, and classify the properties of, and the relation ships between, plane and solid geometric figures:**- 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).
- 2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.
- 2.3 Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.

### 6th Grade

**1.0 Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems:**- 1.1 Understand the concept of a constant such as π; know the formulas for the circumference and area of a circle.
- 1.2 Know common estimates of π (3.14; 22⁄7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
- 1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.

**2.0 Students identify and describe the properties of two-dimensional figures:**- 2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
- 2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
- 2.3 Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).

## RESEARCH THEORIES OF COGNITION AND LEARNING

### Constructivist Learning: Experience with Measurement

- Researchers have found that students feel disconnected from their math instruction and do not view math as relevant in their lives. The cause of this has been found to be the style of teaching. As a result, a constructivist approach to teaching has begun to take part in mathematics instruction (Clark et al., 2008).
- Constructivism has been found to be essential in generating greater success in mathematics classrooms
- Constructivism places greater responsibility of discovering and learning information on the students and therefore allows them to better understand the relevance of mathematical concepts and become more motivated and interested in mathematics. This then results in the students ability to improve math performance and meet the standards.

- Constructivism is an approach in which the students learn skills that are relevant to the students’ backgrounds and experiences, motivation problems are addressed, students learn to work together, and higher-level skills are utilized (Roblyer et al, 1997).
- Relate abstract concepts to students’ real-world, everyday experiences (build background knowledge) (Weinberg, 2001; Greenes, 2004)
- The most fundamental aspect of measurement and the best way to teach it is by creating situations in which students experience measuring things. (Martinie, 2004; Preston & Thompson, 2004)
- According to Wilson and Osborne (1988, in Reys et al., 2004), “Students should be given frequent opportunities to use measurement in their school experience, most preferably through real-life work projects that involve doing and experimenting rather than by passively observing” (p. 109).
- In order for understanding of the concepts of measurement to be fully developed, the procedures of measuring should be taught in unison with real contexts(Barrett & Clements, 2003).
- Constructivism: students actively measure objects in classroom in order to build their knowledge schemas about measurement (Weinberg, 2001)

### Suggestions for Teachers

- A case study performed by Morris and Hiebert (2009) showed the effects of teacher instruction on measurement.
- The study found that in order to teach effectively, teachers must study the effects of their own practice.
- Unless teachers are clear about what they intend students to learn, it is difficult to begin examining how instruction might have helped students learn it. As a result, teachers must set the learning goal for the students prior to the instruction (Morris & Hiebert, 2009).

- O'Keefe and Bobis (2008) found that teachers must understand in great depth the concept of measurement in order to be effective teachers of mathematics.
- It is critical for students to learn to reason mathematically about measurement and not merely measure accurately (Tower & Hunter, 2010).

### Suggestions for Teaching

- Towers and Hunter (2010) suggest that teaching measurement estimation can be thought of as the use of units of measure in a mental way rather than with the aid of measurement tools
- Build on children’s knowledge and interests on measurement (Greenes, 2004)
- With adult assistance, children can achieve a greater understanding of measurement concepts (scaffolding) (Greenes, 1999; Vygotsky, 1978; Zvonkin, 1992)
- Promote student metacognition: provide opportunities for children to talk about what they did and why; they should also use information from previous experiences to make decisions (Sierpinska, 1998)
- The way students construct, interpret, think about, and make sense of mathematical ideas is determined by the elements and organization of mental structures the students are using to process math (Battista, 2004).
- Researchers have found that students’ development of conceptualizations and reasoning can be characterized in terms of “levels of sophistication” (Batista & Clements, 1996). These levels include
- a) the informal, preinstructional reasoning
- b) the formal mathematical concepts targeted by instruction
- c) the cognitive plateaus reached by students in moving from (a) to (b).

- Assessments should be linked to research on student learning and cognition because this linkage is rarely made in traditional assessment and, in those cases, both instruction and assessment suffer. (Masters & Mislevy, 1993).
- Children will begin to measure accurately when they start to understand part-whole relationships because they think about a unit as one of many equal parts, making up the whole. (Reece & Kamii, 2001)
- Van de Walle (2010) provides a sequence of experiences for measurement instruction.

1. Making Comparisons-students will understand the attribute to be measured

2. Using Models of Measuring Units-students will understand how filling, covering, matching, or making other comparisons of an attribute with measuring units produces a

number called a measure

3. Using Measuring Instruments-students will use common measuring tools with understanding and flexibility

## COGNITIVE OBSTACLES AND MISCONCEPTIONS

### Cognitive Obstacles

- “Accounting for the whole during unit-forming”: distinguishing between
*half of*a number line and where the*number half*goes (Mitchell & Horne, 2008). - Distinguishing between reading already-marked partitions on a number line and creating own partitions (Mitchell & Horne, 2008).
- Recognizing length measurements in bent perimeters and other complex and irregular formations (Mitchell & Horne, 2008).
- When finding perimeter, finding the length of a side not explicitly stated due to an inadequate understanding of perimeter (Yeo, 2008).
- Using the appropriate units when computing perimeter and area (Yeo, 2008).
- Defining area as a concept rather than a formula (Yeo, 2008).
- Conceptual understanding of height and base (Van de Walle, 2010).

### Common Misconceptions

- Attending to the vertical lines (on a number line) instead of the parts: counting lines instead of spaces (Mitchell & Horne, 2008).
- Counting lines and including the zero-point (Mitchell & Horne, 2008).
- Overgeneralizing number lines by assuming the end number forms the whole rather than “using the scale to help with unit forming” (Mitchell & Horne, 2008).
- Misreading the number line inscriptions by decimalizing the count: reading each sub-unit mark as if the whole was divided into tenths (point one, point two, point three…) (Mitchell & Horne, 2008).
- Confusing the
*concepts*of area and perimeter (even when able to reiterate the correct formulas) (Yeo, 2008). This confusion is largely due to an overemphasis on formulas with little or no conceptual background. - Confusing the formulas for area and perimeter (Van de Walle, 2010).
- Confusing a figure’s height and slanted side (e.g. cylinder) (Van de Walle, 2010).

## PEDAGOGICAL TOOLS AND STRATEGIES

### Begin By Building Conceptual Knowledge

- Base primary measurement activities on what it means to measure, rather than how to measure (Castle & Needham, 2007).
- The skill of measuring with a unit must be directly linked to the concept of measuring as a process of comparing attributes, using measuring units, and using measuring instruments (Van de Walle, 2010).

### Nonstandard Units: Body Benchmarks

- Compare standard and nonstandard units – nonstandard units are inexact but give us a frame of reference about the length, weight, etc. of an object (Weinberg, 2001).
- Nonstandard units make it easier to focus directly on the attribute being measured (Van de Walle, 2010).
- However, the benefits of nonstandard measuring units may last only a day or two.

- A student needs to first understand nonstandard measurement before they can grasp and be taught actual units of measurement (Clement, 1999).
- However, if there is no good instructional reason for nonstandard units, use standard units to increase students' experience and familiarity with the unit (Van de Walle, 2010).

- A good nonstandard unit: a student’s foot (trace and cut out) (Weinberg, 2001).
- Personal benchmarks (nonstandard units that are used to represent standard units) are useful references for estimating measurement (Muir, 2005; Weinberg, 2001).
- Students should be encouraged to think of their own benchmarks. Teacher imposed benchmarks are not as useful as those the students create, because the ones students come up with are more memorable.
- Below are some questions that challenge children to think about the relationships of nonstandard measurement to things that they are sure of.

1. Is it reasonable for an apple to be 1 foot tall?

2. Is it reasonable for your foot to be about 30 centimeters long?

3. Is it reasonable for your finger to be more than one penny long?

4. Is it possible for you to hop over a hole that is 30 inches across?

5. Is it possible for you to hop over a puddle that is 7 feet across?

6. Is it possible to measure your length in paperclips?

7. Is it reasonable for a chicken to be three pennies long?

8. Is it reasonable ____________________________________________?

### Estimation

- Estimating is a measurement technique that should be developed throughout the school years and should focus on helping children better understand the process of measuring and the role of the size of the unit (NCTM, 2000).
- Students must know where to segment the item and have the ability to recall and then mentally manipulate images of the unit when estimating (Towers & Hunter, 2010).
- The significance of estimation should be conveyed to students as an ordinary, everyday task (Muir, 2005).
- Relate to students how much estimation affects their own lives! Estimation experiences should be purposeful. For example, ask students to think about how much food and drink will be needed for the class party.
- Many students think of an estimation as a "guess" before they find the actual measurement, and fail to see its importance. Estimation skills should be practiced and explained as a measuring tool.
- To get students thinking about how often estimation is used and why it is important, ask students to brainstorm times where they used estimation rather than an "exact" measurement, or challenge the class to think about that the world would be like if everything had to be measured exactly.
- Researchers have found that the "guess and check" method of having students first making an estimate and then measuring accurately develop accuracy estimation skills (Towers & Hunter, 2010).
- When teaching estimation, keep the following tips in mind:
- Help students learn strategies by having them use a specific approach.
- Discuss how different students made their estimates.
- Accept a range of estimates.
- Encourage students to give a range of measures that they believe includes the actual measure.
- Make measurement estimation an ongoing activity.

- Always have students estimate a measurement before they make it for both nonstandard and standard units because of the following reasons: (Van de Walle, 2010).
- Estimation helps students focus on the attribute being measured and the measuring process
- Estimation provides intrinsic motivation to measurement activities
- Estimation helps develop familiarity with standard units
- the use of a benchmark to make an estimate promotes multiplicative reasoning

### Creative Teaching Ideas

- Use activities and children’s literature to develop ideas about measurement in young children (Greenes, 2004).
- Literature helps students see that measurement is all around in the world (Ward, 2005).
- Construct model hot air balloons to teach metric measurements (Kuhl & Shaffer, 2008).
- Use sewing projects to teach measurement (Goral & Gilderbloom, 2008).
- Use a variety of objects to measure and measurement tools at a “measurement center” in the classroom (Castle & Needham, 2007).
- Use measurement problems involving indirect comparisons (Castle & Needham, 2007).
- Interview children about their understanding of measurement (Castle & Needham, 2007).
- Keep measurement journals (Castle & Needham, 2007).
- Use calculators to check conversion work (Weinberg, 2001).
- Promote discussion of and reflection of new measurement discoveries (Greenes, 2004).
- Integrate mathematics into routine class activities (Greenes, 2004).
- Provide visual representations of amounts of things to visualize and conceptualize how great a number can be (Ward, 2005).

### Strategies for Students with Special Needs

- For students with vision impairments: use a rough piece of paper like sandpaper to measure (Weinberg, 2001).
- For students with hearing impairments: measure after their partner so they can see an example, teacher write up findings on board for students to read (Weinberg, 2001).
- For slower workers, make some of the measurements optional (Weinberg, 2001).

### Pedagogical Recommendations

- Using squares to make “length x width” understandable (Yeo, 2008).
- Connecting the concept of finding the area to a real life problem such as finding the area of a basketball court (Yeo, 2008).
- Interweaving conceptual areas of measurement for optimal understanding (Mitchell & Horne, 2008):

- Concept of proportion: using number lines with unequal spacings.

- Concept of the unit: using number lines with improper fractions.

- Concept of additivity: using number lines in general because the zero point is a crucial point.

- Concept of proportion: using number lines with unequal spacings.
- Review the concept of multiplication as seen in arrays. Display rows and columns of objects and discuss why multiplication tells what the total amount is (Van de Walle, 2010).

### Measurement Activities

Activities derived from Van de Walle (2010).

**Longer, Shorter, Same (Pre-K to K)**

Make several sorting-by-length learning stations at which students sort objects as longer, shorter, or about the same as a specified object. The reference object can be changed to produce different sorts. A similar task involves putting objects in order from shortest to longest.

**Length (or Unit) Hunt (Pre-K to K)**

Give pairs of students a strip of tagboard, a stick, a length of rope, or some other object with an obvious length dimension. The task on one day might be to find five things in the room that are shorter than, longer than or about the same length as their target unit. They can draw pictures or write the names of the things they find.

By making the target length a standard unit (e.g., a meter stick or a 1-meter length of rope), the activity can be repeated to provide familiarity with important standard units.

**Estimate and Measure (K-3)**

Make lists of items in the room to measure. Run a piece of masking tape along the dimension of objects to be measured. On the list, designate the units to be used. Do not forget to include curves or other distances that are not straight lines. Include estimates before the measures. Remember that young children have probably had limited experiences with estimating distances.

**Changing Units (3-6)**

Have students measure a length with a specified unit. Then provide them with a different unit that is either twice as long or half as long as the original unit. Their task is to predict the measure of the same length using the new unit. Students should write down their estimations and explanations of how they were made. Stop and have a discussion about their estimations and then ahve them make the actual measurement. Cuisenaire rodes are excellent for this activity. Older students can be challeneged with units that are more difficult multiples of the original unit.

**Make Your Own Ruler (1-6)**

Precut narrow strips of construction paper 5 cm long and about 2 cm wide. Use two different colors. Discuss how the strips could be used to measure by laying them end to end. Provide long strips of tagboard about 3 cm wide. Without explicit guided direction, have students make their own ruler by gluing the units onto the tagboard. Have a list of a few things to measure. Students use their new rulers to measure the items on the list. Discuss the results. It is possible that there will be discrepancies due to rulers that were not made properly or to a failure to understand how a ruler works.

The same activity can be done using larger nonstandard units such as tracings of students' fooprints glued onto strips of adding machine tape. Older children can use a standard unit (centimeter, inch, foot) to make marks on the strips and color in the spaces with alternating colors.

**Rectangle Comparison-Square Units (4-6)**

Students are given a pair of rectangles that are either the same or very close in area. THey are also given a model or drawing of a single square unit and a ruler that measures the appropriate unit. The students are not permitted to cut out the rectangles. They may draw on them if they wish. The task is to use their rulers to determine, in any way that they can, which rectangle is larger or whether they are the same. They should use words, pictures, and numbers to explain their conclusions. Some suggested pairs are as follows:

4x10 and 5x8

5x10 and 7x7

4x6 and 5x5

__Fixed Volume: Comparing Prisms (3-6)__

Give each pair of students a supply of centimeter cubes or wooden cubes. Their task is, for a fixed number of cubes, to build different rectangular prisms and record the surface area for each prism formed. A good number of cubes to suggest is 64, since a minimal surface area will occur with a 4 x 4 x 4 cube. With 64 cubes a lot of prisms can be made. However, i fyou are short of cubes, other good choices are 24 or 36 cubes. Using the tables students construct, they should observe any patterns that occur. In particular, what happens to the surface area as the prism becomes less like a tall, skinny box and more like a cube?

** Box Comparison-Cubic Units** (

__5-6)__Provide students with a pair of small boxes that you have folded up from poster board. Use unit dimensions that match the blocks that you have for units. Students are given two boxes, exactly one block, and an appropriate ruler. (If you use 2-cm cubes, make a ruler with the unit equal to 2 centimeters.) The students' task is to decide which box has the greater volume or if they have the same volume.

Here are some suggested box dimensions (LxWxH):

6x3x4

5x4x4

6x6x2

Students should use words, drawings, and numbers to explain their conclusions

__One-Handed Clocks__**(K-2)**

Prepare a page of clock faces. On each clock draw an hour hand. Include placements that are approximately a quarter past the hour, a quarter til the hour, half past the hour, and some that are close to but not on the hour. For each clock face, the students' task is to write the digital time and draw a minute hand on the clock where they think it would be.

**About One Unit (3-6)**

Give students a model of a standard unit, and have them search for objects that measure about the same as that one unit. For example, to develop familiarity with the meter, give students a piece of rope 1 meter long. Have them make lists of things that are about 1 meter. Keep separate lists for things that are a little less (or more) or twice as long (or half as long). Encourage students to find familiar items in their daily lives. In the case of lengths, be sure to include curved or circular lengths. Later, students can try to predict whether a given object is more than, less than, or close to 1 meter.

__Familiar References (1-3)__

Use the book *Measuring Penny *(Leedy, 2000) to get students interested in the variety of ways familiar items can be measured. In this book, the author bridges between nonstandard (e.g., dog biscuits) and standard units to measure Penny the pet dog. Have your students use the idea of measuring Penny to find something at home (or in class) to measure in as many ways as they can think using standard units. The measures should be rounded to whole numbers (unless children suggest adding a fractional unit to be more precise). Discuss in class the familiar items chosen and their measures so that different ideas and benchmarks are shared.

__Guess the Unit (1-4)__

Find examples of measurements of all types in newspapers, on signs, or in other everyday situations. Present the context and measures but without units. The task is to predict what units of measure were used. Have students discuss their chioces.

**Estimation Scavenger Hunt (3-6)**

Conduct estimation scavenger hunts. Give teams a list of measurements, and have them find things that are close to having those measurements. Do not permit the use of measuring instruments. A list might include the following items:

- A length of 3.5 m
- Something that weights more than 1 kg but less than 2 kg
- A container that holds about 200 ml
- An angle of 45 degrees or 135 degrees

Let students suggest how to judge results in terms of accuracy.

## CURRICULA AND TECHNOLOGICAL RESOURCES

### Strategies

- Incorporate technology by using spreadsheets to compare the average length of different measurements on the body in the nonstandard unit of someone’s foot (Weinberg, 2001)
- Introduce and reinforce mathematical topics with popular children’s literature (Long, 2000):
- [Sir Cumference and the First Round Table] used for 5th graders to introduce, reinforce and review the concepts of perimeter and area (Long, 2000)
- Have students create geometrical shapes out of paper as story is read

- Read the book [How Big is a Foot?] (Myller, 1962) to illustrate why we need standard measurements

- [Sir Cumference and the First Round Table] used for 5th graders to introduce, reinforce and review the concepts of perimeter and area (Long, 2000)

### Websites

- [Cabri] Software to help conceptualize 3D geometry.
- [Measurement] This site teaches students how to measure.
- [Measurement Lessons] This site provides several worksheets and tools for teaching units of measurement.
- [Teaching Measures] This website is an interactive program for students to practice working with measurements.
- [The Geometer's Sketchpad] The Geometer’s Sketchpad: This interactive site provides 3D geometric figures.
- [Wingeom] An interactive program where you can construct 2D and 3D geometric figures.
- [AAA Math] A website dedicated to lesson plans and ideas for measurement. Topics vary from time, mass, volume, the metric system, and more.
- [Funbrain] This site is more directed towards primary grades and has a few simple games involving measurement with a ruler.
- [World of Measurement] A great website for upper elementary students with tests and activities. It was also awarded the Education Planet "Math Top Site Award" by their teacher/reviewers for it's quality content and usefulness to Math educators and students.
- [UNC Dictionary of Measurement] This site is a dictionary of any measurement terms teachers and students may encounter. It also has conversion tools for metric and English systems of measurement as well as tables and charts ranging from international hat sizes to nutritional daily value.

## ANNOTATED REFERENCES

- Barett, J.E. & Clements, D.H. (2003). Quantifying Path Length: Fourth-Grade Children's Developing Abstractions for Linear Measurement. Cognition and Instruction, 21(4), 475-520.
- Battista, Michael A. (2004). Applying cognition-based assessment to elementary school students' development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185-204.
- Battista, M. T.,&Clements, D. H. (1996). Students’understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27, 258–292.
- Castle, K. & Needham, J. (2007). First graders’ understanding of measurement. Early Childhood Education Journal, 35, 215-221.
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