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Chapter 19: Developing Measurement Concepts

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Chapter 19: Developing Measurement Concepts

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By: Jessica Fowler, Isabelle Jimenez, Kate Griner, Randi Figueroa

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“a measurement is a number that indicates a comparison between the attribute of the object (or situation, or event) being measured and the same attribute of a given unit of measure.”

The process of measuring:

FIRST students decide on the attribute to be measured.

SECOND Select a unit that has that attribute.

THIRD Compare the units—by filling, covering, matching, or using some other method—with the attribute of the object being measured. The number of units required to match the object is the measure.

***It is essential to make meaning of the measuring process or activity students are partaking in.

“The skill of measuring with a unit must be explicitly linked to the concept of measuring as a process of comparing attributes, using measuring units and using measuring instruments.”

It is important to keep in mind that often when measuring length, width, height, weight, etc. direct comparisons can be made more easily, but when dealing with measurements such as volume or mass other methods of comparison may need to be done in order to compare to objects or things.

Standard units of measurement:

Standard units are the units we usually use to measure the weight, length or capacity of objects.

Non- standard units of measurement:

often idea of non-standard measures is to focus the child on the concept of heavier, lighter, longer, shorter, etc. before they go move onto the next step of measuring using standard units.

EXAMPLES:

|   Standard     |   Nonstandard
|   One Foot      |     long paper strips
|   Ruler             |   straws
|   Yard stick     |   Tissue boxes
|   Shoe

Teaching tips for estimating measurement from the textbook (page 460):

1. Help students learn strategies by having them first try a specified approach. Later activities should permit students to choose whatever techniques they wish.

2. Discuss how different students made their estimates. This will confirm that there is no single right way to estimate while reminding students of other useful approaches.

3. Accept a range of estimates. Think in relative terms about what is a good estimate. Within 10 percent for length is reasonable. Even 30 percent “off” may be reasonable for weights or volumes.

4. Do not promote a “winning” estimate. This discourages estimation and promotes only seeking the exact answer.

5. Encourage students to give a range of estimates that they believe includes the actual measure. For example, the door is between 7 and 8 feet tall. This not only is a practical approach in real life but also helps focus on the approximate nature of estimation.

6. Make measurement estimation an ongoing activity. Post a daily measurement to be estimated. Students can record their estimates and discuss them in a five-minute period. Older students can take turns determining the daily measurements to estimate, with individuals or a team assigned this task each week.

7. Be precise with your language. Do not use the word measure interchangeably with the word estimate (Towers & Hunter, 2010). Randomly substituting one word for the other will cause uncertainty and possibly confusion in students.

Van de Walle, John A.; Karp, Karen S.; Bay-Williams, Jennifer M. (2015-02-11). Elementary and Middle School Mathematics: Teaching Developmentally (9th Edition) (Teaching Student-Centered Mathematics Series) (Page 460). Pearson. Kindle Edition.

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Length is defined in the textbook as an attribute of an object that is found by locating two endpoints and examining how far it is between those points.

It tends to be the first attribute students learn to measure.

Begin teaching students with direct comparison models, once students have a good understanding you can move on to indirect comparison models.

Common misconceptions students may have when learning to measure length:

·      Measuring from the wrong end of the ruler or beginning at 1 instead of 0

·       Counting the hash marks rather than the spaces (units)

·      Not aligning two objects when comparing them

·      Switching between cm and inches on a ruler

There are four important principles of iterating units of length:

1.     Units must be equal in length or you cannot iterate them by counting.

2.      Units must align with the length being measured or a different quantity is measured.

3.      Units must be placed without gaps or a part of the length is not measured.

4.      Units must be placed without overlaps or the length has portions that are measured more than one time.

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Area is the measure of a two-dimensional space inside a region. Data from the 2011 NAEP suggest that only 24% of fourth grade students were able to find the area of a square with a perimeter of 12 units. Estimating and measuring area begin in third grade. Things that can be done to help gain further knowledge and understanding area are:

Comparison activities: Comparison activities help students distinguish between size (area) and shape, length, and other dimensions.

Using physical models of area  units: "covering the surface" by filling in shapes with manipulatives.

The relationship between area and perimeter are a continual source of confusion for students. Perimeter is a third grade standard, but only 71% of eighth graders accurately identified the perimeter of a given shape. When developing formulas for area it is important to recognize these common two misconceptions:

Confusing linear and square units: if a shape has no distinguished units, then there is no area for that shape.

Difficulty in conceptualizing the meaning of height and base: Any side can be a base to a shape, and for every base there is a corresponding height perpendicular to the base.

The formulas for rectangles, parallelograms, triangles and trapezoids are as follows:

Rectangle: Base x Height

Parallelogram: Base x Height

Triangles: 1/2 Base x Height

Trapezoids: Base 1 + Base 2/2 x Height

The relationship between the circumference of a circle and the length of the diameter is one of the seventh grade state standards. The circumference of every circle is about 3.14 times as long as the diameter. This introduces pi (3.14...) to students, and allows them to discover the areas of circles. The formula for finding the area of a circle is: π(r)^2.

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Weight is a measure of the pull of gravity on an object, and mass is the amount of matter inside an object. The most common way to compare weights are by physically holding them. Also by using spring scales and balance scales. Using physical models of weight and mass units help students conceptualize these topics, but are harder to do than with area or volume because you need to reference a weight.

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Understanding attributes of angles and measuring them is first introduced beginning in the fourth grade, continuing on into middle school. Using comparison activities, physical models, and tools like protractors are all helpful tools for students to have to solve problems, but also develop a firm understanding of angles and their attributes.

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Students need to learn that in time there are units such as seconds, minutes, and hours, and develop concepts of how long these units are.  Reading clocks, and solving problems with time are important to do when introducing concepts of time. They can discuss what happens to the big hand as the little hand goes from one hour to the next, predict reading digital clocks when shown a time on an analog clock, and relate telling time after hours to the time before the next hour, and discuss a.m. and p.m.

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The money skills typically required in primary grades are:

Recognizing coins

identifying and using the 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

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Volume and capacity are both terms for the "size" of three dimensional shapes, a topic first introduced in the fifth grade. For a further understanding of this, using physical models of volume and capacity really help students physically see what they are finding. The formulas developed for volumes of common solid shapes are:

Cylinders: π(r^2) x height

Cones, pyramids, and spheres: (π(r)^2) x height/3

All of these formulas can be related back to the single concept of base x height.

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Overall this chapter discussed all aspects of measurement for elementary school students and provided readers with many strategies to use when teaching measurement. The biggest takeaway I had from this chapter would be the common misconceptions students may have while measuring. This chapter supports teachers in measurement instruction and explains a student view in a sense as to how they may approach certain measurements.

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Resources for Math chapter

Ixl.com

Pre-k – comparing height, weight and capacity

https://www.ixl.com/math/pre-k/compare-height-weight-and-capacity

fourth grade- measuring using an inch ruler

https://www.ixl.com/math/grade-4/measure-using-an-inch-ruler

Other games with measurement

Grade unspecified

http://www.childu.com/sample_act/math_k/grade/subject/mak_04_03_03.html

1-2 grade

http://www.topmarks.co.uk/r.aspx?sid=5035

grade unidentified

http://www.xpmath.com/forums/arcade.php?do=play&gameid=11