Elementary Mathematics Benchmarks, Measurement Strand
M.K.1 Compare the length, weight, and capacity (volume) of objects.
a. Make direct comparisons between objects (e.g., recognize which is shorter, longer, taller, lighter, heavier, or holds more).
b. Estimate length, weight, and capacity, and check estimates with actual measurements.
- Select and use appropriate measurement tools (rulers, tape measures, scales, containers, clocks, thermometers).
- Relate direct comparisons of objects to comparisons of numerical measurements or estimates.
Example: Tom, who is four feet tall, is shorter than Jose, who is five feet tall, because 4 is smaller than 5.
M.1.1 Measure length, weight, capacity, time, and money.
a. Use rulers, scales, and containers to measure and compare the dimensions, weight, and capacity (volume) of classroom objects.
Note: The concepts of addition and order are intrinsic in quantities such as length, weight, and volume (capacity). So measuring such quantities provides an independent empirical basis for understanding the properties of numbers that is different from simple counting.
b. Tell time from analog (round) clocks in half-hour intervals.
- Use the expressions "o’clock" and "half past."
c. Count, speak, write, add, and subtract amounts of money in cents up to $1 and in dollars up to $10.
- Know the values of U.S. coins (penny, nickel, dime, quarter, dollar bill).
- Use the symbols $ and ¢ separately (e.g., $4, 35¢ instead of $PS.4.35).
- Use coins to decompose monetary amounts given in cents.
Example: 17¢ = one dime, one nickel, and two pennies = 10 + 5 + 1 + 1; or 17¢ = three nickels and two pennies = 5 + 5 + 5 + 1 + 1.
- Understand and solve money problems expressed in a different ways, including how much more or how much less.
Note: Avoid problems that require conversion from cents to dollars or vice versa.
M.2.1 Add, subtract, compare, and estimate measurements.
a. Estimate, measure, and calculate length in meters, centimeters, yards, feet, and inches.
- Recognize and use standard abbreviations: m, cm, yd, ft, and in, as well as the symbolic notation 3'6".
- Understand and use units appropriate to particular situations.
Example: Standard U.S .school notepaper is sized in inches, not centimeters.
- Add and subtract mixed metric units (e.g., 8m,10cm + 3m,5cm) but defer calculation with mixed English units (e.g., 3ft,1in + 1ft,8in) until third grade.
Note: Conversion between systems awaits a later grade.
b. Measure the lengths of sides and diagonals of common two-dimensional figures such as triangles, rectangles (including squares), and other polygons.
- Measure to the nearest centimeter or half inch using meter sticks, yardsticks, rulers, and tape measures marked in either metric or English units.
Note: Measure within either system without conversion between systems.
- Create and use hand-made rulers by selecting an unconventional unit length (e.g., a hand-width), marking off unit and half-unit lengths.
- Explore a variety of ways to measure perimeter and circumference.
Examples: Encircle with a tape measure; measure and sum various pieces; wrap with a string and then measure the length of the string. Compare answers obtained by different strategies and explain any differences.
Note: Comparing the result of a direct measurement (encircling) with that of adding component pieces underscores the importance of accuracy and serves as a prelude to understanding the significance of significant digits.
c. Estimate and measure weight and capacity in common English and metric units.
- Recognize, use, and estimate common measures of volume (quarts, liters, cups, gallons) and weight (pound, kilogram).
- Understand and use common expressions such as half a cup or quarter of a pound that represent fractional parts of standard units of measurement.
d. Compare lengths, weights, and capacities of pairs of objects.
- Demonstrate that the combined length of the shorter pieces from two pairs of rods is shorter than the combined lengths of the two longer pieces.
- Recognize that the same applies to combined pairs of weights or volumes.
Note: Even though this relation may seem obvious, it is an important demonstration of the fundamental relation between addition and order, namely, that if a ≤ b and c ≤ d, then a + c ≤ b + d.
M.3.1 Recognize why measurements need units and know how to use common units.
a. Understand that all measurements require units and that a quantity accompanied by a unit represents a measurement.
- Know and use the names and approximate magnitudes of common units:
For length: kilometer, meter, centimeter; mile, yard, foot, inch.
For capacity: liter, milliliter; gallon, quart, pint, cup.
For time: year, month, week, day, hour, minute, second.
For money: pennies, nickels, dimes, quarters, dollars.
Note: Many of these units have been introduced in prior grades; others will be introduced in later grades. Here some are pulled together for reinforcement and systematic use. Each year in grades 2-6 some new measures should be introduced and previous ones reinforced. Which are done in which grades is of lesser importance.
b. Know common within-system equivalences:
- 1 meter = 100 centimeters, 1 yard = 3 feet, 1 foot = 12 inches.
- 1 liter = 1,000 milliliters, 1 gallon = 4 quarts, 1 quart= two pints.
- 1 year = 12 months, 1 week = 7 days, 1 hour = 60 minutes, 1 minute = 60 seconds.
- 1 dollar = 4 quarters = 10 dimes = 100 pennies, 1 quarter= 5 nickels = 25 pennies, 1 dime = 2 nickels = 10 pennies, 1 nickel = 5 pennies.
c. Choose reasonable units of measure, estimate common measurements, use appropriate tools to make measurements, and record measurements accurately and systematically.
d. Use decimal notation to express, add, and subtract amounts of money.
Note: Dealing with money enables students to become accustomed to decimal notation, i.e., $1.49 + $0.25 = $1.74.
e. Solve problems requiring the addition and subtraction of lengths, weights, capacities, times, and money.
- Include use of common abbreviations: m, cm, kg, g, l, ml, hr, min, sec, in, ft, lb, oz, $, ¢.
Note: Add and subtract only within a single system, using quantities within students' experience. Use real data where possible, but limit the size and complexity of numbers so that problem solving, not computation, is the central challenge of each task.
M.4.1 Understand and use standard measures of length, area, and volume.
a. Know and use common units of measure of length, area, and volume in both metric and English systems.
- Always use units when recording measurements.
- Know both metric and English units: centimeter, square centimeter, cubic centimeter; meter, square meter, cubic meter; inch, square inch, cubic inch; foot, square foot, cubic foot.
- Use abbreviations: m, cm, in, ft, yd; m2, cm2, in2, ft2, yd2; sq m, sq cm, sq in, sq ft, sq yd; m3, cm3, in3, ft3 and yd3.
b. Convert measurements of length, weight, area, volume, and time within a single system.
Note: Emphasize conversions that are common in daily life. Common conversions typically involve adjacent units—for example, hours and minutes or minutes and seconds, but not hours and seconds. Know common within-system equivalences.
- Use unit cubes to build solids of given dimensions and find their volumes.
- 1 square foot = 122 square inches; 1 square meter as 1002 square centimeters; 1 cubic foot = 123 cubic inches; 1 cubic meter as 1003 cubic centimeters.
c. Visualize, describe, and draw the relative sizes of length, area, and volume units in the different measurement systems.
- Estimate areas of rectangles in square inches and square
Note: Avoid between-system conversions.
Examples: Centimeter vs. inch, foot and yard vs. meter; square centimeter vs. square inch; square yard vs. square meter; cubic foot vs. cubic meter.
d. Recognize that measurements are never exact.
- Both recorded data and answers to calculations should be rounded to a degree of precision that is reasonable in the context of a given problem and the accuracy of the measuring instrument.
Note: All measurements of continuous phenomena such as length, capacity, or temperature are approximations. Measurements of discrete items such as people or bytes can be either exact (e.g., size of an athletic team) or approximate (e.g., size of a city).
e. Solve problems involving area, perimeter, surface area, or volume of rectangular figures.
M.5.1 Make, record, display, and interpret measurements of everyday objects.
a. Select appropriate units to make measurements and include units in answers.
b. Recognize and use measures of weight, information, and temperature.
- For information: bytes, kilobytes (K or Kb), megabytes (M), gigabytes (G). 1G = 1,000M, 1M = 1,000K, 1K = 1,000 bytes.
Note: Literally, the multiplier is 1,024 = 210, but for simplicity in calculation, 1,000 is generally used instead.
- For weight: kilogram (kg), gram (g), pound (lb), ounce (oz). 1 kg= 1,000 g, 1 lb = 16 oz.
- For temperature: Centigrade and Fahrenheit degrees. 32°F = 0°C; 212°F = 100°C.
c. Record measurements to a reasonable degree of accuracy, using fractions and decimals as needed to achieve the desired detail.
d. When needed, use a calculator to find answers to questions associated with measurements.
e. Create graphs and tables to present and communicate data.
Note: Measurement is not a focus in grade 6 of these expectations.