# 5: Measurement - Mathematics

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## Measure (mathematics)

In mathematics, a measure on a set is a systematic way to assign a number to subsets of a set, intuitively interpreted as the size of the subset. Those sets which can be associated with such a number, we call measurable sets. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space. This assigns the usual length, area, or volume to certain subsets of the given Euclidean space. For instance, the Lebesgue measure of an interval of real numbers is its usual length, but also assigns numbers to other kinds of sets in a way that is consistent with the lengths of intervals.

Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see § Definition, below). A measure must further be countably additive: if a 'large' subset can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets that are measurable, then the 'large' subset is measurable, and its measure is the sum (possibly infinite) of the measures of the "smaller" subsets.

In general, if one wants to associate a consistent size to all subsets of a given set, while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets the so-called measurable subsets, which are required to form a σ -algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.  Indeed, their existence is a non-trivial consequence of the axiom of choice.

Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

## 5: Measurement - Mathematics The distance between two objects or places is measured as length.
The standard unit of length according to the metric system is meter (m).
Based on the length that needs to be measured, meter can be converted into different units like millimetre (mm), centimetre (cm) and kilometre (km).
&bull 1 km = 1000 m
&bull 1 m = 100 cm
&bull 1 cm = 10 mm
For example the length of a pencil is measured in centimetres, while the distance between two places is measured in kilometres. View Lessons & Exercises for Measurement of Length &rarr
• Introduction to Measurement of Length in Tenths and Hundredths
• Centimetre and Millimetre
• Difference in Length
• Writing in decimal and fraction
• Review of Measurement of Length in Tenths and Hundredths ### Measurement of Length -- Summary

Learnhive Lesson on Measurement of Length

### Factoids

The other units of measurement of length are inches, feet, yard and miles. In the United States these measuring units are used.

A parent comment on the new Common Core State Standards
http://edsource.org/2014/common-core-standards-bring-dramatic-changes-to-elementary-school-math-2

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## Measurement

MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.

A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

## 5: Measurement - Mathematics

The student will determine an amount of elapsed time in hours and minutes within a 24-hour period.

Computation and Estimation

Probability, Statistics, Patterns, Functions, and Algebra

Words and Definitions

2.12 Time to the nearest 5 minutes

3.11 Time to the nearest minute and elapsed time in hour increments

4.9 Elapsed time hours and minutes within a 12 hour time period

Elapsed Time – the amount of time between a start time and an end time.

“Sir Cumference” series (by Cindy Neuschwander)

Teaching Student-Centered Mathematics (by John Van de Walle)

Dinah Zike’s Notebook Foldables (by Dinah Zike)

Dinah Zike’s Big Book of Math (by Dinah Zike)

Mathematical Art-o-Facts (by Catherine Jones Kuhns)

Object Lessons: Teaching Math Through the Visual Arts (by Caren Holtzman)

Power Point Activities

UNDERSTANDING THE STANDARD

ESSENTIAL UNDERSTANDINGS

ESSENTIAL KNOWLEDGE AND SKILLS

· Elapsed time is the amount of time that has passed between two given times.

· Elapsed time can be found by counting on from the beginning time to the finishing time.

– Count the number of whole hours between the beginning time and the finishing time.

– Count the remaining minutes.

Add the hours and minutes. For example,

tofind the elapsed time between 10:15 a.m. and 1:25 p.m., count on as follows:

from 10:15 a.m. to 1:15 p.m., count 3 hours

from 1:15 p.m. to 1:25 p.m., count 10 minutes and then

add 3 hours to 10 minutes to find
the total elapsed time of 3
hours and 10 minutes.

· Understand that elapsed time can be found by counting on from the beginning time to the finishing time.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

## Measurement: Level 5

The key idea of measurement at level 5 is that all measurements are approximate.

Because measurement involves continuous quantities even the most accurate measurements are only approximations. As students develop their ability to measure a variety of attributes using a variety of units, they need appreciate that measurements are never exact, and that all measurements contain errors.

For any measurement, the level of precision required will depend on the way the information is going to be used. For example, when purchasing fertiliser the number of litres required is probably sufficient but when purchasing medicine the number of millilitres required is likely to be more appropriate.

At level 5 students are also able to split complex shapes into component parts in order to calculate their length, area, or volume. For example, the surface area of a cylinder can be calculated as sum of the area of two circles and a rectangle. At this level students need to develop the ability to compose and decompose shapes in order to find the lengths, areas and volumes of various complex objects.

This key idea develops from the key idea of measurement at level 4 which involves the application of multiplicative thinking to measurement.

This key idea is extended to the key idea of measurement at level 6 where students apply abstract mathematical formula in measurement problems.

## 5TH GRADE MEASUREMENT AND DATA This page provides sample 5th Grade Measurement and Data Centers from our  5 th Grade Math Centers ਎ Book.  Try out the samples listed in blue under each Common Core State Standard or download the 5th Grade Math Centers eBook and have all the 5th Grade Number, Geometry, Measurement and Data Centers you’ll need for the entire school year in one convenient digital file.  With over 170 easy-prep, engaging centers this resource will simplify your lesson planning and make hands-on math instruction an integral part of your classroom.
Teaching in a state  that is implementing their own specific math standards?  Download ourਅth  Grade Correlations document  for cross-referenced tables outlining the alignment of each state's standards with the CCSS-M, as well as the page numbers in our 5th Grade Math Centers eBook related to each standard.

Convert like measurement units within a given measurement system

5.MD.A.1 ਌onvert among different-sized standard measurement units within a given measurement system (e.g., convert 5cm to 0.05m), and use these conversions in solving multi-step, real world problems.

Represent and interpret data

5.MD.B.2 Make a line plot to display a set of measurements in fractions of a unit (1/2, ¼, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.   Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition

5.MD.C.3  Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

Build a Cubic Meter

5.MD.C.4 Measure volume by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Build Rectangular Prisms
What's the Volume?   5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

Exploring Volume

b. Apply the formulas V= l x w x h and V = b x h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Roll a Rectangular Prism

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Find the Volume

## Conclusion

We are constantly measuring the world around us and using that information to make decisions. From the casual decision on the type of snack to enjoy to the important one of how much medicine to take, we quantify and measure values. And we’ve been measuring the world since very early times, making adjustments and new discoveries of how to measure continuously. With all of these measurements there is a margin of error included in even the most precise measurement. But through awareness of these errors and careful attention to the values and units, we can approach very high levels of accuracy in our measurements. And that is the ultimate goal of measurement – to provide accurate information that everyone can understand and use.

### Summary

In almost every facet of modern life, values – measurements – play an important role. We count calories for a diet, stores measure the percentage of tax on our purchases, and our doctors measure important physiological indicators, like heart rate and blood pressure. From the earliest documented days in ancient Egypt, systems of measurement have allowed us to weigh and count objects, delineate boundaries, mark time, establish currencies, and describe natural phenomena. Yet, measurement comes with its own series of challenges. From human error and accidents in measuring to variability to the simply unknowable, even the most precise measures come with some margin of error.

### Key Concepts

Since their earliest days, systems of measurement have provided a common ground for individuals to describe and understand their world. Measurement helps to give context to observations and a means to describe phenomena.

A measurement consists of two parts – the amount present or numeric measure, and the unit that the measurement represents within a standardized system.

When direct measurement is not possible, scientists can estimate parameters through indirect measurement.

While errors do occur in measurement, measurement error generally refers to the uncertainty or variability around a measure that occurs naturally due to the limitations of the tool we are using to measure the quantity.