Measurement, Time, & Money

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Developing

Measurement Concepts

Dr. Gwen Deger

Youngstown State University

Class 15

Math Joke of the Day:

How do you maintain heat in

a cold room?

Head to the corner. It's

probably 90 degrees there.

Today's Agenda

Teaching Measurement

Upcoming Assignments

CRA Work Samples & Draft Due 3/251.

DIM Multiplication/Division Videos Due 3/25

2.

DIM Fractions, Decimals, & Percentages Videos Due
4/1

3.

DIM Measurement, Geometry, Analysis & Probability
Videos Due 4/8

4.

Teaching

Measurement

The Meaning & 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.

Step

Goal

Type of Activity

Notes

Decide on the attribute
to be measured.

1.
Identify measurable
attributes of an object,
situation, or event.

See Activity 18.1 for an

example of deciding which

attribute(s) will be measured.

In this case, it is weight,
volume, length, or area

Make sure students realize

that you can measure objects

in a variety of ways.

2. Select a unit that has that

attribute.

Recognize that the unit must

have length if measuring
length, if measuring weight
the unit must have weight,

and so on.

Give students cards with a
variety of objects and units
and have them match the

unit to the measurable

attribute.

Recognizing the correct unit
is essential in grasping the
meaning of measurement.

Measurement Instruction: Progression of

Experiences

Step

Goal

Type of Activity

Notes

3. Make Comparisons
Students will understand the
attribute to be measured.

Make comparisons based on the
attribute (e.g., longer/shorter). Use

direct comparisons whenever

possible.

When the attribute is

understood, there is no

further need for

comparison activities.

4.Use concrete
representations 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.

Use concrete representations (think
C S A) of individual measuring units to
fill, cover, match, or make the desired

comparison of the attribute of the
object with the unit. The number of
units required to match the object is

its measure.

Begin with nonstandard

units and then progress to
the use of standard units.

Measurement Instruction: Progression of

Experiences

Step

Goal

Type of Activity

Notes

5. Use Measuring Tools

Students will use common
measuring tools with
understanding and flexibility.

Make measuring tools (grouped units)
and compare them with the concrete
representations of an individual unit

to see how the measurement tool

performs the same function.

Without a careful

comparison of the non-
standard (or informal)
tools with the standard

tools, the connection can

be lost.

6. Use Formulas

Students should identify the

formulas for measuring
attributes such as area,

perimeter (length), or volume.

After many experiences with individual

units, have students identify the
patterns of 2 times the length + 2

times the width for finding the

perimeter of a rectangle.

Formulas are only

presented after students
have enough experiences
to identify the connections
between the formula and
the act of matching units.

Measurement Instruction: Progression of

Experiences

Multiple Choice

Which term means, “a number that indicates a comparison between the attribute of the object (or situation, or event) being measured and a given unit of measure with the same attribute?”

• Simulation

• Situation

• Formula

• Measurement

The United States is one of three nations (with Myanmar and Liberia) that have
not adopted the International System of Units (SI) a synonym for the metric
system.
Because of its international importance, the U.S. students must know SI(metric
system) for such things as product design, manufacturing, marketing, and
labeling to participate in the global marketplace.
The customary system uses inches, feet, and so on. Customary units continue
to be important in the United States for various careers (e.g., carpentry) and
contexts, so students need to develop familiarity with both systems.
Select a Unit

Knowing how big units are and determining the precision needed for the
context

Select a Unit

Based on the attribute and includes what is longer/ shorter or
heavier/lighter
Tiling

involves equal partitioning, use as many copies of the unit
as are needed to fill or match the attribute measured

Iteration

using a single copy of the unit

Transitivity principle for indirect measurement

when we use a third object to make a comparison

Use precise language

Make Comparisons

Use Concrete Representations of

Measuring Units

1. Focus directly on the attribute being measured- irregular shapes
can be measured with square tiles or circle counters.

2. Provide a rationale for using standard units. Standard units have
more meaning after students have experienced measuring with
their own collection of nonstandard units.

Introduce Nonstandard Units

Three Broad Goals:

Familiarity with the Unit1.

Ability to select an appropriate unit

2.

Knowledge of the relationship between units

3.

Develop familiarity with standard units:

Comparisons that focus on a single unit
Activities that develop personal referents or benchmarks
for single units or easy multiples of units

Introduce Standard Units

Introduce Standard Units

Use Measuring Tools

Research shows that many students are challenged to give the correct
measure of an object that wasn’t aligned with the end of a scale.
Students also experienced difficulty when the increments on the tool
were not one unit (i.e., two units or half a unit).
Let students construct measurement tools, like their own personal ruler!
Comparison and discussion are important:

Students discuss how measurement with iterating individual units
compares with measurement using a tool.

Multiple Choice

What careers display the importance for students to learn the metric system?

• Education & Politics

• Careers in Medicine

• Construction, Farming, & Interior Design

• Product Design, Manufacturing, Marketing, & Labeling

Use Formulas

Estimation & Approximation

Measurement estimation is the process of using mental and visual information
to measure or make comparisons without using measuring tools.
Reasons for estimation:

Helps students focus on the attribute and the measuring process.
Provides intrinsic motivation.
Develops familiarity with standard units.
Promotes multiplicative reasoning through using a benchmark.

Approximation of a quantity is one that is close to but not exactly the same as
the quantity.
Emphasize the use of approximate language, because measurements are not
exact.

Strategies for Estimating

Strategies for Estimating Measurements

Where in this picture do you

see

Benchmarks or referents

1.

Chunking or using sub

dividing

2.

Iterate units

3.

Measurement Estimation Activity

Activity 18.6 Estimation Scavenger Hunt

Conduct estimation scavenger hunts by giving teams a list of either
nonstandard or standard measurements and having them find things that
are close to those measurements. At first, do not permit the use of
measuring tools. Use the Estimation Scavenger Hunt activity page for
possible ideas and add your own. Let students suggest how to judge results
in terms of accuracy. Students with special needs in math may benefit from
having a reference such as an example of 1 square inch or 1 milliliter.
Multilingual learners are likely to have stronger metric measurement sense
and native U.S. students of customary measures, so group heterogeneously
to build on the strengths of both.

Length

Partitioning with Length Models

Activity 18.10 Estimate and Measure

Make lists of items in the room to measure or use the
Estimating and Measuring with Nonstandard Units Recording
Sheet (which also can be done with standard units). Run a
piece of painters’ tape along the dimension of objects to be
measured. Include curves or other distances that are not
straight lines. Have students estimate before they measure.
Students with special needs in math may need a list of
possible strategies. For example, cluster a row or chain of 10
units as a referent to help them visualize. Then they lay 10
units against the object and make their estimate.

Length

Conversion

The customary system has few patterns or
generalizable rules.
The metric system was systematically created
around powers of ten.
Understanding of the role of the decimal point as
indicating the units position is a powerful concept
for making metric conversions.
Avoid mechanical rules that might be misused,
misunderstood, or forgotten.

Multiple Choice

Why is it difficult for students to move from length measurement to the more abstract area measurement?

• Students may think of area as the length of two sides (length × width), rather than the measure of a surface.

• Students use the wrong measurement tool.

• Students use the wrong end of the ruler.

• Students count hash marks rather than spaces when measuring.

Area

The Relationship Between Area &

Perimeter

Relationship Between Area & Perimeter

Activity 18.19 What’s the Rim?

Have students select objects with a perimeter that they would like the whole
class to measure. First, students estimate the perimeter on the What’s the Rim?
activity page. They also need to choose tools strategically (rulers, cash register
tape, or non-stretching string) and measure the perimeter, noting the unit. The
discussion should include class comparisons of at least one common item that
will provide a basis for exploring any measurement differences. Students should
describe how they measured objects that were larger than the tool they were
using and how they knew when to use a flexible measuring tool (i.e., string or cash
register tape). For students with special needs in math, have them trace the
perimeter with their finger prior to measuring. For a challenge, students can
measure curved perimeters such as the top edge of a wastebasket.

Develop Formulas for Perimeter & Area

Area of Rectangles & Other Parallelograms

Areas of Triangles & Trapezoids

Multiple Choice

As students move to thinking about formulas it supports their conceptual knowledge of how the perimeter of rectangles can be put into general form. What formula below displays a common student error for finding the perimeter?

• P = l + w + l + w

• P = l + w

• P = 2l + 2w

• P = 2(l + w)

Surface Area

Circumference & Area of Circles

Volume & Capacity

Developing Formulas for Volume

Relationship Between Volume & Surface Area

Activity 18.29 Which Silo Holds More?

Give pairs of students two sheets of equal-sized paper. With one sheet, they
make a tube shape (cylinder) by taping the two long edges together. They make a
shorter, fatter cylinder from the other sheet by taping the short edges together.
Then ask, “If these were two silos, would they hold the same amount, or would one
hold more than the other?” To test the conjectures, use a filler such as beans,
popcorn, or pasta. Place the skinny cylinder inside the fat one. Fill the skinny tube
and then lift it up, allowing the filler to empty into the fat cylinder. What do they
find?

Multiple Choice

Volume and capacity are both terms for measures of the “size” of three-dimensional regions. What statement is true of volume but not of capacity?

• Refers to the amount a container will hold.

• Refers to the amount of space of occupied by three-dimensional region.

• Refers to the measure of only liquids.

• Refers to the measure of surface area.

Weight & Mass

Angles

Time

Reading Clocks

Elapsed Time

Multiple Choice

Which approach provides a foundation for problems involving elapsed time?

• Calculating Double Digits within 100

• Word Problem Key Word Strategies

• Using Pencil & Paper to Subtract from 60

• The Mental Process of Counting On in Multiples of 5 Minutes

Money

Activity 18.37 Hundred Chart Money Count
Give students a bottom-up hundred chart and a collection of play
money. Begin with two different coins—for example, a quarter and a
dime. Use place value to represent the 25 cents in the same way
students have previously used the chart (count two rows up and over
five spaces to the right). Place the quarter on the 25 space and then
count 10 more (up one row) and place the dime on 35. The total of the
two coins is 35 cents. Use other coin collections and what students
already know about patterns on the hundred chart to calculate the
total value.

Common Challenges in Measurement & How to Help

Common Challenges in Measurement & How to Help

Common Challenges in Measurement & How to Help

Upcoming Assignments

CRA Work Samples & Draft Due Tonight1.

DIM Multiplication/Division Videos Due Tonight2.

DIM Fractions, Decimals, & Percentages Videos Due 4/13.

DIM Measurement, Geometry, Analysis & Probability Videos
Due 4/8

4.

Error Pattern Analysis & Findings Summary Draft Due 4/155.

DIM Foundations of Algebra Videos Due 4/226.

Final Critical Task Due 4/267.