Sixth Grade
Measurement
Standard 6-5: The student will
demonstrate through the mathematical processes an understanding of surface
area; the perimeter and area of irregular shapes; the relationships among the
circumference, diameter, and radius of a circle; the use of proportions to
determine unit rates; and the use of scale to determine distance.
The indicators for this standard
are grouped by the following major concepts:
- Circumference and Area
- Perimeter and Area
- Proportional Reasoning
The indicators that support each of those major concepts and an explanation of the essential learning for each major concept follows.
Circumference and
Area
Indicators
6-5.1 Explain the relationships among the circumference, diameter, and
radius of a circle.
6-5.2 Apply strategies and formulas with an approximation of pi (3.14,
or 
) to find the circumference and area of a circle.
6-5.3 Generate strategies to determine the surface area of a rectangular
prism and a cylinder.
Third grade students identified the
specific attributes of circles: center, radius, circumference, and diameter. Fourth
grade students generated strategies to determine the area of rectangles and
triangles. In the fifth grade, students applied formulas to determine perimeters
and areas of triangles, rectangles, and parallelograms.
The focus for sixth grade is on
explaining the relationship among the circumference, diameter, and radius of a
circle. As a result of exploring those relationships, students should develop
strategies to find the circumference and area of a circle. It is important to
note that in sixth grade, students use the approximation for pi, that is 3.14, or . In eight grade students will apply formulas to determine
the exact circumference and area of a circle.
Students should spend time
investigating problem situations
involving surface area of a rectangular prism and a cylinder as well as
circumference and area of circles and be given the opportunity to generate
strategies to that lead to conceptual understanding of those concepts. In other
words, sixth grade students should not be simply given formulas and asked to
apply them. Application of the formal formulas for those concepts will occur in
seventh grade. “Memorizing” measurement formulas becomes unnecessary when the
mathematics makes sense to students and they understand the concepts. This builds on experiences students had in
fourth and fifth grades when they generated strategies to determine the area of
triangles, rectangles, and parallelograms.
Since sixth grade is the first time
students are introduced to pi. They
should use concrete or computer generated models involving circumference and
diameter to derive approximations for pi.
Students can easily discover the relationship (pi) between the
circumference and diameter of a circle by measuring numerous circles and
computing the ratio of the circumference to the diameter. In addition, students should be able to create
and solve problems that involve circumference and area of a circle when given
the diameter or radius.
Students should be given the
opportunity to develop conceptual understanding of surface area. This can be accomplished by having students
construct models, measure dimensions, estimate the areas. Teachers can have students use their
knowledge about two-dimensional shapes to help with developing an understanding
of the concepts related to three-dimensional objects. In fifth grade students translated between
two-dimensional representations and three-dimensional objects. In other words,
students worked extensively with nets. As a result, it should be easy for
students to use that knowledge, link it to their fifth grade knowledge of area
of a rectangle, and combine with their sixth grade experiences to find the surface
area of a cylinder, for example. The same type of experience can be used for
the rectangular prism.
Measurement is closely tied to many
topics in geometry and algebra and should not be taught in isolation. For example the relationship of measurement
and geometry involves areas and volumes where students find areas or volumes
from lengths or to find lengths from
volumes or areas and lengths and with algebra, the student are provided the
opportunity to develop formulas through the use of patterns relationships and
use the formulas (equations) to find areas and volumes.
Teacher
Notes: Measurement and geometry are
usually presented together. Their
relationship is most evident in the development of formulas for measures of
geometric figures. Students should never
use formulas without participating in the development of those formulas. Developing the formulas and seeing how they
are connected and interrelated is significantly more important than having
students apply formulas to a series of figures.
If students just plug numbers into formulas, the activity becomes
computational and does not aid in the development of measurement sense or
geometry and impedes the student’s conceptual understanding of area and
perimeter when working with regular and irregular shapes.
Premature use of formulas can lead to
a use of words with little or no meaning.
Connections
To:
Literature
Sir
Cumference and the Dragon of Pi: A Math Adventure, Cindy Neuschwander,
Charlesbridge Publishing (When Sir Cumference drinks a potion that turns him
into a dragon, his son, Radius, searches for a magic number known as pi which
will restore him to his former shape.)
Perimeter and Area
Indicators
6-5.4 Apply strategies and procedures to estimate the perimeters and
areas of irregular shapes.
6-5.5 Apply strategies and
procedures of combining and subdividing to find the perimeters and areas of
irregular shapes.
Second grade students predicted the
results of combining and subdividing polygons and circles.
Third grade students analyzed the
results of combining and subdividing circles, triangles, quadrilaterals,
pentagons, hexagons, and octagons as well as generating strategies to determine
the perimeters of polygons.
Fourth grade students analyzed the
perimeter of a polygon and generated strategies to determine the area of
rectangles and triangles.
Fifth grade students applied formulas
to determine the perimeters and areas of triangles, rectangles, and
parallelograms and applied strategies and formulas to determine the volume of
rectangular prisms.
Students should be provided with
drawings of irregular shapes drawn on grid paper of various units (U.S.
Customary and Metric Units). Students
should relate how they found perimeter of regular shapes to how they would find
the perimeter of irregular shapes. Perimeter
of irregular shapes is an estimate. As
students determine the perimeter ask them to justify their answer. Students can use block letters to find
perimeter of irregular shapes by placing them on grid paper to find the perimeter
and then asked to justify their answers.
Other representations can be made by using square tiles, geoboards, and
graphic representations.
As students move towards fluency in
finding perimeters and areas of irregular shapes, students should decompose (subdivide)
the irregular shapes into familiar shapes such as squares, rectangles, and
triangles and find the perimeter and area of each subdivided shape. They can determine the area of the irregular
shape by summing the areas of its composite shapes. To find the perimeter, students could use
string to trace the outline of the shape then place the straighten string on a
ruler to find the perimeter or depending on the shape, they could measure each
side and then sum up the lengths to find the perimeter.
Teacher
Notes: Although students are being
introduced to the concepts of perimeter and area of irregular shapes for the
first time, the goal is to have students progress to fluency.
Connections
to:
Technology
Web Site
Proportional
Reasoning
Indicators
6-5.6 Use proportions to determine unit rates.
6-5.7 Use a
scale to determine distance.
From the elementary concept of
simplifying a fraction, students should be able to make the connection to unit
rates. Simple unit rates with which students are more familiar include miles
per hour, dollars per pound, feet per second. Through graphing various ratio
tables, students should discover unit rates and apply the concept to solve a
variety of problems including finding a better price, currency conversion,
predictions based on samples, similarity, and scale drawing.
The concept of proportional reasoning
should be included throughout many of the other major concepts. Students should be given opportunities to
make connections between ratios, proportional reasoning, and other major
mathematical concepts. Obvious
opportunities include probability (a ratio of favorable outcomes to total
possible outcomes); odds (a ratio of possible outcomes to not possible
outcomes); fractions (equivalent fractions can be found by the same method used
to find equivalent ratios- proportions); graphs (graphs of equivalent ratios is
a straight line); similarity/scale (equal ratios of linear measurement); rate
(a comparison of two different things or values); percent (part to whole
comparison); etc.
They also apply the concept of ratio
when using a scale to determine distance. Students should determine and explain if there
is a relationship between ratios and use proportional reasoning to solve
problems. One of the conclusions that
students must reach when working with equal ratios is that they result from
multiplication or division, not addition or subtraction. The emphasis should be on collecting,
organizing, and analyzing proportional data rather than on rules or formulas.
There are
three ways to express the same scale:
Representive Fraction (R.F.) 1/500,000
commonly written as 1:500,000. One unit
of length on the map represents 500,000 of that same unit on the ground. For example, 1 inch on the map = 500,000
inches on the ground. One cm on the map
represents 500,000 cm on the ground
and so on.
Statement of Equivalency One
inch = 7.9 miles It may be useful to
know how many miles on the ground one inch on the map represents. For this map, the R.F. is 1/500,000, so 1
inch on the map = 500,000 inches on the ground.
To find out how many miles 500,000 inches equals, divide 500,000 inches
by 63,360 inches per mile. The answer is
7.89 miles (or approximately 7.9 miles).
So, one inch on the map = about 7.9 miles on the ground.
Similarly,
to find out how many kilometers on the ground 1 cm on the map represents,
divide 500,000 cm by 100,000 cm per km.
The answer is 1 cm on the map = 5 km on the ground.
Bar Scale A
bar scale is a graphic representation of distance. It shows, visually, what distance on the map
represents a unit of distance on the ground. To use a scale, the student must measure the
distance between two places and use the scale to find the distance by
multiplying. For example, on the map,
the distance between
Map scale
is the same thing as a mathematical ratio.
A ratio is the relationship between two numbers and scale is the
relationship between the distance between the measurement on the map and the
actual distance between the two real places.
Teacher
Notes: Ratios can be used when examining
part-to-whole relationships (fractions), part-to-part relationships, and rates.
Students should have experience with and understand the difference between all
types.
A unit rate is a rate expressed
with a denominator of 1.
miles 180 miles 60 miles
hour
3 hours 1 hour
60
miles per hour
Example 1: Mr. Williams drove 130
miles from
130 miles (÷2) = 65
miles = 65 m.p.h.
2 hours (÷2) 1
hour
Example 2: Julie bought 3 lbs. of peaches
for $1.00. How much did she pay per
pound (cost/lb)?
$1.00 (÷3) = $0.33 = $0.33 per lb. or 33˘ per lb.
3 lb. (÷3)
1 lb.
Extension
Strategies
Use multiplication or division to
find the missing term in a ratio of a proportion (unit rates).
Division: Unit rates (If Mary
drives 150 miles in 3 hours, how far will she drive in 5 hours?)
150
mi (¸ 3) = 50 mi = 50 m.p.h.
3 hr (¸ 3) 1
hr
50 mi x 5 hr = 250 mi = 250
miles in 5 hours
1 hr
x 5 hr 5 hr
Connections
To:
Other
Sixth Grade Indicators
6-2.6 Understand the relationship between ratio/rate and multiplication/ division.