Seventh Grade
Measurement
Standard 7-5: The student will
demonstrate through the mathematical processes an understanding of how to use
ratio and proportion to solve problems involving scale factors and rates and
how to use one-step unit analysis to convert between and within the U.S.
Customary System and the metric system.
The indicators for this standard
are grouped by the following major concepts:
- Perimeter, Circumference,
Area, and Volume
- Proportional Reasoning
- Equivalencies
- Conversions
The
indicators that support each of those major concepts and an explanation of the
essential learning for each major concept follows.
Perimeter,
Circumference, Area, and Volume
Indicators
7-5.2
Apply strategies and formulas to determine the surface area and
volume of the three-dimensional shapes prism, pyramid, and cylinder.
7-5.3 Generate strategies to determine the
perimeters and areas of
trapezoids.
Fifth grade
students applied strategies and formulas to determine the volume of rectangular
prisms and worked with nets – the two-dimensional representations of both
rectangular prisms and cylinders. Sixth
grade students generated strategies to determine the surface area of a
rectangular prism and a cylinder. Seventh grade students should now apply
formulas to determine the surface area and volume of prisms and cylinders.
Now for the first time they should apply strategies and formulas to determine
the surface area and volume of pyramids. That means seventh grade
students should be fluent applying formulas to find the surface area and volume
of prisms, pyramids, and cylinders.
Fifth grade
students applied formulas to determine the perimeters and areas of triangles,
rectangles, and parallelograms. Sixth grade students focused on the perimeter
and area of irregular shapes. Seventh grade students should use that knowledge
to generate strategies to determine the perimeters and areas of trapezoids. In
fourth grade one of the quadrilaterals students analyzed was trapezoid. As a
result they should have familiarity with that geometric shape. However, a quick
review of the characteristics of trapezoids may be necessary prior to work on
perimeter and area, especially the concept of “height” of a trapezoid.
When generating strategies to
determine the perimeters and areas of trapezoids, students should analyze and
discuss the formula for finding the area of a rectangle, A = l x w. When area formula is changed into an
equivalent form of A = b x h, this form can be useful in developing the area
formula for trapezoids. One strategy to
determine area of a trapezoid is to have students cut out two identical
trapezoids, put them together to form a parallelogram, and relate the area of
the parallelogram formed to the area of the trapezoid. The Pythagorean Theorem
will be introduced in eighth grade and is not a requirement of seventh grade
indicators.
base 2
base 1 base = base 1 + base 2
A
= height x (base 1 + base 2)
Two trapezoids make a parallelogram
with the same height and a base
equal to the sum of bases of the trapezoid. So the formula for are of trapezoid is:
A = ½
h(base 1 + base 2)
Another strategy is to use the area
formulas for a rectangle and a triangle to see why the formula for a trapezoid
works.
Students should spend time investigating problem situations involving perimeters
of trapezoids and be given the opportunity to discover the formulas for
themselves using concrete materials and computer models. The Pythagorean
Theorem is not introduced until the eighth grade. Therefore, when finding
height of a trapezoid other strategies should be used.
As stated in sixth grade; “memorizing”
measurement formulas becomes unnecessary when the mathematics makes sense to
students and they understand the concepts.
Measurement is closely tied to many
topics in geometry and algebra and should not be taught in isolation.
Teacher
Notes: A common error made by students when using formulas comes from no
conceptual understanding of the meaning of height in geometric figures. Before using formulas involving height, students
should discuss the meaning of height of a geometric figure and be able to
identify where a height could be measured.
Students should solve a variety of
problems (including multi-step) involving volumes of prisms, pyramids, and
cylinders.
Premature use of formulas can lead to
a use of words with little or no meaning.
Proportional
Reasoning
Indicators
7-5.2
Use ratio and proportion to solve problems involving scale factors
and rates.
In sixth grade students used
proportions to determine unit rates. Sixth grade students also used a scale to
determine distance. In seventh grade, students extend their understanding of
and use of proportional reasoning (unit rates and scale) to solve problems
using ratio and proportion involving scale factors and rates.
Proportional reasoning involves the
ability to compare ratios as well as predict or produce equivalent ratios. It also requires the students to compare
mentally different pieces of information and to make comparisons of the
quantities involved and the relationships between quantities.
To help in the development of
proportional reasoning, students should be given a variety of tasks which might
include situations involving measurements, prices, geometric and other visual
contexts, and rates. Students should be able to distinguish between
proportional and nonproportional situations or comparisons by giving examples
of each and describing their differences; and they need to be able to relate
proportional reasoning to existing processes such as the concept of unit
fraction to concept of unit rate.
Students need experiences with the
different forms of representing proportional reasoning;
- numeric representation,
- graphic representation, displays
proportional relationship as a line;
- symbolic representation, a
mathematical statement generalizing the relationship of the circle’s
circumference (C) to the circle’s diameter would be written: C = (3.14)(d)
Students
should connect their work on proportionality with their work on area and volume
as they investigate similar objects. They
should understand that if a scale factor describes the relationship of
corresponding lengths in two similar objects then the square of the scale
factor describes how corresponding areas are related and the cube of the scale
factor describes how corresponding volumes are related. Students should also apply proportionality
when they make scale drawings.
Students should be given problems that
involve the construction or interpretation of scale drawings. Problems can be created from maps,
blueprints, science, and literature. For
example, Gulliver’s Travels, by
Jonathan Swift has many passages from which problems can be created for scaling
and proportionality. Another use of
literature is the poem by Shel Silverstein, “One Inch Tall”. Here students can use ratios and
proportionality to make decisions about the statements made in the poem based
on mathematics. They can also explore scale
factors when examining a line in the poem that states that if you were one inch
tall you could wear a thimble on your head. (Other lines can be examined as
well.)
The use of technology can expand the
set of measurement experiences. For
example, using CBLs or other devices to provide additional experiences with
rate and derived measurements. Students
could use CBL to measure a student’s distance from an object as he walks from
or towards it and plot the corresponding points on a distance-time graph. Students would experience many problem
situations since there would be many different paths to look at, different
start-end points, and variations in speed to name a few variations. (This would be an excellent opportunity to
collaborate with the science teacher.)
The introduction and use of the
procedural concept of the cross-product algorithm should only be introduced
after students have had considerable amount of time with developing and
experimenting with more intuitive and conceptual methods for solving
proportions.
Teacher
Notes: For students to be able to
understand and work with proportional relationships, it requires proportional
reasoning. Proportional reasoning is
described as a way of thinking about and recognizing multiplicative
relationships in realistic situations.
Proportional reasoning is a critical element of the middle grades
curriculum. Students need to see many
problem situations and then be able to solve the problem situations through
proportional reasoning. Proportional
situations are based on multiplicative relationships. Proportional reasoning involves quantitative
thinking as well as qualitative thinking.
Connections
To:
Other Seventh
Grade Indicators
7-2.5.
Apply
ratios, rates, and proportions to discounts, taxes, tips,
interest, unit costs, and similar
shapes.
7-3.2.
Analyze
tables and graphs to describe the rate of change between and among quantities.
7-3.3 Understand slope as a constant rate of change.
7-3.6.
Represent
proportional relationships with graphs, tables, and equations.
7-3.7
Classify
relationships as either directly proportional, inversely proportional, or
nonproportional.
7-4.6.
Compare
the areas of similar shapes and the areas of congruent shapes
7-4.7. Explain the proportional relationship among attributes of
similar shapes.
7-4.8 Apply proportional reasoning to find missing attributes of
similar shapes.
Literature
- If You Hopped Like a Frog, David M. Schwartz, Scholastic
Press (A children’s book that compares human feats like jumping with those
same actions in animals. For example, if you hopped like a frog, you could
get to first base in one hop!)
- Actual
Size, Steve
Jenkins, Houghton Mifflin (A book that not only introduces students to a
variety of animals, but also presents those animals, or their body parts,
at their actual size.)
- Gulliver’s Travels, Jonathan Swift (passages
from which problems can be created for scaling and proportionality.)
- “One Inch Tall”, Shel
Silverstein (Here students can use ratios and proportionality to make
decisions about the statements made in the poem based on mathematics.)
Technology
Web Sites
for the concept of Rates
- http://www.utdanacenter.org/mathtoolkit/downloads/support/proportionality.pdf
- http://www.glenbrook.k12.il.us/gbssci/phys/Class/1DKin/U1L1d.html
- http://illuminations.nctm.org/LessonDetail.aspx?id=L584
Equivalencies
Indicators
7-5.4 Recall equivalencies associated with length,
mass and weight, and liquid volume: 1
square yard = 9 square feet, 1 cubic meter = 1 million cubic centimeters, 1
kilometer = 
mile, 1 inch = 2.54 centimeters; 2.2 kilograms = 1 pound; and 1.06
quarts = 1 liter.
In grades two through four, students
have been asked to recall equivalencies with length, time, liquid volume, and
weight within the U.S. Customary System.
In grade five, students recalled equivalencies within the metric system
associated with length, liquid volume, and mass.
Seven grade students extend their learning
of equivalencies to include equivalencies between the U.S. Customary System and
the metric system.
Connections as well as associations
should be made with familiar equivalencies such as from linear units of measure
to square and cubic units of measure.
For example, there are 3 feet in 1 yard so to find the number of square
feet in 1 square yard, have students draw a square with side lengths of 3 feet
(1 yard), then find the area of the square.
The area will be 9 square feet which is equivalent to 1 square yard.
The use of pictures (visual images) to
recall equivalencies is another strategy to teach this indicator. For example, rulers with both customary and
metric units can help with the relationship between cm and inches. By examining the ruler students will be able
to see that 2.54 cm = 1 inch.
The strategies above are used to
provide visual images to develop measurement equivalencies. These visual images will help student be able
to recall the required equivalencies.
Connections
To:
Other Seventh
Grade Indicators
7-5.5
Use one-step unit analysis to convert between and within the U.S.
Customary System and the metric system.
The connection is self explanatory. For details about essential
learning related to 7-5.5 see the major concept of “Conversions” below.
Conversions
Indicators
7-5.5
Use one-step unit analysis to convert between and within the U.S.
Customary System and the metric system.
In grades two through four, students
have been asked to recall equivalencies with length, time, liquid volume, and
weight within the U.S. Customary System.
In grade five, students recalled equivalencies within the metric system
associated with length, liquid volume, and mass.
In order to convert between and within
the measurement systems, students need an understanding of unit equivalencies
such as 12 inches = 1 foot, 60 seconds = 1 minute, 2.54 cm = 1 inch, 1.06
quarts = 1 liter, 1000 ml = 1L to name a few
Students are asked to convert a
quantity expressed in one set of units to another equivalent quantity expressed
in a different set of units. In other
words, using unit (dimensional) analysis means that you will keep the units of
measure throughout the problem. For
example, if there are 2.54 cm to 1 inch, how many centimeters are in a foot? By writing each in fraction format
(horizontally), students can see that all units cancel except the final ones in
the answer.
2.54 cm x 12 in. = 30.48 cm = 30.48 cm in
a foot.
1 in. 1 ft. 1
ft.
Another example: 10 yd to feet
10 yd
x 3 ft = 10 x 3 ft = 30
ft
1 1 yd
1
In other words, you can cancel units just like you do factors when dealing with fractions. This is called “unit cancellation”.
Teacher
Notes: In science unit analysis is
typically called dimensional analysis. Therefore, students should be familiar
with both unit and dimensional analysis terms.
Connections
To:
Other
Seventh Grade Indicators
- As was stated above the
seventh grade indicators under the major concept of “Equivalencies” are related
to the indicators under the major concept of “Conversions” because students
should have recall of basic equivalencies in order to make conversions.