Enhanced Instructional Transition Guide
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Unit 07:
Measurement: Two- and Three-Dimensional (19 days)
Possible Lesson 01 (19 days)
POSSIBLE LESSON 01 (19 days)
This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with districtapproved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time
frame to meet students’ needs. To better understand how your district is implementing CSCOPE lessons, please contact your child’s teacher. (For your convenience, please
find linked the TEA Commissioner’s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.)
Lesson Synopsis:
Students review measurement concepts of two-dimensional figures, which include perimeter and area. Students investigate lateral surface area and total surface area of prisms,
pyramids, and cylinders with the use of concrete models and nets. Students examine models of prisms, cylinders, pyramids, spheres, and cones and connect these models to
formulas for volume. Students connect the visual and geometric representations of three-dimensional figures with the numeric and algebraic representations, while validating and
justifying conclusions and solutions. Students use dimensional analysis to complete conversions within and between measurement systems. Students describe the resulting
effects on perimeter, area, and volume when the dimensions of a shape or solid are changed proportionally.
TEKS:
The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas law. Any standard
that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit.
The TEKS are available on the Texas Education Agency website at http://www.tea.state.tx.us/index2.aspx?id=6148
8.2
Number, operation, and quantitative reasoning.. The student selects and uses appropriate operations to solve problems and justify
solutions. The student is expected to:
8.2B
Use appropriate operations to solve problems involving rational numbers in problem situations.
Readiness Standard
8.2C
Evaluate a solution for reasonableness.
Supporting Standard
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
8.2D
Use multiplication by a given constant factor (including unit rate) to represent and solve problems involving proportional
relationships including conversions between measurement systems.
Supporting Standard
8.7
Geometry and spatial reasoning.. The student uses geometry to model and describe the physical world. The student is expected to:
8.7A
Draw three-dimensional figures from different perspectives.
Supporting Standard
8.7B
Use geometric concepts and properties to solve problems in fields such as art and architecture.
Supporting Standard
8.8
Measurement.. The student uses procedures to determine measures of three-dimensional figures. The student is expected to:
8.8A
Find lateral and total surface area of prisms, pyramids, and cylinders using concrete models and nets (two-dimensional models).
Supporting Standard
8.8B
Connect models of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of these objects.
Supporting Standard
8.8C
Estimate measurements and use formulas to solve application problems involving lateral and total surface area and volume.
Readiness Standard
8.9
Measurement.. The student uses indirect measurement to solve problems. The student is expected to:
8.9B
Use proportional relationships in similar two-dimensional figures or similar three-dimensional figures to find missing
measurements.
Readiness Standard
8.10
Measurement.. The student describes how changes in dimensions affect linear, area, and volume measures. The student is expected
to:
8.10A
Describe the resulting effects on perimeter and area when dimensions of a shape are changed proportionally.
Supporting Standard
8.10B
Describe the resulting effect on volume when dimensions of a solid are changed proportionally.
Supporting Standard
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Underlying Processes and Mathematical Tools:
8.14
Underlying processes and mathematical tools.. The student applies Grade 8 mathematics to solve problems connected to everyday
experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to:
8.14A
Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with
other mathematical topics.
8.14B
Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating
the solution for reasonableness.
8.14C
Select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking
for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards
to solve a problem.
8.14D
Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation,
and number sense to solve problems.
8.15
Underlying processes and mathematical tools.. The student communicates about Grade 8 mathematics through informal and
mathematical language, representations, and models. The student is expected to:
8.15A
Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or
algebraic mathematical models.
Performance Indicator(s):
Grade 08 Mathematics Unit 07 PI 01
Create a presentation (e.g., design plan, blueprint, etc.) that includes scale models and perspective views of several real-life three-dimensional models (e.g., prisms, pyramids,
cylinder, spheres, cones, etc.). Use the proportional relationships of the scale models to estimate and find missing measurements needed to solve formulas to find the surface
area and volume of each scale model. Evaluate the solutions for reasonableness, and justify each solution with a calculator. Explain, in writing, the solution process used to
find missing measures, surface area, and volume, and describe the resulting effect on the measures (e.g., perimeter, area, volume, etc.) when the dimensions of the scale
models are changed proportionally.
Sample Performance Indicator:
As a group project for architectural design class, Karen, Whitney, and Adrienne decided to create a scale model of notable structures in the
United States with a scale of 50 ft = 1 in. They must paint the sides and stabilize each structure by filling it with plaster. (1 foot = 0.3048 meters)
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Create a design plan for each structure that includes a sketch of the three-dimensional building, two-dimensional model of the net, and the formulas needed to determine the
amount of paint (square inches) and plaster needed (cubic inches) for each scale model. Evaluate the solutions for reasonableness, and justify each solution with a calculator.
In addition, explain, in writing, the solution process used to find the dimensions, surface area, and volume, and describe the resulting effect on the amount of paint and plaster
needed if a larger scale model was requested three times larger than Karen, Whitney, and Adrienne’s scale models.
Standard(s): 8.2B , 8.2C , 8.2D , 8.7A , 8.7B , 8.8A , 8.8B , 8.8C , 8.9B , 8.10A , 8.10B , 8.14A , 8.14B , 8.14C , 8.14D , 8.15A
page 4 of 205 Enhanced Instructional Transition Guide
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
ELPS ELPS.c.1H , ELPS.c.4J , ELPS.c.5F , ELPS.c.5G
Key Understanding(s):
Real-life problems may be modeled and measurement application problems may be solved using three-dimensional models built from twodimensional models called nets.
The relationship between the sum of individual areas and the formulas for surface area can be communicated with mathematical language.
Concrete models and nets of three-dimensional figures, such as prisms, pyramids, and cylinders, may be used to communicate the lateral and total
surface area of these objects.
Formulas for lateral and total surface area of prisms, cylinders, and pyramids may be applied to solve problem situations and communicate the
appropriate unit of measure.
Concrete models of three-dimensional figures, such as prisms, pyramids, and cylinders, may be used to communicate the connection between these
objects and formulas for volume.
Formulas for volume of prisms, cylinders, pyramids, spheres, and cones may be applied to solve problem situations and communicate the
appropriate unit of measure.
Real-life problem situations involving lateral surface area, total surface area, and volume are solved using estimation and formulas.
If all dimensions of a figure are changed proportionally, the perimeter ratio is equivalent to the scale factor, the area ratio is equivalent to the scale
factor squared, and the volume ratio is equivalent to the scale factor cubed. If all dimensions are not changed, the perimeter, area, and volume is
affected, but must be recalculated.
The process of evaluating a formula that must be rewritten to solve for another variable involves using a plan or strategy to keep the values on both
sides of the formula equally balanced.
Misconception(s):
Some students may think that the side which rests on the bottom is the base of the prism instead of identifying the base as the two parallel faces
which are polygons.
Some students may think the resulting effects on area, when dimensions are changed proportionally by a scale factor, would be to multiply the area
by “two times” the scale factor instead of multiplying the area by the “square of” the scale factor.
Some students may think the resulting effects on volume, when dimensions are changed proportionally by a scale factor, would be to multiply the
volume by “three times” the scale factor, instead of multiplying the volume by the “cube of” the scale factor.
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Some students may confuse the context of lateral surface area in problem situations.
Underdeveloped Concept(s):
Some students may think the lateral surface of a cylinder is a rectangle, instead of a curved surface because of the net model.
Some students may think that “B” is synonymous with “b”, the length of the base, instead of “B”, which represents the area of the base of a three­
dimensional figure.
Vocabulary of Instruction:
composite figure
dimensional analysis
lateral surface area
total surface area
volume
Materials List:
calculator (scientific) (1 per student, 1 per teacher)
cardstock (1 sheet per teacher)
cardstock (3 sheets per 2 students)
cardstock (4 sheets per 6 students)
chart paper (1 per 2 students)
color tiles (12 red, 12 blue) (1 set per 2 students)
Geometric Match Up Cards Chart (1 per 2 students) (previously created)
linking cubes or centimeter cubes (150 cubes per 3 students)
map pencils (1 yellow, 1 blue, 1 green) (1 set per student)
markers (1 set per 2 students)
math journal (1 per student)
plastic zip bag (sandwich sized) (1 per 2 students)
plastic zip bag (sandwich sized) (1 per student)
rectangular prism (real)
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
ruler (standard) (1 per student)
scissors (1 per student)
scissors (1 per teacher)
STAAR Grade 8 Mathematics Reference Materials (1 per student)
tape (clear) (1 roll per 2 students)
tape (clear) (1 roll per 2 students)
tape (clear) (1 roll per teacher)
Three-Dimensional Figures (1 set per 2 students) (previously created)
three-dimensional geometric model (cube) (1 per teacher)
wrapping paper (1 roll per teacher)
Attachments:
All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments
that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website.
Sample of Mathematics and Measurement KEY
Sample of Mathematics and Measurement
Irregular Measurement and Nets KEY
Irregular Measurement and Nets
Composite Figures KEY
Composite Figures
Dimensional Analysis Practice KEY
Dimensional Analysis Practice
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Dimensional Analysis KEY
Dimensional Analysis
Geometric Match Up Cards KEY
Geometric Match Up Cards
Geometric Questions KEY
Geometric Questions
Templates for Three-Dimensional Figures KEY
Templates for Three-Dimensional Figures
Nets for Three-Dimensional Figure Challenge KEY
Nets for Three-Dimensional Figure Challenge
Centimeter Grid Paper
Surface Area KEY
Surface Area
Net Puzzle KEY
Net Puzzle
Volume Notes
Volume KEY
Volume
page 8 of 205 Enhanced Instructional Transition Guide
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Problem Solving with Measurement KEY
Problem Solving with Measurement
Surface Area and Volume KEY
Surface Area and Volume
Jane’s Yard
Exploring Measurements KEY
Exploring Measurements
My Garden KEY
My Garden
Geometric Gravel Garden KEY
Geometric Gravel Garden
The Empire State Building
Dimensional Changes on Volume KEY
Dimensional Changes on Volume
Dimensional Changes KEY
Dimensional Changes
Measurement and Dimensional Change KEY
Measurement and Dimensional Change
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Measurement and Dimensional Change Practice KEY
Measurement and Dimensional Change Practice
GETTING READY FOR INSTRUCTION
Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to
teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using
the Content Creator in the Tools Tab. All originally authored lessons can be saved in the “My CSCOPE” Tab within the “My Content” area. Suggested
Day
1
Suggested Instructional Procedures
Notes for Teacher
Topics:
Spiraling Review
Area and perimeter
Using formulas found on the STAAR Grade 8 Mathematics Reference Materials
Engage 1
Students use experience and reasoning skills to review order of operations and formulas for the area and
perimeter or circumference of two-dimensional figures.
Instructional Procedures:
1. Prior to instruction, create a class resource: Sample of Mathematics and Measurement
for each teacher by copying on cardstock and cutting apart.
2. Place students in 8 groups. Distribute 1 card from class resource: Sample of Mathematics
and Measurement to each group and the STAAR Grade 8 Mathematics Reference Materials
ATTACHMENTS
Teacher Resource: Sample of
Mathematics and Measurement
KEY (1 per teacher)
Class Resource: Sample of
Mathematics and Measurement
(1 set per teacher)
Teacher Resource: Sample of
Mathematics and Measurement
(1 per teacher)
to each student. Instruct students to analyze their problem, identify which formula from the
STAAR Grade 8 Mathematics Reference Materials can be used to solve the problem, and
page 10 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
record their solution process in their math journal. Allow 2 minutes for students to complete
the activity. Monitor and assess students to check for understanding. Facilitate group
discussions about selecting formulas and developing a solution process, as needed.
Ask:
Where have you seen these formulas before? Answers may vary. On the STAAR
Grade 8 Mathematics Reference Materials; in class; etc.
What do you notice about the chart? Answers may vary. There are four cells that are
related: problem situation, formulas that can be used to solve the situation, a pictorial
representation of the situation, and a place to calculate the solution for the problem
situation; etc.
Are there any formulas that require the use of order of operations? If so, which
ones? (The formula for finding the area of a trapezoid is an example of order of
operations.)
What is the correct order of operations you would use for the formula used to
calculate the area of a trapezoid:
? (Parentheses: b1 + b2; then multiply this
Notes for Teacher
MATERIALS
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
cardstock (1 sheet per teacher)
scissors (1 per teacher)
math journal (1 per student)
TEACHER NOTE
Students tend to confuse perimeter and area. As
you ask the questions regarding these two
concepts physically motion with your forefinger
and outline the region’s edges for perimeter
(distance around the region) and move the palm
of your hand across the region’s surface for area
(covering of the surface of an enclosed region).
sum by the height, h; and finally divide this product by 2.)
TEACHER NOTE
3. Display teacher resource: Sample of Mathematics and Measurement. Facilitate a class
discussion for student groups to explain their problem situation, identify the formula chosen, a
justification for why that formula was chosen, and a description of how the formula was used in
the solution process.
STAAR Grade 8 Mathematics Reference
Materials should be made available to students
at all times.
4. Facilitate a class discussion emphasizing that perimeter is a one-dimensional measure
(length), meaning the solution is labeled using linear units and area is a two-dimensional
page 11 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
measure (length and width), meaning the solution is labeled using square units.
Ask:
What is the difference between perimeter and area? (The perimeter is the distance
(in units) around a region. The area is the covering of the surface (in square units)
enclosed by the perimeter of the region.)
Topics:
Composite two-dimensional figures
Explore/Explain 1
Students identify and use appropriate formulas to find the area and perimeter of composite two-dimensional
figures from problems involving art and architecture.
Instructional Procedures:
1. Display teacher resource: Irregular Measurement and Nets. Facilitate a class discussion
about irregular figures and nets.
Ask:
What two shapes do you see in problem 1? (I see a rectangle and a triangle.)
Where will the line need to be drawn to create a triangle? (Extend the vertical line
created by the 2 ft portion of the height of the figure.)
How would you find the total area of the figure? (Calculate the area of the rectangle
and the triangle and then add both areas together.)
What is the formula for finding the area of a rectangle? (Area = length x width.)
ATTACHMENTS
Teacher Resource: Irregular
Measurement and Nets KEY (1
per teacher)
Teacher Resource: Irregular
Measurement and Nets (1 per
teacher)
Handout: Irregular Measurement
and Nets (1 per student)
Teacher Resource: Composite
Figures KEY (1 per teacher)
Teacher Resource: Composite
Figures (1 per teacher)
Handout: Composite Figures (1
per student)
MATERIALS
page 12 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
What is the formula for finding the area of a triangle? (Area = one-half x base x
height.)
How would you find the height of the triangle? (Subtract two from ten.)
Describe how you would calculate the perimeter? (Add each length around the
figure.)
How would you find the length of the slanted side of the figure? (The slanted side
of the figure is the hypotenuse of a right triangle, so I would use the Pythagorean
Theorem: a 2 + b 2 = c 2. c =
=
= 10.)
How would you use math symbols to describe how to calculate the perimeter? (16
ft + 10 ft + 10 ft + 2 ft + 10 ft = 48 ft)
Notes for Teacher
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
calculator (scientific) (1 per
student)
scissors (optional) (1 per student)
TEACHER NOTE
Irregular figures, composite figures, and nets
(two-dimensional models for three-dimensional
figures) are created by combining two or more
2. Place students in pairs. Distribute the STAAR Grade 8 Mathematics Reference Materials,
calculator, and handout: Irregular Measurement and Nets to each student. Instruct
student pairs to use their STAAR Grade 8 Mathematics Reference Materials to identify the
formula(s) that can be used to solve the problem, use the formula(s) to find the solution to
each problem, and justify all solutions with a calculator. Allow time for students to complete the
activity. Monitor and assess student pairs to check for understanding. Facilitate a class
discussion to debrief student solutions.
polygons or circles. Finding the area of an
irregular figure, composite figure, or a net
requires adding the areas of all the associated
figures together. The solution process may also
require subtracting the area of the figure that
may not be included in the requirements of the
problem. Working with irregular figures,
composite figures, and nets provides an
3. Display teacher resource: Composite Figures. Facilitate a class discussion about composite
figures.
Ask:
opportunity for review of calculating the area of a
rectangle, square, triangle, circle, and trapezoid.
page 13 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
How is problem 1 different from the problems you just completed? Answers may
vary. There is a rectangle within a rectangle. The problem wants us to take away a portion
of the area. We will need to subtract areas, rather than adding them; etc.
What two shapes are in this problem? (There are two rectangles.)
How can you find the area not covered by the picture? (Find the total area of the
rectangular board and the area of the picture and then subtract the two to find the
difference.)
What problem do you notice with the measurements units? (The dimensions are
listed in both inches and feet.)
What do you need to do before finding the areas? (Convert 1 ft to 12 in.)
What written description, using math symbols, could be used to describe how to
solve the problem? ((15 x 12) – (12 x 9) = 180 – 108 = 72 square inches)
Notes for Teacher
TEACHER NOTE
For struggling students, allow them to create the
figure on a sheet of paper and cut out the part(s)
of the figure that may not be needed.
TEACHER NOTE
Stress to students that square units are used to
measure area.
State Resources
MTR 6 – 8: Semantic Feature Analysis Charts –
And That’s a Wrap
4. Distribute handout: Composite Figures to each student as independent practice and/or
homework.
2 – 3
Topics:
Spiraling Review
Dimensional analysis
Conversions within and between customary and metric measuring systems
Explore/Explain 2
Students discuss dimensional analysis as a method for conversions within and between customary and metric
measuring systems.
ATTACHMENTS
Teacher Resource: Dimensional
Analysis Practice KEY (1 per
teacher)
Teacher Resource: Dimensional
page 14 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Instructional Procedures:
1. Prior to instruction, display the following for the class to see:
1 ≈ 2.54 cm
2. Display teacher resource: Dimensional Analysis Practice.
3. Distribute a calculator and handout: Dimensional Analysis Practice to each student.
Instruct students to complete problems 1 – 3 on their handout. Allow time for students to
complete the activity. Monitor and assess student pairs to check for understanding. Facilitate
a class discussion to debrief student solutions.
Ask:
According to the diagram, how many centimeters are equivalent to 1 inch?
Explain. Answers may vary. There are between 2 and 3 cm since the diagram shows the
length of an inch ending between 2 and 3 cm. There are 2.5 cm since the diagram shows
the length of an inch ending approximately halfway between 2 and 3 cm; etc.
4. Using the displayed teacher resource: Dimensional Analysis Practice, facilitate a class
discussion about problems 4 – 8 by posing the problem and instructing students to work on
the problem individually. Allow 2 minutes for students to complete each problem. Monitor and
assess students to check for understanding. Facilitate a class discussion to debrief student
solutions and model the “cancelling of units” by striking through the units.
Ask:
Notes for Teacher
Analysis Practice (1 per teacher)
Handout: Dimensional Analysis
Practice (1 per student)
Teacher Resource: Sample of
Mathematics and Measurement
KEY (1 per teacher)
Teacher Resource: Sample of
Mathematics and Measurement
(1 per teacher)
MATERIALS
calculator (scientific) (1 per student,
1 per teacher)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
TEACHER NOTE
Students were first introduced to dimensional
analysis in Grade 7 where dimensional analysis
was used as a method to perform conversions
within the same measurement system.
page 15 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Notes for Teacher
How can you set up 1 inch = 2.54 centimeters as a ratio?
How can you use that ratio to determine the number of inches in 32 cm? (By
setting up a proportion,
)
TEACHER NOTE
Dimensional analysis is a conceptual tool often
used in science and math to complete
computations and justify reasonableness of
5. Using the displayed teacher resource: Dimensional Analysis Practice and a calculator,
demonstrate how these ratios can be used to make conversions, depending on which units
need to cancel. Explain to students that dimensional analysis allows you to determine which
way to write the ratio so that units will cancel, leaving only the desired unit in the answer.
results by analyzing the dimensions or units.
This is also called the Factor-Label method or
the Unit Factor method. Dimensional analysis is
a problem-solving technique that uses the fact
that any number or expression can be multiplied
by 1 without changing its value.
6. Using the displayed teacher resource: Dimensional Analysis Practice and a calculator,
demonstrate that sometimes dimensional analysis requires the multiplication by more than
one ratio that represents 1.
page 16 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
7. Display teacher resource: Sample of Mathematics and Measurement.
8. Place students in pairs. Instruct student pairs to use the displayed information to complete
problems 9 – 14 on their handout: Dimensional Analysis Practice and verify all solutions
with a calculator. Allow time for students to complete the activity. Monitor and assess student
pairs to check for understanding. Facilitate a class discussion to debrief student solutions and
clarify dimensional analysis.
Ask:
What unit is given in the problem? Answers may vary. Inches; etc.
What is the desired unit the problem is asking us to find? Answers may vary.
Centimeters; etc.
How does the desired unit of measure compare to the original unit of measure?
Answers may vary. I am given meters and need to convert to feet. A foot is shorter than a
meter, so the conversion should have more feet than the original unit; etc.
What resource(s) may you use to gather other information? Answers may vary. I
could use the STAAR Grade 8 Mathematics Reference Materials; etc.
How can you set up the problem for dimensional analysis so only the desired unit
is in the answer? Answers may vary. As a proportional relationship where inches will
cancel out; etc.
What units will you cancel so only the desired unit is in the answer? Answers may
vary. Inches; etc.
How can you determine if the converted unit of measure is reasonable? Answers
may vary. Since it takes more centimeters to equal an inch, when I convert to inches, there
should be a fewer number of inches; etc.
Why do you convert each individual dimension and not the area? Answers may
page 17 of 205 Enhanced Instructional Transition Guide
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
vary. In area calculations, I multiply the dimensions. If I just convert the area answer, I have
not taken into consideration that each dimension affects the area and needs to be
converted; etc.
Topics:
Dimensional analysis
Conversions within and between customary and metric measuring systems
Explore/Explain 3
ATTACHMENTS
Teacher Resource: Dimensional
Analysis KEY (1 per teacher)
Handout: Dimensional Analysis (1
per student)
Students demonstrate an understanding of dimensional analysis as they complete a set of problems requiring
conversions within and between customary and metric measuring systems.
MATERIALS
Instructional Procedures:
1. Place students in pairs. Distribute a calculator, STAAR Grade 8 Mathematics Reference
Materials, and handout: Dimensional Analysis to each student. Instruct student pairs to use
dimensional analysis to convert the given unit and verify all solutions with a calculator. Allow
time for students to complete the activity. Monitor and assess student pairs to check for
understanding. Facilitate a class discussion to debrief student solutions, solidify vocabulary,
and clarify misconceptions regarding dimensional analysis.
Ask:
What unit is given in the problem? Answers may vary. Yards; etc.
What is the desired unit the problem is asking us to find? Answers may vary.
Meters; etc.
calculator (scientific) (1 per
student)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
State Resources
TEXTEAMS: MS Proportionality – Centimeters:
Inches; Lost and Gained
page 18 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
Is there other information you need to set up the problem? Answers may vary. How
many meters are in a yard; etc.
How can you set up the problem for dimensional analysis so only the desired unit
is in the answer? Answers may vary. You set up the problem so that centimeters and
inches will cancel, leaving only yards; etc.
What units will you cancel so only the desired unit is in the answer? Answers may
vary. Centimeters and inches; etc.
4
Topics:
Spiraling Review
Attributes of three-dimensional figures
Engage 2
Students use logic and reasoning skills to review the attributes and characteristics of three-dimensional figures.
Instructional Procedures:
1. Prior to instruction, create a card set: Geometric Match Up Cards for every 2 students by
copying on cardstock, cutting apart, and placing in a plastic zip bag.
2. Display a model of a cube. Instruct students to write down everything they know about the
figure in their math journal. Facilitate a class discussion about the attributes of a cube.
Ask:
What do you know about this geometric figure? Answers may vary. A cube has six
square sides, with right angles; etc.
ATTACHMENTS
Teacher Resource: Geometric
Match Up Cards KEY (1 per
teacher)
Teacher Resource: Geometric
Match Up Cards (1 per teacher)
Card Set: Geometric Match Up
Cards (1 per 2 students)
Teacher Resource: Geometric
Questions KEY (1 per teacher)
Handout: Geometric Questions
(1 per student)
page 19 of 205 Enhanced Instructional Transition Guide
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
What is one thing you can state concerning this geometric figure? Answers may
vary. It has 12 congruent edges; etc.
3. Display page 1 of teacher resource: Geometric Match Up Cards and facilitate a class
discussion about three-dimensional figures.
Ask:
Is there anything listed in this chart that is not included in your journal list?
Answers may vary. The cube had 8 vertices; etc.
How does this chart help display the information about this geometric figure?
Answers may vary. The chart organizes the items listed and makes the information easier
to read; etc.
4. Place students in pairs. Distribute a sheet of chart paper, markers, roll of tape, and a card
set: Geometric Match Up Cards to each pair. Instruct student pairs to match and organize
cards for the three-dimensional figures, create a table on their sheet of chart paper to display
the name, definition, three-dimensional model of the figure, and two-dimensional net for each
three-dimensional figure. Remind students that there will be some unused cards. Allow time
for students to complete the activity. Monitor and assess student pairs to check for
understanding. Facilitate a class discussion to debrief student solutions and clarify
vocabulary.
Ask:
Notes for Teacher
MATERIALS
cardstock (3 sheets per 2 students)
scissors (1 per teacher)
plastic zip bag (sandwich sized) (1
per 2 students)
three-dimensional geometric model
(cube) (1 per teacher)
math journal (1 per student)
chart paper (1 per 2 students)
markers (1 set per 2 students)
tape (clear) (1 roll per 2 students)
TEACHER NOTE
Display charts on the walls creating “word walls”
for ELL students.
What part of a three-dimensional figure represents the faces, vertices, and
edges? Answers may vary.
What do you know about a prism? A pyramid? (A prism is a solid formed by polygons
page 20 of 205 Enhanced Instructional Transition Guide
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Suggested Instructional Procedures
Notes for Teacher
connected at their edges. They have two bases that are congruent parallel polygons and
lateral faces that form rectangles connected by corresponding vertices to the bases.) (A
pyramid is a solid formed by polygons connected at their edges with one base that is a
polygon. The lateral faces form triangles connecting the vertices of the base to the
common vertex of the lateral faces.)
What do you know about a cylinder? A cone? (A cylinder is a solid with two congruent
circular bases that are parallel and one curved lateral surface.) (A cone is a solid with one
circular base and a curved lateral surface that meets at a vertex point not on the base.)
How is the shape of the base used to determine the name of the threedimensional figure? (The name of the figure describes the base of the threedimensional figure.)
What is the difference between a prism and a pyramid? (A prism has two congruent,
parallel bases and rectangular lateral faces. A pyramid has one polygonal base and
triangular lateral faces.)
5. Distribute handout: Geometric Questions to each student. Instruct students to use the
information on their chart paper to identify the attributes of each three-dimensional figure.
Allow time for students to complete the activity. Monitor and assess students to check for
understanding. Facilitate a class discussion to debrief student solutions and clarify any
misconceptions and/or vocabulary terms.
6. Instruct students to post their Geometric Match Up Cards Chart around the classroom for
future instruction.
5 – 6
Topics:
Spiraling Review
page 21 of 205 Enhanced Instructional Transition Guide
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Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
Creating three-dimensional models from two-dimensional nets
Explore/Explain 4
Students measure and record dimensions of the following two-dimensional nets: cube, rectangular prism,
triangular prism, pentagonal prism, hexagonal prism, octagonal prism, triangular pyramid, rectangular pyramid,
pentagonal pyramid, hexagonal pyramid, cylinder, cone, and trapezoidal prism. Students create a threedimensional model of each two-dimensional net.
Instructional Procedures:
1. Place students in pairs. Distribute a ruler and pair of scissors to each student, and a roll of
tape and handout: Templates for Three-Dimensional Figures to each student pair.
Instruct students to measure the dimensions of each three-dimensional figure, record each
measure on the net, cut out each two-dimensional net, and use tape to assemble each threedimensional figure. Allow time for students to complete the activity. Monitor and assess
student pairs to check for understanding and ensure that measurements are clearly recorded
on the face edge of each model, not on the tab. Facilitate a class discussion about the
models of the three-dimensional figures, referencing the previously created Geometric Match
Up Charts posted to clarify any misconceptions and/or vocabulary.
Ask:
ATTACHMENTS
Teacher Resource: Templates for
Three-Dimensional Figures KEY
(1 per teacher)
Handout: Templates for ThreeDimensional Figures (1 per 2
students)
MATERIALS
ruler (standard) (1 per student)
scissors (1 per student)
tape (clear) (1 roll per 2 students)
Geometric Match Up Cards Chart (1
per 2 students) (previously created)
TEACHER NOTE
What geometric figures make up the three-dimensional figure? Answers may vary.
Circle, rectangle, triangle, trapezoid; etc.
How does a circle differ from a rectangle? A triangle? (A circle has no vertices. A
circle is not a polygon. A rectangle and a triangle are polygons.)
What are some similarities and differences between the three-dimensional
Some of the two-dimensional nets have tabs
and some do not. The tabs are not part of the
total surface area. Tabs are added to make it
easier for students to create the threedimensional model. Students need to be able to
page 22 of 205 Enhanced Instructional Transition Guide
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Suggested Instructional Procedures
figures posted on the Geometric Match Up Cards? Answers may vary. Some of the
three-dimensional figures are prisms; some are pyramids, cylinders, and cones. Prisms
have rectangular, lateral faces. Pyramids have triangular, lateral faces. Cylinders and
cones have a curved, lateral surface; etc.
What is the relationship between the net (two-dimensional model) and the model
(three-dimensional model) of a figure? (The net is a flattened view of the threedimensional model if it is disassembled along certain edges of the model.)
How do you create a net for a three-dimensional figure? Answers may vary. Trace
the outside of each lateral face, rolling the three-dimensional figure in order to trace each
face; etc.
How can you verify your two-dimensional model is a net for your threedimensional figure? Answers may vary. Cut out the model and fold it to see if it makes
the desired three-dimensional figure; etc.
Is there a different net you can draw for this same three-dimensional figure? Yes,
nets will vary; etc.
Notes for Teacher
recognize the nets of three-dimensional figures
with and without tabs.
TEACHER NOTE
In order to reproduce materials requiring linear
measure that are consistent with intended
measurements noted on the KEY, set the print
menu to print the handout at 100% by selecting
“None” or “Actual size” under the Page
Scaling/Size option.
2. Collect the set of Three-Dimensional Figures created by each group for future instruction.
Topics:
Creating two-dimensional nets from three-dimensional figures
Explore/Explain 5
Students illustrate various two-dimensional nets that can be used to produce a specified three-dimensional
figure. Students justify illustrated nets by creating a three-dimensional model from their two-dimensional net.
ATTACHMENTS
Teacher Resource: Nets for
Three-Dimensional Figure
Challenge KEY (1 per teacher)
Handout: Nets for ThreeDimensional Figure Challenge
(1 per student)
page 23 of 205 Enhanced Instructional Transition Guide
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Instructional Procedures:
1. Place students in pairs. Distribute a pair of scissors, handout: Nets for Three-Dimensional
Figure Challenge, and handout: Centimeter Grid Paper to each student. Instruct students
to illustrate and identify the various two-dimensional nets of a cube and other threedimensional figures. Allow time for students to complete the activity. Monitor and assess
student pairs to check for understanding. Facilitate a class discussion to debrief student
solutions.
Ask:
Notes for Teacher
Handout: Centimeter Grid Paper
(1 per student)
MATERIALS
scissors (1 per student)
TEACHER NOTE
What geometric figures make up the net for this prism? (rectangles)
How can you verify your net will fold into this three-dimensional figure? (Fold the
net to see if it creates the model.)
How does the net help you name the three-dimensional figure? (Prism: Look for
two congruent bases and look for rectangles for the lateral faces. Pyramids: Look for 1
polygon base and look for triangles for the lateral faces. Cylinder: Look for 2 circular
bases and a rectangular shape that will fold into a curved lateral surface. Cone: Look for 1
circular base and a curved lateral surface.)
Provide centimeter grid paper for students to use
to brainstorm different ways to draw nets for
problems 1 and 2 on handout: Nets for ThreeDimensional Figure Challenge. Students should
cut out the nets to prove their nets fold into the
given three-dimensional figures.
TEACHER NOTE
In order to reproduce materials requiring linear
measure that are consistent with intended
measurements noted on the KEY, set the print
menu to print the handout at 100% by selecting
“None” or “Actual size” under the Page
Scaling/Size option.
7 – 9
Topics:
page 24 of 205 Enhanced Instructional Transition Guide
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Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
Spiraling Review
Introduction to lateral versus total surface area
Engage 3
Students use logic and reasoning skills with class discussion to determine the difference between lateral and
total surface area by analyzing a real-life object and the formulas on the STAAR Grade 8 Mathematics Reference
Materials.
Instructional Procedures:
1. Prior to instruction wrap a rectangular prism with wrapping paper to represent a present.
2. Facilitate a class discussion about rectangular prisms.
Ask:
What type of polygon forms the base of this wrapped gift? (rectangle)
How do you calculate the area of a rectangle? (Multiply the length times the width: A
=lw.)
What type of polygon forms the lateral faces of the wrapped gift? (rectangle)
How many faces does this rectangular prism have? (Six faces: 2 rectangular bases
and 4 lateral faces.)
How do you find the area of each face/side of a rectangular prism? (Multiply the
ATTACHMENTS
Teacher Resource: Nets for
Three-Dimensional Figure
Challenge KEY (1 per teacher)
Handout: Nets for ThreeDimensional Figure Challenge
(1 per student)
MATERIALS
rectangular prism (real)
wrapping paper (1 roll per teacher)
scissors (1 per teacher)
tape (clear) (1 roll per teacher)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
page 25 of 205 Enhanced Instructional Transition Guide
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Suggested Instructional Procedures
length and width of each face: A = lw.)
How do you find the total surface area of a rectangular prism? Lateral surface
area? (Sum the areas of the six rectangular faces of the rectangular prism or use the
formula from the STAAR Grade 8 Mathematics Reference Materials: Area of lateral faces +
Area of the two bases: S = Ph + 2B, where P represents the perimeter of the Base of the
three-dimensional figure and B represents the Area of the Base of the three-dimensional
figure. Lateral surface area is the sum of the areas of the side rectangular faces of the
rectangular prism: S = Ph.)
What are the dimensions of the rectangular prism? (length, width, and height)
What dimensions are used to calculate the lateral area? The total surface area?
(Each side of the base and the height of the prism are used to calculate the lateral area.
The dimensions of the bases and each side of the base and the height of the prism are
used to calculate the total surface area.)
What is the perimeter of the base? (2l + 2w = P)
What is the area of the base? (lw = B)
What is the difference between the lateral area and the total surface area? (The
lateral area is the sum of the areas of the lateral faces. The total surface area is the sum
of the areas of the two bases and the sum of the areas of the lateral faces.)
Notes for Teacher
TEACHER NOTE
Some students may have difficulty calculating
the surface area by using a formula or by finding
the area of the composite figure formed by the
net.
TEACHER NOTE
The teacher may choose to explore why “Ph” in
the formula for the total surface area of a prism:
S = Ph + 2B calculates the lateral surface area.
When finding the area of each lateral face
(rectangle) of a prism, the formula “lw” could be
used to calculate the area of each rectangle.
Each lateral face of a prism has the height of the
rectangular prism as one of its dimensions and
the length of a side of the Base of the
rectangular prism as its other dimension.
Calculating the perimeter of the Base and then
multiplying by the height may be shown
mathematically as: h(Side 1 + Side 2 + Side 3,
etc.)
Using the Distributive Property of Multiplication
with respect to Addition yields: h(Side 1) + h(Side
2) + h(Side 3), etc. This method would be like
page 26 of 205 Enhanced Instructional Transition Guide
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Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
finding the area of each separate rectangular
lateral face and then adding the areas together.
State Resources
TEXTEAMS: MS Proportionality – A Real Cover
Up
Topics:
Calculating lateral and total surface area
Explore/Explain 6
Students use previously constructed three-dimensional models of a rectangular prism, triangular prism,
hexagonal prism, cylinder, rectangular pyramid, and trapezoidal prism to examine relationships between the
attributes of the figure and formulas to find the lateral and total surface area. Students use a three-dimensional
model to create a sketch and identify the correct name of the figure, figures that form the base(s) and lateral
ATTACHMENTS
Teacher Resource: Surface Area
KEY (1 per teacher)
Handout: Surface Area (1 per
student)
Teacher Resource (optional): Nets
for Three-Dimensional Figures
KEY (1 per teacher)
surface of the figure, area formulas for each figure that forms the base(s) and lateral surfaces, and a calculation
of the lateral and total surface areas.
MATERIALS
Instructional Procedures:
1. Place students in pairs. Distribute the STAAR Grade 8 Mathematics Reference Materials and
the rectangular prism, triangular prism, hexagonal prism, cylinder, rectangular pyramid, and
trapezoidal prism, from the previously created set of Three-Dimensional Figures, to each pair.
Instruct students to use the measurements previously recorded on each figure to find both the
Three-Dimensional Figures (1 set
per 2 students) (previously created)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
math journal (optional) (1 per
page 27 of 205 Enhanced Instructional Transition Guide
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
lateral and total surface areas for each three-dimensional figure. Allow time for students to
complete the activity. Monitor and assess student pairs to check for understanding.
2. Distribute handout: Surface Area to each student. Instruct students to name each threedimensional figure, create a sketch of each three-dimensional figure, identify the figures that
form the base, and the lateral and total surface area for each figure. Allow time for students to
complete the activity. Monitor and assess student pairs to check for understanding. Facilitate
a class discussion about pyramids and prisms, as needed.
Ask:
What figure(s) is the base of the model? (Pyramids and prisms: a type of polygon;
cylinder and cone: circle.)
What is the relationship between the name of the prism/pyramid and the shape of
the base? (The name of the base is part of the name of the three-dimensional figure.)
If the formula to calculate the area of the base for your three-dimensional model
is not shown on your STAAR Grade 8 Mathematics Reference Materials, how can
you divide the base of the three-dimensional model into a figure with a formula
provided on the formula chart? Answers may vary. For a pentagon: A diagonal line can
be drawn to connect opposite vertices in the pentagon, creating a triangle and a trapezoid;
etc.
After dividing the base of your three-dimensional model, what formula(s) would
you use from your STAAR Grade 8 Mathematics Reference Materials to calculate
the area? Explain. Answers may vary. For a pentagon, I divided it into a triangle and a
trapezoid. Use the following formulas to calculate the two regions, then add the two areas
student)
TEACHER NOTE
As an optional activity, the teacher may allow
each pair of students to select 6 of the 13 threedimensional models from their previously
created set of Three-Dimensional Models to
complete the handout: Surface Area. Students
may record the information in their math journal
using the blank handout: Surface Area, as a
guide.
TEACHER NOTE
Teacher resource: Surface Area KEY shows
solutions for a rectangular prism, triangular
prism, hexagonal prism, cylinder, rectangular
pyramid, and trapezoidal prism. Teacher
resource: Nets for Three-Dimensional Figures
KEY shows the dimensions for each net,
measured to the nearest half-centimeter. When
3.14 was used for
, the rounding occurred after
the calculations.
page 28 of 205 Enhanced Instructional Transition Guide
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Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
together to get the area of the pentagon’s base:
TEACHER NOTE
Triangle’s area: A = bh +Trapezoid’s area: A= (b1 + b2)h; etc.
The ruler on the STAAR Grade 8 Mathematics
Reference Materials can be used in place of a
ruler to familiarize students to its use during the
STAAR test.
TEACHER NOTE
In order to produce rulers that are consistent
with the rulers on the STAAR Mathematics
Reference Materials, follow these steps:
1. Set the print menu to print the pages
at 100% by selecting "None" or "Actual
size" under the Page Scaling/Size
option.
2. Print on paper that is wider than 8 ½
inches, such as 11 by 17 inch paper.
3. Trim the paper to 8 ½ by 11 inches so
that the rulers will be on the edge of
the paper.
Topics:
Lateral and total surface area
ATTACHMENTS
Teacher Resource: Net Puzzles
page 29 of 205 Enhanced Instructional Transition Guide
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Suggested Duration: 19 days
Suggested Instructional Procedures
Elaborate 1
Students arrange two-dimensional figures to form a net of a specified three-dimensional figure. Students find
Notes for Teacher
KEY (1 per teacher)
Card Set: Net Puzzles (1 set per 6
students)
the lateral and total surface areas of each three-dimensional figure.
Instructional Procedures:
1. Prior to instruction, create a card set: Net Puzzles for every 6 students by copying on
cardstock, cutting apart, and placing each shape with pieces in a plastic zip bag.
2. Distribute a calculator, STAAR Grade 8 Mathematics Reference Materials, and puzzle from
card set: Net Puzzles to each student. Each student will receive a set for a cone, cylinder,
triangular pyramid, rectangular pyramid, triangular prism, or rectangular prism.
3. Instruct students to use the picture of the three-dimensional figure in their puzzle set to
arrange the cut out pieces to create a net (two-dimensional model) of the three-dimensional
figure, trace the net in their math journal, calculate and record the area of each piece using
the given dimensions, then add the area of each piece to find the total surface area, and
MATERIALS
cardstock (4 sheets per 6 students)
scissors (1 per teacher)
plastic zip bag (sandwich sized) (1
per student)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
calculator (1 per student)
math journal (1 per student)
page 30 of 205 Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Enhanced Instructional Transition Guide
Suggested
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Suggested Instructional Procedures
Notes for Teacher
verify solutions with a calculator.
State Resources
MTR 6 – 8: Semantic Feature Analysis Charts –
And That’s a Wrap
TEACHER NOTE
In order to reproduce materials requiring linear
measure that are consistent with intended
measurements noted on the KEY, set the print
Allow 10 minutes for students to complete the activity. Remind students that square units are used to
menu to print the handout at 100% by selecting
measure area. Monitor and assess students to check for understanding. Facilitate a class discussion
"None" or "Actual size" under the Page
about total surface area.
Scaling/Size option.
Ask:
TEACHER NOTE
What geometric figures make up the net of this figure? Answers may vary.
Triangular pyramid; etc.
What formulas may be used to find the total surface area of this figure? Answers
may vary. Region A + 3 • Region B; etc.
What must you do with all the individual areas of the net to find the total surface
area? (Add all the different areas of each piece.)
What relationship exists between the sum of the individual areas and the formula
r 2: 2
for surface area? (The formula for surface area of a cylinder: 2 rh + 2
rh
represents the area of the lateral region of the cylinder which when the curved surface is
flattened forms a rectangular shape
formula to calculate the area of a rectangle: lw
The focus of this part of the lesson is for
students to understand that irregular figures and
nets are created by combining two or more
polygons or circles, creating an irregular
composite figure or a net (two-dimensional
model for a three-dimensional figure). Finding
the area of an irregular figure or a net requires
adding the areas of all the associated figures
together.
page 31 of 205 Enhanced Instructional Transition Guide
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Suggested Instructional Procedures
Notes for Teacher
where 2 π r = l and h = w. There are two circular bases so, 2 r 2 represents the area of
the two bases. The formula for the surface area of a prism: Ph + 2B: P represents the
perimeter of the base and B represents the area of the base. Since each lateral face is a
rectangle and one dimension of each lateral face is part of the perimeter of the base, to
calculate the lateral area I can do lw + lw + lw + … for each lateral face or I can do (l + l + l
+ …)w which translates to Ph where (l + l + l + …) = P and w = h. Since I have two bases, I
calculate the area of one base and double it which is represented by 2B. The formula for
the surface area of a pyramid:
Pl + B: where P represents the perimeter of the base and
TEACHER NOTE
Grade 8 only requires students to calculate the
volume of a cone, but students should be able to
calculate the surface area of the net of a cone
since the two-dimensional model represents a
composite figure consisting of a circular area
and half of a circular area.
TEACHER NOTE
B represents the area of the base. Since each lateral face is a triangle and the base of
In Grade 7, students made a net (two-
each lateral face is part of the perimeter of the base of the pyramid, to calculate the lateral
dimensional model) of the surface area of a
area, I can do
bh +
h, which translates to
bh +
bh + … for each lateral face or I can do (b + b + b + …)
Pl where (b + b + b + …) = P and h =
three-dimensional figure (7.8C).
l. Since there is one
State Resources
base, I calculate the area of the base which is represented by B.)
MTR 6 – 8: Semantic Feature Analysis Charts –
Volume and Surface Area
10
Topics:
Spiraling Review
Volume
Explore/Explain 7
Students use prior knowledge to discuss volume of prisms and cylinders. Students are introduced to the
formulas for volume of pyramids and cones from the STAAR Grade 8 Mathematics Reference Materials.
ATTACHMENTS
Teacher Resource: Volume Notes
(1 per teacher)
page 32 of 205 Enhanced Instructional Transition Guide
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Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Students use appropriate formulas to calculate the volume of previously constructed three-dimensional models.
Instructional Procedures:
1. Facilitate a class discussion about volume.
Ask:
What is volume? Answers may vary. The number of cubic units that can fit inside a threedimensional figure; etc.
What are the dimensions of a two-dimensional figure? (length and width)
What are the dimensions of a three-dimensional figure? (length, width, and height)
What dimensions are used to calculate the volume of a rectangular prism? (length
x width x height)
What does “length x width” represent in the formula: “length x width x height”?
(It represents the calculated area of the base of the rectangular prism.)
Why do you multiply the area of the rectangular base by the height of the
rectangular prism? (The volume of a rectangular prism is “stacks” or “layers” of cubes
that cover the area of the base. By multiplying the area of the base by the height, I am
“counting” the number of cubes that will fit in the rectangular prism.)
What does “B” represent? (The area of the base.)
Why is the formula: V = Bh another way you can calculate the volume of a
rectangular prism? (I am still counting the number of cubes in each layer and then
multiplying by the height, number of layers of cubes, of the rectangular prism.)
What formula can you use to calculate the number of cubes to fill any prism, if the
base area of any prism tells you how many cubes are in each layer of the volume?
(Volume of a Prism: Area of Base x height: V = Bh.)
Notes for Teacher
Teacher Resource: Volume KEY
(1 per teacher)
Teacher Resource: Volume (1 per
teacher)
Handout: Volume (1 per student)
MATERIALS
math journal (1 per student)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
Three-Dimensional Figures (1 set
per 2 students) (previously created)
TEACHER NOTE
Finding volume using a formula is introduced in
Grade 7. Finding the volume of cones and
spheres is a new concept for Grade 8.
TEACHER NOTE
Some students may not be aware that units
represent measure for perimeter, square units
represent measure for area, and cubic units
represent measure for volume.
page 33 of 205 Enhanced Instructional Transition Guide
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Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
2. Display teacher resource: Volume Notes. Facilitate a class discussion about how to find the
volume of various three-dimensional figures. Instruct students to summarize the discussion as
notes in their math journal.
TEACHER NOTE
Some students may not realize the “B” in the
formula for volume stands for the area of the
base of the three-dimensional figure and usually
3. Display teacher resource: Volume.
has to be calculated using the formula for area
4. Place students in pairs. Distribute the STAAR Grade 8 Mathematics Reference Materials and
handout: Volume to each student and a set of previously created Three-Dimensional Figures
to each student pair. Instruct student pairs to use their set of Three-Dimensional Figures, as
needed, to identify the base, draw the base, calculate the area of the base, record the height
of the three-dimensional figure, record the formula needed to calculate the volume of the
three-dimensional figure, and then calculate the volume of the three-dimensional figure. Allow
time for students to complete the activity. Monitor and assess student pairs to check for
understanding. Facilitate a class discussion to debrief student solutions and clarify any
misconceptions and/or vocabulary terms.
Ask:
of the base. The shape of the base determines
the formula used to find area.
TEACHER NOTE
Students commonly are confused when using
the formula for triangular prisms by the two
different heights. For V = (
)h, the height in (
) is the height of the triangle. The other
height in the formula is the height of the prism.
What is the shape of the base? Answers may vary. A square; etc.
If you know the area of the base of the prism, how will this help you find the
volume? Pyramid? (The volume of a prism is stacks of cubes covering the base of the
three-dimensional model. By multiplying the number of cubes in a layer by the height of the
three-dimensional figure, I can count the total number of cubes to fill the prism. The
volume of a pyramid is one-third the Area of Base times height: V =
Bh.)
What does “B” represent in the volume formula? (The area of the base.)
How many dimensions are calculated in the volume formula? (3)
page 34 of 205 Enhanced Instructional Transition Guide
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Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
How does this affect the units in the answer? (The answer to a volume question is
always written with cubic units and since I am multiplying 3 dimensions, I put cubic units.)
11
Topics:
Spiraling Review
Surface area
Volume
Explore/Explain 8
Students determine if volume or surface area is most appropriate to solve a problem situation. Students use
formulas for volume and surface area to solve problems that involve conversions with and between customary
and metric measuring systems.
ATTACHMENTS
Teacher Resource: Problem
Solving with Measurement KEY
(1 per teacher)
Handout: Problem Solving with
Measurement (1 per student)
Instructional Procedures:
1. Distribute a calculator, STAAR Grade 8 Mathematics Reference Materials, and handout:
Problem Solving with Measurement to each student. Instruct students to use their STAAR
Grade 8 Mathematics Reference Materials to solve each problem and verify all solutions with
a calculator. Allow time for students to complete the activity. Monitor and assess students to
check for understanding. Facilitate individual discussions to debrief student solutions, as
needed.
Ask:
MATERIALS
calculator (scientific) (1 per
student)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
What is the problem asking you to calculate: surface area or volume? Explain.
Answers may vary. Surface Area; etc.
What formula (lateral area, total surface area, or volume) would you use to solve
page 35 of 205 Enhanced Instructional Transition Guide
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Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
this problem? Answers may vary. Total Surface Area; etc.
How did you determine what formula to use? Answers may vary. I looked on the
STAAR Grade 8 Mathematics Reference Materials formula chart; etc.
How will you use this formula to calculate the lateral area? Total surface area?
Volume? Answers may vary. Use the dimensions shown in the formula, either measure to
find the dimension or use the dimensions shown on the diagram, then substitute the values
for the dimensions and evaluate the formula; etc.
12
Topics:
Spiraling Review
Two-dimensional nets
Three-dimensional models
Surface area
Volume
Elaborate 2
Students demonstrate an understanding of two-dimensional nets, three-dimensional models, lateral and total
ATTACHMENTS
Teacher Resource: Surface Area
and Volume KEY (1 per teacher)
Handout: Surface Area and
Volume (1 per student)
surface area, and volume.
Instructional Procedures:
1. Place students in pairs. Distribute a calculator, STAAR Grade 8 Mathematics Reference
Materials, and handout: Surface Area and Volume to each student. Instruct students to use
their STAAR Grade 8 Mathematics Reference Materials to solve each problem and verify all
solutions with a calculator. Allow time for students to complete the activity. Monitor and assess
student pairs to check for understanding. Facilitate a class discussion to debrief student
MATERIALS
calculator (scientific) (1 per
student)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
page 36 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Notes for Teacher
solutions.
TEACHER NOTE
In order to reproduce materials requiring linear
measure that are consistent with intended
measurements noted on the KEY, set the print
menu to print the handout at 100% by selecting
"None" or "Actual size" under the Page
Scaling/Size option.
TEACHER NOTE
In order to produce rulers that are consistent
with the rulers on the STAAR Mathematics
Reference Materials, follow these steps:
1. Set the print menu to print the pages
at 100% by selecting "None" or "Actual
size" under the Page Scaling/Size
option.
2. Print on paper that is wider than 8 ½
inches, such as 11 by 17 inch paper.
3. Trim the paper to 8 ½ by 11 inches so
that the rulers will be on the edge of
the paper.
13
Topics:
page 37 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
Spiraling Review
Dimensional change, two-dimensional figures
Engage 4
Students use logic and reasoning skills to discuss predictions of the effects on perimeter and area when the
dimensions of a two-dimensional figure are changed proportionally.
ATTACHMENTS
Teacher Resource: Jane’s Yard (1
per teacher)
Instructional Procedures:
MATERIALS
1. Display teacher resource: Jane’s Yard. Instruct students to read the problem and record a
prediction of the effect the dimensional change will have on the perimeter and area in their
math journal. Allow time for students to complete the activity. Monitor and assess students to
check for understanding. Facilitate a class discussion for students to share their predictions,
without validating any given prediction.
Ask:
math journal (1 per student)
How do you predict the dimensional change will affect the perimeter of Jane's
yard? Answers may vary. The perimeter will double; etc.
How do you predict the dimensional change will affect the area of Jane’s yard?
Answers may vary. The area will double; etc.
Topics:
Dimensional change, two-dimensional figures
Explore/Explain 9
ATTACHMENTS
Teacher Resource: Exploring
Measurements KEY (1 per
teacher)
page 38 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Students investigate and record the effects on perimeter and area when the dimensions of a two-dimensional
figure are changed proportionally.
Instructional Procedures:
1. Place students in pairs. Distribute 12 red color tiles, 12 blue color tiles, the STAAR Grade 8
Mathematics Reference Materials, and handout: Exploring Measurements to each student.
Instruct student pairs to complete problem 1. Allow time for students to complete the activity.
Monitor and assess student pairs to check for understanding. Facilitate a class discussion to
debrief two-dimensional change in problem 1.
Ask:
How many tiles are needed for the length of the original rectangle in problem 1?
(3)
How many tiles are needed for the width of the original rectangle in problem 1?
(2)
What is the perimeter? (10 inches)
How do you find the perimeter? (Add the length of all sides.)
How do you find the area? (Area: length times width)
Notes for Teacher
Handout: Exploring
Measurements (1 per student)
Teacher Resource: My Garden
KEY (1 per teacher)
Handout: My Garden (1 per
student)
MATERIALS
color tiles (12 red, 12 blue) (1 set
per 2 students)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
TEACHER NOTE
In Grade 8, students change the dimensions of
figures proportionally. In other words, all
What is the area? (6 in2)
measurements must change by the same scale
What is a dimension? (A dimension is a length, width, or height of a figure.)
What does it mean to double something? (Multiply by two.)
How would you describe the effect on the dimensions if each dimension of the
rectangle is doubled? (Each dimension is multiplied by 2: 2l and 2w.)
If the dimensions of the above rectangle are doubled, how many tiles are needed
for the length of the new rectangle? (2 • 3: 6 tiles)
factor. In high school Geometry, students will
investigate non-proportional change and will use
a scale factor for one dimension and a different
scale factor for another dimension.
page 39 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
How many tiles are needed for the width of the new rectangle? (2 • 2: 4 tiles)
What is the perimeter of the new rectangle? (20 inches: 2(2•3) + 2(2•2))
What is the area of the new rectangle? (24 in2: (2•3)(2•2))
Compare and contrast the perimeters and areas. What happened? (The perimeter
of the original rectangle doubled or was multiplied by a factor of 2: 2(2l + 2w).The area of
the original rectangle quadrupled or was multiplied by a factor of 4: (2l)(2w) = 4(lw). Note:
the factor of 4 to quadruple the area of the original rectangle is the square of the scale
factor 2: 22.)
When the dimensions of a figure are changed proportionally, how is the
perimeter affected? (The new perimeter = scale factor • original perimeter.)
When the dimensions of a figure are changed proportionally, how is the area
affected? (The new area = (scale factor) 2 • original area.)
How can you prove these figures are similar? (Similar figures have corresponding
angles that are congruent and the ratios of corresponding sides are proportional. Since
the dimensions were changed proportionally, the ratios of the corresponding sides will be
proportional and the angles remain the same degrees, so the corresponding angles are
congruent.)
2. Instruct student pairs to complete the remainder of their handout: Exploring
Measurements. Allow time for students to complete the activity. Monitor and assess student
pairs to check for understanding. Facilitate a class discussion to debrief student solutions.
3. Distribute handout: My Garden to each student as independent practice and/or homework.
14
Topics:
page 40 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
Spiraling Review
Percent
Related measurement concepts
Dimensional change, two-dimensional figures
Elaborate 3
Students solve a real-life problem situation by extending concepts of measurement and numerical
understanding that include percents, proportions, area, perimeter, Pythagorean theorem, and the effects of
dimensional change.
Instructional Procedures:
1. Display teacher resource: Jane’s Yard. Facilitate a class discussion reviewing student’s
predictions (recorded earlier in their math journal) on the effect of dimensional change and
clarifying misconceptions.
2. Display teacher resource: Geometric Gravel Garden. Facilitate a class discussion to review
calculations involving area.
Ask:
What are the dimensions of the overall garden? Explain. (length: 18 ft + 4 ft + 18 ft =
40 ft. width: 13 ft + 4 ft + 13 ft = 30 ft.)
What formula(s) can you use to calculate the area of the pathways? (Area of a
rectangle: lw.)
What formula(s) can you use to calculate the area of the figures in Quadrant I?
(Area of a triangle: bh ÷ 2. Area of a trapezoid: (b1 + b2) • h ÷ 2.)
ATTACHMENTS
Teacher Resource: Jane’s Yard (1
per teacher)
Teacher Resource: Geometric
Gravel Garden KEY (1 per
teacher)
Teacher Resource: Geometric
Gravel Garden (1 per teacher)
Handout: Geometric Gravel
Garden (1 per student)
MATERIALS
math journal (1 per student)
map pencils (1 yellow, 1 blue, 1
green) (1 set per student)
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
TEACHER NOTE
To help students identify the location of a figure
page 41 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
How will you determine the base of the right triangle? (Use the Pythagorean
in the gravel garden, a coordinate grid is used
Theorem: a 2 + b 2 = c2. Base =
as a reference tool.
.)
3. Place students in pairs. Distribute handout: Geometric Gravel Garden and 1 yellow, 1
green, and 1 blue map pencil to each student. Instruct students to shade each region of the
gravel garden with the color indicated on the diagram and use their STAAR Grade 8
Mathematics Reference Materials to solve each problem. Allow time for students to complete
the activity. Monitor and assess student pairs to check for understanding. Facilitate a class
discussion to debrief student solutions. Facilitate individual group discussions about
perimeter, area, and dimensional change, as needed.
Ask:
TEACHER NOTE
Instruct students to only shade one region at a
time in the diagram on page 1. As students
complete the chart on page 2, instruct students
to shade the region on page 1 that corresponds
to the area being calculated in the chart on page
2.
What formula(s) can you use to calculate the area of the figures in Quadrant II?
(Area of a triangle: bh ÷ 2. Area of a circle: r2.)
What formula(s) can you use to calculate the area of the figures in Quadrant III?
(Area of a rectangle: lw. Area of a circle: r2.)
What formula(s) can you use to calculate the area of the figures in Quadrant IV?
(Area of a triangle: bh ÷ 2. Area of a trapezoid: (b1 + b2) • h ÷ 2.)
How can you calculate the total area for the white gravel? (Subtract the sum of the
areas for the yellow, green, and blue colors from the overall area of the entire garden.)
Is the area covered by white gravel more or less than 50% of the garden?
Explain. (More than 50% because the white gravel covers 751.48 ft 2 of the garden and
the total garden area is 1200 ft 2. 50% of 1200 = 600. 751.48 > 600.)
How can you calculate the percent for each color of gravel? (Set up the proportion:
page 42 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
and solve for x.)
How can you calculate the amount of gravel in tons needed for each color of
gravel? (Set up the proportion:
and solve for x.)
How can you check to see if your responses for the chart in problem 3 are
correct? (Find the sum of each column. Total for the area column = 1200 ft 2. Total for %
column = 100%. Total for amount of gravel needed = 5 tons.)
What equation can you record to represent the effects of dimensional change on
the perimeter? (New perimeter: (scale factor) • original perimeter.)
What equation can you record to represent the effects of dimensional change on
the area? (New area: (scale factor)2 • original area.)
15
Topics:
Spiraling Review
Dimensional change, three-dimensional figures
Explore/Explain 10
Students discuss, investigate, and record the effects on volume when the dimensions of a three-dimensional
figure are changed proportionally.
Instructional Procedures:
1. Display teacher resource: Empire State Building. Instruct students to record a prediction in
their math journal of how the volume will be affected when the dimensions are changed
proportionally. Allow students 2 – 3 minutes to complete the activity.
ATTACHMENTS
Teacher Resource: The Empire
State Building (1 per teacher)
Teacher Resource: Dimensional
Changes on Volume KEY (1 per
teacher)
Handout: Dimensional Changes
on Volume (1 per student)
page 43 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
2. Place students into groups of 3 and distribute 150 linking cubes or centimeter cubes to each
group. Instruct students to use the cubes to build a rectangular prism with a length of 4 units,
a width of 2 units, and a height of 1 unit. Allow time for students to complete the activity.
Monitor and assess student groups to check for understanding. Facilitate a class discussion
about dimensional changes.
Ask:
What do you think will happen to the volume if you double just one of the
dimensions? Answers may vary. The volume will be two times greater than the original
volume; etc.
Notes for Teacher
MATERIALS
math journal (1 per student)
linking cubes or centimeter cubes
(150 cubes per 3 students)
TEACHER NOTE
Students begin this phase with an example
where only one dimension is changed, then two
dimensions, and finally all three dimensions are
3. Instruct students to prove their responses by doubling one of the dimensions, building the
new figure, and finding the new volume. Allow time for students to complete the activity.
Monitor and assess student groups to check for understanding. Facilitate a class discussion
about the effect of the dimensional change to the volume.
Ask:
What happened to the volume when you doubled one of the dimensions? (The
volume doubled.)
Why do you think the volume doubled? Answers may vary. Since only one dimension
was doubled, the original volume was doubled or multiplied by 2: Original Volume = l x w x
h. Double one dimension
new volume = 2l x w x h = 2(l x w x h)
This represents the
original volume being doubled; etc.
Does it matter which dimensions you chose to double? (no)
changed so students see non-examples and
examples of figures that are changed
proportionally.
TEACHER NOTE
In Grade 8, students will change the dimensions
of figures proportionally. In other words, all
measurements must change by the same scale
factor. In high school Geometry, students will
investigate non-proportional change and will use
a scale factor for one dimension and a different
scale factor for another dimension.
page 44 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
What do you think will happen to the volume if you double two of the
dimensions? Answers may vary. The volume will be 4 times greater; etc.
Notes for Teacher
TEACHER NOTE
It is important for students to understand the
4. Instruct students to prove their response by doubling both of the dimensions, building the new
figure, and finding the new volume. Allow time for students to complete the activity. Monitor
and assess student groups to check for understanding. Facilitate a class discussion about the
effect of the dimensional change to the volume.
Ask:
Were you correct? Answers may vary. No; etc.
What actually happened? (The new volume is quadruple the original volume or the new
volume is 4 times greater than the original volume.)
Why do you think the volume quadrupled? Answers may vary. Since two of the
dimensions were doubled and 2 x 2 = 4, then the volume was affected by being multiplied
by 4: Original volume = l x w x h. Double two dimensions
new volume = 2l x 2w x h = 2 x
2 x (l x w x h)
This represents the original volume being quadrupled; etc.
Does it matter which two dimensions you chose to double? (no)
What do you think will happen to the volume if you double all three of the original
dimensions? Answers may vary. The volume will be 8 times greater; etc.
effects on perimeter, area, and volume when all
linear dimensions of a figure are changed
proportionally by the same scale factor.
The three examples below show how perimeter,
area, and volume are affected.
Perimeter of a Rectangle
P = 2(l + w)
If each of the two linear dimensions are doubled,
the new perimeter is:
New P = 2(2l + 2w) = 2 x 2(l + w)
By doubling each dimension (scale factor of 2),
the new perimeter is double the original
perimeter.
Area of a Rectangle
A = lw
If each of the two linear dimensions are doubled,
the new area is:
5. Instruct students to prove their response by doubling all three dimensions of the original
figure, building the new figure, and finding the new volume. Allow time for students to
complete the activity. Monitor and assess student groups to check for understanding.
Facilitate a class discussion about the effect of the dimensional change on the volume.
Ask:
New A = (2l)(2w) = 2•2(lw) = (2)2 (lw) = (scale
factor) 2(lw)
By doubling each dimension (scale factor of 2),
the new area is the (scale factor)2 times the
original area.
page 45 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Were you correct? Answers may vary. Yes; etc.
What actually happened? (The new volume is 8 times greater than the original volume.)
Why do you think that the volume was multiplied by 8? Answers may vary. Since all
the dimensions were doubled and 2 x 2 x 2 = 8, then the volume was affected by being
multiplied by 8: Original volume = l x w x h. Double all the dimensions
new volume = 2l x
2w x 2h = 2 x 2 x 2 x (l x w x h) = 8 x (l x w x h)
Notes for Teacher
Volume of Rectangular Prism
V = l xwx h
If each of the three linear dimensions are
doubled, the new volume is:
New V = 2l x 2w x 2h
= 2 x 2 x 2 x l x wx h
This represents the original volume being
multiplied by 8 or (2) 3: (scale factor) 3; etc.
= (2)3 x l x w x h
= (scale factor)3 x l x w x h
By doubling each dimension (scale factor of 2),
6. Distribute handout: Dimensional Changes on Volume to each student. Facilitate a class
discussion to help clarify the instructions. Instruct student groups to use their linking cubes or
centimeter cubes to build the prism, build the new figure, and find the new volume. Allow time
for students to complete the activity. Monitor and assess student groups to check for
understanding. Facilitate a class discussion about the effect of the dimensional change on the
volume.
Ask:
the new volume is the (scale factor)3 times the
original volume.
How was the volume affected when you doubled the dimensions of the
rectangular prism? (The volume got 8 times greater.)
How is this change in volume related to the change of doubling the linear
dimensions of length, width, and height? Explain. (The original volume = l x w x h.
After doubling each dimension, the new volume = 2l x 2w x 2h = 2 x 2 x 2 x l x w x h = (2)3 x
original volume = 8 x original volume.)
How was the volume affected when you tripled the dimensions? Explain. (The
volume was 27 times greater because the original volume = l x w x h. The new volume = 3l x
page 46 of 205 Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Notes for Teacher
3w x 3h = 3 x 3 x 3 x l x w x h = (3)3 x original volume = 27 x original volume.)
How was the volume affected when you quadrupled the dimensions? (The volume
was 64 times greater because the original volume = l x w x h. The new volume = 4l x 4w x
4h = 4 x 4 x 4 x l x w x h = (4)3 x original volume = 64 x original volume.)
How was the volume affected when you halved the dimensions? (The volume was
times smaller because the original volume = l x w x h. The new volume = l x w x h =
x
x
x l x w x h = ( )3 x original volume =
x original volume.)
What scale factor are you using when you double the dimensions? (2) Triple the
dimensions? (3) Quadruple the dimensions? (4) Halve the dimensions? (
)
How does the scale factor affect how the volume changes when the dimensions
are changed proportionally? (The original volume is multiplied by the cube of the scale
factor to get the new volume.)
16
Topics:
Spiraling Review
Dimensional change, three-dimensional figures
Explore/Explain 11
Students summarize the effects on perimeter, area, and volume when linear dimensions are changed
proportionally.
Instructional Procedures:
ATTACHMENTS
Teacher Resource: Dimensional
Changes KEY (1 per teacher)
Handout: Dimensional Changes
(1 per student)
1. Place students in groups of 4 and distribute handout: Dimensional Changes to each
page 47 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
student. Instruct students to complete the handout. Allow time for students to complete the
activity. Monitor and assess student groups to check for understanding. Facilitate a class
discussion about the effects proportional changes on linear dimensions have on perimeter,
area, and volume.
Ask:
When each dimension of a two-dimensional figure is changed proportionally, how
is the perimeter changed? (The perimeter of the similar figure = original perimeter
multiplied by the scale factor of the dimensional change.)
When each dimension of a two-dimensional figure is changed proportionally, how
is the area changed? (The area of the similar figure = original area multiplied by (scale
factor) 2 of the dimensional change.)
When each dimension of a three-dimensional figure is changed proportionally,
how is the volume changed? (The volume of the similar figure = original volume
multiplied by the (scale factor) 3 of the dimensional change.)
17 – 18
Topics:
Spiraling Review
Related measurement concepts
Dimensional change, two- and three-dimensional figures
Elaborate 4
Students summarize the effects on perimeter, area, and volume when linear dimensions are changed
proportionally.
ATTACHMENTS
Teacher Resource: Measurement
and Dimensional Change KEY (1
per teacher)
Handout: Measurement and
page 48 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Instructional Procedures:
1. Distribute the STAAR Grade 8 Mathematics Reference Mateirals and handout:
Measurement and Dimensional Change to each student. Instruct students to complete the
handout. Allow time for students to complete the activity. Monitor and assess students to
check for understanding. Facilitate a class discussion to summarize measurement and
dimensional change.
Notes for Teacher
Dimensional Change (1 per
student)
Teacher Resource: Measurement
and Dimensional Change
Practice KEY (1 per teacher)
Handout: Measurement and
Dimensional Change Practice (1
per student)
MATERIALS
STAAR Grade 8 Mathematics
Reference Materials (1 per student)
ruler (standard) (1 per student)
TEACHER NOTE
In order to reproduce materials requiring linear
measure that are consistent with intended
measurements noted on the KEY, set the print
menu to print the handout at 100% by selecting
"None" or "Actual size" under the Page
Scaling/Size option.
TEACHER NOTE
page 49 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Notes for Teacher
In order to produce rulers that are consistent
with the rulers on the STAAR Mathematics
Reference Materials, follow these steps:
1. Set the print menu to print the pages
at 100% by selecting "None" or "Actual
size" under the Page Scaling/Size
option.
2. Print on paper that is wider than 8 ½
inches, such as 11 by 17 inch paper.
3. Trim the paper to 8 ½ by 11 inches so
that the rulers will be on the edge of
the paper.
ADDITIONAL PRACTICE
Handout (optional): Measurement and
Dimensional Change Practice may be used as
additional practice if needed.
19
Evaluate 1
Instructional Procedures:
1. Assess student understanding of related concepts and processes by using the Performance
MATERIALS
calculator (scientific) (1 per
student)
STAAR Grade 8 Mathematics
page 50 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Suggested Instructional Procedures
Indicator(s) aligned to this lesson.
Notes for Teacher
Reference Materials (1 per student)
Performance Indicator(s):
Grade 08 Mathematics Unit 07 PI 01
Create a presentation (e.g., design plan, blueprint, etc.) that includes scale models and perspective views of
several real-life three-dimensional models (e.g., prisms, pyramids, cylinder, spheres, cones, etc.). Use the
proportional relationships of the scale models to estimate and find missing measurements needed to solve
formulas to find the surface area and volume of each scale model. Evaluate the solutions for
reasonableness, and justify each solution with a calculator. Explain, in writing, the solution process used to
find missing measures, surface area, and volume, and describe the resulting effect on the measures (e.g.,
perimeter, area, volume, etc.) when the dimensions of the scale models are changed proportionally.
Sample Performance Indicator:
As a group project for architectural design class, Karen, Whitney, and Adrienne decided
to create a scale model of notable structures in the United States with a scale of 50 ft =
1 in. They must paint the sides and stabilize each structure by filling it with plaster. (1
foot = 0.3048 meters)
page 51 of 205 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Notes for Teacher
Create a design plan for each structure that includes a sketch of the three-dimensional building, twodimensional model of the net, and the formulas needed to determine the amount of paint (square inches)
page 52 of 205 Grade 8/Mathematics
Unit 07:
Suggested Duration: 19 days
Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Notes for Teacher
and plaster needed (cubic inches) for each scale model. Evaluate the solutions for reasonableness, and
justify each solution with a calculator. In addition, explain, in writing, the solution process used to find the
dimensions, surface area, and volume, and describe the resulting effect on the amount of paint and plaster
needed if a larger scale model was requested three times larger than Karen, Whitney, and Adrienne’s scale
models.
Standard(s): 8.2B , 8.2C , 8.2D , 8.7A , 8.7B , 8.8A , 8.8B , 8.8C , 8.9B , 8.10A , 8.10B , 8.14A
, 8.14B , 8.14C , 8.14D , 8.15A
ELPS ELPS.c.1H , ELPS.c.4J , ELPS.c.5F , ELPS.c.5G
05/08/13
page 53 of 205 Grade 8
Mathematics
Unit: 07 Lesson: 01
Sample of Mathematics and Measurement KEY
Air
Conditioning
Unit
Perimeter: Rectangle
P = 2l + 2w
or
P = 2(l + w)
1 meter
Perimeter: Square
P = 4s
1 meter
Circumference: Circle
C = 2 π r or
C= πd
Solution:
P = 4s
P = 4(1)
P = 4 meters
Yolanda wants to frame
her tomato garden with
barbed wire. How much
barbed wire is needed?
Area: Square
A = s2
1.5 inches
Solution:
P = 2(1.5) + 2(1) or
P=3+2
P = 5 meters
Air
Conditioning
Unit
1 meter
Panfilo wants to place a
fence around his air
conditioning unit. His unit
measures 1 meter on
both sides. How much
fencing is needed?
Rubber Stamp
1 meter
Solution:
C = 2π r
C ≈ 2(3.14)(0.75)
C ≈ 4.71 inches
What is the area
occupied by the air
conditioning unit?
Area: Rectangle
A = lw or A = bh
What is the area of
Yolanda’s tomato
garden?
Area: Triangle
bh
1
A = bh or A =
2
2
Solution:
A = lw
A = (1.5)(1)
A = 1.5 m 2
What is the area of the
triangular sand pit?
Area: Trapezoid
1
A = (b1 + b2)h
2
(b + b )h
or A = 1 2
2
Area: Circle
A = π r2
©2012, TESCCC
Solution:
1
A = (1)(1)
2
1
A=
m2
2
6 meters
Circus Ring
Solution:
Sergio built a cage for his
rabbit with the
dimensions shown. What
is the area of the cage?
Solution:
A = s2
A = (1) 2
A = 1 m2
1 meter
What is the
circumference of
Matthew’s circular rubber
stamp?
1
A = (b1 + b2)h
2
1
A = (0.75 + 1)0.5
2
A = 0.4375 m 2
What is the area of the
circus ring?
10/11/12
Solution:
A= πr 2
A ≈ (3.14) (3) 2
A ≈ (3.14)(9)
A ≈ 28.26 m 2
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Sample of Mathematics and Measurement
Air
Conditioning
Unit
Perimeter: ________
P = 2l + 2w
or
P = 2(l + w)
1 meter
1 meter
Circumference:
________
C = 2 π r or C = π d
What is the
circumference of
Matthew’s circular rubber
stamp?
Solution:
Solution:
Yolanda wants to frame
her tomato garden with
barbed wire. How much
barbed wire is needed?
Area: ________
A = s2
1.5 inches
Rubber Stamp
1 meter
Solution:
Solution:
What is the area
occupied by the air
conditioning unit?
Area: ________
bh
1
A = bh or A =
2
2
Area: ________
A = lw or A = bh
Solution:
Solution:
What is the area of
Yolanda’s tomato
garden?
What is the area of the
triangular sand pit?
Area: ________
1
A = (b1 + b2)h
2
(b + b )h
or A = 1 2
2
Area: ________
A = π r2
Solution:
6 meters
Circus Ring
Solution:
Sergio built a cage for his
rabbit with the
dimensions shown. What
is the area of the cage?
©2012, TESCCC
Air
Conditioning
Unit
1 meter
Panfilo wants to place a
fence around his air
conditioning unit. His unit
measures 1 meter on
both sides. How much
fencing is needed?
1 meter
Perimeter: ________
P = 4s
What is the area of the
circus ring?
10/11/12
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Irregular Measurement and Nets KEY
Write down the formula(s) that is used to work each problem. Work the problem and justify your
response.
1. The following diagram is a model of Miranda’s bedroom. How many square feet of carpeting will
be needed for Miranda’s bedroom? How much border will be needed to go around the perimeter
of the bedroom?
Formulas: Area of a Square: s 2 and Area of a Triangle: bh ÷ 2
Perimeter: 10 + 10 + 2 + (10  2)2 + (16  10)2 + 16 = 48 ft
Area: Area of Square + Area of Triangle:
A = 10 • 10 + (6 • 8) ÷ 2 = 124 ft 2
10 ft
2 ft
10 ─ 2 = 8
10 ft
16 ─ 10 = 6
16 ft
2. The following diagram is a model of a small closet Mark uses at home. What is the area of this
closet? What is the perimeter of the closet?
Formulas: Area of a Rectangle: lw and Area of a Triangle: bh ÷ 2
Perimeter: 12 + 16 + 7 + 4 + (16  4)2 + (12  7)2 = 52 ft
Area: Area of Rectangle + Area of Rectangle + Area of Triangle:
A = 7 • 12 + 4 • 7 + (5 • 12) ÷ 2 = 142 ft2
16 ft
12 ft
7 ft
7 ft
12 ─ 7 = 5
©2012, TESCCC
16 ─ 4 = 12
4 ft
05/01/13
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Irregular Measurement and Nets KEY
3. The area shown below will be used for the elementary school playground. What is this area? How
much fencing is needed to enclose the playground? Use 3.14 for the value of pi.
Formulas: Area of a Circle:  r 2 and Area of a Rectangle: lw and Area of a Triangle bh ÷ 2
Perimeter = 10 + (4)2 + (3)2 + 3 + 10 + (3.14 • 4 ÷ 2) = 34.28 yd
Area: Area of Semi-Circle + Area of Rectangle + Area of Triangle:
A = (3.14 • (4 ÷ 2) 2 ÷ 2) + (4 • 10) + (3 • 4 ÷ 2) = 52.28 yd2
4 yd
4 yd
10 yd
10 yd
3 yd
Use a calculator to answer problems 4 and 5 using the figure below.
1204.2 ft
1315.3 ft
1200 ft
1200 ft
1400 ft
1200 ft
1300 ft
4. Mr. Scherer alternates his cattle between two trapezoid shaped pastures. What is the area of the
two pastures?
Area: Area of left trapezoid + Area of right trapezoid: (b1 + b2)h ÷ 2
((1200 + 1400)1300 ÷ 2) + ((1200 + 1400)1200 ÷ 2)
Area = 3,250,000 ft2
5. How much fencing is needed to enclose the two pastures?
Perimeter of left trapezoid + 2 legs and top base of right trapezoid
Perimeter: 1200 + 1315.3 + 1400 + 1300 + 1204.2 + 1204.2 + 1200
Perimeter = 8, 823.7 ft
©2012, TESCCC
05/01/13
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Irregular Measurement and Nets
Write down the formula(s) that is used to work each problem. Work the problem and justify your
response.
1. The following diagram is a model of Miranda’s bedroom. How many square feet of carpeting will
be needed for Miranda’s bedroom? How much border will be needed to go around the perimeter
of the bedroom?
10 ft
2 ft
10 ft
16 ft
2. The following diagram is a model of a small closet Mark uses at home. What is the area of this
closet? What is the perimeter of the closet?
12 ft
7 ft
16 ft
©2012, TESCCC
4 ft
05/01/13
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Irregular Measurement and Nets
3. The area shown below will be used for the elementary school playground. What is this area? How
much fencing is needed to enclose the playground? Use 3.14 for the value of pi.
4 yd
10 yd
3 yd
Use a calculator to answer problems 4 and 5 using the figure below.
1204.2 ft
1315.3 ft
1200 ft
1200 ft
1400 ft
1200 ft
1300 ft
4. Mr. Scherer alternates his cattle between the trapezoid shaped pastures. What is the area of the
two pastures?
5. How much fencing is needed to enclose the two pastures?
©2012, TESCCC
05/01/13
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Composite Figures KEY
For each problem, shade the area you need to find in green and shade the area you need to remove
in red.
1. Cheryl mounts a rectangular shaped picture
a) Use a written description in conjunction with
in the center of a rectangular board according
math symbols to describe how to solve the
problem.
to the diagram below. How many square
inches of the board are not covered by the
Convert 1 foot to inches.
Subtract the area of the rectangular picture
picture?
from the area of the rectangular board:
1 foot: 1 ft • 12 in. per ft = 12 in.
(lw)large rectangle ─ (lw)small rectangle
b) Find the indicated area.
A = (12 in. • 15 in.) ─ (9 in. • 12 in.)
A = 180 in2 ─ 108 in2
A = 72 in2
9 ft
2. Mr. Garcia is going to paint a mural on one
wall in his son’s room. There is a rectangular
shaped window that is 24 inches wide and 36
inches high on the wall. How many square
feet of the wall will be painted?
a) Use a written description in conjunction with
math symbols to describe how to solve the
problem.
Convert the dimensions of the rectangular
window to feet. Subtract the area of the
rectangular window from the area of the
rectangular wall:
24 inches: 24 in. ÷ 12 in. per ft = 2 ft
36 inches: 36 in. ÷ 12 in. per ft = 3 ft
(lw)wall ─ (lw)window
b) Find the indicated area.
A = (12 ft • 9 ft) ─ (2 ft • 3 ft)
A = 108 ft 2 ─ 6 ft 2
A = 102 ft 2
3. Mr. Callaway is a farmer. He recently rented a a) Use a written description in conjunction with
new field and needs to calculate its area in
math symbols to describe how to solve the
order to make plans for planting a crop next
problem.
season. The field is rectangular, but there is a
Subtract the rectangular area for the oil pump
small rectangular area inside of the field that
from the area of the total rectangular field:
(lw)total field ─ (lw)oil pump region
is fenced off for an oil pump. The fenced off
area inside the field is 240 feet by 120 feet.
b) Find the indicated area.
How many square feet of the field will Mr.
A = (960 ft •480 ft) – (240 ft •120 ft)
Callaway use to plant a crop?
A = 460,800 ft 2 – 28,800 ft 2
A = 432,000 ft2
©2012, TESCCC
10/11/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Composite Figures KEY
For each problem, shade the area you need to find in green and shade the area you need to remove
in red.
4. Mr. Gonzalez is planting carpet grass in the
a) Use a written description in conjunction with
backyard of his new home. The grass is sold
math symbols to describe how to solve the
problem.
in pallets of square “grass tiles.” The
backyard is rectangular. There are two oak
Calculate the area of the square planter’s
box. Double this area and subtract this
trees in the backyard; each tree has a square
planter’s box around its base. The planter’s
product from the area of the total rectangular
box around each tree has a side of 6 feet.
backyard:
(lw)total backyard ─ 2(s 2)planter’s box
How many square feet of grass does Mr.
Gonzalez need for his new backyard?
b) Find the indicated area.
A = (93 ft •72 ft) ─ 2(6 ft • 6 ft)
A = 6,696 ft 2 ─ 72 ft 2
A = 6,624 ft2
78 in.
5. Mrs. Santos plans to cover the outside of her
rectangular door with wallpaper. The window
inside the door is 1.5 feet by 9 inches. How
many square feet of wallpaper does she need
to cover the door, excluding the window?
a) Use a written description in conjunction with
math symbols to describe how to solve the
problem.
Convert the dimensions of the rectangular
window and door to feet. Subtract the area of
the rectangular window from the area of the
rectangular door:
9 inches: 9 in. ÷ 12 in. per ft = 0.75 ft
36 inches: 36 in. ÷ 12 in. per ft = 3 ft
78 inches: 78 in. ÷ 12 in. per ft = 6.5 ft
(lw)door ─ (lw)window
b) Find the indicated area.
A = (3 ft • 6.5 ft) ─ (0.75 ft • 1.5 ft)
A = 19.5 ft 2 ─ 1.125 ft 2
A = 18.375 ft.2
6. Mrs. Pierce’s husband built the following
a) Use a written description in conjunction with
backdrop for her drama class. Mr. Pierce cut
math symbols to describe how to solve the
out a square hole with a side of 3 feet and a
problem.
Calculate the radius of the circular hole. Calculate
circular hole with a diameter of 2 feet from a
the area of the square hole and the circular hole.
piece of plywood. Mrs. Pierce is going to paint
Subtract the sum of the area of the square and
the front of the piece of rectangular plywood.
circle from the area of the rectangular piece of
How many square feet of the plywood is to be
plywood:
painted?
9 ft
radius: diameter ÷ 2 = 2 ft ÷ 2 = 1 ft
(lw)plywood ─ (s 2square hole + π r 2)
©2012, TESCCC
b) Find the indicated area.
A ≈ (12 ft • 9 ft) ─ (3 ft • 3 ft + 3.14(1 ft • 1 ft))
A ≈ 108 ft 2 ─ (9 ft 2 + 3.14 ft 2)
A ≈ 108 ft 2 ─ 12.14 ft 2
A ≈ 95.86 ft2
10/11/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Composite Figures
For each problem, shade the area you need to find in green and shade the area you need to remove
in red.
a) Use a written description in conjunction with
1. Cheryl mounts a rectangular shaped picture
in the center of a rectangular board according
math symbols to describe how to solve the
problem.
to the diagram below. How many square
inches of the board are not covered by the
picture?
b) Find the indicated area.
2. Mr. Garcia is going to paint a mural on one
wall in his son’s room. There is a rectangular
shaped window that is 24 inches wide and 36
inches high on the wall. How many square
feet of the wall will be painted?
a) Use a written description in conjunction with
math symbols to describe how to solve the
problem.
b) Find the indicated area.
3. Mr. Callaway is a farmer. He recently rented a a) Use a written description in conjunction with
new field and needs to calculate its area in
math symbols to describe how to solve the
order to make plans for planting a crop next
problem.
season. The field is rectangular, but there is a
small rectangular area inside of the field that
is fenced off for an oil pump. The fenced off
area inside the field is 240 feet by 120 feet.
How many square feet of the field will Mr.
Callaway use to plant a crop?
b) Find the indicated area.
©2012, TESCCC
10/11/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Composite Figures
For each problem, shade the area you need to find in green and shade the area you need to remove
in red.
4. Mr. Gonzalez is planting carpet grass in the
a) Use a written description in conjunction with
backyard of his new home. The grass is sold
math symbols to describe how to solve the
in pallets of square “grass tiles.” The
problem.
backyard is rectangular. There are two oak
trees in the backyard; each tree has a square
planter’s box around its base. The planter’s
box around each tree has a side of 6 feet.
How many square feet of grass does Mr.
Gonzalez need for his new backyard?
b) Find the indicated area.
5. Mrs. Santos plans to cover the outside of her
rectangular door with wallpaper. The window
inside the door is 1.5 feet by 9 inches. How
many square feet of wallpaper does she need
to cover the door, excluding the window?
a) Use a written description in conjunction with
math symbols to describe how to solve the
problem.
b) Find the indicated area.
9 ft
6. Mrs. Pierce’s husband built the following
a) Use a written description in conjunction with
backdrop for her drama class. Mr. Pierce cut
math symbols to describe how to solve the
out a square hole with a side of 3 feet and a
problem.
circular hole with a diameter of 2 feet. Mrs.
Pierce is going to paint the front of the piece
of the rectangular plywood. How many square
feet of the plywood is to be painted?
©2012, TESCCC
b) Find the indicated area.
10/11/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis Practice KEY
1. According to the diagram above, approximately how many centimeters are equivalent to 1
inch? Explain.
Answers may vary. Possible responses may include: There are between 2 cm to 3 cm since
the diagram shows the length of an inch ending between 2 and 3 cm. There are 2.5 cm since
the diagram shows the length of an inch ending approximately halfway between 2 and 3 cm.
2. Record the following information:
1 inch is approximately equivalent to 2.54 centimeters.
3. Write two ratios that represent the information from problem 2.
1 inch
and
2.54 cm
2.54 cm
1 inch
4. A problem-solving method known as dimensional analysis may be used to convert inches to
centimeters and centimeters to inches. Dimensional analysis allows you to determine which
way to write the ratio so that units will cancel and leave only the desired unit in the answer.
a) Fill in the missing information to convert 7 inches to centimeters: 7 inches = 17.78 cm.
(We will cancel the units of inches and leave only cm.)
 2.54 cm 
7 in. 
 = 7( 2.54 ) cm
 1 in. 
b) Fill in the missing information to convert 24 centimeters to inches: 24 cm ≈ 9.45 in.
(We will cancel the units of cm and leave only inches.)
1 in.   24 

24 cm 
 = 
 in.
 2.54 cm   2.54 
©2012, TESCCC
10/11/12
page 1 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis Practice KEY
Dimensional analysis may be used to convert other measurement units. Use the ratios below, the
STAAR Grade 8 Mathematics Reference Materials, and dimensional analysis to perform the
conversions for problems 5 through 8.
2.2 lbs
1 kg
or
1 kg
2.2 lbs
3.79 L
1 gal
or
1 gal
3.79 L
? cm
1 in.
or
1 in.
? cm
5. How many liters of lemonade are equivalent to 5.5 gallons of lemonade?
(We will cancel the units of gallons and leave only liters.)
 3.79 L 
5.5 gal 
 = 5.5(3.79) L = 20.845 L
 1 gal 
6. How many pounds are equivalent to 15 kilograms?
(We will cancel the units of kilograms and leave only pounds.)
 2.2 lbs 
15 kg 
 = 15(2.2) lbs = 33 lbs
 1 kg 
7. How many feet are equivalent to 39 centimeters?
(We will cancel the units of centimeters and leave only feet.)
(39 i 1 i 1)
 1 in.   1 ft 
39 cm 
≈ 1.3 ft
 
 =
(2.54 i 12)
 2.54 cm   12 in. 
8. How many fluid ounces of soda are equivalent to 1.5 liters of soda?
(We will cancel the units of liters and leave only fluid ounces.)
 1 gal   128 fl oz  (1.5 i 1 i 128)
1.5 L 
≈
 =
 
(3.79 i 1)
 3.79 L   1 gal 
©2012, TESCCC
10/11/12
50.7 fl oz
page 2 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis Practice KEY
Use the given information and dimensional analysis to convert the given dimensions for problems 9
through 14.
9. Panfilo wants to place a fence around his square air conditioning unit. His unit measures 1
meter on each side. If the dimensions are converted to inches, how much fencing is needed?
What is the area occupied by the air conditioning unit?
Answer: convert meters to inches
(1 i 100 i 1)
 100 cm   1 in. 
1m 
≈ 39.4 in.

 =
(1 i 2.54)
 1 m   2.54 cm 
Perimeter = 4(39.4 in.) = 157.6 in.
Area = (39.4 in.)2 = 1,552.36 sq. in.
10. Yolanda wants to frame her tomato garden with barbed wire. The length and width of the
garden are 1.5 meters and 1 meter. If the dimensions are converted to yards, how much
barbed wire is needed? What is the area of Yolanda’s tomato garden?
Answer: convert meters to yards
(1.5 i 100 i 1 i 1)
 100 cm   1 in.   1 yd 
Length: 1.5 m 
≈ 1.6 yds
 
 
 =
(1 i 2.54 i 36)
 1 m   2.54 cm   36 in. 
(1 i 100 i 1 i 1)
 100 cm   1 in.   1 yd 
Width: 1 m 
≈ 1.1 yds

 
 =
(1 i 2.54 i 36)
 1 m   2.54 cm   36 in. 
Perimeter = 2(1.6 yds + 1.1 yds) = 5.4 yds
Area = (1.6 yds)(1.1 yds) = 1.76 sq. yds
11. Matthew designed a logo for a rubber stamp he plans to use in his office. The diameter of the
rubber stamp is 1.5 inches. If the dimensions are converted to centimeters, what is the
circumference of Matthew’s circular rubber stamp?
Answer: convert inches to centimeters
 2.54 cm   1.5 i 2.54 
1.5 in. 
 = 
 = 3.81 cm
1
 1 in.  

Circumference = π (3.81) ≈ 11.96 cm
©2012, TESCCC
10/11/12
page 3 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis Practice KEY
12. Sam wants to build a frame for a triangular sand pit. The base and height of the sand pit are 1
meter each. If the dimensions are converted to feet, what is the area of the triangular sand pit?
Answer: convert meters to feet
(1 i 100 i 1 i 1)
 100 cm   1 in.   1 ft 
1m 
≈ 3.28 ft
 
 
 =
(1 i 2.54 i 12)
 1 m   2.54 cm   12 in. 
(3.28 ft)(3.28 ft)
Area =
≈ 5.38 sq ft
2
13. Sergio built a cage with the floor shaped like a trapezoid for his rabbit. The two bases measure
0.75 meters and 1 meter. The height of the trapezoid floor is 0.5 meters. If the dimensions are
converted to yards, what is the floor area of the cage?
Answer: convert meters to yards
(0.75 i 100 i 1 i 1)
 100 cm   1 in.   1 yd 
Top Base: 0.75 m 
≈ 0.8 yds
 
 
 =
(1 i 2.54 i 36)
 1 m   2.54 cm   36 in. 
(1 i 100 i 1 i 1)
 100 cm   1 in.   1 yd 
Bottom Base: 1 m 
≈ 1.1 yds

 
 =
(1 i 2.54 i 36)
 1 m   2.54 cm   36 in. 
(0.5 i 100 i 1 i 1)
 100 cm   1 in.   1 yd 
Height: 0.5 m 
≈ 0.5 yds
 
 
 =
(1 i 2.54 i 36)
 1 m   2.54 cm   36 in. 
(0.8 yds + 1.1 yds) i 0.5 yds
Area =
= 0.475 sq yds
2
14. A circus ring has a diameter of 6 meters. If the dimensions are converted to feet, what is the
area of the circus ring?
Answer: convert meters to feet
(6 i 100 i 1 i 1)
 100 cm   1 in.   1 ft 
6m 
≈ 19.7 m
 
 
 =
(1 i 2.54 i 12)
 1 m   2.54 cm   12 in. 
Radius = 19.7 m ÷ 2 = 9.85 m
Area = π (9.85 m)2 ≈ 304.65 sq m
©2012, TESCCC
10/11/12
page 4 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis
Use dimensional analysis to convert the given units of measure.
2.2 lbs
1 kg
or
1 kg
2.2 lbs
3.79 L
1 gal
or
1 gal
3.79 L
2.54 cm
1 in.
or
1 in.
2.54 cm
1. Sam walked 230 yards to the cafeteria from his math class. How many meters did Sam walk?
2. Jane belongs to a motorcycle club. Over the weekend, she drove 325 kilometers. How many
miles did Jane travel?
3. The area of a square is 36 square centimeters. How many centimeters long is one side of the
square? How many inches long is one side of the square? What is the area of the square
expressed as square inches?
©2012, TESCCC
05/01/13
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis
4. In example below, a square’s area of 36 square centimeters was converted directly to an area
of a square expressed in square inches. How does the result compare to the answer in
problem 3? Why is the result below incorrect?
 1 in.  (36 1)
36 sq cm 
 = 2.54  14.17 sq in.
 2.54 cm 
5. The area of a triangle is 6 square feet. If the base of the triangle is 3 feet, how many
centimeters long is the height of the triangle?
6. A rectangular piece of land has a length and width of 1.5 miles and 0.75 miles, respectively.
What is the area of the land in square kilometers?
©2012, TESCCC
05/01/13
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis Key
Use dimensional analysis to convert the given units of measure.
2.2 lbs
1 kg
or
1 kg
2.2 lbs
3.79 L
1 gal
or
1 gal
3.79 L
2.54 cm
1 in.
or
1 in.
2.54 cm
1. Sam walked 230 yards to the cafeteria from his math class. How many meters did Sam walk?
Answer: convert yards to meters
 36 in.   2.54 cm   1 m  (230 i 36 i 2.54 i 1)
= 210.312 m
230 yd 
 
 
 =
(1 i 1 i 100)
 1 yd   1 in.   100 cm 
2. Jane belongs to a motorcycle club. Over the weekend, she drove 325 kilometers. How many
miles did Jane travel?
Answer: convert kilometers to miles
 1,000 m   100 cm   1 in.   1 yd   1 mi 
325 km 
 =
 
 
 
 
 1 km   1 m   2.54 cm   36 in.   1,760 yd 
(325 i 1,000 i 100 i 1 i 1 i 1)
≈ 201.9 mi
(1 i 1 i 2.54 i 36 i 1,760)
3. The area of a square is 36 square centimeters. How many centimeters long is one side of the
square? How many inches long is one side of the square? What is the area of the square
expressed as square inches?
Answer: convert centimeters to inches after finding the length of the square in centimeters
Side = 36 = 6 cm
 1 in.   6 i 1 
6 cm 
 = 
 ≈ 2.36 in.
 2.54 cm   2.54 
Area = (2.36 in.)2 ≈ 5.6 sq in.
©2012, TESCCC
10/11/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis Key
4. In example below, a square’s area of 36 square centimeters was converted directly to an area
of a square expressed in square inches. How does the result compare to the answer in
problem 3? Why is the result below incorrect?
(36 i 1)
 1 in. 
36 sq cm 
≈ 14.17 sq in.
 =
2.54
 2.54 cm 
Answer:
The area of the square expressed as square inches does not match the area of the square
from problem 3.
The conversion of 36 sq cm was treated as a single dimension instead of the product of two
1 inch
linear dimensions. The ratio
is a conversion for a single linear dimension.
2.54 cm
5. The area of a triangle is 6 square feet. If the base of the triangle is 3 feet, how many
centimeters long is the height of the triangle?
Answer: convert feet to centimeters after calculating the height of the triangle in feet
bh
Area =
2
3h
6 sq ft =
2
 3h 
2(6) = 2 

 2 
12 = 3h
4 ft = h
 12 in.   2.54 cm  (4 i 12 i 2.54)
4 ft 
= 121.92 cm
 
 =
1i1
 1 ft   1 in. 
6. A rectangular piece of land has a length and width of 1.5 miles and 0.75 miles, respectively.
What is the area of the land in square kilometers?
Answer: convert miles to kilometers
 1,760 yd   36 in.   2.54 cm   1 m   1 km 
1.5 mi 
 
 
 
 
 =
 1 mi   1 yd   1 in.   100 cm   1,000 m 
(1.5 i 1,760 i 36 i 2.54 i 1 i 1)
≈ 2.41 km
(1 i 1 i 1 i 100 i 1,000)
 1,760 yd   36 in.   2.54 cm   1 m   1 km 
0.75 mi 
 
 
 
 
 =
 1 mi   1 yd   1 in.   100 cm   1,000 m 
(0.75 i 1,760 i 36 i 2.54 i 1 i 1)
≈ 1.21 km
(1 i 1 i 1 i 100 i 1,000)
Area = (2.41 km)(1.21 km) ≈ 2.92 sq km
©2012, TESCCC
10/11/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis
Use dimensional analysis to convert the given units of measure.
1. Sam walked 230 yards to the cafeteria from his math class. How many meters did Sam walk?
2. Jane belongs to a motorcycle club. Over the weekend, she drove 325 kilometers. How many
miles did Jane travel?
3. The area of a square is 36 square centimeters. How many centimeters long is one side of the
square? How many inches long is one side of the square? What is the area of the square
expressed as square inches?
©2012, TESCCC
10/11/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Analysis
4. In example below, a square’s area of 36 square centimeters was converted directly to an area
of a square expressed in square inches. How does the result compare to the answer in
problem 3? Why is the result below incorrect?
 1 in.  (36 i 1)
36 sq cm 
≈ 14.17 sq in.
 =
2.54
 2.54 cm 
5. The area of a triangle is 6 square feet. If the base of the triangle is 3 feet, how many
centimeters long is the height of the triangle?
6. A rectangular piece of land has a length and width of 1.5 miles and 0.75 miles, respectively.
What is the area of the land in square kilometers?
©2012, TESCCC
10/11/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Match Up Cards KEY
Title
Cube
Definition
Model
Net
A three-dimensional figure with
six square faces, twelve
congruent edges, and eight
vertices.
A three-dimensional figure with
Rectangular
two congruent rectangular bases
Prism
and four rectangular lateral faces.
Triangular
Prism
Pentagonal
Prism
©2012, TESCCC
A three-dimensional figure with
two congruent triangular bases
and three rectangular lateral
faces.
A three-dimensional figure with
two congruent pentagonal bases
and five rectangular lateral faces.
10/11/12
page 1 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Match Up Cards KEY
Title
Definition
Hexagonal
Prism
A three-dimensional
figure with two congruent
hexagonal bases and six
rectangular lateral faces.
Octagonal
Prism
A three-dimensional
figure with two congruent
octagonal bases and
eight rectangular lateral
faces.
Triangular
Pyramid
A three-dimensional
figure with one triangular
base and three triangular
lateral faces.
Model
Net
A three-dimensional
figure with one
Rectangular
rectangular base and
Pyramid
four triangular lateral
faces.
©2012, TESCCC
10/11/12
page 2 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Match Up Cards KEY
Title
Definition
Model
Net
A three-dimensional figure with
Pentagonal
one pentagonal base and five
Pyramid
triangular lateral faces.
Hexagonal
Pyramid
A three-dimensional figure with
one hexagonal base and six
triangular lateral faces.
Octagonal
Pyramid
A three-dimensional figure with
one octagonal base and eight
triangular lateral faces.
Cylinder
A three-dimensional figure with
two circular bases, no vertices,
and a curved lateral surface.
Cone
©2012, TESCCC
A three-dimensional figure with
one circular base, one vertex, and
a curved lateral surface.
10/11/12
page 3 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Match Up Cards
Cube
Rectangular
Prism
Triangular
Prism
©2012, TESCCC
A three-dimensional figure with
six square faces, twelve
congruent edges, and eight
vertices.
A three-dimensional figure with
two congruent rectangular bases
and four rectangular lateral faces.
A three-dimensional figure with
two congruent triangular bases
and three rectangular lateral
faces.
Pentagonal
Prism
A three-dimensional figure with
two congruent pentagonal bases
and five rectangular lateral faces.
Hexagonal
Prism
A three-dimensional figure with
two congruent hexagonal bases
and six rectangular lateral faces.
Octagonal
Prism
A three-dimensional figure with
two congruent octagonal bases
and eight rectangular lateral
faces.
Triangular
Pyramid
A three-dimensional figure with
one triangular base and three
triangular lateral faces.
10/11/12
page 1 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Match Up Cards
Rectangular
Pyramid
A three-dimensional figure with
one rectangular base and four
triangular lateral faces.
Pentagonal
Pyramid
A three-dimensional figure with
one pentagonal base and five
triangular lateral faces.
Hexagonal
Pyramid
A three-dimensional figure with
one hexagonal base and six
triangular lateral faces.
Octagonal
Pyramid
A three-dimensional figure with
one octagonal base and eight
triangular lateral faces.
Cylinder
A three-dimensional figure with
two circular bases, no vertices,
and a curved lateral surface.
Cone
©2012, TESCCC
A three-dimensional figure with
one circular base, one vertex, and
a curved lateral surface.
10/11/12
page 2 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
1.5 in.
Geometric Match Up Cards
1.
5
in
.
©2012, TESCCC
10/11/12
page 3 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Match Up Cards
©2012, TESCCC
10/11/12
page 4 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Match Up Cards
©2012, TESCCC
10/11/12
page 5 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Questions KEY
1. Draw a three-dimensional model of a rectangular prism on the grid below. Shade each face
yellow, outline each edge green, and highlight each vertex red.
Refer to the Geometric Match Up Cards chart for problems 2 through 16. In problems 5-15, tell the
shape of the base(s) and faces.
2. How many different geometric models can be defined as a prism?
Answer: 6
3. How many different geometric models can be defined as a pyramid?
Answer: 5
4. How many vertices, faces and edges are on a triangular prism?
Vertices: 6
Faces: 5
Edges: 9
5. The net of a cube consists of six square faces.
6. The net of a square prism consists of two squares and four rectangles.
7. The net of a triangular prism consists of two triangles and three rectangles.
©2012, TESCCC
10/11/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Questions KEY
8. The net of a pentagonal prism consists of two pentagons and five rectangles.
9. The net of a hexagonal prism consists of two hexagons and six rectangles.
10. The net of an octagonal prism consists of two octagons and eight rectangles.
11. The net of a triangular pyramid consists of four triangles.
12. The net of a square pyramid consists of four triangles and one square.
13. The net of a pentagonal pyramid consists of five triangles and one pentagon.
14. The net of a hexagonal pyramid consists of six triangles and one hexagon.
15. The net of a cylinder consists of one rectangle and two circles.
16. Lateral rectangle face is to prism as lateral triangle face is to pyramid.
©2012, TESCCC
10/11/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Questions
1. Draw a three-dimensional model of a rectangular prism on the grid below. Shade each face
yellow, outline each edge green, and highlight each vertex red.
Refer to the Geometric Match Up Cards chart for problems 2 through 16. In problems 5-15, tell the
shape of the base(s) and faces.
2. How many different geometric models can be defined as a prism?
3. How many different geometric models can be defined as a pyramid?
4. How many vertices, faces and edges are on a triangular prism?
Vertices: ____________
Faces: ____________
Edges: ____________
5. The net of a cube consists of ________________ faces.
6. The net of a square prism consists of ________________ and ________________.
7. The net of a triangular prism consists of ________________ and ________________.
©2012, TESCCC
10/11/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Questions
8. The net of a pentagonal prism consists of ________________ and ________________.
9. The net of a hexagonal prism consists of ________________ and ________________.
10. The net of an octagonal prism consists of ________________ and ________________.
11. The net of a triangular pyramid consists of ________________.
12. The net of a square pyramid consists of ________________ and ________________.
13. The net of a pentagonal pyramid consists of ________________ and ________________.
14. The net of a hexagonal pyramid consists of ________________ and ________________.
15. The net of a cylinder consists of ________________ and ________________.
16. Lateral rectangle face is to ________________ as lateral triangle face is to ______________.
©2012, TESCCC
10/11/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures KEY
5 cm
Note: The dimensions shown on each net were measured to the nearest half of a centimeter.
Changes in the dimensions may occur when the nets are copied. A small diagram of each net
with the measured dimensions given in centimeters is provided below.
Cube
Rectangular Prism
Base
Base
lateral
face
Base: 2 congruent squares – 4.5 cm x 4.5 cm
Lateral rectangular faces – 4 rectangles:
4 congruent rectangles – 4.5 cm x 6 cm
Bases and lateral faces: 6 congruent squares
Each side of the cube = 5 cm
Triangular Prism
Pentagonal Prism
lateral
face
Base: 2 congruent right triangles:
Base of triangle = 5 cm height of triangle = 5 cm
Lateral rectangular faces – 3 rectangles:
2 congruent squares: 5 cm x 5 cm
1 rectangle: 5 cm x 7 cm
©2012, TESCCC
4 cm
Base
3.5 cm
Base
4 cm
6 cm
2 cm
lateral
face
Base: 2 congruent pentagons
Divided pentagon into a triangle and trapezoid:
Base of triangle = 6 cm height of triangle = 2 cm
Trapezoid: b1 = 4 cm; b2 = 6 cm; h = 3.5 cm
Lateral rectangular faces – 5 rectangles:
5 congruent squares: 4 cm x 4 cm
10/11/12
page 1 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures KEY
Note: The dimensions shown on each net were measured to the nearest half of a centimeter.
Changes in the dimensions may occur when the nets are copied. A small diagram of each net
with the measured dimensions given in centimeters is provided below.
Hexagonal Prism
Octagonal Prism
2 cm
3 cm
3 cm
7 cm
7 cm
3.5 cm
3.5 cm
Base
3 cm
3 cm
3 cm
lateral
face
lateral
face
Base: 2 congruent octagons
Divided octagon into 2 congruent trapezoids and
1 rectangle:
Trapezoid: b1 = 3 cm; b2 = 7 cm; h = 2 cm
Rectangle: 3 cm x 7 cm
Lateral rectangular faces – 8 rectangles:
8 congruent squares: 3 cm x 3 cm
Rectangular Pyramid
Base: 2 congruent hexagons
Divided hexagon into 2 congruent trapezoids:
Trapezoid: b1 = 3.5 cm; b2 = 7 cm; h = 3 cm
Lateral rectangular faces – 6 rectangles:
6 congruent squares: 3.5 cm x 3.5 cm
Triangular Pyramid
lateral
Base face
5.5 cm
lateral
face
6.5 cm
Base
5 cm
4.5 cm
6 cm
Base: 1 equilateral triangle:
Base = 6.5 cm; height = 5.5 cm
Lateral triangular faces – 3 congruent triangles:
Base = 6.5 cm; height = 6 cm
©2012, TESCCC
Base
Base: rectangle:
1 square: 5 cm x 5 cm
Lateral triangular faces – 4 congruent triangles:
Base = 5 cm; height = 4.5 cm
10/11/12
page 2 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures KEY
Note: The dimensions shown on each net were measured to the nearest half of a centimeter.
Changes in the dimensions may occur when the nets are copied. A small diagram of each net
with the measured dimensions given in centimeters is provided below.
Pentagonal Pyramid
Hexagonal Pyramid
lateral
face
6.5 cm
2.5 cm
3.5 cm
6.5 cm 3.5 cm
4 cm Base
3 cm
7 cm
Base
lateral
face
6.5 cm
Base: 1 pentagon
Divided pentagon into a triangle and trapezoid:
Base of triangle = 6.5 cm
Height of triangle = 2.5 cm
Trapezoid: b1 = 4 cm; b2 = 6.5 cm; h = 3.5 cm
Lateral triangular faces – 5 triangles:
5 congruent triangles: b = 4 cm; h = 6.5 cm
Cylinder
Base: 1 hexagon
Divided hexagon into 2 congruent trapezoids:
Trapezoid: b1 = 3.5 cm; b2 = 7 cm; h = 3 cm
Lateral triangular faces – 6 triangles:
6 congruent triangles:
Base = 3.5 cm; height = 6.5 cm
Cone
2.5 cm
2.5 cm
13.5 cm
8 cm
Lateral
surface
Base
Base: 2 circles
Radius = 2.5 cm
Lateral surface: curved surface flattened into a
rectangle
©2012, TESCCC
Base: 1 circle
Radius = 2.5 cm
Lateral surface: curved surface
Perpendicular height of cone = 8 cm
10/11/12
page 3 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures KEY
lateral
face
2.5 cm
Note: The dimensions shown on each net were measured to the nearest half of a centimeter.
Changes in the dimensions may occur when the nets are copied. A small diagram of each net with
the measured dimensions given in centimeters is provided below.
Trapezoidal Prism
lateral
face
lateral
face
2.5 cm
3.5 cm
Base
2.5 cm
7.5 cm
lateral
face
2.5 cm
Base
Base: 2 congruent trapezoids
Trapezoid: b1 = 2.5 cm; b2 = 7.5 cm; h = 2.5 cm
Lateral rectangular faces – 4 rectangles:
2 congruent rectangles: 2.5 cm x 3.5 cm
1 square: 2.5 cm x 2.5 cm
1 rectangle: 2.5 cm x 7.5 cm
©2012, TESCCC
10/11/12
page 4 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Cube
©2012, TESCCC
10/11/12
page 1 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Rectangular Prism
©2012, TESCCC
10/11/12
page 2 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Triangular Prism
©2012, TESCCC
10/11/12
page 3 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Pentagonal Prism
©2012, TESCCC
10/11/12
page 4 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Hexagonal Prism
©2012, TESCCC
10/11/12
page 5 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Octagonal Prism
©2012, TESCCC
10/11/12
page 6 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Triangular Pyramid
©2012, TESCCC
10/11/12
page 7 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Rectangular Pyramid
©2012, TESCCC
10/11/12
page 8 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Pentagonal Pyramid
©2012, TESCCC
10/11/12
page 9 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Hexagonal Pyramid
©2012, TESCCC
10/11/12
page 10 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Cylinder
©2012, TESCCC
10/11/12
page 11 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Cone
©2012, TESCCC
10/11/12
page 12 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Templates for Three-Dimensional Figures: Trapezoidal Prism
©2012, TESCCC
10/11/12
page 13 of 13
Grade 8
Mathematics
Unit: 07 Lesson: 01
Nets for Three-Dimensional Figure Challenge KEY
Three-Dimensional Figure
Net
1. Illustrate four other ways the net could be drawn and still be folded to produce the threedimensional figure shown above.
Example A
Example B
Example C
©2012, TESCCC
Example D
10/11/12
page 1 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Nets for Three-Dimensional Figure Challenge KEY
2. Use grid paper to cut out the nets shown below (a – f). Indicate which of the nets (a – f) can be
folded to produce a cube.
a) Does the net shown below
fold into a cube? YES
b) Does the net shown below
fold into a cube? YES
c) Does the net shown below
fold into a cube? YES
d) Does the net shown below
fold into a cube? YES
e) Does the net shown below
fold into a cube? NO
f) Does the net shown below
fold into a cube? YES
3. Show other nets that will fold into a cube. Draw the nets below.
Answer: Answers will vary. Some samples are shown below
©2012, TESCCC
10/11/12
page 2 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Nets for Three-Dimensional Figure Challenge KEY
4. Complete the following table by drawing the three-dimensional figure that is formed by the given
net or by drawing a net for the given three-dimensional figure.
Three-Dimensional Figure
©2012, TESCCC
Net
10/11/12
page 3 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Nets for Three-Dimensional Figure Challenge
Three-Dimensional Figure
Net
1. Illustrate four other ways the net could be drawn and still be folded to produce the threedimensional figure shown above.
Example A
Example B
Example C
©2012, TESCCC
Example D
10/11/12
page 1 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Nets for Three-Dimensional Figure Challenge
2. Use grid paper to cut out the nets shown below (a – f). Indicate which of the nets (a – f) can be
folded to produce a cube.
a) Does the net shown below
fold into a cube?
b) Does the net shown below
fold into a cube?
c) Does the net shown below
fold into a cube?
d) Does the net shown below
fold into a cube?
e) Does the net shown below
fold into a cube?
f) Does the net shown below
fold into a cube?
3. Show other nets that will fold into a cube. Draw the nets below.
©2012, TESCCC
10/11/12
page 2 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Nets for Three-Dimensional Figure Challenge
4. Complete the following table by drawing the three-dimensional figure that is formed by the given
net or by drawing a net for the given three-dimensional figure.
Three-Dimensional Figure
©2012, TESCCC
Net
10/11/12
page 3 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Centimeter Grid Paper
©2012, TESCCC
10/12/12
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Rectangular Prism KEY
1. Use the net from the handout: Nets for Three-Dimensional Figures for a rectangular prism to
find the surface area of the rectangular prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Rectangular Prism
Rectangles
Sketch of Three-Dimensional Figure
6 rectangular faces form this
rectangular prism: 2 rectangular bases and
4 rectangular lateral faces
Formula: Total Surface Area
SA = Ph + 2B
SA = (4 • 4.5)(6) + 2(4.5 • 4.5)
SA = 148.5 cm2
Rectangle: A = l • w
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
05/01/13
OR
SA = 2(4.5 · 4.5) + 4(4.5 · 6) = 148.5 cm2
Total Surface Area
page 1 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Triangular Prism KEY
2. Use the net from the handout: Nets for Three-Dimensional Figures for a triangular prism to find
the surface area of the triangular prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Triangular Prism
Bases are triangles
Lateral faces are rectangles
Sketch of Three-Dimensional Figure
2 congruent triangles form the bases plus
1 non-congruent rectangle and 2 congruent
rectangles form lateral faces.
Formula Total Surface Area: SA = Ph + 2B
SA = 5(5 + 5 + 7) + 2(5 • 5 ÷ 2) = 110 cm2
1
bh
bh or A =
2
2
Rectangle: A = l • w
Triangles: A =
Area Formulas for Each Figure that Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
05/01/13
OR
1
2( bh) + ( l • w) + ( l • w) + ( l • w)
2
1
2( • 5 • 5) + ( 7 • 5) + 2( 5 • 5) = 110 cm2
2
Total Surface Area
page 2 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Hexagonal Prism KEY
3. Use the net from the handout: Nets for Three-Dimensional Figures for a hexagonal prism to find
the surface area of the hexagonal prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Hexagonal Prism
Bases are hexagons
Lateral faces are rectangles
Sketch of Three-Dimensional Figure
This hexagonal prism is a 3-D figure with 8 faces: 2
congruent hexagonal bases and 6 congruent
rectangular lateral faces. Area of a hexagon can be
calculated by drawing a diagonal of the hexagon to
create congruent trapezoids.
Formula: SA = Ph + 2B
SA = 6 •(3.5 • 3.5) + 2(2 •
Rectangle: A = l • w
(b  b )h
1
Trapezoid: A = (b1 + b2)h or A = 1 2
2
2
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
05/01/13
1
(3.5 + 7) • 3) =
2
136.5 cm2 OR
Total Surface Area = 4(
= 4(
1
(b1 + b2)h) + 6(l • w)
2
1
(3.5 + 7)3) + 6(3.5•3.5)
2
=136.5 cm2
Total Surface Area
page 3 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Cylinder KEY
4. Use the net from the handout: Nets for Three-Dimensional Figures for a cylinder to find the
surface area of the cylinder.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Cylinder
Bases are circles
Curved lateral surface when cut apart is a
rectangle
Sketch of Three-Dimensional Figure
A cylinder has 2 circular congruent bases
and a curved lateral surface that when cut
apart is a rectangle.
Total Surface Area = 2(r2) + (l • w)
= 2(3.14•(2.5) 2) + (8 •13.5)
≈ 147.25 cm2
Use 3.14 for 
Circle: A = r2
Rectangle: A = l • w
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
05/01/13
Total Surface Area
page 4 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Rectangular Pyramid KEY
5. Use the net from the handout: Nets for Three-Dimensional Figures for a rectangular pyramid to
find the surface area of the rectangular pyramid.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Rectangular Pyramid
Base is a rectangle
Lateral faces are triangles
Sketch of Three-Dimensional Figure
Rectangle: A = l • w
1
bh
Triangle: A = bh or A =
2
2
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
05/01/13
This rectangular pyramid has 5 faces with
1 rectangular base and 4 congruent
triangular lateral faces.
1
Formula Surface Area: SA = P(l) + B
2
1
SA =
• 4 • 5 • 4.5 + 5 • 5 = 70 cm2
2
OR
1
Total Surface Area = (l • w) + 4( bh)
2
1
SA = (5 • 5) + 4( • 5• 4.5) = 70 cm2
2
Total Surface Area
page 5 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Trapezoidal Prism KEY
6. Use the net from the handout: Nets for Three-Dimensional Figures for a trapezoidal prism to
find the surface area of the trapezoidal prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Trapezoidal Prism
Bases are trapezoids
Lateral faces are rectangles
Sketch of Three-Dimensional Figure
This Trapezoidal Prism has 6 faces with 2
congruent trapezoidal bases and 4 lateral
rectangular faces.
Surface Area Formula: SA = Ph + 2B
1
Trapezoid: A = (b1 + b2)h
2
or
(b1 + b2 )h
A=
2
Rectangle: A = l • w
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
05/01/13
1
SA=(2.5 + 3.5 + 7.5 + 3.5)2.5 + 2( (2.5 +
2
7.5)2.5) = 67.5 cm2 OR
Total Surface Area =
1
2( (b1 + b2)h) + 2(l • w) + (l • w )+ (l • w)
2
1
SA = 2( (2.5 + 7.5)2.5) + 2(3.5•2.5) +
2
(2.5•2.5 )+ (7.5•2.5) = 67.5 cm2
Total Surface Area
page 6 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Rectangular Prism
1. Use the net from the handout: Nets for Three-Dimensional Figures for a rectangular prism to
find the surface area of the rectangular prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Sketch of Three-Dimensional Figure
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
10/12/12
Total Surface Area
page 1 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Triangular Prism
2. Use the net from the handout: Nets for Three-Dimensional Figures for a triangular prism to find
the surface area of the triangular prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Sketch of Three-Dimensional Figure
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
10/12/12
Total Surface Area
page 2 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Hexagonal Prism
3. Use the net from the handout: Nets for Three-Dimensional Figures for a hexagonal prism to find
the surface area of the hexagonal prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Sketch of Three-Dimensional Figure
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
10/12/12
Total Surface Area
page 3 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Cylinder
4. Use the net from the handout: Nets for Three-Dimensional Figures for a cylinder to find the
surface area of the cylinder.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Sketch of Three-Dimensional Figure
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
10/12/12
Total Surface Area
page 4 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Rectangular Pyramid
5. Use the net from the handout: Nets for Three-Dimensional Figures for a rectangular pyramid to
find the surface area of the rectangular pyramid.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Sketch of Three-Dimensional Figure
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
10/12/12
Total Surface Area
page 5 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: Trapezoidal Prism
6. Use the net from the handout: Nets for Three-Dimensional Figures for a trapezoidal prism to
find the surface area of the trapezoidal prism.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Sketch of Three-Dimensional Figure
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
10/12/12
Total Surface Area
page 6 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area: ________________________
Use the Nets for 3-Dimensional Figures activity to find the surface area of the following threedimensional figure.
Name of Three-Dimensional Figure
Figures That Form Base(s) and Lateral
Surface of the 3-Dimensional Figure
Sketch of Three-Dimensional Figure
Area Formulas for Each Figure That Forms
Base(s) and Lateral Surfaces of the
3-Dimensional Figure
©2012, TESCCC
10/12/12
Total Surface Area
page 7 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Net Puzzles KEY
1. Cone:
Surface Area: Region A + Region B
Region A:
A = π r 2 ÷ 2; 3.14(7) 2 ÷ 2 = 76.93
Region B:
A = π r 2; 3.14(3.5) 2 = 38.465
Surface Area: 76.93 + 38.465 = 115.395 u 2
2. Cylinder:
Surface Area: 2 • Region A + Region B
Region A:
A = π r 2; 3.14(3.5) 2 = 38.465
Region B:
A = (2 π r)(7); (2 • 3.14 • 3.5)(7) = 153.86
Surface Area: 2(38.465) + 153.86 = 230.79 u 2
Note: The length of the lateral surface is the
circumference of the circular base.
3. Triangular Pyramid:
Surface Area: Region A + 3 • Region B
Region A:
A = (6 • 5.2) ÷ 2 = 15.6
Region B:
A = (6 • 7) ÷ 2 = 21
Surface Area: 15.6 + 3 • (21) = 78.6 u 2
4. Rectangular Prism
Surface Area: 2 • Region A + 2 • Region B + 2 •
Region C
Region A:
A = 8 • 4 = 32
Region B:
A = 4 • 12 = 48
Region C:
A = 8 • 12 = 96
Surface Area: 2 • 32 + 2 • 48 + 2 • 96 = 352 u 2
©2012, TESCCC
10/12/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Net Puzzles KEY
5. Rectangular Pyramid:
Surface Area: Region A + 4 • Region B
Region A:
A = 8 • 8 = 64
Region B:
A = (8 • 10) ÷ 2 = 40
B
Surface Area: 64 + 4 • 40 = 224 u 2
B
A
B
B
6. Triangular Prism:
B
A
C
A
Surface Area: 2 • Region A + 2 • Region B +
Region C
Region A:
A = (4 • 6) ÷ 2 = 12
Region B:
A = 5 • 11 = 55
Region C:
A = 6 • 11 = 66
Surface Area: 2 • 12 + 2 • 55 + 66 = 200 u 2
B
©2012, TESCCC
10/12/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Net Puzzles
Copy on cardstock and cut apart along the solid lines. Place one set in a plastic zip bag.
Triangular Prism Set
Triangular Prism
Triangular Pyramid Set
Triangular Pyramid
©2012, TESCCC
10/12/12
page 1 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Net Puzzles
Copy on cardstock and cut apart along the solid lines. Place one set in a plastic zip bag.
Cone Set
Cone
Cylinder Set
7 units
Cylinder
©2012, TESCCC
10/12/12
page 2 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Net Puzzles
Copy on cardstock and cut apart along the solid lines. Place one set in a plastic zip bag.
Rectangular Prism Set
Rectangular Prism
©2012, TESCCC
10/12/12
page 3 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Net Puzzles
Copy on cardstock and cut apart along the solid lines. Place one set in a plastic zip bag.
Rectangular Pyramid Set
Rectangular Pyramid
©2012, TESCCC
10/12/12
page 4 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Volume Notes
1. Look at a SQUARE PRISM (3-dimensional figure with two square bases and four lateral
rectangular faces). Stand it on one square base. Imagine that this square prism is a single stack of
square shaped crackers. In general, the volume of the prism would be modeled by the area of one
cracker, the square base, times the number of stacked crackers. Mathematically, we state: the
volume of a square prism = area of the square base x height of the prism which is represented by
the formula: Volume square prism = B x h.
2. This formula, V = Bh, will apply to ANY PRISM or CYLINDER, where “B” represents the area of
the base and “h” represents the height of the prism or cylinder.
3. RECTANGULAR PRISM: Consider a package of construction paper. The volume of the package
is the area of the bottom rectangle, 1 sheet, times the number of sheets in the package of
construction paper. Mathematically, we state: Volume of a rectangular prism = Area of the
Rectangular Base x height. This is represented by the formula: V = Bh. Since the area of the
base, B, is the area of a rectangle (A = lw), we can also write the formula as: V = lwh.
4. TRIANGULAR PRISM: Consider a stack of triangular shaped crackers. The volume of the stack is
the area of the bottom triangle, 1 cracker, times the number of crackers in the stack.
Mathematically we state: Volume of a triangular prism = Area of the Triangular Base x Height.
This is represented by the formula: V = Bh. Since the area of the base, B, is the area of a triangle,
bh
bh
(A =
), we can also write the formula as: V = ( )h .
2
2
5. CYLINDER: Consider a package of flour tortillas. The volume of the package is the area of a
circular tortilla times the number of flour tortillas in the package. Mathematically we state: Volume
of a cylinder = Area of the Circular Base x Height. This is represented by the formula:
V = Bh. Since the area of the base, B, is the area of a circle (A = πr 2), we can also write the
formula as: V = πr 2 h.
©2012, TESCCC
05/01/13
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Volume KEY
Use the three-dimensional models constructed from the handout: Templates for Three-Dimensional
Figures. For each three-dimensional figure shown below:
• Identify the base and draw the base.
• Calculate the area of the base.
• Write the height of the three-dimensional figure.
• Write the formula to calculate the volume of the three dimensional figure.
• Calculate the volume of the three-dimensional figure.
• Use 3.14 for ̟. Round answers using ̟ to the nearest hundredth.
Figure
Square Prism
(Cube)
Rectangular
Prism
Draw the Base
Area of the
Base
Length x width
5 x 5 = 25
Height of
Figure
Process to
Find Volume
Volume
5
Length x width
x height
25 x 5
125 cm3
6
Length x width
x height
20.25 x 6
121.5 cm3
5
[(base x
height) ÷ 2] x
height
12.5 x 5
62.5 cm3
Length x width
4.5 x 4.5 =
20.25
Triangular Prism
(base x
height) ÷ 2
(5 x 5)÷ 2 =
12.5
Cylinder
π x radius2
3.14 x 2.52 ≈
19.625
©2012, TESCCC
10/12/12
8
π x radius2 x
height
19.625 x 8
157 cm3
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Volume
Use the three-dimensional models constructed from the handout: Templates for Three-Dimensional
Figures. For each three-dimensional figure shown below:
• Identify the base and draw the base.
• Calculate the area of the base.
• Write the height of the three-dimensional figure.
• Write the formula to calculate the volume of the three dimensional figure.
• Calculate the volume of the three-dimensional figure.
Figure
Draw the Base
Area of the
Base
Height of
Figure
Process to
Find Volume
Volume
Square Prism
(Cube)
Rectangular
Prism
Triangular Prism
Cylinder
©2012, TESCCC
10/11/12
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement KEY
For problems 1-7:
 Read the problem and determine if the problem is asking for SURFACE AREA or VOLUME.
 Circle the correct word to indicate the problem type based on the question.
 Write down the correct formula.
 Find the needed dimensions from the problem or by measuring with the ruler from your STAAR
Grade 8 Reference Materials. Use 3.14 for , round your answers to the nearest hundredth.
 Calculate the surface area or volume. Keep your work neat and organized.
 Write a complete sentence to answer the question. Include the correct unit of measurement in
your answer.
Some problems contain measurement conversions.
1. Before this tissue box was opened, the opening in the top was filled with cardboard. Deanna is
going to wrap an empty tissue box like this to use for a ballot box in the Junior High Spring Fling
Sweetheart Contest. How much gift-wrapping paper does Deanna need to wrap the entire tissue
box?
Circle: Surface Area or Volume
Formula: S = 2(lw)+2(lw)+2(lw)
OR
S = Ph + 2B
S = (9 + 4 + 9 + 4) • 3 + 2(9 • 4)
Answer: 150 in2
3 in.
9 in.
4 in.
2. When this vanilla scented candle was new, it was 1 foot tall and the distance across the top of
the candle through the wick, which was centered, was 8 inches. How many cubic inches of wax
were used to form this candle?
Formula: V = Bh = (r 2) • h
V = (3.14 • 42) • 12 ≈ 602.88
Circle: Surface Area or Volume
Answer: 602.88 in3
©2012, TESCCC
05/01/13
page 1 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement KEY
3. This type of tractor was used to pack road surfaces in the mid 1900’s. Look at the front wheel of
the tractor and determine the approximate amount of road surface that came in contact with the
front wheel in one complete rotation.
Circle: Surface Area or Volume
fee
5
.
20 inches 2
Formula: Lateral Surface Area: 2rh
Note: Convert 2.5 feet to inches
h = 2.5 feet • 12 inches = 30 inches
S = 2 • 3.14 • 10 • 30 ≈
Answer: 1,884.00 in2
t
4. Holly works in the Product Analysis division of Fun in Sun Sporting Equipment, Inc. The tent
shown below includes a material floor. As part of Holly’s cost analysis, she needs to determine
the amount of material used to make the tent. Use the dimensions in the picture to calculate the
amount of material used to make this tent. Round your answer to the nearest hundredth.
Hint: There is a missing dimension you will need to calculate using the Pythagorean Theorem.
Note: Find the hypotenuse of the right triangle with legs of length 5 ÷ 2 and 4.
2.52  42  4.716990566
Formula: 2(bh ÷ 2) + 2(lw) + lw
Front and Back of Tent: 2(bh ÷ 2)
Left and Right Sides of Tent: 2(lw)
Bottom of Tent: lw
2(5 • 4 ÷ 2) + 2(7 • 4.71699) + 5 • 7 ≈
Answer: 121.04 ft.2
4 feet
Circle: Surface Area or Volume
5 feet
©2012, TESCCC
7 feet
05/01/13
page 2 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement KEY
5. As part of cross-curricular applications, students must tie their research project in Language
Arts to all four core subjects. As one part of the relation to math, Nick is going to find out how
much paper it took to fill this English-Japanese dictionary in terms of cubic feet. Use the
dimensions in the picture for your calculations.
Circle: Surface Area or Volume
Formula: V = lwh
Convert 18 inches to feet: 18 ÷ 12 = 1.5
Convert 42 inches to feet: 42 ÷ 12 = 3.5
V = 3 • 1.5 • 3.5
42 inches
Answer: 15.75 ft.3
3 feet
18 in
ch e s
6. There is a large cylindrical aquarium in the center of the new bank. The aquarium’s diameter is
120 inches and its height extends from the floor to the 12-foot ceiling. What is the approximate
capacity of the aquarium in terms of cubic feet? Round your answer to the nearest hundredth.
Formula: V = r 2 h
Convert 120 inches to feet: 120 ÷ 12 = 10
V = 3.14(5)2 • 12 ≈
Answer: 942.00 ft3
Circle: Surface Area or Volume
©2012, TESCCC
05/01/13
page 3 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement KEY
7. Mrs. Kirkland is opening a giant can of corn to make a casserole for a family reunion. She
wonders how much paper was used to make the label for the can. Represent your response in
terms of inches. Round your answer to the nearest hundredth.
Lateral Surface Area
Circle: Surface Area or Volume
Formula: Lateral Surface Area = 2 rh
S = 2 • 3.14 • 4 • 12 ≈
Answer: 301.44 in2
8 inches
CSCOPE
Corn
1 foot
CORN
8. Describe how you calculate the volume of a tennis ball. Write the formula and indicate what
measurements you would use to calculate the volume.
4
Formula:  r 3 We need to know the radius of the tennis ball to calculate the volume. We
3
can measure the circumference of the tennis ball and solve for the radius by dividing the
circumference by 2 and by . By substituting the value of the radius and following the
4
correct order of operations in the formula  r 3 , we are able to calculate the volume of
3
the tennis ball.
9. Describe how you calculate the volume of a snow cone cup. Write the formula and indicate what
measurements you would use to calculate the volume. For this special snow cone cup, the
height is the same length as the radius.
1
1
1
Formula: Bh = ( r 2 )r =  r 3 We need to know the radius of the circular base of the
3
3
3
cone and the height of the cone to calculate the volume. We can measure the radius of
the cone and turn the cone upside down and measure how tall the cone is to get the
height. By substituting in the measurement of the radius and height and following the
1
1
1
correct order of operations in the formula Bh = ( r 2 )r =  r 3 , we are able to calculate
3
3
3
the volume of the snow cone cup.
©2012, TESCCC
05/01/13
page 4 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement KEY
10. Gary is installing a seawater aquarium in his home. The aquarium is shaped like a rectangular
prism with the following dimensions: length of 4.5 feet, width of 2 feet, and a height of 2.5 feet.
The tank is filled with water. Gary needs to constantly monitor the salinity of the water; therefore,
he needs to know how many gallons of water are in the tank. Use this information to answer the
questions below.
Note: There are several methods to do conversions. Dimensional analysis was used in Unit 12
Lesson 01. As a review, dimensional analysis is used in the solutions for problem 10.
a) If 231 cubic inches is equivalent to 1 gallon of water, how many gallons of water are in the
aquarium if the water line is 3 inches below the top of the tank?
Convert feet to inches and then calculate volume of aquarium – for the height, remove 3
inches.
4.5 12
 12 in. 
=
= 54 in.
Length: 4.5 ft 

Gallons of water in aquarium:
1
 1 ft 
2 12
 12 in. 
=
= 24 in.
Width: 2 ft 

 1 gal 
1
34,992 cu. in. 
 1 ft 

 231 cu.in. 
2.5 12
 12 in. 
= 30 in.
Height: 2.5 ft 
151.5 gallons
 =
1
 1 ft 
V = lwh
length = 54 in.; width = 24 in.; height = 30 in. ─ 3 in. = 27 in.
V = (54)(24)(27) = 34,992 cu. in.
b) All the information Gary uses to adjust the levels of salinity in the aquarium uses liters and
kilograms. How many liters of water are in the aquarium if 1 gallon is equivalent to 3.79 liters?
Convert gallons to liters:
 3.79 L  151.5 3.79
= 574.185 L
151.5 gal 
 =
1
 1 gal 
c) Gary buys 4 pound bags of salt pellets to adjust the salinity levels in the aquarium. How many
kilograms is equivalent to 4 pounds if 2.2 pounds is equivalent to 1 kilogram?
Convert pounds to kilograms:
4 1
 1 kg 
=
 1.82 kg
4 lbs 

2.2
 2.2 lbs 
d) Gary wants to cover the lateral faces of the aquarium with a special coating to help maintain
the water temperature of the aquarium. The sheets needed to cover the lateral faces are sold
by square meters. How many square meters of the covering does Gary need for the lateral
faces?
Convert feet to meters and then calculate lateral surface area:
4.5 12 2.54 1
 12 in.   2.54 cm   1 m 
=
 1.37 m
Length: 4.5 ft 





1 1 100
 1 ft   1 in.   100 cm 
2 12 2.54 1
 12 in.   2.54 cm   1 m 
=
 0.61 m
Width: 2 ft 





1 1 100
 1 ft   1 in.   100 cm 
2.5 12 2.54 1
 12 in.   2.54 cm   1 m 
=
 0.76 m
Height: 2.5 ft 





1 1 100
 1 ft   1 in.   100 cm 
Lateral Surface Area: 2(lh) + 2(wh) = 2(1.37 • 0.76) + 2(0.61 • 0.76)  3.01 sq m
©2012, TESCCC
05/01/13
page 5 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement
For problems 1-7:
 Read the problem and determine if the problem is asking for SURFACE AREA or VOLUME.
 Circle the correct word to indicate the problem type based on the question.
 Write down the correct formula.
 Find the needed dimensions from the problem or by measuring with the ruler from your STAAR
Grade 8 Reference Materials. Use 3.14 for ; round your answers to the nearest hundredth.
 Calculate the surface area or volume. Keep your work neat and organized.
 Write a complete sentence to answer the question. Include the correct unit of measurement in
your answer.
1. Before this tissue box was opened, the opening in the top was filled with cardboard. Deanna is
going to wrap an empty tissue box like this to use for a ballot box in the Junior High Spring Fling
Sweetheart Contest. How much gift-wrapping paper does Deanna need to wrap the entire tissue
box?
Circle: Surface Area or Volume
Formula: ______________________
3 in.
9 in.
4 in.
2. When this vanilla scented candle was new, it was 1 foot tall and the distance across the top of
the candle through the wick, which was centered, was 8 inches. How many cubic inches of wax
were used to form this candle?
Circle: Surface Area or Volume
©2012, TESCCC
Formula: ________________________
05/01/13
page 1 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement
3. This type of tractor was used to pack road surfaces in the mid 1900’s. Look at the front wheel of
the tractor and determine the approximate amount of road surface that came in contact with the
front wheel in one complete rotation.
Circle: Surface Area or Volume
fee
5
.
20 inches 2
Formula: ________________________
t
4. Holly works in the Product Analysis division of Fun in Sun Sporting Equipment, Inc. The tent
shown below includes a material floor. As part of Holly’s cost analysis, she needs to determine
the amount of material used to make the tent. Use the dimensions in the picture to calculate the
amount of material used to make this tent. Round your answer to the nearest hundredth.
Hint: There is a missing dimension you will need to calculate using the Pythagorean Theorem.
Formula: ________________________
4 feet
Circle: Surface Area or Volume
5 feet
©2012, TESCCC
7 feet
05/01/13
page 2 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement
5. As part of cross-curricular applications, students must tie their research project in Language
Arts to all four core subjects. As one part of the relation to math, Nick is going to find out how
much paper it took to fill this English-Japanese dictionary in terms of cubic feet. Use the
dimensions in the picture for your calculations.
Formula: ________________________
42 inches
Circle: Surface Area or Volume
3 feet
18 in
ch e s
6. There is a large cylindrical aquarium in the center of the new bank. The aquarium’s diameter is
120 inches and its height extends from the floor to the 12-foot ceiling. What is the approximate
capacity of the aquarium in terms of cubic feet? Round your answer to the nearest hundredth.
Circle: Surface Area or Volume
©2012, TESCCC
Formula: ________________________
05/01/13
page 3 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement
7. Mrs. Kirkland is opening a giant can of corn to make a casserole for a family reunion. She
wonders how much paper was used to make the label for the can. Represent your response in
terms of inches. Round your answer to the nearest hundredth.
Circle: Lateral Surface Area or Volume
Formula: ________________________
8 inches
CSCOPE
Corn
1 foot
CORN
8. Describe how you calculate the volume of a tennis ball. Write the formula and indicate what
measurements you would use to calculate the volume.
Formula: ________________________
9. Describe how you calculate the volume of a snow cone cup. Write the formula and indicate what
measurements you would use to calculate the volume. For this special snow cone cup, the
height is the same length as the radius.
Formula: ________________________
©2012, TESCCC
05/01/13
page 4 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Problem Solving with Measurement
10. Gary is installing a seawater aquarium in his home. The aquarium is shaped like a rectangular
prism with the following dimensions: length of 4.5 feet, width of 2 feet, and a height of 2.5 feet.
The tank is filled with water. Gary needs to constantly monitor the salinity of the water; therefore,
he needs to know how many gallons of water are in the tank. Use this information to answer the
questions below.
a) If 231 cubic inches is equivalent to 1 gallon of water, how many gallons of water are in the
aquarium if the water line is 3 inches below the top of the tank?
b) All the information Gary uses to adjust the levels of salinity in the aquarium uses liters and
kilograms. How many liters of water are in the aquarium if 1 gallon is equivalent to 3.79 liters?
c) Gary buys 4 pound bags of salt pellets to adjust the salinity levels in the aquarium. How many
kilograms is equivalent to 4 pounds if 2.2 pounds is equivalent to 1 kilogram?
d) Gary wants to cover the lateral faces of the aquarium with a special coating to help maintain
the water temperature of the aquarium. The sheets needed to cover the lateral faces are sold
by square meters. How many square meters of the covering does Gary need for the lateral
faces?
©2012, TESCCC
05/01/13
page 5 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume KEY
Match each three-dimensional figure with the appropriate net from the choices given below. There
are extra answer choices.
1
2
3
4
5
Answer:
Answer:
Answer:
Answer:
Answer:
Letter G
Letter C
Letter B
Letter F
Letter A
A.
B.
C.
D.
E.
F.
G.
H.
6. Write a paragraph to describe the model of a prism and a pyramid.
A prism has lateral faces that are rectangles and two parallel bases that are polygons. A
pyramid has lateral faces that are triangles and one base that is a polygon. The name of
prisms and pyramids are based upon their base(s) and lateral faces. The first name of a prism
or pyramid will be the name of the polygon that is the base(s). If the lateral faces are
rectangles, the second name is prism. If the lateral faces are triangles, the second name is
pyramid.
©2012, TESCCC
05/01/13
page 1 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume KEY
For problems 7 – 14 use the ruler on the STAAR Grade 8 Reference Materials to measure the
dimensions when the dimensions are not given. Write your measurements to the nearest half of a
centimeter. Use 3.14 for . You will round your answers after doing the calculations for the problem.
7. The pep squad is making spirit sticks to sell out of heavy cardboard. The spirit sticks will be
cylindrical noisemakers that are completely covered in material printed with the school’s
mascot. Measure this pattern in centimeters to determine the approximate amount of material
needed for 1 spirit stick. To the nearest square centimeter, what is the surface area of the
cylinder?
The following dimensions were used to calculate the surface area of the cylinder’s net:
diameter = 2.5 cm
radius = 2.5 cm ÷ 2 = 1.25 cm
height of cylinder (base of rectangle) = 6 cm
Calculate height of rectangle using circumference of the circle = (3.14 • 2.5 cm) = 7.85 cm
OR measure the height = 7.9 cm, use 8 cm since we measure to the nearest half of a cm.
Area of Circle = 3.14(1.25)2 = 4.90625
Due to changes of dimensions when the page is copied, the teacher will need to check the
measurements on the copied page.
Surface Area of Cylinder = 2(area of circle) + area of rectangle = 2(4.90625) + (6 • 7.85) =
56.9125 ª 57 cm2 OR 2(4.90625) + (6 • 8) = 57.8125 ª 58 cm2.
©2012, TESCCC
05/01/13
page 2 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume KEY
3.5 cm
8. Mikey is using the following net to make a square pyramid model out of cardstock. To the
nearest square centimeter what is the approximate amount of cardstock needed for Mikey’s
pyramid?
1
Answer: (4 • 4 • 3.5) + 4(4) = 44 cm2
2
4 cm
4 cm
4 cm
4 cm
©2012, TESCCC
05/01/13
page 3 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume KEY
9. The following is the net of a mold for a homemade chocolate candy bar that Sheldon ordered
from Chef Pierre. Round your answer to the nearest hundredth to approximate the quantity of
solid chocolate that Sheldon will make with this mold.
Answer: (2.5 • 2.5 • 12.5) = 78.125 cm3 ≈ 78.13 cm3
This answer was calculated using measurements of
12.5 cm and 2.5 cm.
©2012, TESCCC
05/01/13
page 4 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume KEY
14 in.
in
6 in.
10. Jazz is painting the interior of my house. To the nearest square inch, how much wall space will
Jazz cover with one complete rotation of this paintbrush roller? How many square centimeters
will Jazz cover with one complete rotation of this paintbrush roller?
Answer:
Lateral Surface Area of Cylinder = Circumference • height: (3.14 • 6)•14 = 263.76 in2 ª 264 in2
Convert inches to centimeters and then calculate Lateral Surface Area in square centimeters:
 2.54 cm  14  2.54
Height: 14 in. 
= 35.56 cm
 =
1
 1 in. 
6  2.54
 2.54 cm 
Diameter: 6 in. 
=
= 15.24 cm

1
 1 in. 
Lateral Surface Area of Cylinder: (3.14 • 15.24)(35.56) ª 1,701.67 cm2
©2012, TESCCC
05/01/13
page 5 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume KEY
11. Mindy works at Music Mania. They get blank CD’s in tall stacks 3 feet high. What is a
reasonable estimate of the volume of a single stack of these CD’s in cubic inches? Calculate
the volume of a single stack of these CD’s to the nearest cubic inch.
Estimate: Area of Base • height: use 3 for 
1
Estimated Volume: 3 • (2.5) 2 • 36 ─ 3 • ( ) 2 • 36 ª 668.25
4
Convert 3 feet to inches: 3 feet • 12 inches = 36 inches
Volume of whole stack – volume of center hole
3.14 • (2.5)2 • 36 – 3.14 • (0.25)2 • 36 ª 699.435 (The estimate of 668.28 is reasonably close.)
Answer: 699 in3
0.5 in.
2.5 in.
12. What is the volume of a rectangular prism sandbox in cubic feet, if the height is 30 inches,
the length is 72 inches and the width is 72 inches?
Convert inches to feet:
30 inches = 2.5 feet
72 inches = 6 feet
Volume = l • w • h
Volume = 6 • 6 • 2.5
Answer: 90 ft3
13. To the nearest cubic inch, what is the approximate capacity of a cylindrical container if the
diameter is 8 inches and the height is 1 foot?
Volume =  r 2 h
Volume = 3.14(4)2 • 12 ª 602.88
Answer: 603 in.3
14. To the nearest square inch, how much cardboard is needed for the side of the cylindrical
container in problem 13?
Lateral Surface Area = 2  r h
S = 2 • 3.14 • 4 • 12 ª
Answer: 301 in.2
©2012, TESCCC
05/01/13
page 6 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume
Match each three-dimensional figure with the appropriate net from the choices given below. There
are extra answer choices.
1
2
3
4
5
Answer:
Answer:
Answer:
Answer:
Answer:
A.
B.
C.
D.
E.
F.
G.
H.
6. Write a paragraph to describe the model of a prism and a pyramid.
©2012, TESCCC
05/01/13
page 1 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume
For problems 7 – 14 use the ruler on the STAAR Grade 8 Reference Materials to measure the
dimensions when the dimensions are not given. Write your measurements to the nearest half of a
centimeter. Use 3.14 for . You will round your answers after doing the calculations for the problem.
7. The pep squad is making spirit sticks to sell out of heavy cardboard. The spirit sticks will be
cylindrical noisemakers that are completely covered in material printed with the school’s
mascot. Measure this pattern in centimeters to determine the approximate amount of material
needed for 1 spirit stick. To the nearest square centimeter, what is the surface area of the
cylinder?
©2012, TESCCC
05/01/13
page 2 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume
8. Mikey is using the following net to make a square pyramid model out of cardstock. To the
nearest square centimeter what is the approximate amount of cardstock needed for Mikey’s
pyramid?
©2012, TESCCC
05/01/13
page 3 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume
9. The following is the net of a mold for a homemade chocolate candy bar that Sheldon ordered
from Chef Pierre. Round your answer to the nearest hundredth to approximate the quantity of
solid chocolate that Sheldon will make with this mold.
©2012, TESCCC
05/01/13
page 4 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume
14 in.
in
6 in.
10. Jazz is painting the interior of my house. To the nearest square inch, how much wall space will
Jazz cover with one complete rotation of this paintbrush roller? How many square centimeters
will Jazz cover with one complete rotation of this paintbrush roller?
©2012, TESCCC
05/01/13
page 5 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Surface Area and Volume
11. Mindy works at Music Mania. They get blank CD’s in tall stacks 3 feet high. What is a
reasonable estimate of the volume of a single stack of these CD’s in cubic inches? Calculate
the volume of a single stack of these CD’s to the nearest cubic inch.
0.5 in.
2.5 in.
12. What is the volume of a rectangular prism sandbox in cubic feet, if the height is 30 inches,
the length is 72 inches and the width is 72 inches?
13. To the nearest cubic inch, what is the approximate capacity of a cylindrical container if the
diameter is 8 inches and the height is 1 foot?
14. To the nearest square inch, how much cardboard is needed for the side of the cylindrical
container in problem 13?
©2012, TESCCC
05/01/13
page 6 of 6
Grade 8
Mathematics
Unit: 07 Lesson: 01
Jane’s Yard
Jane has a triangular yard with the dimensions of 30 ft by 40 ft by 50 ft. If she
doubles each dimension for the house, how will it affect the perimeter? How will
it affect the area?
©2012, TESCCC
10/12/12
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements KEY
1.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions. Build a new rectangle and find
the new perimeter and area. Remember, both dimensions are being changed
proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
length, in.
3 in.
original rectangle
width, in.
perimeter, in.
2 in.
2(3) + 2(2)
10 in.
area, sq in.
2(3)
6 in.2
ACTION
double
dimensions
length, in.
2•3
6 in.
new rectangle
width, in.
perimeter, in.
2•2
4 in.
2(2•3) + 2(2•2)
20 in.
area, sq in.
(2•3)(2•2)
24 in.2
b.
length, in.
4 in.
original rectangle
width, in.
perimeter, in.
2 in.
2(4) + 2(2)
12 in.
area, sq in.
4(2)
8 in.2
ACTION
triple
dimensions
length, in.
3•4
12 in.
©2012, TESCCC
new rectangle
width, in.
perimeter, in.
3•2
6 in.
2(3•4) + 2(3•2)
36 in.
05/01/13
area, sq in.
(3•4)(3•2)
72 in.2
page 1 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements KEY
2.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions. Build a new rectangle and find
the new perimeter and area. Remember, both dimensions are being changed
proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
length, in.
5 in.
original rectangle
width, in.
perimeter, in.
3 in.
2(5) + 2(3)
16 in.
area, sq in.
5(3)
15 in.2
ACTION
quadruple
dimensions
length, in.
4•5
20 in.
new rectangle
width, in.
perimeter, in.
4•3
12 in.
2(4•5) + 2(4•3)
64 in.
area, sq in.
(4•5)(4•3)
240 in.2
b.
length, in.
3 in.
original rectangle
width, in.
perimeter, in.
2 in.
2(3) + 2(2)
10 in.
area, sq in.
(3)(2)
6 in.2
ACTION
triple
dimensions
length, in.
3•3
9 in.
©2012, TESCCC
new rectangle
width, in.
perimeter, in.
3•2
6 in.
2(3•3) + 2(3•2)
30 in.
05/01/13
area, sq in.
(3•3)(3•2)
54 in.2
page 2 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements KEY
3.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions.
• Build a new rectangle and find the new perimeter and area. Remember, both
dimensions are being changed proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
length, in.
4 in.
original rectangle
width, in.
perimeter, in.
2 in.
2(4) + 2(2)
12 in.
area, sq in.
(4)(2)
8 in2
ACTION
halve
dimensions
length, in.
0.5•4
2 in.
new rectangle
width, in.
perimeter, in.
0.5•2
1 in.
2(0.5•4) + 2(0.5•2)
6 in.
area, sq in.
(0.5•4)(0.5•2)
2 in2
b.
length, in.
2 in.
original rectangle
width, in.
perimeter, in.
2 in.
2(2) + 2(2)
8 in.
area, sq in.
(2)(2)
4 in2
ACTION
halve
dimensions
length, in.
0.5•2
1 in.
©2012, TESCCC
new rectangle
width, in.
perimeter, in.
0.5•2
1 in.
2(0.5•2) + 2(0.5•2)
4 in
05/01/13
area, sq in.
(0.5•2)( 0.5•2)
1 in2
page 3 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements KEY
4.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions.
• Build a new rectangle and find the new perimeter and area. Remember, both
dimensions are being changed proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a. Answers may vary. Sample answers below.
original rectangle
length, in.
width, in.
perimeter, in.
5 in.
7 in.
2(5) + 2(7)
24 in.
area, sq in.
(5)(7)
35 in.2
ACTION
double
dimensions
length, in.
2•5
10 in.
new rectangle
width, in.
perimeter, in.
2•7
14 in.
2(2 • 5) + 2(2 • 7)
48 in.
b. Answers may vary. Sample answers below.
original rectangle
length, in.
width, in.
perimeter, in.
5 in.
6 in.
2(5) + 2(6)
22 in.
area, sq in.
(2 • 5)(2 • 7)
140 in.2
area, sq in.
(5)(6)
30 in2
ACTION
triple
dimensions
length, in.
3•5
15 in.
©2012, TESCCC
new rectangle
width, in.
perimeter, in.
3•6
18 in.
2(3 • 5) + 2(3 • 6)
66 in.
05/01/13
area, sq in.
(3 • 5)(3 • 6)
270 in.2
page 4 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements KEY
5.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions.
• Build a new rectangle and find the new perimeter and area. Remember, both
dimensions are being changed proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a. Answers may vary. Sample answers below.
original rectangle
length, in.
width, in.
perimeter, in.
10 in.
10 in.
2(10) + 2(10)
40 in.
area, sq. in.
(10)(10)
100 in.2
ACTION
quadruple
dimensions
length, in.
4 • 10
40 in.
new rectangle
width, in.
perimeter, in.
4 • 10
40 in.
2(4 • 10) + 2(4 • 10)
160 in.
b. Answers may vary. Sample answers below.
original rectangle
length, in.
width, in.
perimeter, in.
6 in.
8 in.
2(6) + 2(8)
28 in.
area, sq in.
(4 • 10)(4 • 10)
1600 in.2
area, sq in.
(6)(8)
48 in.2
ACTION
halve
dimensions
length, in.
0.5 • 6
3 in.
©2012, TESCCC
new rectangle
width, in.
perimeter, in.
0.5 • 8
4 in.
2(0.5 • 6) + 2(0.5 • 8)
14 in.
05/01/13
area, sq in.
(0.5 • 6)(0.5 • 8)
12 in.2
page 5 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements KEY
6. How is the perimeter of the original rectangle affected when we proportionally:
a. doubled the dimensions?
The perimeter of the original rectangle is doubled:
Original Perimeter: 2l + 2w; Double each dimension: 2•l and 2•w
New Perimeter: 2(2•l + 2•w) = 2(Original Perimeter)
b. tripled the dimensions?
The perimeter of the original rectangle is tripled:
Original Perimeter: 2l + 2w; Triple each dimension: 3•l and 3•w
New Perimeter: (2(3•l) + 2(3•w)) = (3(2l) + 3(2w)) = 3(2l + 2w) = 3(Original Perimeter)
c. quadrupled the dimensions?
The perimeter of the original rectangle is quadrupled:
Original Perimeter: 2l + 2w; Quadruple each dimension: 4•l and 4•w
New Perimeter: (2(4•l) + 2(4•w)) = (4(2l) + 4(2w)) = 4(2l + 2w) = 4(Original Perimeter)
d. halved the dimensions?
The perimeter of the original rectangle is halved:
1
1
Original Perimeter: 2l + 2w; half each dimension: •l and •w
2
2
1
1
1
1
1
1
New Perimeter: (2( •l) + 2( •w)) = ( (2l) +
(2w)) =
(2l + 2w) =
(Original Perimeter)
2
2
2
2
2
2
7. How is the area of the original rectangle affected when we proportionally:
a. doubled the dimensions?
The area of the original rectangle is 4 times greater which translates to (scale factor) 2:
Original Area: lw; Double each dimension: 2•l and 2•w
New Area: (2•l)(2•w) = (2 • 2 • l • w) = 4(lw) = 4(Original Area)  (2) 2(Original Area)
b. tripled the dimensions?
The area of the original rectangle is 9 times greater which translates to (scale factor) 2:
Original Area: lw; Triple each dimension: 3•l and 3•w
New Area: (3•l)(3•w) = (3 • 3 • l • w) = 9(lw) = 9(Original Area)  (3) 2(Original Area)
c. quadrupled the dimensions?
The area of the original rectangle is 16 times greater which translates to (scale factor) 2:
Original Area: lw; Quadruple each dimension: 4•l and 4•w
New Area: (4•l)(4•w) = (4 • 4 • l • w) = 16(lw) = 16(Original Area)  (4) 2(Original Area)
d. halved the dimensions?
1
The area of the new rectangle is
the area of the original rectangle which translates to (scale
4
factor) 2:
1
1
Original Area: lw; Half each dimension: •l and •w
2
2
1
1
1 1
1
1
1
New Area: ( •l)( •w) = ( •
• l • w) =
(lw) =
(Original Area)  ( ) 2(Original Area)
2
2
2 2
4
4
2
©2012, TESCCC
05/01/13
page 6 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements KEY
8. What scale factor are we using when we:
a. double the dimensions?
Scale Factor: 2
b. triple the dimensions?
Scale Factor: 3
c. quadruple the dimensions?
Scale Factor: 4
d. halve the dimensions?
1
Scale Factor:
2
9. How does proportional change in the dimensions affect the perimeter of a figure?
The perimeter is changed using the same scale factor as the dimensional change.
(scale factor)(Perimeter)
10. How does the proportional change in the dimensions affect the area of a figure?
The area is changed using the square of the scale factor of the dimensional change.
(scale factor) 2(Area)
11. When both dimensions of a figure are changed using the same scale factor, is the change
proportional or non-proportional? Why?
The change is proportional because the same scale factor is used to change both dimensions. If
we used a different scale factor for each dimension, the resulting figure would not be similar to the
original; thus the sides would not be proportional.
©2012, TESCCC
05/01/13
page 7 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements
1.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions. Build a new rectangle and find
the new perimeter and area. Remember, both dimensions are being changed
proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
original rectangle
length, in.
width, in.
3 in.
2 in.
length, in.
perimeter, in.
area, sq in.
new rectangle
width, in.
perimeter, in.
area, sq in.
b.
original rectangle
length, in.
width, in.
4 in.
2 in.
length, in.
©2012, TESCCC
perimeter, in.
area, sq in.
new rectangle
width, in.
perimeter, in.
area, sq in.
10/12/12
page 1 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements
2.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions. Build a new rectangle and find
the new perimeter and area. Remember, both dimensions are being changed
proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
original rectangle
length, in.
width, in.
5 in.
3 in.
length, in.
perimeter, in.
area, sq in.
new rectangle
width, in.
perimeter, in.
area, sq in.
b.
original rectangle
length, in.
width, in.
3 in.
2 in.
length, in.
©2012, TESCCC
perimeter, in.
area, sq in.
new rectangle
width, in.
perimeter, in.
area, sq in.
10/12/12
page 2 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements
3.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions.
• Build a new rectangle and find the new perimeter and area. Remember, both
dimensions are being changed proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
original rectangle
length, in.
width, in.
4 in.
2 in.
length, in.
perimeter, in.
area, sq in.
new rectangle
width, in.
perimeter, in.
area, sq in.
b.
original rectangle
length, in.
width, in.
2 in.
2 in.
length, in.
©2012, TESCCC
perimeter, in.
area, sq in.
new rectangle
width, in.
perimeter, in.
area, sq in.
10/12/12
page 3 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements
4.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions.
• Build a new rectangle and find the new perimeter and area. Remember, both
dimensions are being changed proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
original rectangle
length, in.
width, in.
perimeter, in.
area, sq in.
24 inches
length, in.
new rectangle
width, in.
perimeter, in.
area, sq in.
b.
original rectangle
length, in.
width, in.
perimeter, in.
area, sq in.
30 in2
length, in.
©2012, TESCCC
new rectangle
width, in.
perimeter, in.
10/12/12
area, sq in.
page 4 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements
5.
• Use color tiles to build the original rectangle. Find the perimeter and area.
• Follow the action arrow for creating the new dimensions.
• Build a new rectangle and find the new perimeter and area. Remember, both
dimensions are being changed proportionally.
• Compare the original perimeter and area with the new perimeter and area.
• Repeat this process for each problem.
a.
original rectangle
length, in.
width, in.
perimeter, in.
area, sq. in.
40 inches
length, in.
new rectangle
width, in.
perimeter, in.
area, sq in.
b.
original rectangle
length, in.
width, in.
perimeter, in.
area, sq in.
48 in2
length, in.
©2012, TESCCC
new rectangle
width, in.
perimeter, in.
10/12/12
area, sq in.
page 5 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements
6. How is the perimeter of the original rectangle affected when we proportionally:
a. doubled the dimensions?
b. tripled the dimensions?
c. quadrupled the dimensions?
d. halved the dimensions?
7. How is the area of the original rectangle affected when we proportionally:
a. doubled the dimensions?
b. tripled the dimensions?
c. quadrupled the dimensions?
d. halved the dimensions?
©2012, TESCCC
10/12/12
page 6 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
Exploring Measurements
8. What scale factor are we using when we:
a. double the dimensions?
b. triple the dimensions?
c. quadruple the dimensions?
d. halve the dimensions?
9. How does proportional change in the dimensions affect the perimeter of a figure?
10. How does the proportional change in the dimensions affect the area of a figure?
11. When both dimensions of a figure are changed using the same scale factor, is the change
proportional or non-proportional? Why?
©2012, TESCCC
10/12/12
page 7 of 7
Grade 8
Mathematics
Unit: 07 Lesson: 01
My Garden KEY
1) I have a rectangular garden with the following dimensions: length = 4 units and width = 3 units.
There is not enough room for all the vegetables I want to plant. Use the grid below to draw a
diagram of the garden. Calculate the perimeter and area of the garden. Begin the diagram in the
top left corner. Perimeter: 2(4) + 2(3) = 14 units. Area: (4)(3) = 12 u 2.
a. If I double the length of my garden, how will
that affect the perimeter and the area of my
original garden? Use the grid to the right to
draw the original garden and the proposed
garden. Begin the diagram in the top left
corner. Calculate the perimeter and area of
the proposed garden.
Perimeter: 2(2•4) + 2(3) = 22 units
Area: (2•4)(3) = 24 u 2
Work Space
b. If I double the width of my garden, how will
that affect the perimeter and the area of my
original garden? Use the grid to the right to
draw the original garden and the proposed
garden. Begin the diagram in the top left
corner. Calculate the perimeter and area of
the proposed garden.
Perimeter: 2(4) + 2(2•3) = 20 units
Area: (4)(2•3) = 24 u 2
Work Space
c. If I double both dimensions of my garden, how
Work Space
will that affect the perimeter and the area of
my garden? Use the grid to the right to draw
the original garden and the proposed garden.
Begin the diagram in the top left corner.
Calculate the perimeter and area of the
proposed garden. How does doubling both
dimensions affect the original perimeter and
area?
Perimeter: 2(2•4) + 2(2•3) = 28 units
Perimeter is doubled: 2(Perimeter)
Area: (2•4)(2•3) = 48 u 2
Area is 4 times greater: (2) 2(Area)
©2012, TESCCC
10/12/12
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
My Garden
1) I have a rectangular garden with the following dimensions: length = 4 units and width = 3 units.
There is not enough room for all the vegetables I want to plant. Use the grid below to draw a
diagram of the garden. Calculate the perimeter and area of the garden. Begin the diagram in the
top left corner.
a. If I double the length of my garden, how will
that affect the perimeter and the area of my
original garden? Use the grid to the right to
draw the original garden and the proposed
garden. Begin the diagram in the top left
corner. Calculate the perimeter and area of
the proposed garden.
Work Space
b. If I double the width of my garden, how will
that affect the perimeter and the area of my
original garden? Use the grid to the right to
draw the original garden and the proposed
garden. Begin the diagram in the top left
corner. Calculate the perimeter and area of
the proposed garden.
Work Space
c. If I double both dimensions of my garden, how
will that affect the perimeter and the area of
my garden? Use the grid to the right to draw
the original garden and the proposed garden.
Begin the diagram in the top left corner.
Calculate the perimeter and area of the
proposed garden. How does doubling both
dimensions affect the original perimeter and
area?
Work Space
©2012, TESCCC
10/12/12
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden KEY
Tina, Andrea, Joel, and Luis are designing a meditation area for school called Geometric Gardens.
The garden is a joint project between the student council and the PTO. The plot for the garden is
rectangular. The students will use white, yellow, green, and blue gravel to create pathways and
geometric designs in the garden. The students plan to use approximately five tons of gravel to fill the
garden.
1. Shade each region of the gravel garden the color indicated in the diagram below.
(5)2 − (4)2 = 3
©2012, TESCCC
10/12/12
page 1 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden KEY
2. Use the diagram of the gravel garden from page 1 to calculate the area in the chart below for each
specified region of the gravel garden:
Location
Area (ft2)
Figure
Calculations
a) Entire grid
Rectangle
overall garden
A = (18 + 4 + 18)(13 + 4 + 13) = (40)(30)
b) Positive y-axis
Rectangle
yellow walkway
A = (4)(13)
52
c) Negative y-axis
Rectangle
yellow walkway
A = (4)(13)
52
d) Positive x-axis
Rectangle
green walkway
A = (4)(18)
72
e) Negative x-axis
Rectangle
green walkway
A = (4)(18)
72
f) Origin: axes
intersection
Square
center blue walkway
A = (4)(4)
16
g) Quadrant I
Right triangle
blue
A = (3 • 4) ÷ 2
6
h) Quadrant I
Right triangle
blue
A = (3 • 4) ÷ 2
6
i) Quadrant I
Trapezoid
yellow
j) Quadrant II
Circle Sections
blue
k) Quadrant II
Right triangle
green
A = (4 • 4) ÷ 2
8
l) Quadrant II
Right triangle
yellow
A = (4 • 4) ÷ 2
8
m) Quadrant III
Semi-circle
blue
A = (3.14)(2) 2 ÷ 2
n) Quadrant III
Rectangle
yellow
A = (6)(4)
24
o) Quadrant IV
Trapezoid
blue
A = (4 + 6)(4) ÷ 2
20
p) Quadrant IV
Isosceles Triangle
green
A = (6 • 6) ÷ 2
18
q) Quadrant IV
2 Right Triangles
yellow
A = 2(3 • 6 ÷ 2)
18
©2012, TESCCC
A = (6 + 12)(4) ÷ 2
A = area of circle ─ (area green ∆ + area
yellow ∆): (3.14)(4) 2 ─ (4•4÷2 + 4•4÷2)
10/12/12
1,200
36
≈ 34.24
≈ 6.28
page 2 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden KEY
3. Complete the chart below to determine the amount of gravel needed for each color.
Total Area of Gravel in
Specified Color
% of Specified Color of
Gravel
in the Garden
Amount of Gravel
Needed of Specified
Color
White
Area of overall rectangle ─
sum of areas of other colors:
1200 ─ (52 + 52 + 72 + 72 +
16 + 6 + 6 + 36 + 34.24 + 8 +
8 + 6.28 + 24 + 20 + 18 + 18)
= 751.48 ft 2
751.48 ÷ 12
62.623
=
1200 ÷ 12
100
≈ 62.6%
62.6 ÷ 20
x
=
100 ÷ 20
5 tons
x ≈ 3.13 tons
Yellow
Sum of Yellow Areas:
52 + 52 + 36 + 8 + 24 + 18 =
190 ft 2
190 ÷ 12
15.83
=
1200 ÷ 12
100
≈ 15.8%
15.8 ÷ 20
x
=
100 ÷ 20
5 tons
≈ 0.79 tons
Green
Sum of Green Areas:
72 + 72 + 8 + 18 = 170 ft 2
170 ÷ 12
14.16
=
1200 ÷ 12
100
≈ 14.2%
14.2 ÷ 20
x
=
100 ÷ 20
5 tons
x ≈ 0.71 tons
Blue
Sum of Blue Areas:
16 + 6 + 6 + 34.24 + 6.28 + 20
= 88.52 ft 2
88.52 ÷ 12
7.376
=
1200 ÷ 12
100
≈ 7.4%
7.4 ÷ 20
x
=
100 ÷ 20
5 tons
x ≈ 0.37 tons
751.48 + 190 + 170 + 88.52 =
1200 ft 2
62.6% + 15.8% + 14.2%
+ 7.4% = 100%
3.13 + 0.79 + 0.71 +
0.37 = 5
Color of Gravel
Totals
4. Use a written description in conjunction with math symbols to describe how to calculate the area
of the blue sections of the circle in quadrant II from the diagram on page 1.
Area of a circle: π r 2; Area of a triangle: bh ÷ 2
Area of blue sections in circle: Area of circle ─ (Area of green triangle + Area of yellow triangle)
Area of blue sections in circle: π r 2 ─ (bh ÷ 2 + bh ÷ 2)
Area of blue sections in circle: (3.14)(4) 2 ─ (4 • 4 ÷ 2 + 4 • 4 ÷ 2)
Area of blue sections in circle is approximately: 50.24 ─ (8 + 8)
Area of blue sections in circle is approximately: 50.24 ─ 16 = 34.24 ft 2
©2012, TESCCC
10/12/12
page 3 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden KEY
5.
a) Complete the table below.
Rectangle 1
length
5 cm
Rectangle 2
width
9 cm
length
8 yds
width
12 yds
Scale Factor
3
Original Perimeter
2(5) + 2(9) = 28 cm
10 + 18 = 28 cm
New Perimeter
2(3 • 5) + 2(3 • 9) = 3(2 • 5) + 3(2 • 9) = 84
3(28 cm) = 84 cm
Scale Factor
0.5
Original Perimeter
2(8) + 2(12) = 40 yds
New Perimeter
2(0.5 • 8) + 2(0.5 • 12) = 0.5(2 • 8) + 0.5(2 • 12)
0.5(40) = 20 yds
Original Area
(5)(9) = 45 cm 2
Original Area
(8)(12) = 96 yds 2
New Area
(3 • 5)(3 • 9) = 45 cm 2
3 • 3 • 5 • 9 = 9(45) = 405 cm 2
New Area
(0.5 • 8)(0.5 • 12) = 24 yds 2
0.5 • 0.5 • 8 • 12 = 0.25(96) = 24 yds 2
b) Describe the effects of the dimensional change on the perimeter of each rectangle.
The new perimeter is the product of the scale factor and the original perimeter.
c) Write an equation that shows the effects of the dimensional change on the perimeter of each
rectangle.
New perimeter = (scale factor) • (original perimeter)
d) Describe the effects of the dimensional change on the area of each rectangle.
The new area is the product of the square of the scale factor and the original area.
e) Write an equation that shows the effects of the dimensional change on the area of each rectangle.
New area = (scale factor) 2 • (original area)
f) Use the equation from part c to determine the new perimeter of a rectangle whose dimensions are
changed proportionally by a scale factor of 1.5 units if the perimeter of the original rectangle is 20
units.
New perimeter = (1.5)(20 units) = 30 units
g) Use the equation from part e to determine the new area of a rectangle whose dimensions are
changed proportionally by a scale factor of 0.6 units if the area of the original rectangle is 15 u 2.
New area = (0.6) 2 (15 u 2) = 5.4 u 2
©2012, TESCCC
10/12/12
page 4 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden
Tina, Andrea, Joel, and Luis are designing a meditation area for school called Geometric Gardens. The
garden is a joint project between the student council and the PTO. The plot for the garden is
rectangular. The students will use white, yellow, green, and blue gravel to create pathways and
geometric designs in the garden. The students plan to use approximately five tons of gravel to fill the
garden.
1. Shade each region of the gravel garden the color indicated in the diagram below.
©2012, TESCCC
10/12/12
page 1 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden
2. Use the diagram of the gravel garden from page 1 to calculate the area in the chart below for each
specified region of the gravel garden:
Location
Figure
a) Entire grid
Rectangle
overall garden
b) Positive y-axis
Rectangle
yellow walkway
c) Negative y-axis
Rectangle
yellow walkway
d) Positive x-axis
Rectangle
green walkway
e) Negative x-axis
Rectangle
green walkway
f) Origin: axes
intersection
Square
center blue walkway
g) Quadrant I
Right triangle
blue
h) Quadrant I
Right triangle
blue
i) Quadrant I
Trapezoid
yellow
j) Quadrant II
Circle Sections
blue
k) Quadrant II
Right triangle
green
l) Quadrant II
Right triangle
yellow
m) Quadrant III
Semi-circle
blue
n) Quadrant III
Rectangle
yellow
o) Quadrant IV
Trapezoid
blue
p) Quadrant IV
Isosceles Triangle
green
q) Quadrant IV
2 Right Triangles
yellow
©2012, TESCCC
Calculations
10/12/12
Area (ft2)
page 2 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden
3. Complete the chart below to determine the amount of gravel needed for each color.
Color of Gravel
Total Area of Gravel in
Specified Color
% of Specified Color of
Gravel
in the Garden
Amount of Gravel
Needed of
Specified Color
White
Yellow
Green
Blue
Totals
4. Use a written description in conjunction with math symbols to describe how to calculate the area of
the blue sections of the circle in quadrant II from the diagram on page 1.
©2012, TESCCC
10/12/12
page 3 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
Geometric Gravel Garden
5.
a) Complete the table below.
Rectangle 1
length
5 cm
Rectangle 2
width
9 cm
length
8 yds
width
12 yds
Scale Factor
3
Original Perimeter
Scale Factor
0.5
Original Perimeter
New Perimeter
New Perimeter
Original Area
Original Area
New Area
New Area
b) Describe the effects of the dimensional change on the perimeter of each rectangle.
c) Write an equation that shows the effects of the dimensional change on the perimeter of each
rectangle.
d) Describe the effects of the dimensional change on the area of each rectangle.
e) Write an equation that shows the effects of the dimensional change on the area of each
rectangle.
f) Use the equation from part c to determine the new perimeter of a rectangle whose dimensions
are changed proportionally by a scale factor of 1.5 units if the perimeter of the original rectangle
is 20 units.
g) Use the equation from part e to determine the new area of a rectangle whose dimensions are
changed proportionally by a scale factor of 0.6 units if the area of the original rectangle is 15 u 2.
©2012, TESCCC
10/12/12
page 4 of 4
Grade 8
Mathematics
Unit: 07 Lesson: 01
The Empire State Building
Justin wants to make a model that is a scaled down version of the Empire State
Building for his math class project. He found the linear dimensions (length, width,
height) of the actual building on the Internet. What do you think will happen to the
volume of the Empire State Building when he changes the dimensions proportionally?
Will the volume be affected like perimeter or area or in a completely different way?
Explain.
©2012, TESCCC
10/12/12
page 1 of 1
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes on Volume KEY
1. For part a and part b:
•
Use cubes to build the original rectangular prism. Find the volume.
•
Follow the action given for the dimensions, build a new rectangular prism and find the new
volume.
•
Compare the original volume with the new volume.
•
Repeat this process for each problem.
•
Discuss the relationship between the original volume and the new volume.
a.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
3
2
1
V = Bh = (Area of Base) • h = l • w • h = 3 • 2 • 1 = 6
ACTION
double
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
3•2=6
2•2=4
1•2=2
Volume
(cubic inches  in3)
3 • 2 • 2 • 2 • 1 • 2 = (2)3 • 3 • 2 • 1  (2)3 • 6 = 48
New volume = (scale factor)3 x original volume
b.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
2
2
2
V = Bh = (Area of Base) • h = l • w • h = 2 • 2 • 2 = 8
ACTION
half
dimensions
new rectangular prism
Length
(inches)
2•
1
2
=1
©2012, TESCCC
Width
(inches)
2•
1
2
=1
Height
(inches)
2•
1
2
=1
Volume
(cubic inches  in3)
2•
1
2
•2•
1
2
•2•
1
2
1
2
1
2
= ( )3 • 2 • 2 • 2  ( )3 • 8 = 1
New volume = (scale factor)3 • original volume
05/01/13
page 1 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes on Volume KEY
2. For part a and part b:
•
Use cubes to build the original rectangular prism. Find the volume.
•
Follow the action given for the dimensions, build a new rectangular prism and find the new
volume.
•
Compare the original volume with the new volume.
•
Repeat this process for each problem.
•
Discuss the relationship between the original volume and the new volume.
a.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
4
2
2
V = Bh = (Area of Base) • h = l • w • h = 4 • 2 • 2 = 16
ACTION
half
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
1
2
1
2
1
2
4•
=2
2•
=1
2•
=1
Volume
(cubic inches  in3)
4•
1
2
•2•
1
2
•2•
1
2
1
2
1
2
= ( )3 • 4 • 2 • 2  ( )3 • 16 = 2
New volume = (scale factor)3 x original volume
b.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
3
3
1
V = Bh = (Area of Base) • h = l • w • h = 3 • 3 • 1 = 9
ACTION
triple
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
3•3=9
3•3=9
1•3=3
©2012, TESCCC
Volume
(cubic inches  in3)
3 • 3 • 3 • 3 • 1 • 3 = (3)3 • 3 • 3 • 1  (3)3 • 9 = 243
New volume = (scale factor)3 • original volume
05/01/13
page 2 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes on Volume KEY
3. For part a and part b:
•
Use cubes to build the original rectangular prism. Find the volume.
•
Follow the action given for the dimensions, build a new rectangular prism and find the new
volume.
•
Compare the original volume with the new volume.
•
Repeat this process for each problem.
•
Discuss the relationship between the original volume and the new volume.
a.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
2
1
1
V = Bh = (Area of Base) • h = l • w • h = 2 • 1 • 1 = 2
ACTION
triple
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
2•3=6
1•3=3
1•3=3
Volume
(cubic inches  in3)
2 • 3 • 1 • 3 • 1 • 3 = (3)3 • 2 • 1 • 1  (3)3 • 2 = 54
New volume = (scale factor)3 • original volume
b.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
3
2
1
V = Bh = (Area of Base) • h = l • w • h = 3 • 2 • 1 = 6
ACTION
quadruple
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
3 • 4 = 12
2•4=8
1•4=4
©2012, TESCCC
Volume
(cubic inches  in3)
3 • 4 • 2 • 4 • 1 • 4 = (4)3 • 3 • 2 • 1  (4)3 • 6 = 384
New volume = (scale factor)3 • original volume
05/01/13
page 3 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes on Volume
•
•
•
•
•
•
For part a and part b:
Use cubes to build the original rectangular prism. Find the volume.
Follow the action given for the dimensions, build a new rectangular prism and find the new
volume.
Compare the original volume with the new volume.
Repeat this process for each problem.
Discuss the relationship between the original volume and the new volume.
a.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
3
2
1
Volume
(cubic inches  in3)
ACTION
double
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
b.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
2
2
2
Volume
(cubic inches  in3)
ACTION
half
dimensions
new rectangular prism
Length
(inches)
©2012, TESCCC
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
05/01/13
page 1 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes on Volume
1. For part a and part b:
•
Use cubes to build the original rectangular prism. Find the volume.
•
Follow the action given for the dimensions, build a new rectangular prism and find the new
volume.
•
Compare the original volume with the new volume.
•
Repeat this process for each problem.
•
Discuss the relationship between the original volume and the new volume.
a.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
4
2
2
Volume
(cubic inches  in3)
ACTION
half
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
b.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
3
3
1
Volume
(cubic inches  in3)
ACTION
triple
dimensions
new rectangular prism
Length
(inches)
©2012, TESCCC
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
05/01/13
page 2 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes on Volume
2. For part a and part b:
•
Use cubes to build the original rectangular prism. Find the volume.
•
Follow the action given for the dimensions, build a new rectangular prism and find the new
volume.
•
Compare the original volume with the new volume.
•
Repeat this process for each problem.
•
Discuss the relationship between the original volume and the new volume.
a.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
2
1
1
Volume
(cubic inches  in3)
ACTION
triple
dimensions
new rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
b.
original rectangular prism
Length
(inches)
Width
(inches)
Height
(inches)
3
2
1
Volume
(cubic inches  in3)
ACTION
quadruple
dimensions
new rectangular prism
Length
(inches)
©2012, TESCCC
Width
(inches)
Height
(inches)
Volume
(cubic inches  in3)
05/01/13
page 3 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes KEY
Refer to these two rectangles for problems 1 and 2.
Small
Rectangle
Large
Rectangle
1. If the perimeter of the small rectangle is 10 feet and the dimensions are doubled, then
what is the perimeter of the large rectangle? Explain.
Answer: 20 feet
Small Rectangle Perimeter 2(l + w)
Large Rectangle Perimeter 2(l + l + w + w) = 2[(l + w) + (l + w)] = 2[2(l + w)]
Large Rectangle Perimeter = 2 x Perimeter of Small Rectangle
Large Rectangle Perimeter = 2 x 10 feet = 20 feet
2. If the area of the small rectangle is 6 square feet and the dimensions are doubled, then
what is the area of the large rectangle? Explain.
Answer: 24 ft2
Small Rectangle Area l • w
Large Rectangle Area 2l • 2w = 2 • 2 • l • w
Large Rectangle Area = (2)2 x Area of Small Rectangle
Large Rectangle Area = 4 x 6 square feet = 24 square feet
3. The volume of a small packing box is 12 ft3. U-Move designed a larger packing box by
doubling the dimensions of the small box. What is the volume of the larger packing box?
Explain.
Answer: 96 ft3
Small Box Volume l • w • h
Large Box Volume 2l • 2w • 2h = 2 • 2 • 2 • l • w • h
Large Box Volume (2)3 x Volume of Small Box
Large Box Volume = 8 x 12 cubic feet = 96 cubic feet
4. Copy Cats is going to take the 8th graders’ class photo and enlarge it for t-shirts by tripling the
dimensions. How many times larger will the enlarged photo be than the original picture?
Explain.
Answer: 9 times larger than original picture
Original Picture Area l • w
Enlarged Picture Area 3l • 3w = 3 • 3 • l • w
Enlarged Picture Area (3)2 • l • w = 9 • (l • w)
©2012, TESCCC
10/12/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes KEY
5. Izza Pizza has a special bagel-bite pizza that has a 3-inch diameter. A personal pan pizza has
a 6-inch diameter. If the crusts are the same thickness, how many special bagel-bite pizzas
must Harry Lee eat to equal one personal pan pizza? Explain.
Answer: 4 times as many special bagel-bite pizzas
Special Bagel-Bite Pizza Area π • r 2 = π • (1.5)2 = π • 2.25
Personal Pan Pizza Area π • r 2 = π • (3)2 = π • 9; Note: 3 = 2 • 1.5
9 is four times larger than 2.25 4 • 2.25
6. The area of the larger rectangle is 216 cm2. If the two rectangles are similar, find the area of
the smaller rectangle. Explain. What is the area of the larger rectangle in square inches?
Answer: 24 cm2
Since the 2 rectangles are similar, corresponding sides are proportional:
length of small rectangle
length of large rectangle
=
width of small rectangle
width of large rectangle
6 cm
18 cm
6 cm
18 cm
=
width of small rectangle
width of large rectangle
6 cm
x3
18 cm
=
width of small rectangle x 3
width of large rectangle
2
Scale Factor = 3 (Scale Factor) • Area of Small Rectangle = Area of Large Rectangle
(3)2 • Area of Small Rectangle = 216 cm2
Area of Small Rectangle = 216 cm2 ÷ 9 = 24 cm2
Area Large Rectangle Square Inches: Length = 18 cm and Width = 216 cm2 ÷ 18 cm = 12 cm
 1 in.  18 i 1
≈ 7.09 in.
Length: 18 cm 
Or
 =
2.54
 2.54 cm 
2
1 in. 
2
 1 in.  12 i 1
= 33.5 in2
216
cm


=
≈
Width: 12 cm 
4.72
in.

 2.54 cm 
2.54
 2.54 cm 
Area in Square Inches: (7.09)(4.72) ≈ 33.5 in2
7. The volume of a cube is 27cm3. If the dimensions are quadrupled, then what is the
volume of the new cube? Explain. What is the volume of the original cube in cubic inches?
Answer: 1,728 cm3
Volume of Cube l • w • h = 27 cm3
Volume of New Cube 4 • l • 4 • w • 4 • h = 4 • 4 • 4 • l • w • h = (4)3 • l • w • h = 64 • 27 cm3 =
1,728 cm3
Volume Original Cube in Cubic Inches:
Length and Width and Height = 3 cm each side
 1 in.  3 i 1
≈ 1.18 in.
Or
Length, Width, Height: 3 cm 
 =
3
 2.54 cm  2.54
1 in. 
3
3
3
Volume in Cubic Inches: (1.18) ≈ 1.64 cubic in.
27 cm 
 = 1.65 in
2.54
cm


©2012, TESCCC
10/12/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes
Refer to these two rectangles for problems 1 and 2.
Small
Rectangle
Large
Rectangle
1. If the perimeter of the small rectangle is 10 feet and the dimensions are doubled, then what is
the perimeter of the large rectangle? Explain.
2. If the area of the small rectangle is 6 square feet and the dimensions are doubled, then what is
the area of the large rectangle? Explain.
3. The volume of a small packing box is 12 ft3. U-Move designed a larger packing box by
doubling the dimensions of the small box. What is the volume of the larger packing box?
Explain.
4. Copy Cats is going to take the 8th graders’ class photo and enlarge it for t-shirts by tripling the
dimensions. How many times larger will the enlarged photo be than the original picture?
Explain.
©2012, TESCCC
10/12/12
page 1 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Dimensional Changes
5. Izza Pizza has a special bagel-bite pizza that has a 3-inch diameter. A personal pan pizza has
a 6-inch diameter. If the crusts are the same thickness, how many special bagel-bite pizzas
must Harry Lee eat to equal one personal pan pizza? Explain.
6. The area of the larger rectangle is 216 cm2. If the two rectangles are similar, find the area of
the smaller rectangle. Explain. What is the area of the larger rectangle in square inches?
6 cm
18 cm
7. The volume of a cube is 27cm3. If the dimensions are quadrupled, then what is the volume of
the new cube? Explain. What is the volume of the original cube in cubic inches?
©2012, TESCCC
10/12/12
page 2 of 2
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change KEY
1. James cut a piece of granite to use as a counter top in a kitchen. The counter top is rectangular
with dimensions of length = 4 feet and width = 36 inches. He needs to cut a circular hole in the
center of the counter top. The diameter of the hole is 18 inches. What will be the area of the
counter top after James cuts out the hole?
18 in.
36 in.
a) Draw and label a diagram to model the problem.
4 ft = 48 in.
b) Use a written description in conjunction with math symbols to describe how to solve the
problem.
Calculate the area of the rectangle and the area of the circle. Subtract the area of the hole
from the area of the rectangular counter top to calculate the area of the counter top with the cut
out hole.
Formula for rectangle area: lw; formula for circle area:  r 2
Area of rectangle ─ area of circle: lw ─  r 2
(48 in.)(36 in.) ─ 3.14(9 in.) 2 = 1728 in 2 ─ 254.34 in 2  1,473.66 in. 2
3.
Original
Figure
New
Figure
6 units
12 units
1 unit
2.
3 units
Use the diagrams of similar figures to complete the table.
2 units
8 units
Original
Perimeter
P
Scale
Factor
(SF)
P = 2(6) + 2(3)
P = 12 + 6
P = 18 units
6  2 = 12
P = 2(2) + 2(1)
P=4+2
P = 6 units
24=8
(SF) = 2
(SF) = 4
Process
Step
New
Perimeter
(SF)  P
2(12 + 6)
2  18
36 units
(SF) x P
4(4 + 2)
46
24 units
4. What is the relationship between the perimeters of similar figures?
To get the new perimeter: Multiply the original perimeter by the scale factor for the similar figures.
©2012, TESCCC
05/01/13
page 1 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change KEY
6.
Original
Figure
New
Figure
6 units
12 units
1 unit
5.
3 units
Use the diagrams of similar figures to complete the table.
2 units
Original
Area
A
Scale
Factor
(SF)
A = (6)  (3)
A = 18 u2
6  2 = 12
A = (2)  (1)
A = 2 u2
24=8
8 units
(SF) = 2
(SF) = 4
Process
Step
New
Area
(SF)2  A
(2)2  18
4  18
72 u2
(SF)2  A
(4)2  2
16  2
32 u2
What is the relationship between the areas of similar figures?
To get the new area: Multiply the original area by the (scale factor)2  (use the scale factor for the
similar figures).
Use the diagrams of similar figures to complete the table.
Original
Figure
New
Figure
Original
Volume
V
Scale
Factor
(SF)
Process
Step
New
Volume
(SF)3  V
(2)3  27
8  27
216 cm3
(cube)
(cube)
V=333
V = 27 cm3
7.
3 cm
32=6
(SF) = 2
6 cm
What is the relationship between the volumes of similar figures?
To get the new volume: Multiply the original volume by the (scale factor)3  (use the scale factor for
the similar figures).
©2012, TESCCC
05/01/13
page 2 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change KEY
Measures will vary due to copy machine
8. Use the ruler on the Grade 8 STAAR Reference Materials to measure the needed dimensions to
the nearest half centimeter. Complete the chart below for the camera.
Figure
Area Formula
Measured Dimensions
Area in cm2
Large
Rectangle
A=l•w
A = 10 x 14
A = 140cm2
Small Square
A=s
A = 1.5 x 1.5
A = 2.25 cm2
Large Circle
(entire circle)
A= r
2
A = (3.14)(4)
Small Circle
A= r
2
A = (3.14)(2.5)
Trapezoid
A = 0.5(b1+ b2)h
©2012, TESCCC
2
A = 50.24
cm2
2
2
A = .5(5.5 + 7.5)(2.5)
05/01/13
A = 19.625
cm2
A = 16.25
cm2
page 3 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change
1. James cut a piece of granite to use as a counter top in a kitchen. The counter top is rectangular
with dimensions of length = 4 feet and width = 36 inches. He needs to cut a circular hole in the
center of the counter top. The diameter of the hole is 18 inches. What will be the area of the
counter top after James cuts out the hole?
a) Draw and label a diagram to model the problem.
b) Use a written description in conjunction with math symbols to describe how to solve the
problem.
3.
Original
Figure
New
Figure
6 units
12 units
1 unit
2.
3 units
Use the diagrams of similar figures to complete the table.
Original
Perimeter
P
Scale
Factor
(SF)
Process
Step
New
Perimeter
2 units
8 units
4. What is the relationship between the perimeters of similar figures?
©2012, TESCCC
05/01/13
page 1 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change
6.
Original
Figure
New
Figure
6 units
12 units
1 unit
5.
3 units
Use the diagrams of similar figures to complete the table.
Original
Area
A
Scale
Factor
(SF)
Process
Step
New
Area
Scale
Factor
(SF)
Process
Step
New
Volume
2 units
8 units
What is the relationship between the areas of similar figures?
Use the diagrams of similar figures to complete the table.
Original
Figure
New
Figure
Original
Volume
V
(cube)
(cube)
7.
3 cm
6 cm
What is the relationship between the volumes of similar figures?
©2012, TESCCC
05/01/13
page 2 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change
8. Use the ruler on the Grade 8 STAAR Reference Materials to measure the needed dimensions to
the nearest half centimeter. Complete the chart below for the camera.
Figure
Area Formula
Measured Dimensions
Area in cm2
Large
Rectangle
Small Square
Large Circle
(entire circle)
Small Circle
Trapezoid
©2012, TESCCC
05/01/13
page 3 of 3
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice KEY
Use the diagrams of similar figures to complete the table.
Original
Figure
New
Figure
Original
Perimeter
P
24 units
Scale
Factor
(SF)
1
2
24 • = 12
P = 72 units
1.
1
(SF) =
2
P = 2(18) + 2(12)
P = 36 + 24
P = 60 units
2.
1
3
18 • = 6
1
(SF) =
3
Process
Step
New
Perimeter
(SF) • P
1
x 72
2
36 units
(SF) • P
1
(36 + 24)
3
20 units
1
• 60
3
3. What is the relationship between the perimeters of similar figures?
To get the new perimeter: Multiply the original perimeter by the scale factor for the similar figures.
Use the diagrams of similar figures to complete the table.
Original
Figure
New
Figure
Original
Area
A
A = 288 u2
4.
Scale
Factor
(SF)
1
24 • = 12
2
5.
12 units
(SF) =
A = (18) • (12)
A = 216 u2
1
2
1
18 • = 6
3
(SF) =
1
3
Process
Step
New
Area
(SF)2 • A
1
( )2 • 288
2
1
• 288
4
72 u2
(SF)2 • A
1
( )2 • 216
3
1
• 216
9
24 u2
6. What is the relationship between the areas of similar figures?
To get the new area: Multiply the original area by the (scale factor)2 (use the scale factor for the
similar figures).
©2012, TESCCC
10/12/12
page 1 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice KEY
Original
Figure
New
Figure
Original
Volume
V
V = 128 in3
7.
Scale
Factor
(SF)
8 • 3 = 24
(SF) = 3
length = 24 inches
12 •
8.
V = 3,744 in3
1
=6
2
(SF) =
1
2
Process
Step
New
Volume
(SF)3 • V
(3)3 • 128
27 • 128
3,456 in3
(SF)3 • V
1
2
( )3(3744)
468 in3
1
(3744)
8
9. What is the relationship between the volumes of similar figures?
To get the new volume: Multiply the original volume by the (scale factor)3 (use the scale factor
for the similar figures).
©2012, TESCCC
10/12/12
page 2 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice KEY
10. Use the ruler on the Grade 8 STAAR Reference Materials to measure the needed dimensions to
the nearest half-centimeter. Complete the chart below for the party hat.
Measures will vary due to copy machine. Sample answers are given.
Figure
Area Formula
Measured Dimensions
Area in cm2
Triangle
(entire)
0.5bh
0.5(8)(10) = 40
40 cm2
Square
(one square)
s2
Circle
©2012, TESCCC
π
r2
s=1
s2 = 1
3.14(1.5)2 = 7.065
10/12/12
1 cm2
7.065 cm2
page 3 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice KEY
11. Use the net shown below and the appropriate formulas to model and find the total surface area.
Write a paragraph to describe how to determine the surface area of the figure and relate the
written description to the formulas.
Formulas and Net Model
Description and Formulas
Each region of the net is a rectangle. Calculate
the area for each rectangle and then add the
rectangular areas together. The formula would
be: lw + lw + lw + lw + lw + lw =
(3•1.5) + (4•1.5) + (1.5•3) + (4•3) + (4•1.5) + (4•3)
= 4.5 + 6 + 4.5 + 12 + 6 + 12 = 45 u 2.
Surface Area: 4.5 + 4.5 + 6 + 6 + 12 + 12 = 45u 2
12. Rectangle ABCD is similar to rectangle WXYZ. If the area of rectangle ABCD is 90 cm 2 and the
length is 15 cm, what is the area of rectangle WXYZ and what is the width if the length of
rectangle WXYZ is 5 cm? Complete the table below.
length
width
Rectangle ABCD
15 cm
Rectangle ABCD
90 ÷ 15 = 6 cm
Rectangle WXYZ
5 cm
Rectangle WXYZ
1
6•
= 2 cm
3
©2012, TESCCC
scale factor
length of new
length of original
5
1
=
15
3
width of new
width of original
2
1
=
6
3
10/12/12
perimeter
area
Rectangle ABCD
2(15) + 2(6)
30 + 12
42 cm
Rectangle ABCD
90 cm 2
Rectangle WXYZ
1
• 42 = 14 cm
3
Rectangle WXYZ
1
( ) 2 • 90
3
10 cm 2
page 4 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice KEY
13. Write an equation that describes the effects of dimensional change on the perimeter. Use this
equation and the information from the table in problem 12 to find the length of rectangle EFGH if it
is similar to rectangle ABCD and its perimeter is 210 cm.
New perimeter = (scale factor) • (original perimeter)
210 = x • 42: 210 ÷ 42 = x
x = 5: scale factor = 5
New length = (scale factor) • (original length): new length = 5 • 15 = 75 cm
14. Write an equation that describes the effects of dimensional change on the area. Use this equation
and the information from the table in problem 12 to find the length of rectangle RSTU if it is similar
to rectangle ABCD and its area is 810 cm 2.
New area = (scale factor) 2 • (original area)
810 = x 2 • 90: 810 ÷ 90 = x 2
x 2 = 9; x = 3: scale factor = 3
New length = (scale factor) • original length: new length = 3 • 15 = 45 cm
15. Express the area of a square in square meters. The area of the square is currently expressed as
625 square inches.
Answer: convert inches to meters after calculating the side length of the square in inches.
Side length = 625 = 25 in.
 2.54 cm   1 m  (25 i 2.54 i 1)
25 in. 
= 0.635 m
 
 =
(1 i 100)
 1 in.   100 cm 
Area = (0.635 m)2 ≈ 0.40 sq m
©2012, TESCCC
10/12/12
page 5 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice
Use the diagrams of similar figures to complete the table.
Original
Figure
New
Figure
Original
Perimeter
P
Scale
Factor
(SF)
Process
Step
New
Perimeter
Process
Step
New
Area
P = 72 units
1.
2.
3. What is the relationship between the perimeters of similar figures?
Use the diagrams of similar figures to complete the table.
Original
Figure
4.
New
Figure
Original
Area
A
Scale
Factor
(SF)
A = 288 u2
5.
6. What is the relationship between the areas of similar figures?
©2012, TESCCC
04/16/12
page 1 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice
Original
Figure
New
Figure
Original
Volume
V
Scale
Factor
(SF)
Process
Step
New
Volume
V = 128 in3
7.
length = 24 inches
8.
V = 3,744 in3
9. What is the relationship between the volumes of similar figures?
©2012, TESCCC
04/16/12
page 2 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice
10. Use the ruler on the Grade 8 STAAR Reference Materials to measure the needed dimensions to
the nearest half-centimeter. Complete the chart below for the party hat.
Figure
Triangle
(entire)
Area Formula
Measured Dimensions
Area in cm2
Square
(one square)
Circle
©2012, TESCCC
04/16/12
page 3 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice
11. Use the net shown below and the appropriate formulas to model and find the total surface area.
Write a paragraph to describe how to determine the surface area of the figure and relate the written
description to the formulas.
Formulas and Net Model
Description and Formulas
12. Rectangle ABCD is similar to rectangle WXYZ. If the area of rectangle ABCD is 90 cm 2 and the
length is 15 cm, what is the area of rectangle WXYZ and what is the width if the length of rectangle
WXYZ is 5 cm? Complete the table below.
length
width
Rectangle ABCD
Rectangle WXYZ
©2012, TESCCC
scale factor
perimeter
area
Rectangle ABCD
Rectangle ABCD
Rectangle ABCD
Rectangle WXYZ
Rectangle WXYZ
Rectangle WXYZ
04/16/12
page 4 of 5
Grade 8
Mathematics
Unit: 07 Lesson: 01
Measurement and Dimensional Change Practice
13. Write an equation that describes the effects of dimensional change on the perimeter. Use this
equation and the information from the table in problem 12 to find the length of rectangle EFGH if it is
similar to rectangle ABCD and its perimeter is 210 cm.
14. Write an equation that describes the effects of dimensional change on the area. Use this equation
and the information from the table in problem 12 to find the length of rectangle RSTU if it is similar to
rectangle ABCD and its area is 810 cm 2.
15. Express the area of a square in square meters. The area of the square is currently expressed as
625 square inches.
©2012, TESCCC
04/16/12
page 5 of 5