9.3.1A Calculating Measurements
Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate.
For example: Measure the height and radius of a cone and then use a formula to find its volume.
Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures.
For example: Find the volume of a regular hexagonal prism by decomposing it into six equal triangular prisms.
Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert between measurement systems.
For example: 60 miles/hour = 60 miles/hour × 5280 feet/mile × 1 hour/3600 seconds = 88 feet/second.
Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements.
For example: Suppose the sides of a rectangle are measured to the nearest tenth of a centimeter at 2.6 cm and 9.8 cm. Because of measurement errors, the width could be as small as 2.55 cm or as large as 2.65 cm, with similar errors for the height. These errors affect calculations. For instance, the actual area of the rectangle could be smaller than 25 cm2 or larger than 26 cm2, even though 2.6 × 9.8 = 25.48.
Overview
Standard 9.3.1 Essential Understandings
The study of Geometry is often motivated by the search for answers to perimeter, area and volume problems. The practical solutions to these problems are often useful beyond the classroom walls. In this standard, students work with two-dimensional and three-dimensional figures and objects.
It is important for students to realize that there are different measurements involved in three-dimensional objects. Using a cone as an example, students might measure lengths such as radius of the base, height or slant height, or area such as lateral area or total surface area or volume such as the volume of a cone.
With the change in measurements is an accompanying change in units. Lengths are typically measured in linear units, area is typically measured in square units, and volume is typically measured in cubic units. Students must understand that, for example, a square measuring one inch on a side is a square inch, and a cube measuring one centimeter on an edge is a cubic centimeter.
As in many areas of geometry, similarity is found in this standard as well. Students will find that similar objects have ratios of areas and volumes that relate to, but are not equal to, the ratios of their linear measures.
All Standard Benchmarks
9.3.1.1
Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate.
9.3.1.2
Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures.
9.3.1.3
Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert between measurement systems.
9.3.1.4
Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively.
9.3.1.5
Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements.
Benchmark Group A Calculating Measurements
9.3.1.1
Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate.
9.3.1.2
Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures.
9.3.1.3
Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert between measurement systems.
9.3.1.5
Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Determine the surface area and volume of pyramids with a variety of bases, including both right and oblique. Computations should be made using both physical models with measurement tools and diagrams or descriptions with appropriate formulas.
- Determine the surface area and volume of cones, including both right and oblique. Computations should be made using both physical models with measurement tools and diagrams or descriptions with appropriate formulas.
- Relate the slant height, height and radius of a cone using the Pythagorean Theorem.
- Determine the surface area and volume of spheres and hemispheres. Computations should be made using both physical models with measurement tools and diagrams or descriptions with appropriate formulas.
- Compose individual 3-dimensional shapes into compound shapes to model real objects.
- Decompose compound figures into single figures by breaking them apart along appropriate faces. Use appropriate formulas to calculate their perimeters, areas, surface areas and volumes of the needed portions of the single figures.
- Use correct units when expressing solutions to length, area and volume problems.
- Convert solutions to length, area and volume problems between measurement systems.
- Estimate solutions to length, area and volume problems.
Work from previous grades that supports this new learning:
- Calculate surface areas and volumes of cylinders and prisms.
- Calculate areas of triangles, quadrilaterals and circles.
- Compose and decomposed 3-D figures using nets.
- Estimate areas and perimeters of irregular figures on a grid.
- Work with formulas and variables.
- Use arithmetic skills to evaluate functions using fractions and decimals.
NCTM Standards: Measurement
Understand measurable attributes of objects and the units, systems, and processes of measurement
In grades 9-12 all students should-
- make decisions about units and scales that are appropriate for problem situations involving measurement.
Apply appropriate techniques, tools, and formulas to determine measurements
In grades 9-12 all students should-
- analyze precision, accuracy, and approximate error in measurement situations;
- understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders;
- apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations;
- use unit analysis to check measurement computations.
Common Core State Standards (CCSS)
HS.N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
HS.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
HS.G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Misconceptions
Student Misconceptions and Common Errors
- Students believe that the volumes of pyramids and cones are one-half the volumes of their associated prisms and cylinders.
- Students include areas of faces where composed figures adjoin when computing surface area.
- Students get confused about which type of units should be used to label their solutions.
- Students invert conversion fractions when changing units.
Vignette
In the Classroom
In this vignette, students have been studying finding surface area and volume of individual figures: prisms, pyramids, cylinders, cones, and spheres. The next step is to have them consider compound figures. Their teacher, Ms. Johnson, will have them study a variety of storage containers and determine the surface area and volume of each figure.
She has posted this Question of the Day (QOD) in the corner of the board: "Which is best?"
Ms. Johnson put the following sketches on the board:
T: These sketches are all types of silos. Can anyone tell me what a silo is?
S1: It's on a farm.
T: It can be.
S2: The farmers put corn in them.
T: That is one of their uses.
S3: I've seen ones with sand in them.
T: That could be correct too. Silos are used to store things like grain, sand, and stone. You can find them on farms and in shipping yards. What geometric figures do you see in these drawings?
S1: Cylinders
T: Would you label them for us?
(Student goes to the board and labels the two cylinders.)
T: What else?
S2: Hemisphere
T: Label it.
S3: Cone
T: Label.
S4: Prism
T: What kind of prism?
S4: Hexagonal
T: Label that too. What's the last figure?
S5: A hexagonal pyramid.
(Waits until the final student has finished labeling.)
T: All of these storage silos can be called composite figures because they are all composed of more than one figure. These ones are composed of exactly two different figures joined along a common face. A more complicated design could include more figures. Or a composite figure could just have the same figure repeated more than once.
S3: Like two cones joined along the circles.
T:Yes. If we are going to work with these figures today, we need to remember where we've been. What do we know about them?
S1: We can find their volumes.
T: Are they all the same?
S2: No. The cylinder and the prism volumes are area of base times height, but for the pyramid and the cone you need to divide by 3.
S3: And the half sphere is that formula with pi, 2$\pi$r2
T: Do we know anything else?
S4: We know how to find surface area by breaking apart all of the faces.
T: These are all ideas that we will use for the activity. Today we are going to focus on the efficiency of these storage silos. Who can explain what efficiency means?
S1: It's how well something works.
S2: Like gas mileage.
T: Exactly. Efficiency is a ratio that describes how well an object works. For a storage silo, we will use the ratio of its volume to its surface area for efficiency. What does that mean we will need to figure out?
S1: The surface area of each silo.
S2: And the volume too.
T: Then what?
S3: We will need to turn it into a ratio, like a fraction or decimal.
T: OK. I think you have the idea. Work with your partner on this project. You will need to turn in work that shows four things for each silo: First, a labeled sketch of its net.
S5: How do we draw the net of a hemisphere again?
S4: It can't be made from a flat piece of paper.
S5: Would we just do a net of the rest of the figure and indicate where the hemisphere attaches?
T: Good solution. The next two items for each silo are calculations for surface area and calculations for volume. Both of these need to ....
Class: Have appropriate units!
T: Yep. The last part is your calculations for efficiency. When you get this done, you will be ready to answer the QOD. Be sure to use complete sentences and explain your final solution. Get to it.
The classroom is arranged so that students typically sit in pairs, so everyone knows who their partner is for class activities for this week. Ms. Johnson circulates through the room to get everyone on track to begin their work. She stops at each pair to answer questions about this activity and about the key concepts from the unit.
Many of the students ask her if their work is correct. Instead of answering them directly, Ms. Johnson suggests that they do a quick estimate of the area or volume and see if their result seems reasonable. She listens as they perform this work to ensure that they are using correct formulas and measurements. If a pair is struggling with the key ideas, she works with them until she is confident that the activity can be completed.
Several groups are unsure about how to approach the regular hexagon. Ms. Johnson anticipated that this would be the most difficult of the figures. She knows that there are two main approaches that are most successful and asks the students questions to determine which approach they are taking. She then asks guiding questions either to lead them to decompose the hexagon into more familiar shapes or to help them recall how to work with the apothem. She also suggests drawing a side view for determining the height of the pyramid and focuses their attention on the right triangles that can be created.
At the close of class, she gives a brief homework assignment to either focus on converting units or to engage in additional practice with composite figures.
[Activity Answers: Silo 1: SA = 496$\pi$ft2 ; V = 1536$\pi$ft3 ; eff = 3.10 . Silo 2: SA = 975$\pi$ft2 ; V =4500$\pi$ft3 ; eff = 4.61 . Silo 3: SA = 2210.13 ft2 ; V = 7856.58 ft3 ; eff = 3.55.]
Resources
Teacher Notes
- Teachers may show the relationship of the volumes of pyramids to prisms and of cones to cylinders using fillable models with water, sugar or rice.
- Teachers need to focus on decomposing figures into their associated nets to determine all of the faces that form the exterior and comprise the surface area.
- Teachers may stress that linear units are for lengths (one-dimensional), square units are for areas (two-dimensional), and cubic units are for volumes (three-dimensional).
- Teachers may use the Factor-Label method (also known as Dimension Analysis) from chemistry to teach converting units. This method focuses on matching the units in conversion factors so that each unit that is in a numerator also occurs in a denominator until only the desired units remain.
- Sophia Lesson consisting of several Powerpoints to both teach and practice area and volume formulae with pyramids and cones
Additional Instructional Resources
- National Council of Teachers of Mathematics. (2010). Focus in high school Mathematics: Reasoning and sense making in geometry. Reston, VA: NCTM.
- SEN Teacher website of teaching resources. This page includes printable nets for several polyhedra.
- To build quick models, consider getting a set of plastic reusable building pieces like on of these:
Polydrons
Zometool
Magformers
Geofix
- Khan Academy has videos on multiple topics; this one is on cylinders.
- Dan Meyer's blogsite includes his version of a complete Geometry course. Many of the activities can be incorporated for use with any textbook:
regular polygon: an equilateral and equiangular polygon.
pyramid: polyhedron with a polygonal base and triangular lateral faces that meet at a common vertex (called the apex).
cone: a figure with a circular base and a point (the vertex) in another plane.
sphere: a figure formed as the set of all points in space equidistant from one fixed point.
Reflection - Critical questions regarding teaching and learning of these benchmarks
- What other instructional strategies can I use to engage my students with finding lengths, areas and volumes?
- How can I use manipulatives to help students visualize abstract geometric figures?
- How do I scaffold my instruction for my students?
- What additional scaffolding do I need to provide ELL students?
- Do the tasks I've designed connect to underlying concepts or focus on memorization?
- How can I tell if students have reached this learning goal?
- How did I differentiate this lesson?
Wolfram Alpha is a computational knowledge engine that answers a variety of math questions, including unit conversions, and creates plots and visualizations. Free online access to the Wolfram|Alpha computational knowledge engine:
answer questions; do math; instantly get facts, calculators, unit conversions, and real-time quantitative data and statistics; create plots and visualizations; and access vast scientific, technical, chemical, medical, health, business, financial, weather, geographic, dictionary, calendar, reference, and general knowledge-and much more.
Kimberling, C. (2003). Geometry in action. Emeryville, CA: Key College Publishing.
Marzano, R.J. & Pickering, D.J. (2005). Building academic vocabulary. Alexandria, VA: ASCD.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Assessment
Teacher Note: Give several examples citing Webb's DOK model (performance based, multiple choice, short answer, free response (released MCA II & III items as a potential resource).
Option 1: Follow up assignment to the Vignette that works with conversion factors.
- The silos from the activity have been filled with corn. Each bushel of corn weighs approximately 56 pounds and each bushel has a volume of 1.25 cubic feet. Calculate the weight of the contents of each silo in metric tons. (Note: 1 tonne = 2204.622 pounds)
- A rule of thumb to find the capacity in tonnes for a silo is to square the radius multiply by the total height and divide by twenty. How good of an estimate is this rule of thumb for these silos?
- Each silo is going to be manufactured in Canada. Redo all of your calculations from the activity after converting all needed lengths to meters.
Answers:
1. Silo 1 -- 98.06 tonnes; Silo 2 -- 287.28 tonnes; Silo 3 -- 159.65 tonnes
2. Silo 1 -- 89.6 tonnes; Silo 2 -- 281.25 tonnes; Silo 3 -- 194.4 tonnes. The rule of thumb works best with cylindrical silos.
3. As 1 foot = 0.3048 meters, multiply all areas by (0.3048)2 and all volumes by (0.3048)3 .
Option 2: Multiple choice and constructed response problems. Questions 1-3 are from the MCA-II Item Sampler.
1. Find the area of the figure. (Use 3.14 for $\pi$.)
A. 67.71 square feet
B. 70.07 square feet
C. 72 square feet
D. 91.26 square feet
2. Elena inherited three small spherical beads from her grandmother. They had radii of 2 mm, 3mm, and 4 mm. She wanted to have them melted and recast to form one larger sphere. Its radius would be closest to:
A. 3 mm.
B. 5 mm.
C. 6 mm.
D. 9 mm.
3. A family is carpeting two rectangular rooms. They have chosen carpeting that costs the same amount per square yard for each room. A 12-foot by 15-foot carpet for the bedroom costs $600. If the dimensions of the living room are 20 feet by 18 feet, what will it cost to carpet the living room?
A. $624
B. $720
C. $1000
D. $1200
4. The formula for finding the volume of a right circular cylinder is V = .$\pi$r2. What is the approximate volume of this right circular cylinder?
A. 265.5 m3
B. 326.7 m3
C. 1036.9 m3
D. 1061.9 m3
5. Match each solid with its net.
A. cone
B. cylinder
C. triangular prism
D. triangular pyramid
E. rectangular pyramid
6. A right pyramid has a height of 5 cm and a square base with side length 3 cm. Sketch the pyramid, then calculate its surface area and volume.
7. A right cylinder has radius 4 ft and height 10 ft. Sketch the cylinder, then calculate its surface area and volume.
Answers:
1. B
2. B
3. D
4. D
5. A. V
B. I
C. IV
D. II
E. III
6.
Surface Area = 40.32 cm2
Volume = 15 cm3
7.
Surface Area = 112$\pi$ ft2
Volume = 160$\pi$ ft3
Option 3: Your math teacher is a finalist for Extreme Makeover: Geometry Edition. The show producers need to know the areas of the floor for carpeting and the walls for painting. They also need to know the volume of the room to determine the needed amount of lighting. Your task is to write a report for the producers that include a scale diagram, details your calculations, and determines how much of each supply is needed. Carpeting is measured in square yards, paint in gallons and lighting in lumens.
Teacher Note: This performance task works best in an irregularly shaped room. Students will need access to measuring devices and should work in groups to complete this assessment.
Differentiation
There is often a tendency for teachers to do too much, too soon with concepts such as these. Although students may have seen these formulas in the past, their study at this level provides an opportunity to make connections between like figures. Resist the temptation to rush through formulas too quickly, before true understanding has taken place among students. Teachers should take more time and use hands-on materials as much as possible.
Once vocabulary is set, this sort of activity can give a great deal of confidence to an ELL student. They are able to see and touch many of the objects they're working with and are able to follow up the discussion with formulas for calculations.
Have students build Platonic and Archimedean Solids, then develop surface area and volume formulas for these solids.
Bring familiar objects to the classroom, physically or by using pictures or diagrams, and calculate measurements of the objects. This can be differentiated for many levels by using simple 3-D objects for struggling students and complex ones to challenge others.
Parents/Admin
Administrative/Peer Classroom Observation
Students are... | Teachers are... |
classifying three-dimensional objects. | starting the discussion about the objects. |
calculating areas and volumes of these objects. | circulating around the room, answering questions and helping when necessary. |
measuring needed lengths of objects. | checking that appropriate measurement tools are being used. |
choosing formulae to complete their work. | asking guiding questions to help students choose correct formulae. |
Parent Resources
- Sophia and Khan Academy are websites with uploaded lessons teaching multiple topics. Most of these lessons were developed and reviewed by teachers.
- Teacher Tube and You Tube include multiple uploaded lessons on most school topics.
- Many textbook publishers have websites with additional resources and tutorials. Check children's textbooks for web links.