9.3.1B Scale Factors
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.
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
Dtermine 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 B Scale Factors
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.What students should know and be able to do [at a mastery level] related to these benchmarks:
- Students should be able to calculate lengths, areas and volumes of similar figures.
- Students should be able to recognize scale factors.
- Students should be able to square ratios (take them to the second power) and cube ratios (take them to the third power).
- Students should be able to take the square root of ratios and the cube root of ratios.
- Students should recognize which scale ratio is needed to solve the given problem (length, area, or volume).
Work from previous grades that supports this new learning:
- Students have used scale factors to determine lengths on maps and models.
- Students have computed and used scale factors with similar geometric shapes.
- Students have found equivalent ratios.
- Students have evaluated expressions with exponents and roots, in particular working with powers (and roots) of two ("squaring") and three ("cubing").
Misconceptions
Student Misconceptions and Common Errors
- Students believe all dimensions have the same similarity ratio.
- Students confuse squaring ratios and using the square root of ratios when converting between linear and area ratios.
- Students confuse cubing ratios and using the cube root of ratios when converting between linear and volume and ratios.
Vignette
In the Classroom
In this vignette, student's work with scale factors related to length, area and volume. Supplies needed: handouts of Michael Jordan's traced hand, rulers, soda pop cans, string, compasses, scissors and paper (scratch paper will work). Optional: bring in balls of various sizes for students to compare to "their basketball." This is a two-day activity. Students should work on completing the calculations on the first day, with an opportunity to finish the remaining work and a follow-up discussion on the second day.
The image below is a traced copy of Michael Jordan's hand. The image needs to be enlarged to full-page size (no margins) to be correct.
Teacher: We will be working on an activity involving measurement and scale factor for the next two days. Each of you has a packet to fill out and each person's results will be slightly different. The materials manager for your group should get rulers for everyone to begin and other supplies as you need them.
NOTE: The students are arranged in groups of 3-4. This is a hands-on activity where the teacher's role is to provide support, ask guiding questions, and keep everyone on task.
Teacher: Michael Jordan had an extraordinary career in professional basketball. One factor that contributed to his success was the size of his hands. In this activity, you will compare your hand with MJ's and determine how big some ordinary objects seem to MJ.
1. Estimate the ratio of your hand to Michael Jordan's hand.
A. Trace your right hand on a sheet of paper.
B. Measure the following 12 lengths on your traced hand and MJ's hand. Use centimeters.
i. Length of index finger You __________ MJ ___________
ii. Length of middle finger You __________ MJ ___________
iii. Length of ring finger You __________ MJ ___________
iv. Length of pinky finger You __________ MJ ___________
v. Width of index finger You __________ MJ ___________
vi. Width of middle finger You __________ MJ ___________
vii. Width of ring finger You __________ MJ ___________
viii.Width of pinky finger You __________ MJ ___________
ix. Base of thumb to base of pinky You __________ MJ ___________
x. Width of palm You __________ MJ ___________
xi. Tip of thumb to tip of middle finger You __________ MJ ___________
xii. Tip of thumb to tip of pinky You __________ MJ ___________
xiii.TOTALS You __________ MJ ___________
C. Record the ratio of your hand to Michael Jordan's hand. __________________
2. Change the ratio from 1C to a decimal ____________________
This decimal is the magnitude of the size change (scale factor) from Michael Jordan's hand to your hand.
This decimal is your k.
3. Your hand is _______________ % of the size of Michael Jordan's hand.
4. Determine the circumference of your basketball.
The circumference of a regulation basketball is 74.9 cm.
Use your scale factor (k) from #2 to calculate the circumference of a basketball that is proportionately the same size in your hand as a regulation basketball is in Michael Jordan's hand.
____________________ x 74 cm = ____________________
your k your circumference
5. Determine the surface area of your basketball and a regulation basketball.
A. Find the radius of a regulation basketball. (Recall: 
) Then, find the radius of your basketball.
The radius of a regulation basketball is ____________________
______________ x ________________________ = ________________________
your k radius of regulation basketball radius of your basketball
B. Use the radii you found for a regulation basketball and your basketball to find the Surface Area of each basketball. (Recall: 
)
i. Regulation basketball's SA: ii. Your basketball's SA:
C. Record the ratio of your SA to MJ's SA: ____________________
D. Change this ratio to a decimal: ____________________
E. Find 
: ____________________
F. What do you notice?
6. Determine the volume of your basketball and a regulation basketball.
A. Use the radii from #5A to find the Volume of each basketball. (Recall: 
)
i. Regulation basketball's V: ii. Your basketball's V:
B. Record the ratio of your V to MJ's V: ____________________
C. Change this ratio to a decimal: ____________________
D. Find 
: ____________________
E. What do you notice?
7. Determine the height of your basketball hoop.
Michael Jordan is 6' 6" tall.
A. The height of a regulation basketball hoop is 10'. Determine the height of a basketball hoop that is proportionately the same height to you as a regulation hoop is to Michael Jordan.
B. The diameter of a regulation basketball hoop is 18". Determine the diameter of a basketball hoop that is proportionately the same diameter to you as a regulation hoop is to Michael Jordan.
8. Determine the dimensions and build a soda pop can.
Use your scale factor (k) from #2 to determine the dimensions of a soda pop can that is proportionately the same size in your hand as a regular soda pop can is in Michael Jordan's hand.
A. Find the height of your soda pop can
i. Actual height of can ii. Height of your can
B. Find the circumference of your soda pop can
i. Actual circumference of can ii. Circumference of your can
C. Find the radius of your soda pop can
ii. Actual radius of can ii. Radius of your can
D. Neatly sketch and label a net of your soda pop can
E. Construct a 3-D model of your soda pop can
9. Determine the number of ounces in your soda pop can.
A regular soda pop can has 12 ounces. How many ounces are in your soda pop can?
REMEMBER
*********************************
Comparing lengths (1-dimensional) leads to values of k.
Comparing areas (2-dimensional) leads to values of .
Comparing volumes (3-dimensional) leads to values of .
Questions for Day 2:
- Why did we take 12 different hand measurements?
- Could we have used inches instead of centimeters?
- What type of ball would be the closest in size to your basketball? (i.e.--volleyball, softball, baseball, tennis ball, golf ball)
- For what other everyday objects would it be interesting to find proportionate sizes?
Resources
Teacher Notes
- Teachers should emphasize squaring the scale factor when working with area and cubing the scale factor when working with volume.
- Teachers may choose to emphasize always converting to the linear version of scale factor before beginning any other work with similar figures.
- Teachers may model computing linear, area, and volume conversion ratios at the start of every 3-dimensional similarity problem.
- NCTM Illuminations: NCTM Figure This
This NCTM's Illuminations activity is titled "Linking Length, Perimeter, Area and Volume." There are four lessons which guide students through the process of understanding that for two similar solids, if the ratio of their corresponding lengths is a:b, then the ratio of their corresponding surface areas is a2:b2 and the ratio of their corresponding volumes is a3:b3.
- This NCTM Illuminations activity uses scale factor to find distances using maps of Manhattan.
- This Sophia Lesson is on similarity and scale factors for length, area, and volume.
Additional Instructional Resources
Dan Meyer's blogsite includes his version of a complete Geometry course. Many of the activities can be incorporated for use with any textbook.
scale factor: the ratio between corresponding edge lengths of similar geometric shapes or figures.
Reflection - Critical questions regarding teaching and learning of these benchmarks:
- What other instructional strategies can I use to engage my students with scale factor ratios?
- How can I use manipulatives to help my students visualize these abstract concepts?
- 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
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).
http://www.bbc.co.uk/apps/ifl/schools/gcsebitesize/maths/quizengine?quiz=congruencysimilarity&templateStyle=maths
The above is an assessment from the BBC. It's a ten-question "quiz," and a person can easily check her/his answers. For a correct answer, the student is told, "Well done," and for an incorrect answer, the student is told, "Hard luck." Both contain brief explanations of the answer, and why it is correct or incorrect.
On the quiz, there are a couple of questions relating to congruent triangles. One method of proving triangles congruent will be unfamiliar to most students in the United States. That method is "RHS," or "Right-angled, hypotenuse, side." The equivalent in the United States would be "Hypotenuse Leg," or "HL."
For the beginning of the lesson (in case a student or teacher would like instruction and/or review before taking the quiz), go to:
http://www.bbc.co.uk/schools/gcsebitesize/maths/shapes/congruencysimilarityrev1.shtml
Also view the assessment for 9.3.3D (benchmark 9.3.3.6) for more work with similarity.
Differentiation
Special emphasis should be placed on keeping "squaring" (as in ratio of areas) separate from "cubing" (as in ratio of volumes) as long as possible. It is only after students feel comfortable with the two concepts separately that they should be combined.
Consider starting by having students use unit blocks to build a rectangular prism with dimensions 4 x 6 x 8. Then have them build models with dimensions 2 x 3 x 4 and 8 x 12 x 16. Calculate surface areas for each of these and predict surface area for 6 x 9 x 12. Repeat this activity for volume.
Emphasize the vocabulary, especially the difference between area (amount of surface required) and volume (amount of space required). Suggest that students use the units attached to any given numbers to help determine what is being measured. Linear units indicate lengths, square units indicate areas and cubic units indicate volumes.
Bring in model cars, planes or boats (or have students put together the models themselves), and calculate scale factor, ratios of lengths, areas and volumes, and actual lengths, areas and volumes. For example, the blueprints for a house give an excellent opportunity to discuss scale factor, areas of walls and floors, and volumes of rooms.
Parents/Admin
Administrative/Peer Classroom Observation
Students are... | Teachers are... |
completing a computer lab which works with length, area and volume. | setting the stage for the computer lab. |
discussing findings and results with a partner. | walking around the lab, answering questions and helping when necessary. |
correctly using measurement tools to measure lengths, heights and circumferences of objects. | asking students if their measurements are reasonable, checking for correct use of measurement tools. |
Parent Resources
- Sophia and Khan Academy are websites with uploaded lessons teaching multiple topics. Most of these lessons were developed by teachers and reviewed.
- 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.