9.3.3D Congruent & Similar Figures
Know and apply properties of congruent and similar figures to solve problems and logically justify results.
For example: Analyze lengths and areas in a figure formed by drawing a line segment from one side of a triangle to a second side, parallel to the third side.
Another example: Determine the height of a pine tree by comparing the length of its shadow to the length of the shadow of a person of known height.
Another example: When attempting to build two identical 4-sided frames, a person measured the lengths of corresponding sides and found that they matched. Can the person conclude that the shapes of the frames are congruent?
Overview
Standard 9.3.3 Essential Understandings
The study of geometric figures might be considered one of the "main topics" in geometry, along with, among others, proportional thinking, reasoning and sense making. The focus of this standard is on geometric figures, with the others listed above also being involved to a great extent.
In this standard, students will work with a transversal that intersects two lines. They will study angles formed by the three lines, and determine whether or not the two intersected lines are parallel. They will see many types of triangles, including "special" triangles such as isosceles, equilateral, right, 30-60-90 and 45-45-90 triangles, focusing on the properties of each of these figures.
Students will study similarity and understand that congruence is a special case of similarity. They will work with scale factors in "real-life" applications of similar figures. They will look at quadrilaterals and put them into a hierarchy, with "special" quadrilaterals being "nested" within others. Also, in this standard, students will study circles and their properties, with special focus on their applications.
Benchmark Group D - Congruent and Similar Figures
9.3.3
9.3.3.6 Know and apply properties of congruent and similar figures to solve problems and logically justify results.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Recognize similar figures in "real life;"
- Calculate lengths using proportional reasoning;
- Calculate scale factors seen in similar figures.
Work from previous grades that supports this new learning includes:
- Represent proportional relationships with equations;
- Use proportional reasoning to solve problems involving ratios in various contexts.
NCTM Standards
Geometry
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
- Analyze properties and determine attributes of two- and three-dimensional objects;
- Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
- Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
- Use trigonometric relationships to determine lengths and angle measures.
- Use visualization, spatial reasoning, and geometric modeling to solve problems:
- Use geometric models to gain insights into, and answer questions in, other areas of mathematics;
- Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Common Core State Standards (CCSM)
- HS.G-CO (Congruence) Prove geometric theorems.
- HS.G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
- HS.G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
- HS.G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals
- HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Define trigonometric ratios and solve problems involving right triangles.
- HS.G-SRT.7. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Apply trigonometry to general triangles.
- HS.G-SRT.10. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
- HS.G-C (Circles) Understand and apply theorems about circles.
- HS.G-C.1. Prove that all circles are similar.
- HS.G-C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- HS.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
- HS.F-TF (Trigonometric Functions) Extend the domain of trigonometric functions using the unit circle.
- HS.F-TF.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6.
Misconceptions
- Students sometimes misinterpret "scale factor," or use it incorrectly.
- Students sometimes set up proportions incorrectly.
Vignette
In the Classroom
This vignette lets students use measurements of features on the faces on Mount Rushmore to determine the scale of other features.
The teacher shows this graphic on a screen in the classroom.
Teacher: What is this picture showing?
Student: Mount Rushmore?
Teacher: Right, it's Mount Rushmore before the carving. This is what it looks like now.
The teacher shows the next graphic.
The teacher can provide any of the additional "fun facts" about Mount Rushmore, as desired.
- Originally, the idea was to carve the mountain showing the presidents from head-to-toe, but that idea was quickly scrapped, because the size of the heads would have been much less than was desired.
- It was decided then to carve the presidents head-to-waist.
- Gutzon Borglum started the carving in 1927 and passed away in 1941. At that point, his son, Lincoln Borglum, completed the sculpture, finishing President Lincoln's head. The construction stopped there, instead of continuing head-to-waist, in honor of Gutzon Borglum (and also due to a possible lack of funding).
- 90% of the "carving" was done using dynamite.
- Approximately 400 people worked on the construction site, and yet, during the entire project, no lives were lost in the construction.
- In the Sculptor's Studio on the grounds of Mount Rushmore stands this 15-foot-tall model, which shows what the carving on the mountain would have looked like if the presidents would have been carved head-to-waist:
- Also in the Sculptor's Studio are two of the four plaster masks that were brought up to the mountain by the workers. Using these masks, the workers blasted the rock and formed the faces. Each of these masks is five feet tall. Lincoln's mask and Jefferson's mask still exist. Washington's and Roosevelt's were broken.
Teacher: In your groups, solve this problem: On Mount Rushmore, President Lincoln's mouth is 18 feet wide. Using the same scale factor, if President Lincoln had been carved head-to-toe, how tall would his carving be on the mountain?
Students will work in groups of 2-4, discussing the best way to proceed. They will soon find that they need more information.
Student: How tall was President Lincoln?
Teacher: He was six feet, four inches tall.
Student: How wide was his mouth?
Teacher: I'm not sure there is any data available about the width of President Lincoln's mouth.
Student: Well, how can we answer the question?
Student 2: We could measure our mouths. President Lincoln's mouth must have been about the same width as ours.
The teacher hands a ruler to each group and the students measure their mouths. At this point, the class could use an average width from the group, or use a class average, or use individual data to work with the proportion.
Student: Was President Lincoln smiling? That would make a difference in our measurements.
Teacher: No, you can assume he was stone-faced.
Let's say that the students determine that a 2.5-inch-wide mouth is deemed acceptable. Then they'll solve the proportion
$\frac{2.5}{18}=\frac{76}{h}$
It turns out that h = 547.2 feet tall, or approximately the height of a 54-story building.
Student: I noticed that our units aren't in agreement. The 2.5 and the 76 are in inches, and the 18 is in feet. Does that matter?
Teacher: That's a good question. How many used a different proportion?
Some will have converted 18 feet to 216 inches, and some used the following proportion:
$\frac{2.5}{216}=\frac{76}{h}$
They will notice that h = 6566.4 inches. Of course, most people will then convert that into inches by dividing by 12, getting the same 547.2 feet that was found in the original.
Teacher: Does it matter which units you used originally?
Student: Nope.
Teacher: Now that we have solved that, here's another question: What scale factor would take President Lincoln's head in real life to his head on the plaster mask? How about as it is carved on the mountain?
Student: How tall was President Lincoln's head?
Teacher: How could you estimate the height of his head?
Student: We could measure our heads, but does it matter that we're not six feet, four inches tall? Would that make a difference?
After some discussion, it's decided that a nine-inch (or .75 feet) head would be used in the equation, as follows:
(.75)(h) = 5
Teacher: What did you find?
Student: We found that h = 6.666..., or $6\frac{2}{3}$.
Teacher: Good. What does that mean?
Student: That's the scale factor.
Teacher: Yes, it is. What does that mean?
Student: That means that President Lincoln's head on the model is $6\frac{2}{3}$ times as tall as his head in real life.
Teacher: Excellent. How about the next question?
Student: We don't have enough information. How tall is his head on the mountain?
Teacher: It's 60 feet tall.
Student: OK, so we can use (.75)(h) = 60, and find that h = 80.
Teacher: Very good. What does that mean?
Student: It means that 80 is the scale factor, so that President Lincoln's head on the mountain is 80 times as tall as his head in real life.
Resources
Teacher Notes
- Emphasize that a "scale factor" is just a multiplier.
- Have students explain their proportional reasoning as they solve the problems.
How Far Can You Go in a New York Minute?
In this lesson, students use proportions and similar figures to adjust the size of the New York City Subway Map so that it is drawn to scale. Students are asked to evaluate whether these changes are necessary improvements.
Additional Instructional Resources
National Council of Teachers of Mathematics (NCTM). (2010). Focus in High School Mathematics: Reasoning and Sense Making in Geometry. Reston, VA: National Council of Teachers of Mathematics.
New Vocabulary
scale factor: a number which multiplies (or "scales") some quantity.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- What other instructional strategies can be used to engage students with properties of triangles?
- How can manipulatives be used to help students visualize these abstract concepts?
- How can the instruction be scaffolded for students?
- What additional scaffolding should be provided to ELL students?
- Do the tasks that have been designed connect to underlying concepts or focus on memorization?
- How can it be determined if students have reached this learning goal?
- How can the lesson be differentiated?
Assessment
This is a picture of Spaceship Earth, an attraction in Epcot at Walt Disney World. Spaceship Earth is 180 feet tall and looks somewhat like a golf ball.
The following questions might serve as an assessment for this standard, but teachers should be careful how these questions are posed, as students will need more information than is currently in the questions.
1. If Spaceship Earth were a golf ball, how tall would be the golfer who hits it?
2. If the golfer uses a driver, how long would be the club the golfer uses to hit the "ball"?
3. How wide and how deep would be the hole on the green, in order for this "ball" to drop?
4. How much dirt would have to be moved in order to make the hole?
5. What is the scale factor that takes a regulation golf ball to the size of Spaceship Earth?
6. When the golfer is done with the round of golf and gets into the car to drive home, how long is that car?
7. If a golfer can hit a regulation golf ball 250 yards with a driver, how far could the golfer described above hit Spaceship Earth?
DOK Level: 4
Assessment hints/solutions
1. Assume a 6-foot-tall (or, 72-inch tall) golfer and a ball with diameter 1.68 inches, per USGA rules.
$\frac{1.68}{180}=\frac{72}{g}$
Solving this equation for g, you get a golfer who is 7,714 feet tall. This is almost 1.5 miles in height.
2. Students will have to know that a 44-inch driver is considered "standard length" for a male golfer.
$\frac{1.68}{180}=\frac{44}{c}$
Solving this for c, you get a driver that is approximately 4,714 feet tall.
3. The standard golf hole has a diameter of 4.25 inches and a minimum depth of 4 inches.
For the diameter:
$\frac{1.68}{180}=\frac{4.25}{d}$
So the diameter of the hole would be approximately 455 feet, or greater than the length of a football field.
For the depth:
$\frac{1.68}{180}=\frac{4}{h}$
The depth of the hole would be approximately 429 feet, or about the height of a 42-story building.
4. How much dirt would have to be moved in order to make the hole?
The formula to find the volume of a cylinder is (π)(radius^{2})(height).
Since the diameter is about 455 feet, the radius would be about 227.5 feet. The height would be 428 feet, so the volume of the cylindrical hole would be approximately 69,591,539 cubic feet.
This is approximately the same amount of dirt that it would take to fill the Hubert H. Humphrey Metrodome in Minneapolis, Minnesota, shown here:
5. If f represents the scale factor, then:
(1.68 inches)(f) = 180 feet or (1.68 inches)(f) = 2160 inches, and f is approximately 1,286, so Spaceship Earth is approximately 1,286 times as tall as a regulation golf ball.
6. This is a picture of a Buick Lucerne, as an example of the car the golfer might drive:
The Buick Lucerne is about 17 feet or 204 inches, long, so:
$\frac{1.68}{180}=\frac{204}{b}$
Solving for b, we have a car that is 21,857 feet, or more than four miles, long.
7. 250 yards is the same as 9000 inches, so:
$\frac{1.68}{180}=\frac{9000}{d}$
Solving for d, we have a drive that is 964,286 feet, or a little more than 182 miles long.
Differentiation
The Vignette activity (and accompanying Assessment) might need to be split into two days, with review given at the start of the second day.
- There are a few issues with the Vignette activity as it relates to ELL students. First, they might not have any idea what Mount Rushmore is, nor why there are faces carved into it. For this reason, it's a good idea to put them into a group in which there are students who are familiar with Mount Rushmore and the Presidents. Also, ELL students might have difficulty with feet as a unit of measurement, as they might be much more familiar with metric distances. gain, putting them into a group with students who are familiar with this unit of measurement might help.
- Regarding the Assessment, explain what Epcot is, as many ELL students will not know. Also, you will probably have to explain golf terms such as "driver," "green" and "regulation" golf ball. Having pictures of other points of reference (the Metrodome, a golf hole, a Buick Lucerne, etc.) will help.
Extend the Vignette activity to other types of similar figures, or to the changes in height of Alice in "Alice in Wonderland."
Parents/Admin
Students are: (descriptive list) | Teachers are: (descriptive list) |
calculating lengths relative to similar figures. | asking guiding questions for students to answer for use in their calculations. |
calculating scale factors, given similar figures. | observing and helping as needed. |
Parent Resources
- Sophia geometry lessons
This social learning community website includes lessons on multiple topics. Most of these lessons were developed by teachers and reviewed.
- Khan Academy geometry lessons
This website offers a library of videos and practice exercises on multiple topics.
- Teacher Tube, YouTube
These websites include multiple uploaded lessons on most school topics.
- Mathematics textbooks
Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.