ALEX Lesson Plan


Falling Down a Rabbit Hole Can Lead to a King Sized Experience - Exploring Similar Figures Using Proportions

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  This lesson provided by:  
Author:Lillie Coleman
System: Fairfield City
School: Fairfield High Preparatory School
  General Lesson Information  
Lesson Plan ID: 30067


Falling Down a Rabbit Hole Can Lead to a King Sized Experience - Exploring Similar Figures Using Proportions


In this lesson, students will explore the concept of similarity.  Students will first simplify ratios, solve proportions using the cross product property, and use properties of proportions to solve real-world problems.  Similar polygons will then be introduced and their properties used to solve real-world problems.  

 Associated Standards and Objectives 
Content Standard(s):
MA2015 (7)
2. Recognize and represent proportional relationships between quantities. [7-RP2]
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. [7-RP2a]
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. [7-RP2b]
c. Represent proportional relationships by equations. [7-RP2c]
Example: If total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. [7-RP2d]
MA2015 (9-12) Geometry
15. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. [G-SRT2]
Example 1:
Given the two triangles above, show that they are similar.
4/8 = 6/12
They are similar by SSS. The scale factor is equivalent.
Example 2:
Show that the two triangles are similar.
Two corresponding sides are proportional and the included angle is congruent. (SAS similarity)

Local/National Standards:

Numbers and Operations

Understand meanings of operations and how they relate to one another

Compute fluently and make reasonable estimates


Understand patterns, relations, and functions

Use mathematical models to represent and understand quantitative relationships


Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships


Apply appropriate techniques, tools, and formulas to determine measurements 

Primary Learning Objective(s):

Students will be able to:

Apply proportional reasoning to solving problems.

Find measures of corresponding parts of similar figures.

Deduce the measure of an angle in a polygon from given assumptions.

Deduce the length of a side of a polygon from given assumptions.

Identify geometric models that represent problem situations.


Additional Learning Objective(s):

 Preparation Information 

Total Duration:

61 to 90 Minutes

Materials and Resources:

Students will complete all activities in groups of 4.

Group members will be determined by student performance during the background review.

Each student will need a measuring tool, pencil, paper, and four function calculator.

Teacher should request maps from the local motor club.  Maps can also be obtained from social studies colleagues or printed from the Internet.  Please note that all maps must contain a scale.

Teacher and students should be familiar with the Internet. 

Technology Resources Needed:

One Internet capable laptop per group of 4 students.

Document camera, Internet capable laptop, projector and whiteboard combination for demonstrations and presentations by teacher and students. 


Teacher will review the problem-solving skill of translating words to algebra symbols by setting up and solving proportions.

The Multiplication Property of Equality and cross product recognition should be retaught.

Students will also be reminded that when expressing a ratio a/b; b cannot be zero because you cannot divide by zero. (the denominator)

Teacher will use the web link below to facilitate the proportion review:

Then click on 'Solving Proportions'.


Background review and examples are complete.

Teacher will open the discussion by challenging students to give real-world examples involving ratios or proportions.  (Examples:  basketball statistics, recipes, etc.)

Opening Activity - Reading Connection

Teacher will have students skim Alice's Adventure in Wonderland to find other references to ratios.  The following link should be used for this activity:

Note:  Click Chapters 1, 2, or 5 for this activity.  Depending on the makeup of the groups, each group can be assigned a different chapter.  The same chapter can be assigned to the groups to encourage friendly competition.

The information in the story will be used to make up several problems about ratios that Alice might use  to help her understand her surroundings. Students will share their problems.

The lesson will continue with the introduction and definition of similar figures.

Definition of Similar Polygons

Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

Teacher will present several pairs of similar polygons on the document camera along with several pairs of polygons that are not similar.  Students will be asked to make observations and defend their responses with mathematical processes.  Teacher will point out that a document camera shows an image on a screen that is similar to the image on the document.  Students will be asked to think of other devises that form similar images. (enlarger, photocopier, etc)

The following web link will reinforce this demonstration:

Scale Factor Activity:

Using maps provided by the teacher, students will practice reading scales.  Next, a base location and a destination will be chosen.  Students will then compute the distance between the two points as well as the time it takes to travel between the two points.

Error:  Teacher will remind students to avoid giving one ratio in the wrong order.  Prescription:  Students will be encouraged to write words to name the quantities on both sides.

The following web link will be used to reinforce this demonstration:

Closing Activity

Since this lesson explores similarity rather than verify similarity, the following hands-on activity will be used to check students' understanding.

Teacher will have each student use a straight edge to draw a triangle.  Students will then be asked to measure the sides of the triangle and draw a similar triangle with a scale factor of 1/2.  Students will be asked to explain how they know the triangles are similar.



Assessment Strategies


Students can use the Internet to identify other real-world applications for similar figures.

Students interested in architecture can create a scaled drawing of their bedrooms.

Student athletes can create personal profiles which include vital statistics which show cases their talent.

Students who plan to travel or work in the the international arena can create currency exchange problems.




Students needing extra help will be provided additional practice problems to improve on the prerequisite skills needed for this lesson.

While students are working on the extension activities, students needing extra help will survey their group members about future career plans.  The students will record the  information.  Next the students will query their group members about the proportionality / similarity connection and their chosen career.  The resulting presentation of these surveys will give the students extra real-world application examples. 

Each area below is a direct link to general teaching strategies/classroom accommodations for students with identified learning and/or behavior problems such as: reading or math performance below grade level; test or classroom assignments/quizzes at a failing level; failure to complete assignments independently; difficulty with short-term memory, abstract concepts, staying on task, or following directions; poor peer interaction or temper tantrums, and other learning or behavior problems.

Presentation of Material Environment
Time Demands Materials
Attention Using Groups and Peers
Assisting the Reluctant Starter Dealing with Inappropriate Behavior
Be sure to check the student's IEP for specific accommodations.