Lesson Plan

Congruency, Similarity, and Scale Factor


In this lesson, students will learn methods for identifying congruent and similar figures. They will use the relationship between corresponding parts to make decisions regarding pairs and sets of figures. Students will use the concept of scale factor to consider the changes in figures as they are reduced and enlarged. Students will:

  • identify congruent and similar polygons.
  • explain the characteristics that determine similarity and congruence.
  • discover the relationship between scale factor, perimeter, and area.
  • graph figures and their images (including congruent, similar, and non-similar).
  • write and use rules for reducing and enlarging figures, such as (x, y) → (2x, 2y).
  • consider congruence and similarity of circles.
  • apply congruence and similarity concepts to solve problems with missing values.
  • apply the concept of scale factor to scale drawings and maps.

Essential Questions

  • How can we use the relationship between area and volume to help us draw, construct, model, and represent real situations and/or solve problems of area and volume?


  • Congruent: Having the same shape and size.
  • Corresponding: Parts of a figure, such as sides and angles, which are in the same relative position in two different figures.
  • Similar: Having the same shape, but a different size which is proportional to another figure.
  • Scale Factor: The ratio of the lengths of the corresponding sides in similar figures, the number multiplied by the side length measures of one figure to create a reduced or enlarged image. [IS.1 - Preparation]


180–210 minutes [IS.2 - All Students]

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.


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Formative Assessment

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    • Informally evaluate student understanding through your observation during class discussions of the shape set comparisons, discovering congruence and similarity, and map reading.
    • Formally evaluate students using the Scale Factor graphing activity, Similarity Applications activity, and related presentations.
    • Formally evaluate student understanding using an Exit Ticket at the end of the lesson.

Suggested Instructional Supports

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    Scaffolding, Active Engagement, Modeling, Explicit Instruction
    W:   A question involving similar figures with at least one missing value or a map question involving scale is presented to the class. Students consider ways to solve it. Whether the class figures out how to solve it or not, they learn that this lesson leads to more problems like this and ways to solve them.
    H:   Student groups begin by comparing a set of figures. They describe their observations regarding shape, size, corresponding sides, and corresponding angles. This activity leads into a discussion on congruence and similarity.
    E:   Students practice identifying congruence and similarity using the characteristics of the corresponding parts including ratios. The activities progress to the calculation of the scale factor. Scale factor is used for graphing and problem solving with similar figures. Students apply similarity and scale factor concepts to examples such as missing values problems involving similar figures, shadow problems, mirror reflections, embedded triangles, road map reading, and scale drawing problems.
    R:  Student groups are encouraged to adjust their solutions as necessary during work time and class discussions. You and student peers ask questions to assist partners and presenters in identifying inaccuracies in their thinking and in correcting them.
    E:   Your observation is a key factor as the segments of the lesson progress. The Similar Figure Graphs, Enlargement activity, and Exit Ticket provide methods to evaluate student understanding, progress and proficiency.
    T:   The lesson can be tailored to meet the needs of students using the extension suggestions. The small group activity is suitable for the entire class or students who need additional learning or practice. An Extension activity is designed for students who demonstrate proficiency and are prepared for a challenge. There is a project suggestion for the entire class. This can be individualized to give more challenging parts to students who are ready and less challenging parts to students who need more practice, so all can participate successfully. Additional activities are suggested for classroom review and technology use.
    O:   The lesson begins with a similarity problem to get students thinking about how figures are related and how relationships between them may be useful in life. Students begin comparing figures to make note of similarities and differences in shape and size using the Shape Set and Shape Set Record. The lesson progresses to discussion regarding the specific characteristics of congruent and similar figures. Students practice strategies for working with similar figures such as determination of similarity, comparing measures of corresponding angles, finding ratios of corresponding side lengths, and finding scale factors used for reducing and enlarging figures. The Scale Factor Labsheet gets students graphing similar figures and making more comparisons. These strategies all lead to practice in solving problems involving missing values in similar figures, comparing the relationship between scale factor, perimeter and area of similar figures, using shadows and mirror reflections to create similar triangles, understanding the scale used in scale drawings, and using a scale on a map. The lesson ends with a Similarity Applications activity and an Exit Ticket. A variety of visual and hands-on strategies are used to appeal to different learning styles throughout the lesson, and to assist students in organizing and solidifying a large number of concepts in a short period of time. Several Extension activities are suggested to give additional practice opportunities and extend the learning in the lesson.


    IS.1 - Preparation
    Consider using graphic organizers (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles) to review key vocabulary prior to the lesson.”   
    IS.2 - All Students
    Consider preteaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson (based upon the results of formative assessment), consider the pacing to be flexible to the needs of the students. Also consider the need for reteaching and/or review both during and after the lesson as necessary  
    IS.3 - Struggling Learners
    Consider providing struggling learners with a handout copy of the examples on the board.  
    IS.4 - All Students
    Consider using graphic organizers (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles) to review  key vocabulary prior to and during  the lesson  
    IS.5 - All Students
    Consider providing the concepts on a handout to be referenced during the lesson for some learners. Visual representations can be provided for some concepts.  
    IS.6 - Struggling Learners
    Consider providing struggling students with time to preview/review information prior to the extension activities. 

Instructional Procedures

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    Before the lesson, prepare one shape set for each small group (M-7-6-3_Shape Set.docx). Cut the shapes out and store each set in a zipper bag or envelope. Set them aside.

    When students arrive, place one or two example questions on the board. One example could be the sketch of large and small similar polygons. [IS.3 - Struggling Learners] The figures could be large paper cut-outs or simply drawn on the board. Label some corresponding sides.

    Ask students how they could find the perimeter, or at least one missing side. The second example, if you choose to use one, could be from a map, a school floor plan, or a scale drawing. Ask students if they can figure out a way to find out the size of a specific object on the drawing or in real life. Do not ask students to actually solve the problems. Rather, ask them to explain their ideas on how it could be solved.

    Possible example 1:


    Allow students to think about methods needed to reach a solution to the question(s) you selected. Call on several students to share their ideas. Emphasize correct logic, and use guiding questions to adjust incorrect logic.

    “You probably noticed that most of our responses included discussion of the shape and size of our figures. Sometimes it is difficult to compare two or more objects or situations without knowing what to call the different parts. One way we could make it easy to compare the star figures would be to label the vertex points. We can use any variables (letters) we want. I will use q, r, s, t, and u on the first figure.” Go to the board and label the figure.

    “The second figure will usually have labels that are completely different letters such as b, c, d, e, and f. If the shapes look alike, the second figure may have all the same letters as the first but with the prime symbol on each of them to indicate that the second figure is an image of the first. For our example they would be q', r', s', t', and u'. If the prime symbols are used, the letters need to be placed in the same relative location on both figures and be placed in the same order going around the figures.” Demonstrate on the second star figure.

    “These labels are used to compare the corresponding parts of the two figures. This means when we compare the center top angle of both figures we say ‘angle q corresponds to angle q¢.’ In the same way we could compare corresponding sides. Segment (side) qr corresponds to segment (side) q'r', and so on.” Ask students several questions about identifying other corresponding in this or additional examples to be sure they understand the correspondence concept before moving on.

    “In this lesson we will examine several strategies for comparing figures and solving problems like the star problem. We can also solve many other practical problems involving matters such as graphs, floor plans, scale drawings, and maps.”

    Discovering Congruence and Similarity Activity

    “I would like you to get into groups [of 2–4] for our first activity.” While students are getting into groups, hand out one shape set envelope and one record sheet to each group (M-7-6-3_Shape Set.docx and M-7-6-3_Shape Set Record Sheet.docx).

    “Each group has been given a shape set and a record sheet. I want you to find the different pairs of shapes indicated on your record sheet and group them together. You will be comparing these pairs of figures. As a group you will look at one pair at a time. Brainstorm as many ideas as you can about how the figures compare. Think about how all of the corresponding parts compare. Write a specific comparison statement involving the corresponding sides and angles in the space provided. Repeat these steps until you have compared all of the pairs.” Move around the room listening to and guiding students with questions. Give a warning when work time is nearing an end. Gauge this on when you see most groups finishing the last comparison (around 10–12 minutes).

    As you circulate, pay attention to the variety of statements the groups have written. Make note of statements you want students to share with the class. Some of the statements may be excellent; others may need some work. Refer to the shape set sample responses (M-7-6-3_Shape Set Sample Record.docx).

    As responses are shared for each pair, bring up ideas that may help to more accurately compare the figures. Be sure to mention the need to compare the ratios of corresponding sides.

    After all of the shape pairs have been discussed, bring up congruence and similarity: [IS.4 - All Students]

    • “Some of our shape pairs were identical in shape and size. Can you name them?” (E and F, K and L)
    • “These pairs of figures are called congruent. We have a special symbol to represent congruence; it looks like an equal sign with a squiggle above it.” (Demonstrate drawing the symbol between two figure names like triangle ABC and triangle DEF, or rectangle E and rectangle F.)
    • “On your record sheet, go to the pairs of figures E and F, K and L. Below your comparison statement, add the phrase CONGRUENT FIGURES. Can I have a volunteer to remind us of how we know these figures are congruent?” (same general shape and size, all pairs of corresponding sides are congruent, all pairs of corresponding angles are congruent)
    • Now, look for a pair of figures that are alike but are not the same size. If there are any pairs that are proportional in size they are called mathematically similar. Please label them on your record sheet. The characteristics you are looking for are the same general shape, pairs of corresponding sides that all have the same ratio, and pairs of corresponding angles that are all congruent. Which pairs did you label as SIMILAR FIGURES?” (A and D, C and D, G and H)

    “The special symbol for similarity is just the squiggle.” Demonstrate an example such as Square A ~ Square B. Explain how the vertex letters would be used if present ∆ABC ~ ∆XYZ.

    “We also have two pairs of figures left that are neither congruent nor similar. Can someone explain why?” (They have close to the same general shape, but when you check the corresponding sides they have different ratios, and when you check the corresponding angles they are not equivalent.)

    “Please label these pairs of figures (I and J, M and N) as NEITHER on your record sheet.”

    Checkpoint: Ask students several questions to check for understanding before moving to the next activity.

    Introducing Scale Factor

    “On our record sheets, we looked at pairs of corresponding sides and angles to compare the figures. This gave us the information we needed to determine if the figures were congruent, mathematically similar, or neither. There is another piece of information that is very helpful in comparing figures, solving problems, and making detailed drawings of all sorts including maps. This is the scale factor. We already considered everything we needed to figure out scale factor in our last activity.”


    “We said on our record sheet that square A and square B had corresponding sides whose ratios were all 4:8 or 1:2 (). Did anyone write that your ratios were 8:4 or 2:1?”

    “It depends on your reference point. If you looked at A first and compared it to B, which is larger, you used 4:8 or 1:2. If, on the other hand, you looked at B first and compared it to A, which is smaller, you would have stated 8:4 or 2:1.”

    “With scale factor it is important to know which direction you are comparing. For figures A and B our scale factor is 2 if we look from small (A) to large (B) because each side needs to be multiplied by 2 to create the enlarged figure. But, if we wanted to go from B to A, this would be reducing the size of the square so we would multiply by  to get the reduced figure size.” Go through several more examples of your own or the other pairs of similar figures from the record sheet.

    Emphasize these key concepts: [IS.5 - All Students]

    • Scale factor is the number multiplied by the original figure’s side lengths to create the image, and it may be a fraction or decimal.
    • It is important to know if you are going from small to large (enlargement) or large to small (reduction).
    • When two figures are compared, the figure labeled without prime symbols is the original figure (A, B, C, D). The figure labeled with prime symbols is what you are transforming into, and it is considered the image (A', B', C', D' ). Sometimes different sets of letters are used for the two figures, like triangle ABC transforming into triangle XYZ. In this case, one figure will be labeled “pre-image” and the other will be labeled “image” or “original” and “new.”
    • In congruent figures, the scale factor for all pairs of corresponding sides is 1 because they stay the same size.
    • When a figure is being reduced, it has a scale factor < 1 to “shrink” the side lengths.
    • When a figure is being enlarged, it has a scale factor > 1 to increase the side lengths.
    • In congruent and similar figures, all pairs of corresponding angles are always congruent and never change because of the scale factor of the sides.

    Scale Factor and Similarity Graphing Activity

    For this activity, students should work individually or in pairs. They will plot several points, which are provided on the activity sheet. The points need to be connected in order and labeled with the indicted letters. For each problem, two similar polygons will be formed on each coordinate axis. Students will use the grid squares to label the side lengths all around both figures. They will determine the scale factor of the sides, the perimeter and area of each figure.

    Provide each student a copy of the Scale Factor Labsheet (M-7-6-3_Scale Factor Labsheet and KEY.docx).

    Allow approximately 15–20 minutes to work, but be flexible based on student needs. Monitor student work. Assist students who may need further direction. Ask students who seem very proficient to explain the relationship between the perimeters of one pair of similar figures, or the relationship between the areas of one pair of similar figures.

    When students are finished, call on pairs to present each problem. If students present incorrect information, or use illogical steps, assist them with guiding questions which lead them in the correct direction. In addition, encourage the rest of the class to make necessary corrections on their own papers during the presentations.

    Close this activity by summarizing the characteristics of similar figures. Extend the concept that the pairs of corresponding sides all have the same scale factor to the fact the perimeters of each pair shared the same scale factor as the corresponding sides, while the areas were increased or reduced by the square of the scale factor.

    Partner Activity: Similarity Applications

    Tell students, “We are going to use what we learned about scale factor in the last activity to solve some similarity problems. If you know two figures are mathematically similar, you can find the scale factor for the entire image by using the lengths of just one pair of corresponding sides.”

    “Think back to the star problem at the beginning of the lesson. If you were told that the stars were similar, you could have found the scale using one pair of corresponding sides whose measures were both provided. Once you had the scale factor it could be used to find missing values for pairs of sides where only one length was provided. Let’s practice finding some missing side lengths.”

    Go through several examples. Emphasize that students need to notice whether the image is an enlargement or reduction to determine the scale factor correctly. Include one or two polygon problems with a missing length on the original figure and a missing length on the image.

    Show students how they could separate the triangles to more clearly see the two similar triangles. This will be a challenging task for many students.



    The scale factor is found by dividing a pair of corresponding side lengths we know. For example, 38.5 ÷ 55 = 0.7 (check it: 55 x .7 = 38.5), which means the scale factor 0.7 multiplied by 84 will give us the missing distance across the pond:

    84 x .7 = 58.8 ft

    Do more examples like these if students need additional instructions. Additional examples are available (M-7-6-3_More Work with Similarity and KEY.docx).

    Have students practice finding missing values by completing the Similarity Applications practice sheet (M-7-6-3_Similarity Applications and KEY.docx). This can be completed with a partner. Plan about 15 minutes of work time, or if time is short, send the sheet home as an individual assignment and go over it the next day. Check student work and clarify misconceptions before moving to the map reading segment of the lesson.

    Hand out the Exit Ticket for Lesson 3 (M-7-6-3_Exit Ticket Lesson 3 and KEY.docx) to check students’ understanding.

    Extension: [IS.6 - Struggling Learners]

    • Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following terms should be entered into students’ vocabulary journals: congruence, corresponding, scale factor, similarity. Keep a supply of vocabulary journal pages on hand so students can add pages as needed. Bring up instances where scale factors are needed to solve real-world problems as seen throughout the school year. Use examples related to other subject areas such as maps (social studies), recipes (home skills class), scale drawings (art, drafting) when discussing scale factor. Ask students to bring in maps and scale drawings they find in magazines or newspapers. Distinguish the difference in working with scale factor for reductions and enlargements. Emphasize the difference between similar figures and congruent figures.
    • Small Group: Rethinking Congruence and Similarity: Provide each student with grid paper to graph Figures A-H (M-7-6-3_Grid Size 6.docx). Also give each student or pair of students a list of the figure coordinates below (M-7-6-3_Rethinking Similarity and KEY.docx). Ask students to graph each of the figures, working individually on grid paper.


    Figure A

    Figure B

    Figure C

    Figure D

    ( 1, 2)

    (−1, -5)

    (−4, −5)

    (−1 , −5)

    (4 , 2)

    (5 , −5)

    (1 , −5)

    (1 , −5)

    (4 , 5 )

    (5 , 5)

    (−4 , 5 )

    (2 , −3)

    ( 1, 5)

    (−1 , 5)


    (−2, −3)




    Figure E

    Figure F

    Figure G

    Figure H

    (−2 , −3)

    (2 , 0)

    (5 , −5)

    (3, −3)

    (6 , −3)

    (5, 0)

    (5 , 0)

    (3 , 3)

    (6 , 5)

    (5, 5)

    (−5 , −5)

    (−3 , 6)

    (−2, 5)

    ( 2, 5)


    (−3, −6)

    • Check students’ graphed figures (M-7-6-3_Rethinking Similarity and KEY.docx). Review with students the characteristics of similar and congruent figures. Have students make a comparison statement about the four pairs of figures regarding congruence or similarity. (Example: Figure A is similar to Figure E because all pairs of corresponding sides have the same ratio of 3/8, and all pairs of corresponding angles are equivalent.) Allow students to work with a partner to write their statements. Insist that the statements are clear and specific, including the comparison of corresponding parts. If time permits, allow students to find the ordered pairs necessary to create additional reductions and/or enlargements of the figures which are mathematically similar.
    • Station 1: Ask the Class: Ask students who may be going beyond the standards to create an embedded triangle problem. Request that students create an answer key with an explanation to attach to the problem’s poster. Allow time for students to present their question and solicit responses from the class. Post these around the room or in the station area if there is space.
    • Station 2: Morph Your Character: This activity extends the work of any student who created a character at the Create a Character Station in Lesson 2. Students will use their previous character and the list of coordinate pairs recorded on the record sheet. Students will create a rule using a scale factor which will reduce or enlarge their previous character.

    They may select rules such as () or (2x, 2y) to create a mathematically similar character, or rules such as (2x, 4y) or (1/3x, 3y) to create a non-similar morphed character. Once the rule is determined, they will apply the rule to each ordered pair in the table used for the original character. These ordered pairs should be recorded next to the previous values on the original record sheet or a new record sheet (M-7-6-3_Morph Your Character.docx). Ask students to graph the list of ordered pairs to see what the new version of their character looks like. For the enlargements, a larger chart paper size grid may be needed, and extra grid lines may need to be added. Another option for enlargements is to attach four 8.5″ x 11″ sheets of grid paper on chart paper to make the 4 quadrants of a larger graph.

    If a student has not created a character, s/he can do this activity by going back and completing the Lesson 2 station first.

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