Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 49913
Dilly Dallying with Dilations
Students will understand the concept of dilation by constructing similar polygons on a coordinate grid using coordinate notation of dilation
. Students use similar figures to determine the scale factor. Students use proportions to determine side lengths of similar figures.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Interactive Whiteboard, Basic
Calculators, LCD Projector, Overhead Projector
Instructional Time: 1 Hour(s) 30 Minute(s)
Freely Available: Yes
Keywords: Transformations, Reflections, Translations, Dilations, Shapes, Polygon, Square, Triangle, Quadrilateral,
Rectangle
Instructional Design Framework(s): Direct Instruction
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Guided Practice Worksheet Key Dilations.docx
Practice Worksheet Dilations.docx
Practice Worksheet Key Dilations.docx
Triangle Similarity Intro Activity Dilations.docx
Assessment Dilations.pdf
Assessment Key Dilations.pdf
Guided Practice Worksheet Dilations.pdf
Guided Practice Worksheet Key Dilations.pdf
Practice Worksheet Dilations.pdf
Practice Worksheet Key Dilations.pdf
Triangle Similarity Intro Activity Dilations.pdf
Assessment Dilations.docx
Assessment Key Dilations.docx
Guided Practice Worksheet Dilations.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will use proportions to determine scale factors of dilations.
Students will use proportions to determine a side length of a scaled image.
Students will use scale factors to construct dilations of polygons.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students need a working knowledge of ratios and proportions.
Students need to be able to identify corresponding parts of 2-dimensional figures.
page 1 of 4 Students need to know how to plot points on a coordinate grid.
Students should already have an understanding of the concept of similarity based on reflection, translation, and rotation.
Students need to know the Pythagorean Theorem.
Guiding Questions: What are the guiding questions for this lesson?
What is dilation?
Is the image of an object projected on the wall similar to the object?
How would you find the scale factor?
How can you find the length of a side on a scaled object or image?
Teaching Phase: How will the teacher present the concept or skill to students?
Pair students and have them complete the Triangle Similarity Introduction Activity. (**NOTE** The second page is for teacher reference and should not be
given to students.) Encourage students to discuss and support their ideas. Preferably, there will be some debate about triangles C and D due to size.
After an appropriate amount of discussion, reinforce the idea that triangles C and D are similar but are different sizes. Introduce the term "dilation". Explain that a
dilation is a transformation that stretches or shrinks a figure to create a similar figure. Ask how many students have been to an eye doctor and had their pupils
dilated. Choose students to describe the procedure and discuss the purpose of the process. (Pupils are made larger so that the doctor can look inside the eye.)
Place a flat object (book, Post-It, etc. - something with sides, not round) on the overhead projector and observe the projected image.
Have students discuss with their partners, "Is the projected image similar to the object? Why, or why not?" (2 minutes)
Ask each pair to share their ideas with the whole group. Encourage students to discuss the similarities and differences between the projected image and the object.
Summarize that the projected image is simply an enlarged picture of the object, a dilation. Restate, "A dilation is an enlargement or reduction of something."
Introduce "scale factor": the ratio of a side length of an image to the corresponding side length of the original figure.
Ask students, "How can you find the scale factor of the projected image?" Call on raised hands.
Show students how to identify the ratios between the projected image and the object by measuring and recording the measurements of corresponding sides in
fraction form. Record the measurements from the projected image as the numerators and the measurements from the object as the denominators.
Set up the fractions on the whiteboard to show that you can cross multiply to check that each proportion/ratio is equal. Divide the numerators by the denominators,
and show that each scale factor is equal.(Example:
;
). Enter the measure of each side in its corresponding place, then use cross multiplication to show the each product is equal.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Project the graph from the Guided Practice Worksheet on the white board, and give each student a copy of the worksheet. See the Guided Practice Worksheet
Answer Key.
Pair students and instruct them to fill in the coordinates of the vertices on the worksheet. Have one student work on Image 1 and the other on Image 2 and share
their answers with each other. Monitor groups to ensure that each student remembers how to record the coordinates of the points and provide any necessary
prompts. (2 minutes)
Call on students to give the coordinates of the points and label the points projected on the whiteboard.
Next use the information from Point A and Point E to show how the answer of "3" was obtained in the next column. Have students continue working in pairs to
complete the answers for that column. (1 minute)
Ask students, "What is the relationship between the values of Image 1 and Image 2?" (The values in Image 2 are 3 times the value of the corresponding values in
Image 1).
Write the expression "
." State, "K represents the scale factor. So, in this case
represents the dilation.
Have students continue working in pairs and complete the section, "What are the side lengths?" It is pretty self explanatory after completion of the section using
coordinates. Provide any guidance or prompts as needed. Squares are used in this activity so that all students have to do is count the units to determine side
lengths. (2 minutes)
Ask students, "What is the scale factor?" They should all agree that 3 is the scale factor. Ask, "How do you know?" (Answers should show an understanding that the
values in Image 2 are 3 times the values of the corresponding values in Image 1.)
Ask students, "What would happen to the scale factor if the images are reversed, and EFGH is the original (Image 1) and ABCD is the dilation (Image 2)?" (
)
Explain to students that a scale factor <1 will result in a reduction and >1 in an enlargement.
Ask students to discuss with their partners and be prepared to share, "What conclusions can you make about the relationship between scale factor and
corresponding parts of similar objects/figures?" (2 min.) Have pairs share conclusions with the whole group.
Have students work with their partners to complete the problem on the worksheet, "Construct a dilation of Image 1 with a scale factor of 2 and label it IJKL."
Monitor groups to ensure that students are able to apply the knowledge from the activity to constructing the figure. Provide any prompts as needed.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Have students complete the Dilations Practice Worksheet. This can be assigned as homework. Or, they can be given time at the end of the period to work in class.
Anyone not finished can take it home for homework. Students who finish early can be given a digital copy of the worksheet to create their figures using a computer.
See the Dilations Practice Worksheet Answer Key.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Discussion:
Ask students how they can use the scale factor to construct a dilation of an image on a coordinate grid.
Ask students how to determine if a dilation is going to be a reduction or an enlargement when the scale factor is known.
Ask students to explain how they can use cross multiplication to find the unknown length of a dilated image knowing the lengths of at least one side of the image
and the lengths of all sides in the original image.
Ask students, "In what professions could the skill of constructing dilations be useful, and how would it be used?" Promote discussion. Some examples: Architects,
Vehicle Manufacturing, Toy Models, Advertising, Cartography, any industry which uses scale models and prototypes, etc.)
Summative Assessment
Students will construct similar 2-dimensional polygons (reduction enlargement) by applying the coordinates of vertices and scale factor to the formula
.
Students will use proportions to determine a side length of a dilated figure. Example:
x=5
page 2 of 4 Given a real-world scenario involving dilation, students will determine the scale factor.
Dilation Assessment
Dilation Assessment Answer Key
**NOTE** Students may opt to do the Summative Assessment on a computer so that they can print their artwork.
Formative Assessment
During the Introduction and Guided Practice activities, monitor students to ensure that they possess prerequisite skills to perform operations and functions listed in
the "Prior Knowledge" section
Monitor during paired work to ensure that students are able to determine scale factors.
Use the worksheet from "Independent Practice" to determine if all students have demonstrated understanding of dilation and scale factors to be successful on the
"Summative Assessment."
Feedback to Students
When students are working in pairs, provide minimal prompts and guidance required to scaffold their thinking to arrive at correct answers.
Celebrate success/praise for correct answers and valuable ideas during sharing.
Students receive feedback from peers while collaborating on the Guided Practice Worksheet.
Review errors made on the Independent Practice worksheet with students. Use examples of each error on the board and ask students to determine how/why the
answer is wrong and how to fix it.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations: English Language Learners: Speak slowly using simple language and provide opportunities for collaboration. Use diagrams to show examples of
cross multiplying. Show examples using coordinate notation of dilation
.
Special Education Students: Read instructions and word problems for struggling readers. Collaborative small groups.Use diagrams to show an examples of cross
multiplying. Show examples using coordinate notation of dilation
.
Extensions:
Students could conduct research on the use of dilation in jobs, careers, and industry.
Students could build a scale model of a 3-dimensional object.
Suggested Technology: Document Camera, Computer for Presenter, Interactive Whiteboard, Basic Calculators, LCD Projector, Overhead Projector
Special Materials Needed:
Handouts:
Triangle Introduction Activity Worksheet
Guided Practice Worksheet
Dilations Practice Worksheet
Dilations Assessment
Other materials:
Pencils
Straightedge (for each student)
Basic Calculators (optional)
Student access to computers (optional)
Projector (Overhead or Document Camera) - For "Teaching Phase"
LCD Projector if you want to project the graphs on the whiteboard from the computer instead of on the Overhead or Document Camera.
Additional Information/Instructions
By Author/Submitter
Standards for Mathematical Practice:
MAFS.K12.M.P.2.1 - Reason abstractly and quantitatively
MAFS.K12.M.P.7.1 - Look for and make use of structure
Note: This lesson will only address the dilation portion of MAFS.8.G.1.4
SOURCE AND ACCESS INFORMATION
Contributed by: Brian Bowman
Name of Author/Source: Brian Bowman
District/Organization of Contributor(s): Jackson
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
page 3 of 4 Name
MAFS.8.G.1.4:
Description
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a
sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a
sequence that exhibits the similarity between them.
page 4 of 4