Scale It Up!

1. Create an enlargement or a reduction of an image through the usage of coordinate rules for dilations. 
2. HONORS/CHALLENGE Level: Create a dilation through using compasses, straight edges, and rulers (without the need for using the graph system).

This lesson takes on a “portfolio learning approach” where students build off of vocabulary such as dilation and scale factor. These definitions are developed through concrete and simple examples. Students will develop a sort of algorithm of being able to apply a coordinate system to a pre-image (a picture that is to be dilated) and multiply the coordinates by the scale factor to achieve an enlargement or reduction.

Before the students select an image of their choosing, students will develop the algorithms through their own research and through the development of definitions for dilation, scale factor, enlargement, and reduction. They will record written definitions of these terms so that they have documentation of the set theories and rules of dilations.

As with any coordinate Geometry transformation (i.e. reflection, rotation, translation, dilation), careful attention must be paid to the students’ thinking style. Those who tend to lean towards visual learning, often gravitate towards visual approaches and rules. Our more analytical students often tend to lean towards coordinate rules.

I utilize DeltaMath to provide students exercises in dilations that solidify and further develop the theory of dilation. On DeltaMath, I assign exercises such as identifying the scale factor, performing a dilation on a coordinate, or creating a dilation on a simple shape. More advanced students are quickly moved into exercises concerning dilating figures that have diagonal lines (forcing them to connect dilations with its impact on rise/run) or pushed into the dilation of a triangle construction. Those who do not utilize an online math platform, might want to utilize a worksheet that has a mixture of questions with coordinate rules and dilations of simple shapes like squares and triangles.

After the set of notes are developed, students can then select a logo or a picture of their choice. Usually, I like printing out a bunch of logos like Starbucks, Burger King, IU, Ohio State, Purdue, BMW, etc., so that students can choose an image of interest to perform their dilation on.

The basic level dilation project consists of creating a coordinate system around the logo and creating a dilation. Students do this by identifying reference points and multiplying their coordinates by the scale factor. More advanced students are encouraged to do more of a construction approach WITHOUT a coordinate system.

DeltaMath is able to give custom reports on the problems that each student attempts similar to below:

For instance, this student is excelling at the visual elements of dilations and does well with the support of visual diagrams and graphs. However, on Dilations of a Point, the student has trouble applying the coordinate rule without the support of a graph. Therefore, this student might need more targeted instruction on the dilation of a point.

The rubric for this project assesses not only the implementation of the rules of dilations through the quality of the scale drawing, but also: 

• The clarity of the students’ mathematical communication through correct notation, applications of vocabulary, and ability to use specific examples to illustrate the concepts. 
• A student’s awareness and ability to recognize mistakes and come up with solutions to potential problems faced when working on the scale drawing. 
• A students’ ability to use Geometric precision tools appropriately (this is a process standard that is becoming more and more of an underpinning in the latest Indiana and Common Core Geometry standards).

Dilation Presentation Options: Students get into seeing the scale drawings of their peers. I could envision a variety of formats for the presentations. This project is best presented where students are able to closely examine the work of their peers rather than an oral presentation.

• If you are entirely in person in class and you have a large enough space, you could do a gallery walk of the projects where students write comments on each other’s projects. These projects (along with the portfolios) lend themselves to a sort of science fair exhibit style of format.
• For a more tech savvy approach, I like having the students post their projects on Google Classroom and then fill out a peer feedback form. You can easily send students their feedback in an email. Here is a sample scale drawing project feedback google form.

Opportunities for Connections and Applications:

• If you have a close relationship with your art teacher at your school and your art teacher embraces cross curricular work, you might want to collaborate with the art teacher to enhance this project and introduce more artistic elements. 

This project can also be enhanced by bringing in a guest speaker (or better yet ZOOMING with a guest speaker). I would suggest bringing in an artist to talk about how they use scale in their work. Additionally, someone who does work such as carpentry, architecture or blueprinting could be good if you want to focus on the use of precision in the real world.

• Dilation Project Handout – This is what you should give your students! In essence, the students will build a portfolio with notes through answering the tasks in this packet that will prepare them for the actual scale drawing.
• Dilation DeltaMath Guidance For Teachers – An overview of the types of exercises and why they should be assigned and the differentiation.
• Exemplar Student Work

• For the actual scale drawings themselves: 
  • 1 cm graph paper (both 8.5×11 size and 11×17 poster sized)
  • Rulers
  • Compasses/Protractors
  • Logos – Here is a library of logos for students to use

Day 1

Bellwork – Open Ended question (10 minutes)

Have a picture and a dilated version of the picture. Ask students to brainstorm how one might be able to create an enlargement of a picture. (By this point, students have seen coordinate rules of reflections, rotations and translations). You may also want to include asking students to write down as many coordinate rules as they remember (usually dilations comes in right after a study of reflections, rotations and translations and an introduction to scale factor).

Introduce Project Expectations (10 minutes)

I like to showcase student work and then introduce students to their dilation packet. See attached document for quality exemplar work.

• Suggestions:
  • Have students partner up with a table partner [turn and talk] and analyze some of the exemplary student work. Ask them to explain things like how the exemplar notes and high quality student work shows a high quality understanding of dilations. 
  • Ask students to reflect on the difference between the 3- and the 4 point column of the rubric so that they understand the subtleties in the differences between a good and a great product.

Note Taking/Portfolio Development [Part 1] (20 minutes)

Students should answer questions 1-3 on the document to develop an understanding of the vocabulary. They can watch the video linked to explain some of the basic vocab surrounding dilations, enlargement, reduction and scale factor OR they can search for definitions and examples of these terms through internet searches. [TEACHER SHOULD CHECK THE QUALITY OF THE STUDENTS’ NOTES AT THE END OF THIS PART OF THE PROJECT] – When the students finish their basic note-taking, the students should come up to the teacher (or you come up to them) and you should inspect the quality of the notes based on the rubric. If the students’ notes are insufficient, you can suggest resources for the students to check out and ask them how they can achieve a higher score on the rubric. If the notes are sufficient, they should move onto the DeltaMath exercises.

DeltaMath Practice over Dilations (30 minutes)

Before starting the scale drawings, students should practice the basic techniques with the DeltaMath exercises. See Dilation DeltaMath Guidance For Teachers to see an overview of the types of questions and tasks that are good to assign students. If you do not have access to DeltaMath, a Kuta Software worksheet or something similar on basic dilations might also be effective as well. In my implementation of DeltaMath, I historically find that students who write the problems down on a sheet of paper as they do them, or utilize a Google Doc, are able to practice and internalize the feedback more effectively. For this project, I suggest that the students include the DeltaMath exercises or at least a summary of them in their Notetaking Portfolio.

For this project, I would assign “ Construction of a Dilated Triangle.” In my Geometry Course, I love assigning Geometric constructions through DeltaMath to the students. If you are going this route, I think that it is essential to tell the students that they must watch the video. I tell them to watch one step, stop, and then imitate it (this sort of approach to the video watching is similar to watching an instructional video on how to do a recipe or a do-it-yourself home improvement project).

Project Proposal and Planning (15 minutes)

Students should choose a logo and draft out and sketch their “game plan” before being given a graph paper. It does not need to be perfect but should demonstrate how coordinate rules or Geometric tools are used in the creation of their scale drawing. [TEACHER SHOULD CHECK AT THE END OF THIS POINT EACH STUDENTS’ PROPOSAL- IF THE PROPOSAL LOOKS GOOD, THE STUDENT HAS EARNED THEIR GRAPH PAPER OR POSTER PAPER]

Day 2/Overnight

Depending on whether you assign homework or not, the scale drawing can either be assigned as homework or started the next day in class. Effective scale drawings can take anywhere from 30 minutes to several hours (some students may get so into it that they may spend hours on it at home). Consider assigning 20-30 minutes in class to finish the students’ scale drawings.

Reflection (10-15 minutes)

When students have a final product, they should reflect on their final product by writing a page talking about the processes that they used and the effectiveness of these processes. Students should also think about obstacles they faced and what they would do differently the next time. You could even structure the reflection piece so that students reflect on how they might use precision tools or the concept of scale in a future career.

Presentation (Remaining of the Period- time varies depending on class size)

The presentations could be in the form of a learning event, where students’ scale drawings are posted around the room and students can write Post-it notes to give other students’ feedback on their scale drawings.

From the Indiana Geometry Standards

G.TR.2: Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

PS.5: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Mathematically proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students identify relevant external mathematical resources, such as digital content, and use them to pose or solve problems. They use technological tools to explore and deepen their understanding of concepts and to support the development of learning mathematics. They use technology to contribute to concept development, simulation, representation, reasoning, communication and problem solving. 

PS.6: Attend to precision.  Mathematically proficient students communicate precisely to others. They use clear definitions, including correct mathematical language, in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They express solutions clearly and logically by using the appropriate mathematical terms and notation. They specify units of measure and label axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently and check the validity of their results in the context of the problem. They express numerical answers with a degree of precision appropriate for the problem context.